355
|
1 |
\documentstyle[a4,proof,iman,extra,times]{llncs}
|
|
2 |
%Repetition in the two sentences that begin ``The powerset operator''
|
103
|
3 |
\newif\ifCADE
|
355
|
4 |
\CADEtrue
|
103
|
5 |
|
355
|
6 |
\title{A Fixedpoint Approach to Implementing\\
|
|
7 |
(Co)Inductive Definitions\thanks{J. Grundy and S. Thompson made detailed
|
|
8 |
comments; the referees were also helpful. Research funded by
|
|
9 |
SERC grants GR/G53279, GR/H40570 and by the ESPRIT Project 6453
|
|
10 |
`Types'.}}
|
103
|
11 |
|
355
|
12 |
\author{Lawrence C. Paulson\\{\tt lcp@cl.cam.ac.uk}}
|
|
13 |
\institute{Computer Laboratory, University of Cambridge, England}
|
103
|
14 |
\date{\today}
|
|
15 |
\setcounter{secnumdepth}{2} \setcounter{tocdepth}{2}
|
|
16 |
|
|
17 |
\newcommand\sbs{\subseteq}
|
|
18 |
\let\To=\Rightarrow
|
|
19 |
|
|
20 |
|
355
|
21 |
\newcommand\pow{{\cal P}}
|
|
22 |
%%%\let\pow=\wp
|
|
23 |
\newcommand\RepFun{\hbox{\tt RepFun}}
|
|
24 |
\newcommand\cons{\hbox{\tt cons}}
|
|
25 |
\def\succ{\hbox{\tt succ}}
|
|
26 |
\newcommand\split{\hbox{\tt split}}
|
|
27 |
\newcommand\fst{\hbox{\tt fst}}
|
|
28 |
\newcommand\snd{\hbox{\tt snd}}
|
|
29 |
\newcommand\converse{\hbox{\tt converse}}
|
|
30 |
\newcommand\domain{\hbox{\tt domain}}
|
|
31 |
\newcommand\range{\hbox{\tt range}}
|
|
32 |
\newcommand\field{\hbox{\tt field}}
|
|
33 |
\newcommand\lfp{\hbox{\tt lfp}}
|
|
34 |
\newcommand\gfp{\hbox{\tt gfp}}
|
|
35 |
\newcommand\id{\hbox{\tt id}}
|
|
36 |
\newcommand\trans{\hbox{\tt trans}}
|
|
37 |
\newcommand\wf{\hbox{\tt wf}}
|
|
38 |
\newcommand\nat{\hbox{\tt nat}}
|
|
39 |
\newcommand\rank{\hbox{\tt rank}}
|
|
40 |
\newcommand\univ{\hbox{\tt univ}}
|
|
41 |
\newcommand\Vrec{\hbox{\tt Vrec}}
|
|
42 |
\newcommand\Inl{\hbox{\tt Inl}}
|
|
43 |
\newcommand\Inr{\hbox{\tt Inr}}
|
|
44 |
\newcommand\case{\hbox{\tt case}}
|
|
45 |
\newcommand\lst{\hbox{\tt list}}
|
|
46 |
\newcommand\Nil{\hbox{\tt Nil}}
|
|
47 |
\newcommand\Cons{\hbox{\tt Cons}}
|
103
|
48 |
\newcommand\lstcase{\hbox{\tt list\_case}}
|
|
49 |
\newcommand\lstrec{\hbox{\tt list\_rec}}
|
355
|
50 |
\newcommand\length{\hbox{\tt length}}
|
|
51 |
\newcommand\listn{\hbox{\tt listn}}
|
|
52 |
\newcommand\acc{\hbox{\tt acc}}
|
|
53 |
\newcommand\primrec{\hbox{\tt primrec}}
|
|
54 |
\newcommand\SC{\hbox{\tt SC}}
|
|
55 |
\newcommand\CONST{\hbox{\tt CONST}}
|
|
56 |
\newcommand\PROJ{\hbox{\tt PROJ}}
|
|
57 |
\newcommand\COMP{\hbox{\tt COMP}}
|
|
58 |
\newcommand\PREC{\hbox{\tt PREC}}
|
103
|
59 |
|
355
|
60 |
\newcommand\quniv{\hbox{\tt quniv}}
|
|
61 |
\newcommand\llist{\hbox{\tt llist}}
|
|
62 |
\newcommand\LNil{\hbox{\tt LNil}}
|
|
63 |
\newcommand\LCons{\hbox{\tt LCons}}
|
|
64 |
\newcommand\lconst{\hbox{\tt lconst}}
|
|
65 |
\newcommand\lleq{\hbox{\tt lleq}}
|
|
66 |
\newcommand\map{\hbox{\tt map}}
|
|
67 |
\newcommand\term{\hbox{\tt term}}
|
|
68 |
\newcommand\Apply{\hbox{\tt Apply}}
|
|
69 |
\newcommand\termcase{\hbox{\tt term\_case}}
|
|
70 |
\newcommand\rev{\hbox{\tt rev}}
|
|
71 |
\newcommand\reflect{\hbox{\tt reflect}}
|
|
72 |
\newcommand\tree{\hbox{\tt tree}}
|
|
73 |
\newcommand\forest{\hbox{\tt forest}}
|
|
74 |
\newcommand\Part{\hbox{\tt Part}}
|
|
75 |
\newcommand\TF{\hbox{\tt tree\_forest}}
|
|
76 |
\newcommand\Tcons{\hbox{\tt Tcons}}
|
|
77 |
\newcommand\Fcons{\hbox{\tt Fcons}}
|
|
78 |
\newcommand\Fnil{\hbox{\tt Fnil}}
|
103
|
79 |
\newcommand\TFcase{\hbox{\tt TF\_case}}
|
355
|
80 |
\newcommand\Fin{\hbox{\tt Fin}}
|
|
81 |
\newcommand\QInl{\hbox{\tt QInl}}
|
|
82 |
\newcommand\QInr{\hbox{\tt QInr}}
|
|
83 |
\newcommand\qsplit{\hbox{\tt qsplit}}
|
|
84 |
\newcommand\qcase{\hbox{\tt qcase}}
|
|
85 |
\newcommand\Con{\hbox{\tt Con}}
|
|
86 |
\newcommand\data{\hbox{\tt data}}
|
103
|
87 |
|
|
88 |
\binperiod %%%treat . like a binary operator
|
|
89 |
|
|
90 |
\begin{document}
|
355
|
91 |
%CADE%\pagestyle{empty}
|
|
92 |
%CADE%\begin{titlepage}
|
103
|
93 |
\maketitle
|
|
94 |
\begin{abstract}
|
355
|
95 |
This paper presents a fixedpoint approach to inductive definitions.
|
|
96 |
Instead of using a syntactic test such as `strictly positive,' the
|
|
97 |
approach lets definitions involve any operators that have been proved
|
|
98 |
monotone. It is conceptually simple, which has allowed the easy
|
|
99 |
implementation of mutual recursion and other conveniences. It also
|
|
100 |
handles coinductive definitions: simply replace the least fixedpoint by a
|
|
101 |
greatest fixedpoint. This represents the first automated support for
|
|
102 |
coinductive definitions.
|
130
|
103 |
|
|
104 |
The method has been implemented in Isabelle's formalization of ZF set
|
179
|
105 |
theory. It should be applicable to any logic in which the Knaster-Tarski
|
355
|
106 |
Theorem can be proved. Examples include lists of $n$ elements, the
|
|
107 |
accessible part of a relation and the set of primitive recursive
|
|
108 |
functions. One example of a coinductive definition is bisimulations for
|
|
109 |
lazy lists. \ifCADE\else Recursive datatypes are examined in detail, as
|
|
110 |
well as one example of a {\bf codatatype}: lazy lists. The appendices
|
|
111 |
are simple user's manuals for this Isabelle/ZF package.\fi
|
103
|
112 |
\end{abstract}
|
|
113 |
%
|
355
|
114 |
%CADE%\bigskip\centerline{Copyright \copyright{} \number\year{} by Lawrence C. Paulson}
|
|
115 |
%CADE%\thispagestyle{empty}
|
|
116 |
%CADE%\end{titlepage}
|
|
117 |
%CADE%\tableofcontents\cleardoublepage\pagestyle{headings}
|
103
|
118 |
|
|
119 |
\section{Introduction}
|
|
120 |
Several theorem provers provide commands for formalizing recursive data
|
|
121 |
structures, like lists and trees. Examples include Boyer and Moore's shell
|
|
122 |
principle~\cite{bm79} and Melham's recursive type package for the HOL
|
|
123 |
system~\cite{melham89}. Such data structures are called {\bf datatypes}
|
|
124 |
below, by analogy with {\tt datatype} definitions in Standard~ML\@.
|
|
125 |
|
130
|
126 |
A datatype is but one example of an {\bf inductive definition}. This
|
103
|
127 |
specifies the least set closed under given rules~\cite{aczel77}. The
|
|
128 |
collection of theorems in a logic is inductively defined. A structural
|
|
129 |
operational semantics~\cite{hennessy90} is an inductive definition of a
|
|
130 |
reduction or evaluation relation on programs. A few theorem provers
|
|
131 |
provide commands for formalizing inductive definitions; these include
|
|
132 |
Coq~\cite{paulin92} and again the HOL system~\cite{camilleri92}.
|
|
133 |
|
130
|
134 |
The dual notion is that of a {\bf coinductive definition}. This specifies
|
103
|
135 |
the greatest set closed under given rules. Important examples include
|
|
136 |
using bisimulation relations to formalize equivalence of
|
|
137 |
processes~\cite{milner89} or lazy functional programs~\cite{abramsky90}.
|
|
138 |
Other examples include lazy lists and other infinite data structures; these
|
130
|
139 |
are called {\bf codatatypes} below.
|
103
|
140 |
|
355
|
141 |
Not all inductive definitions are meaningful. {\bf Monotone} inductive
|
|
142 |
definitions are a large, well-behaved class. Monotonicity can be enforced
|
|
143 |
by syntactic conditions such as `strictly positive,' but this could lead to
|
|
144 |
monotone definitions being rejected on the grounds of their syntactic form.
|
|
145 |
More flexible is to formalize monotonicity within the logic and allow users
|
|
146 |
to prove it.
|
103
|
147 |
|
|
148 |
This paper describes a package based on a fixedpoint approach. Least
|
|
149 |
fixedpoints yield inductive definitions; greatest fixedpoints yield
|
355
|
150 |
coinductive definitions. The package has several advantages:
|
103
|
151 |
\begin{itemize}
|
355
|
152 |
\item It allows reference to any operators that have been proved monotone.
|
|
153 |
Thus it accepts all provably monotone inductive definitions, including
|
|
154 |
iterated definitions.
|
|
155 |
\item It accepts a wide class of datatype definitions, though at present
|
|
156 |
restricted to finite branching.
|
130
|
157 |
\item It handles coinductive and codatatype definitions. Most of
|
|
158 |
the discussion below applies equally to inductive and coinductive
|
103
|
159 |
definitions, and most of the code is shared. To my knowledge, this is
|
130
|
160 |
the only package supporting coinductive definitions.
|
103
|
161 |
\item Definitions may be mutually recursive.
|
|
162 |
\end{itemize}
|
|
163 |
The package is implemented in Isabelle~\cite{isabelle-intro}, using ZF set
|
|
164 |
theory \cite{paulson-set-I,paulson-set-II}. However, the fixedpoint
|
|
165 |
approach is independent of Isabelle. The recursion equations are specified
|
|
166 |
as introduction rules for the mutually recursive sets. The package
|
|
167 |
transforms these rules into a mapping over sets, and attempts to prove that
|
|
168 |
the mapping is monotonic and well-typed. If successful, the package
|
|
169 |
makes fixedpoint definitions and proves the introduction, elimination and
|
130
|
170 |
(co)induction rules. The package consists of several Standard ML
|
103
|
171 |
functors~\cite{paulson91}; it accepts its argument and returns its result
|
355
|
172 |
as ML structures.\footnote{This use of ML modules is not essential; the
|
|
173 |
package could also be implemented as a function on records.}
|
103
|
174 |
|
|
175 |
Most datatype packages equip the new datatype with some means of expressing
|
355
|
176 |
recursive functions. This is the main omission from my package. Its
|
|
177 |
fixedpoint operators define only recursive sets. To define recursive
|
|
178 |
functions, the Isabelle/ZF theory provides well-founded recursion and other
|
|
179 |
logical tools~\cite{paulson-set-II}.
|
103
|
180 |
|
355
|
181 |
{\bf Outline.} Section~2 introduces the least and greatest fixedpoint
|
|
182 |
operators. Section~3 discusses the form of introduction rules, mutual
|
|
183 |
recursion and other points common to inductive and coinductive definitions.
|
|
184 |
Section~4 discusses induction and coinduction rules separately. Section~5
|
|
185 |
presents several examples, including a coinductive definition. Section~6
|
|
186 |
describes datatype definitions. Section~7 presents related work.
|
|
187 |
Section~8 draws brief conclusions. \ifCADE\else The appendices are simple
|
|
188 |
user's manuals for this Isabelle/ZF package.\fi
|
103
|
189 |
|
|
190 |
Most of the definitions and theorems shown below have been generated by the
|
|
191 |
package. I have renamed some variables to improve readability.
|
|
192 |
|
|
193 |
\section{Fixedpoint operators}
|
|
194 |
In set theory, the least and greatest fixedpoint operators are defined as
|
|
195 |
follows:
|
|
196 |
\begin{eqnarray*}
|
|
197 |
\lfp(D,h) & \equiv & \inter\{X\sbs D. h(X)\sbs X\} \\
|
|
198 |
\gfp(D,h) & \equiv & \union\{X\sbs D. X\sbs h(X)\}
|
|
199 |
\end{eqnarray*}
|
130
|
200 |
Let $D$ be a set. Say that $h$ is {\bf bounded by}~$D$ if $h(D)\sbs D$, and
|
|
201 |
{\bf monotone below~$D$} if
|
103
|
202 |
$h(A)\sbs h(B)$ for all $A$ and $B$ such that $A\sbs B\sbs D$. If $h$ is
|
|
203 |
bounded by~$D$ and monotone then both operators yield fixedpoints:
|
|
204 |
\begin{eqnarray*}
|
|
205 |
\lfp(D,h) & = & h(\lfp(D,h)) \\
|
|
206 |
\gfp(D,h) & = & h(\gfp(D,h))
|
|
207 |
\end{eqnarray*}
|
355
|
208 |
These equations are instances of the Knaster-Tarski Theorem, which states
|
103
|
209 |
that every monotonic function over a complete lattice has a
|
|
210 |
fixedpoint~\cite{davey&priestley}. It is obvious from their definitions
|
|
211 |
that $\lfp$ must be the least fixedpoint, and $\gfp$ the greatest.
|
|
212 |
|
355
|
213 |
This fixedpoint theory is simple. The Knaster-Tarski Theorem is easy to
|
103
|
214 |
prove. Showing monotonicity of~$h$ is trivial, in typical cases. We must
|
355
|
215 |
also exhibit a bounding set~$D$ for~$h$. Frequently this is trivial, as
|
179
|
216 |
when a set of `theorems' is (co)inductively defined over some previously
|
355
|
217 |
existing set of `formulae.' Isabelle/ZF provides a suitable bounding set
|
|
218 |
for finitely branching (co)datatype definitions; see~\S\ref{univ-sec}
|
|
219 |
below. Bounding sets are also called {\bf domains}.
|
103
|
220 |
|
355
|
221 |
The powerset operator is monotone, but by Cantor's Theorem there is no
|
|
222 |
set~$A$ such that $A=\pow(A)$. We cannot put $A=\lfp(D,\pow)$ because
|
|
223 |
there is no suitable domain~$D$. But \S\ref{acc-sec} demonstrates
|
|
224 |
that~$\pow$ is still useful in inductive definitions.
|
103
|
225 |
|
130
|
226 |
\section{Elements of an inductive or coinductive definition}\label{basic-sec}
|
|
227 |
Consider a (co)inductive definition of the sets $R_1$, \ldots,~$R_n$, in
|
355
|
228 |
mutual recursion. They will be constructed from domains $D_1$,
|
|
229 |
\ldots,~$D_n$, respectively. The construction yields not $R_i\sbs D_i$ but
|
|
230 |
$R_i\sbs D_1+\cdots+D_n$, where $R_i$ is contained in the image of~$D_i$
|
|
231 |
under an injection. Reasons for this are discussed
|
130
|
232 |
elsewhere~\cite[\S4.5]{paulson-set-II}.
|
103
|
233 |
|
|
234 |
The definition may involve arbitrary parameters $\vec{p}=p_1$,
|
|
235 |
\ldots,~$p_k$. Each recursive set then has the form $R_i(\vec{p})$. The
|
|
236 |
parameters must be identical every time they occur within a definition. This
|
|
237 |
would appear to be a serious restriction compared with other systems such as
|
|
238 |
Coq~\cite{paulin92}. For instance, we cannot define the lists of
|
|
239 |
$n$ elements as the set $\listn(A,n)$ using rules where the parameter~$n$
|
355
|
240 |
varies. Section~\ref{listn-sec} describes how to express this set using the
|
130
|
241 |
inductive definition package.
|
103
|
242 |
|
|
243 |
To avoid clutter below, the recursive sets are shown as simply $R_i$
|
|
244 |
instead of $R_i(\vec{p})$.
|
|
245 |
|
|
246 |
\subsection{The form of the introduction rules}\label{intro-sec}
|
|
247 |
The body of the definition consists of the desired introduction rules,
|
|
248 |
specified as strings. The conclusion of each rule must have the form $t\in
|
|
249 |
R_i$, where $t$ is any term. Premises typically have the same form, but
|
|
250 |
they can have the more general form $t\in M(R_i)$ or express arbitrary
|
|
251 |
side-conditions.
|
|
252 |
|
|
253 |
The premise $t\in M(R_i)$ is permitted if $M$ is a monotonic operator on
|
|
254 |
sets, satisfying the rule
|
|
255 |
\[ \infer{M(A)\sbs M(B)}{A\sbs B} \]
|
130
|
256 |
The user must supply the package with monotonicity rules for all such premises.
|
103
|
257 |
|
355
|
258 |
The ability to introduce new monotone operators makes the approach
|
|
259 |
flexible. A suitable choice of~$M$ and~$t$ can express a lot. The
|
|
260 |
powerset operator $\pow$ is monotone, and the premise $t\in\pow(R)$
|
|
261 |
expresses $t\sbs R$; see \S\ref{acc-sec} for an example. The `list of'
|
|
262 |
operator is monotone, as is easily proved by induction. The premise
|
|
263 |
$t\in\lst(R)$ avoids having to encode the effect of~$\lst(R)$ using mutual
|
|
264 |
recursion; see \S\ref{primrec-sec} and also my earlier
|
|
265 |
paper~\cite[\S4.4]{paulson-set-II}.
|
103
|
266 |
|
|
267 |
Introduction rules may also contain {\bf side-conditions}. These are
|
|
268 |
premises consisting of arbitrary formulae not mentioning the recursive
|
|
269 |
sets. Side-conditions typically involve type-checking. One example is the
|
|
270 |
premise $a\in A$ in the following rule from the definition of lists:
|
|
271 |
\[ \infer{\Cons(a,l)\in\lst(A)}{a\in A & l\in\lst(A)} \]
|
|
272 |
|
|
273 |
\subsection{The fixedpoint definitions}
|
|
274 |
The package translates the list of desired introduction rules into a fixedpoint
|
|
275 |
definition. Consider, as a running example, the finite set operator
|
|
276 |
$\Fin(A)$: the set of all finite subsets of~$A$. It can be
|
|
277 |
defined as the least set closed under the rules
|
|
278 |
\[ \emptyset\in\Fin(A) \qquad
|
|
279 |
\infer{\{a\}\un b\in\Fin(A)}{a\in A & b\in\Fin(A)}
|
|
280 |
\]
|
|
281 |
|
130
|
282 |
The domain in a (co)inductive definition must be some existing set closed
|
103
|
283 |
under the rules. A suitable domain for $\Fin(A)$ is $\pow(A)$, the set of all
|
|
284 |
subsets of~$A$. The package generates the definition
|
|
285 |
\begin{eqnarray*}
|
|
286 |
\Fin(A) & \equiv & \lfp(\pow(A), \;
|
|
287 |
\begin{array}[t]{r@{\,}l}
|
|
288 |
\lambda X. \{z\in\pow(A). & z=\emptyset \disj{} \\
|
|
289 |
&(\exists a\,b. z=\{a\}\un b\conj a\in A\conj b\in X)\})
|
|
290 |
\end{array}
|
130
|
291 |
\end{eqnarray*}
|
103
|
292 |
The contribution of each rule to the definition of $\Fin(A)$ should be
|
130
|
293 |
obvious. A coinductive definition is similar but uses $\gfp$ instead
|
103
|
294 |
of~$\lfp$.
|
|
295 |
|
|
296 |
The package must prove that the fixedpoint operator is applied to a
|
|
297 |
monotonic function. If the introduction rules have the form described
|
|
298 |
above, and if the package is supplied a monotonicity theorem for every
|
|
299 |
$t\in M(R_i)$ premise, then this proof is trivial.\footnote{Due to the
|
|
300 |
presence of logical connectives in the fixedpoint's body, the
|
|
301 |
monotonicity proof requires some unusual rules. These state that the
|
130
|
302 |
connectives $\conj$, $\disj$ and $\exists$ preserve monotonicity with respect
|
|
303 |
to the partial ordering on unary predicates given by $P\sqsubseteq Q$ if and
|
103
|
304 |
only if $\forall x.P(x)\imp Q(x)$.}
|
|
305 |
|
355
|
306 |
The package returns its result as an ML structure, which consists of named
|
|
307 |
components; we may regard it as a record. The result structure contains
|
|
308 |
the definitions of the recursive sets as a theorem list called {\tt defs}.
|
|
309 |
It also contains, as the theorem {\tt unfold}, a fixedpoint equation such
|
|
310 |
as
|
103
|
311 |
\begin{eqnarray*}
|
|
312 |
\Fin(A) & = &
|
|
313 |
\begin{array}[t]{r@{\,}l}
|
|
314 |
\{z\in\pow(A). & z=\emptyset \disj{} \\
|
|
315 |
&(\exists a\,b. z=\{a\}\un b\conj a\in A\conj b\in \Fin(A))\}
|
|
316 |
\end{array}
|
|
317 |
\end{eqnarray*}
|
130
|
318 |
It also contains, as the theorem {\tt dom\_subset}, an inclusion such as
|
103
|
319 |
$\Fin(A)\sbs\pow(A)$.
|
|
320 |
|
|
321 |
|
|
322 |
\subsection{Mutual recursion} \label{mutual-sec}
|
130
|
323 |
In a mutually recursive definition, the domain of the fixedpoint construction
|
103
|
324 |
is the disjoint sum of the domain~$D_i$ of each~$R_i$, for $i=1$,
|
|
325 |
\ldots,~$n$. The package uses the injections of the
|
|
326 |
binary disjoint sum, typically $\Inl$ and~$\Inr$, to express injections
|
130
|
327 |
$h_{1n}$, \ldots, $h_{nn}$ for the $n$-ary disjoint sum $D_1+\cdots+D_n$.
|
103
|
328 |
|
|
329 |
As discussed elsewhere \cite[\S4.5]{paulson-set-II}, Isabelle/ZF defines the
|
|
330 |
operator $\Part$ to support mutual recursion. The set $\Part(A,h)$
|
|
331 |
contains those elements of~$A$ having the form~$h(z)$:
|
|
332 |
\begin{eqnarray*}
|
|
333 |
\Part(A,h) & \equiv & \{x\in A. \exists z. x=h(z)\}.
|
|
334 |
\end{eqnarray*}
|
|
335 |
For mutually recursive sets $R_1$, \ldots,~$R_n$ with
|
|
336 |
$n>1$, the package makes $n+1$ definitions. The first defines a set $R$ using
|
|
337 |
a fixedpoint operator. The remaining $n$ definitions have the form
|
|
338 |
\begin{eqnarray*}
|
130
|
339 |
R_i & \equiv & \Part(R,h_{in}), \qquad i=1,\ldots, n.
|
103
|
340 |
\end{eqnarray*}
|
|
341 |
It follows that $R=R_1\un\cdots\un R_n$, where the $R_i$ are pairwise disjoint.
|
|
342 |
|
|
343 |
|
|
344 |
\subsection{Proving the introduction rules}
|
130
|
345 |
The user supplies the package with the desired form of the introduction
|
103
|
346 |
rules. Once it has derived the theorem {\tt unfold}, it attempts
|
130
|
347 |
to prove those rules. From the user's point of view, this is the
|
103
|
348 |
trickiest stage; the proofs often fail. The task is to show that the domain
|
|
349 |
$D_1+\cdots+D_n$ of the combined set $R_1\un\cdots\un R_n$ is
|
|
350 |
closed under all the introduction rules. This essentially involves replacing
|
|
351 |
each~$R_i$ by $D_1+\cdots+D_n$ in each of the introduction rules and
|
|
352 |
attempting to prove the result.
|
|
353 |
|
|
354 |
Consider the $\Fin(A)$ example. After substituting $\pow(A)$ for $\Fin(A)$
|
|
355 |
in the rules, the package must prove
|
|
356 |
\[ \emptyset\in\pow(A) \qquad
|
|
357 |
\infer{\{a\}\un b\in\pow(A)}{a\in A & b\in\pow(A)}
|
|
358 |
\]
|
|
359 |
Such proofs can be regarded as type-checking the definition. The user
|
|
360 |
supplies the package with type-checking rules to apply. Usually these are
|
|
361 |
general purpose rules from the ZF theory. They could however be rules
|
|
362 |
specifically proved for a particular inductive definition; sometimes this is
|
|
363 |
the easiest way to get the definition through!
|
|
364 |
|
130
|
365 |
The result structure contains the introduction rules as the theorem list {\tt
|
|
366 |
intrs}.
|
103
|
367 |
|
355
|
368 |
\subsection{The case analysis rule}
|
|
369 |
The elimination rule, called {\tt elim}, performs case analysis. There is one
|
130
|
370 |
case for each introduction rule. The elimination rule
|
|
371 |
for $\Fin(A)$ is
|
103
|
372 |
\[ \infer{Q}{x\in\Fin(A) & \infer*{Q}{[x=\emptyset]}
|
|
373 |
& \infer*{Q}{[x=\{a\}\un b & a\in A &b\in\Fin(A)]_{a,b}} }
|
|
374 |
\]
|
355
|
375 |
The subscripted variables $a$ and~$b$ above the third premise are
|
|
376 |
eigenvariables, subject to the usual `not free in \ldots' proviso.
|
|
377 |
The rule states that if $x\in\Fin(A)$ then either $x=\emptyset$ or else
|
130
|
378 |
$x=\{a\}\un b$ for some $a\in A$ and $b\in\Fin(A)$; it is a simple consequence
|
|
379 |
of {\tt unfold}.
|
|
380 |
|
355
|
381 |
The package also returns a function for generating simplified instances of
|
|
382 |
the case analysis rule. It works for datatypes and for inductive
|
|
383 |
definitions involving datatypes, such as an inductively defined relation
|
|
384 |
between lists. It instantiates {\tt elim} with a user-supplied term then
|
|
385 |
simplifies the cases using freeness of the underlying datatype. The
|
|
386 |
simplified rules perform `rule inversion' on the inductive definition.
|
|
387 |
Section~\S\ref{mkcases} presents an example.
|
|
388 |
|
103
|
389 |
|
130
|
390 |
\section{Induction and coinduction rules}
|
|
391 |
Here we must consider inductive and coinductive definitions separately.
|
103
|
392 |
For an inductive definition, the package returns an induction rule derived
|
|
393 |
directly from the properties of least fixedpoints, as well as a modified
|
|
394 |
rule for mutual recursion and inductively defined relations. For a
|
130
|
395 |
coinductive definition, the package returns a basic coinduction rule.
|
103
|
396 |
|
|
397 |
\subsection{The basic induction rule}\label{basic-ind-sec}
|
130
|
398 |
The basic rule, called {\tt induct}, is appropriate in most situations.
|
103
|
399 |
For inductive definitions, it is strong rule induction~\cite{camilleri92}; for
|
|
400 |
datatype definitions (see below), it is just structural induction.
|
|
401 |
|
|
402 |
The induction rule for an inductively defined set~$R$ has the following form.
|
|
403 |
The major premise is $x\in R$. There is a minor premise for each
|
|
404 |
introduction rule:
|
|
405 |
\begin{itemize}
|
|
406 |
\item If the introduction rule concludes $t\in R_i$, then the minor premise
|
|
407 |
is~$P(t)$.
|
|
408 |
|
|
409 |
\item The minor premise's eigenvariables are precisely the introduction
|
130
|
410 |
rule's free variables that are not parameters of~$R$. For instance, the
|
|
411 |
eigenvariables in the $\Fin(A)$ rule below are $a$ and $b$, but not~$A$.
|
103
|
412 |
|
|
413 |
\item If the introduction rule has a premise $t\in R_i$, then the minor
|
|
414 |
premise discharges the assumption $t\in R_i$ and the induction
|
|
415 |
hypothesis~$P(t)$. If the introduction rule has a premise $t\in M(R_i)$
|
|
416 |
then the minor premise discharges the single assumption
|
|
417 |
\[ t\in M(\{z\in R_i. P(z)\}). \]
|
|
418 |
Because $M$ is monotonic, this assumption implies $t\in M(R_i)$. The
|
|
419 |
occurrence of $P$ gives the effect of an induction hypothesis, which may be
|
|
420 |
exploited by appealing to properties of~$M$.
|
|
421 |
\end{itemize}
|
130
|
422 |
The induction rule for $\Fin(A)$ resembles the elimination rule shown above,
|
|
423 |
but includes an induction hypothesis:
|
103
|
424 |
\[ \infer{P(x)}{x\in\Fin(A) & P(\emptyset)
|
|
425 |
& \infer*{P(\{a\}\un b)}{[a\in A & b\in\Fin(A) & P(b)]_{a,b}} }
|
|
426 |
\]
|
355
|
427 |
Stronger induction rules often suggest themselves. We can derive a rule
|
|
428 |
for $\Fin(A)$ whose third premise discharges the extra assumption $a\not\in
|
|
429 |
b$. The Isabelle/ZF theory defines the {\bf rank} of a
|
|
430 |
set~\cite[\S3.4]{paulson-set-II}, which supports well-founded induction and
|
|
431 |
recursion over datatypes. The package proves a rule for mutual induction
|
|
432 |
and inductive relations.
|
103
|
433 |
|
|
434 |
\subsection{Mutual induction}
|
|
435 |
The mutual induction rule is called {\tt
|
|
436 |
mutual\_induct}. It differs from the basic rule in several respects:
|
|
437 |
\begin{itemize}
|
|
438 |
\item Instead of a single predicate~$P$, it uses $n$ predicates $P_1$,
|
|
439 |
\ldots,~$P_n$: one for each recursive set.
|
|
440 |
|
|
441 |
\item There is no major premise such as $x\in R_i$. Instead, the conclusion
|
|
442 |
refers to all the recursive sets:
|
|
443 |
\[ (\forall z.z\in R_1\imp P_1(z))\conj\cdots\conj
|
|
444 |
(\forall z.z\in R_n\imp P_n(z))
|
|
445 |
\]
|
355
|
446 |
Proving the premises establishes $P_i(z)$ for $z\in R_i$ and $i=1$,
|
|
447 |
\ldots,~$n$.
|
103
|
448 |
|
|
449 |
\item If the domain of some $R_i$ is the Cartesian product
|
|
450 |
$A_1\times\cdots\times A_m$, then the corresponding predicate $P_i$ takes $m$
|
|
451 |
arguments and the corresponding conjunct of the conclusion is
|
|
452 |
\[ (\forall z_1\ldots z_m.\pair{z_1,\ldots,z_m}\in R_i\imp P_i(z_1,\ldots,z_m))
|
|
453 |
\]
|
|
454 |
\end{itemize}
|
|
455 |
The last point above simplifies reasoning about inductively defined
|
|
456 |
relations. It eliminates the need to express properties of $z_1$,
|
|
457 |
\ldots,~$z_m$ as properties of the tuple $\pair{z_1,\ldots,z_m}$.
|
|
458 |
|
130
|
459 |
\subsection{Coinduction}\label{coind-sec}
|
|
460 |
A coinductive definition yields a primitive coinduction rule, with no
|
103
|
461 |
refinements such as those for the induction rules. (Experience may suggest
|
130
|
462 |
refinements later.) Consider the codatatype of lazy lists as an example. For
|
103
|
463 |
suitable definitions of $\LNil$ and $\LCons$, lazy lists may be defined as the
|
|
464 |
greatest fixedpoint satisfying the rules
|
|
465 |
\[ \LNil\in\llist(A) \qquad
|
|
466 |
\infer[(-)]{\LCons(a,l)\in\llist(A)}{a\in A & l\in\llist(A)}
|
|
467 |
\]
|
130
|
468 |
The $(-)$ tag stresses that this is a coinductive definition. A suitable
|
103
|
469 |
domain for $\llist(A)$ is $\quniv(A)$, a set closed under variant forms of
|
|
470 |
sum and product for representing infinite data structures
|
130
|
471 |
(see~\S\ref{univ-sec}). Coinductive definitions use these variant sums and
|
103
|
472 |
products.
|
|
473 |
|
|
474 |
The package derives an {\tt unfold} theorem similar to that for $\Fin(A)$.
|
355
|
475 |
Then it proves the theorem {\tt coinduct}, which expresses that $\llist(A)$
|
103
|
476 |
is the greatest solution to this equation contained in $\quniv(A)$:
|
130
|
477 |
\[ \infer{x\in\llist(A)}{x\in X & X\sbs \quniv(A) &
|
103
|
478 |
\infer*{z=\LNil\disj \bigl(\exists a\,l.\,
|
355
|
479 |
z=\LCons(a,l) \conj a\in A \conj l\in X\un\llist(A) \bigr)}
|
|
480 |
{[z\in X]_z}}
|
|
481 |
% \begin{array}[t]{@{}l}
|
|
482 |
% z=\LCons(a,l) \conj a\in A \conj{}\\
|
|
483 |
% l\in X\un\llist(A) \bigr)
|
|
484 |
% \end{array} }{[z\in X]_z}}
|
103
|
485 |
\]
|
130
|
486 |
This rule complements the introduction rules; it provides a means of showing
|
|
487 |
$x\in\llist(A)$ when $x$ is infinite. For instance, if $x=\LCons(0,x)$ then
|
355
|
488 |
applying the rule with $X=\{x\}$ proves $x\in\llist(\nat)$. (Here $\nat$
|
|
489 |
is the set of natural numbers.)
|
130
|
490 |
|
103
|
491 |
Having $X\un\llist(A)$ instead of simply $X$ in the third premise above
|
|
492 |
represents a slight strengthening of the greatest fixedpoint property. I
|
130
|
493 |
discuss several forms of coinduction rules elsewhere~\cite{paulson-coind}.
|
103
|
494 |
|
|
495 |
|
130
|
496 |
\section{Examples of inductive and coinductive definitions}\label{ind-eg-sec}
|
103
|
497 |
This section presents several examples: the finite set operator,
|
|
498 |
lists of $n$ elements, bisimulations on lazy lists, the well-founded part
|
|
499 |
of a relation, and the primitive recursive functions.
|
|
500 |
|
|
501 |
\subsection{The finite set operator}
|
|
502 |
The definition of finite sets has been discussed extensively above. Here
|
|
503 |
is the corresponding ML invocation (note that $\cons(a,b)$ abbreviates
|
|
504 |
$\{a\}\un b$ in Isabelle/ZF):
|
|
505 |
\begin{ttbox}
|
|
506 |
structure Fin = Inductive_Fun
|
355
|
507 |
(val thy = Arith.thy addconsts [(["Fin"],"i=>i")]
|
|
508 |
val rec_doms = [("Fin","Pow(A)")]
|
|
509 |
val sintrs = ["0 : Fin(A)",
|
|
510 |
"[| a: A; b: Fin(A) |] ==> cons(a,b) : Fin(A)"]
|
|
511 |
val monos = []
|
|
512 |
val con_defs = []
|
|
513 |
val type_intrs = [empty_subsetI, cons_subsetI, PowI]
|
103
|
514 |
val type_elims = [make_elim PowD]);
|
|
515 |
\end{ttbox}
|
355
|
516 |
We apply the functor {\tt Inductive\_Fun} to a structure describing the
|
|
517 |
desired inductive definition. The parent theory~{\tt thy} is obtained from
|
|
518 |
{\tt Arith.thy} by adding the unary function symbol~$\Fin$. Its domain is
|
|
519 |
specified as $\pow(A)$, where $A$ is the parameter appearing in the
|
|
520 |
introduction rules. For type-checking, the structure supplies introduction
|
|
521 |
rules:
|
103
|
522 |
\[ \emptyset\sbs A \qquad
|
|
523 |
\infer{\{a\}\un B\sbs C}{a\in C & B\sbs C}
|
|
524 |
\]
|
|
525 |
A further introduction rule and an elimination rule express the two
|
|
526 |
directions of the equivalence $A\in\pow(B)\bimp A\sbs B$. Type-checking
|
355
|
527 |
involves mostly introduction rules.
|
|
528 |
|
|
529 |
ML is Isabelle's top level, so such functor invocations can take place at
|
|
530 |
any time. The result structure is declared with the name~{\tt Fin}; we can
|
|
531 |
refer to the $\Fin(A)$ introduction rules as {\tt Fin.intrs}, the induction
|
|
532 |
rule as {\tt Fin.induct} and so forth. There are plans to integrate the
|
|
533 |
package better into Isabelle so that users can place inductive definitions
|
|
534 |
in Isabelle theory files instead of applying functors.
|
|
535 |
|
103
|
536 |
|
|
537 |
\subsection{Lists of $n$ elements}\label{listn-sec}
|
179
|
538 |
This has become a standard example of an inductive definition. Following
|
|
539 |
Paulin-Mohring~\cite{paulin92}, we could attempt to define a new datatype
|
|
540 |
$\listn(A,n)$, for lists of length~$n$, as an $n$-indexed family of sets.
|
|
541 |
But her introduction rules
|
355
|
542 |
\[ \hbox{\tt Niln}\in\listn(A,0) \qquad
|
|
543 |
\infer{\hbox{\tt Consn}(n,a,l)\in\listn(A,\succ(n))}
|
103
|
544 |
{n\in\nat & a\in A & l\in\listn(A,n)}
|
|
545 |
\]
|
|
546 |
are not acceptable to the inductive definition package:
|
|
547 |
$\listn$ occurs with three different parameter lists in the definition.
|
|
548 |
|
|
549 |
\begin{figure}
|
355
|
550 |
\begin{ttbox}
|
103
|
551 |
structure ListN = Inductive_Fun
|
355
|
552 |
(val thy = ListFn.thy addconsts [(["listn"],"i=>i")]
|
|
553 |
val rec_doms = [("listn", "nat*list(A)")]
|
|
554 |
val sintrs =
|
|
555 |
["<0,Nil>: listn(A)",
|
|
556 |
"[| a: A; <n,l>: listn(A) |] ==> <succ(n), Cons(a,l)>: listn(A)"]
|
|
557 |
val monos = []
|
|
558 |
val con_defs = []
|
|
559 |
val type_intrs = nat_typechecks @ List.intrs @ [SigmaI]
|
103
|
560 |
val type_elims = [SigmaE2]);
|
355
|
561 |
\end{ttbox}
|
103
|
562 |
\hrule
|
|
563 |
\caption{Defining lists of $n$ elements} \label{listn-fig}
|
|
564 |
\end{figure}
|
|
565 |
|
355
|
566 |
The Isabelle/ZF version of this example suggests a general treatment of
|
|
567 |
varying parameters. Here, we use the existing datatype definition of
|
|
568 |
$\lst(A)$, with constructors $\Nil$ and~$\Cons$. Then incorporate the
|
|
569 |
parameter~$n$ into the inductive set itself, defining $\listn(A)$ as a
|
|
570 |
relation. It consists of pairs $\pair{n,l}$ such that $n\in\nat$
|
|
571 |
and~$l\in\lst(A)$ and $l$ has length~$n$. In fact, $\listn(A)$ is the
|
|
572 |
converse of the length function on~$\lst(A)$. The Isabelle/ZF introduction
|
|
573 |
rules are
|
103
|
574 |
\[ \pair{0,\Nil}\in\listn(A) \qquad
|
|
575 |
\infer{\pair{\succ(n),\Cons(a,l)}\in\listn(A)}
|
|
576 |
{a\in A & \pair{n,l}\in\listn(A)}
|
|
577 |
\]
|
|
578 |
Figure~\ref{listn-fig} presents the ML invocation. A theory of lists,
|
|
579 |
extended with a declaration of $\listn$, is the parent theory. The domain
|
|
580 |
is specified as $\nat\times\lst(A)$. The type-checking rules include those
|
|
581 |
for 0, $\succ$, $\Nil$ and $\Cons$. Because $\listn(A)$ is a set of pairs,
|
|
582 |
type-checking also requires introduction and elimination rules to express
|
|
583 |
both directions of the equivalence $\pair{a,b}\in A\times B \bimp a\in A
|
|
584 |
\conj b\in B$.
|
|
585 |
|
|
586 |
The package returns introduction, elimination and induction rules for
|
|
587 |
$\listn$. The basic induction rule, {\tt ListN.induct}, is
|
|
588 |
\[ \infer{P(x)}{x\in\listn(A) & P(\pair{0,\Nil}) &
|
|
589 |
\infer*{P(\pair{\succ(n),\Cons(a,l)})}
|
|
590 |
{[a\in A & \pair{n,l}\in\listn(A) & P(\pair{n,l})]_{a,l,n}}}
|
|
591 |
\]
|
|
592 |
This rule requires the induction formula to be a
|
|
593 |
unary property of pairs,~$P(\pair{n,l})$. The alternative rule, {\tt
|
|
594 |
ListN.mutual\_induct}, uses a binary property instead:
|
130
|
595 |
\[ \infer{\forall n\,l. \pair{n,l}\in\listn(A) \imp P(n,l)}
|
103
|
596 |
{P(0,\Nil) &
|
|
597 |
\infer*{P(\succ(n),\Cons(a,l))}
|
|
598 |
{[a\in A & \pair{n,l}\in\listn(A) & P(n,l)]_{a,l,n}}}
|
|
599 |
\]
|
|
600 |
It is now a simple matter to prove theorems about $\listn(A)$, such as
|
|
601 |
\[ \forall l\in\lst(A). \pair{\length(l),\, l}\in\listn(A) \]
|
|
602 |
\[ \listn(A)``\{n\} = \{l\in\lst(A). \length(l)=n\} \]
|
130
|
603 |
This latter result --- here $r``X$ denotes the image of $X$ under $r$
|
103
|
604 |
--- asserts that the inductive definition agrees with the obvious notion of
|
|
605 |
$n$-element list.
|
|
606 |
|
|
607 |
Unlike in Coq, the definition does not declare a new datatype. A `list of
|
130
|
608 |
$n$ elements' really is a list and is subject to list operators such
|
|
609 |
as append (concatenation). For example, a trivial induction on
|
|
610 |
$\pair{m,l}\in\listn(A)$ yields
|
103
|
611 |
\[ \infer{\pair{m\mathbin{+} m,\, l@l'}\in\listn(A)}
|
|
612 |
{\pair{m,l}\in\listn(A) & \pair{m',l'}\in\listn(A)}
|
|
613 |
\]
|
|
614 |
where $+$ here denotes addition on the natural numbers and @ denotes append.
|
|
615 |
|
355
|
616 |
\subsection{A demonstration of rule inversion}\label{mkcases}
|
103
|
617 |
The elimination rule, {\tt ListN.elim}, is cumbersome:
|
|
618 |
\[ \infer{Q}{x\in\listn(A) &
|
|
619 |
\infer*{Q}{[x = \pair{0,\Nil}]} &
|
|
620 |
\infer*{Q}
|
|
621 |
{\left[\begin{array}{l}
|
|
622 |
x = \pair{\succ(n),\Cons(a,l)} \\
|
|
623 |
a\in A \\
|
|
624 |
\pair{n,l}\in\listn(A)
|
|
625 |
\end{array} \right]_{a,l,n}}}
|
|
626 |
\]
|
179
|
627 |
The ML function {\tt ListN.mk\_cases} generates simplified instances of
|
|
628 |
this rule. It works by freeness reasoning on the list constructors:
|
|
629 |
$\Cons(a,l)$ is injective in its two arguments and differs from~$\Nil$. If
|
|
630 |
$x$ is $\pair{i,\Nil}$ or $\pair{i,\Cons(a,l)}$ then {\tt ListN.mk\_cases}
|
355
|
631 |
deduces the corresponding form of~$i$; this is called rule inversion. For
|
|
632 |
example,
|
103
|
633 |
\begin{ttbox}
|
|
634 |
ListN.mk_cases List.con_defs "<i,Cons(a,l)> : listn(A)"
|
|
635 |
\end{ttbox}
|
130
|
636 |
yields a rule with only two premises:
|
103
|
637 |
\[ \infer{Q}{\pair{i, \Cons(a,l)}\in\listn(A) &
|
|
638 |
\infer*{Q}
|
|
639 |
{\left[\begin{array}{l}
|
|
640 |
i = \succ(n) \\ a\in A \\ \pair{n,l}\in\listn(A)
|
|
641 |
\end{array} \right]_{n}}}
|
|
642 |
\]
|
|
643 |
The package also has built-in rules for freeness reasoning about $0$
|
|
644 |
and~$\succ$. So if $x$ is $\pair{0,l}$ or $\pair{\succ(i),l}$, then {\tt
|
|
645 |
ListN.mk\_cases} can similarly deduce the corresponding form of~$l$.
|
|
646 |
|
355
|
647 |
The function {\tt mk\_cases} is also useful with datatype definitions. The
|
|
648 |
instance from the definition of lists, namely {\tt List.mk\_cases}, can
|
|
649 |
prove the rule
|
103
|
650 |
\[ \infer{Q}{\Cons(a,l)\in\lst(A) &
|
|
651 |
& \infer*{Q}{[a\in A &l\in\lst(A)]} }
|
|
652 |
\]
|
355
|
653 |
A typical use of {\tt mk\_cases} concerns inductive definitions of
|
|
654 |
evaluation relations. Then rule inversion yields case analysis on possible
|
|
655 |
evaluations. For example, the Isabelle/ZF theory includes a short proof
|
|
656 |
of the diamond property for parallel contraction on combinators.
|
103
|
657 |
|
130
|
658 |
\subsection{A coinductive definition: bisimulations on lazy lists}
|
|
659 |
This example anticipates the definition of the codatatype $\llist(A)$, which
|
|
660 |
consists of finite and infinite lists over~$A$. Its constructors are $\LNil$
|
|
661 |
and
|
|
662 |
$\LCons$, satisfying the introduction rules shown in~\S\ref{coind-sec}.
|
103
|
663 |
Because $\llist(A)$ is defined as a greatest fixedpoint and uses the variant
|
|
664 |
pairing and injection operators, it contains non-well-founded elements such as
|
|
665 |
solutions to $\LCons(a,l)=l$.
|
|
666 |
|
130
|
667 |
The next step in the development of lazy lists is to define a coinduction
|
103
|
668 |
principle for proving equalities. This is done by showing that the equality
|
|
669 |
relation on lazy lists is the greatest fixedpoint of some monotonic
|
|
670 |
operation. The usual approach~\cite{pitts94} is to define some notion of
|
|
671 |
bisimulation for lazy lists, define equivalence to be the greatest
|
|
672 |
bisimulation, and finally to prove that two lazy lists are equivalent if and
|
130
|
673 |
only if they are equal. The coinduction rule for equivalence then yields a
|
|
674 |
coinduction principle for equalities.
|
103
|
675 |
|
|
676 |
A binary relation $R$ on lazy lists is a {\bf bisimulation} provided $R\sbs
|
|
677 |
R^+$, where $R^+$ is the relation
|
130
|
678 |
\[ \{\pair{\LNil,\LNil}\} \un
|
|
679 |
\{\pair{\LCons(a,l),\LCons(a,l')} . a\in A \conj \pair{l,l'}\in R\}.
|
103
|
680 |
\]
|
|
681 |
|
|
682 |
A pair of lazy lists are {\bf equivalent} if they belong to some bisimulation.
|
130
|
683 |
Equivalence can be coinductively defined as the greatest fixedpoint for the
|
103
|
684 |
introduction rules
|
130
|
685 |
\[ \pair{\LNil,\LNil} \in\lleq(A) \qquad
|
|
686 |
\infer[(-)]{\pair{\LCons(a,l),\LCons(a,l')} \in\lleq(A)}
|
|
687 |
{a\in A & \pair{l,l'}\in \lleq(A)}
|
103
|
688 |
\]
|
130
|
689 |
To make this coinductive definition, we invoke \verb|CoInductive_Fun|:
|
103
|
690 |
\begin{ttbox}
|
130
|
691 |
structure LList_Eq = CoInductive_Fun
|
355
|
692 |
(val thy = LList.thy addconsts [(["lleq"],"i=>i")]
|
|
693 |
val rec_doms = [("lleq", "llist(A) * llist(A)")]
|
|
694 |
val sintrs =
|
|
695 |
["<LNil, LNil> : lleq(A)",
|
|
696 |
"[| a:A; <l,l'>: lleq(A) |] ==> <LCons(a,l),LCons(a,l')>: lleq(A)"]
|
|
697 |
val monos = []
|
|
698 |
val con_defs = []
|
|
699 |
val type_intrs = LList.intrs @ [SigmaI]
|
|
700 |
val type_elims = [SigmaE2]);
|
103
|
701 |
\end{ttbox}
|
|
702 |
Again, {\tt addconsts} declares a constant for $\lleq$ in the parent theory.
|
130
|
703 |
The domain of $\lleq(A)$ is $\llist(A)\times\llist(A)$. The type-checking
|
|
704 |
rules include the introduction rules for lazy lists as well as rules
|
|
705 |
for both directions of the equivalence
|
|
706 |
$\pair{a,b}\in A\times B \bimp a\in A \conj b\in B$.
|
103
|
707 |
|
|
708 |
The package returns the introduction rules and the elimination rule, as
|
130
|
709 |
usual. But instead of induction rules, it returns a coinduction rule.
|
103
|
710 |
The rule is too big to display in the usual notation; its conclusion is
|
130
|
711 |
$x\in\lleq(A)$ and its premises are $x\in X$,
|
|
712 |
${X\sbs\llist(A)\times\llist(A)}$ and
|
|
713 |
\[ \infer*{z=\pair{\LNil,\LNil}\disj \bigl(\exists a\,l\,l'.\,
|
355
|
714 |
z=\pair{\LCons(a,l),\LCons(a,l')} \conj
|
|
715 |
a\in A \conj\pair{l,l'}\in X\un\lleq(A) \bigr)
|
|
716 |
% \begin{array}[t]{@{}l}
|
|
717 |
% z=\pair{\LCons(a,l),\LCons(a,l')} \conj a\in A \conj{}\\
|
|
718 |
% \pair{l,l'}\in X\un\lleq(A) \bigr)
|
|
719 |
% \end{array}
|
|
720 |
}{[z\in X]_z}
|
103
|
721 |
\]
|
130
|
722 |
Thus if $x\in X$, where $X$ is a bisimulation contained in the
|
|
723 |
domain of $\lleq(A)$, then $x\in\lleq(A)$. It is easy to show that
|
103
|
724 |
$\lleq(A)$ is reflexive: the equality relation is a bisimulation. And
|
|
725 |
$\lleq(A)$ is symmetric: its converse is a bisimulation. But showing that
|
130
|
726 |
$\lleq(A)$ coincides with the equality relation takes some work.
|
103
|
727 |
|
|
728 |
\subsection{The accessible part of a relation}\label{acc-sec}
|
|
729 |
Let $\prec$ be a binary relation on~$D$; in short, $(\prec)\sbs D\times D$.
|
|
730 |
The {\bf accessible} or {\bf well-founded} part of~$\prec$, written
|
|
731 |
$\acc(\prec)$, is essentially that subset of~$D$ for which $\prec$ admits
|
|
732 |
no infinite decreasing chains~\cite{aczel77}. Formally, $\acc(\prec)$ is
|
|
733 |
inductively defined to be the least set that contains $a$ if it contains
|
|
734 |
all $\prec$-predecessors of~$a$, for $a\in D$. Thus we need an
|
|
735 |
introduction rule of the form
|
|
736 |
\[ \infer{a\in\acc(\prec)}{\forall y.y\prec a\imp y\in\acc(\prec)} \]
|
|
737 |
Paulin-Mohring treats this example in Coq~\cite{paulin92}, but it causes
|
|
738 |
difficulties for other systems. Its premise does not conform to
|
|
739 |
the structure of introduction rules for HOL's inductive definition
|
|
740 |
package~\cite{camilleri92}. It is also unacceptable to Isabelle package
|
130
|
741 |
(\S\ref{intro-sec}), but fortunately can be transformed into the acceptable
|
103
|
742 |
form $t\in M(R)$.
|
|
743 |
|
|
744 |
The powerset operator is monotonic, and $t\in\pow(R)$ is equivalent to
|
|
745 |
$t\sbs R$. This in turn is equivalent to $\forall y\in t. y\in R$. To
|
|
746 |
express $\forall y.y\prec a\imp y\in\acc(\prec)$ we need only find a
|
|
747 |
term~$t$ such that $y\in t$ if and only if $y\prec a$. A suitable $t$ is
|
|
748 |
the inverse image of~$\{a\}$ under~$\prec$.
|
|
749 |
|
|
750 |
The ML invocation below follows this approach. Here $r$ is~$\prec$ and
|
130
|
751 |
$\field(r)$ refers to~$D$, the domain of $\acc(r)$. (The field of a
|
|
752 |
relation is the union of its domain and range.) Finally
|
|
753 |
$r^{-}``\{a\}$ denotes the inverse image of~$\{a\}$ under~$r$. The package is
|
|
754 |
supplied the theorem {\tt Pow\_mono}, which asserts that $\pow$ is monotonic.
|
103
|
755 |
\begin{ttbox}
|
|
756 |
structure Acc = Inductive_Fun
|
355
|
757 |
(val thy = WF.thy addconsts [(["acc"],"i=>i")]
|
|
758 |
val rec_doms = [("acc", "field(r)")]
|
|
759 |
val sintrs = ["[| r-``\{a\}:\,Pow(acc(r)); a:\,field(r) |] ==> a:\,acc(r)"]
|
|
760 |
val monos = [Pow_mono]
|
|
761 |
val con_defs = []
|
|
762 |
val type_intrs = []
|
103
|
763 |
val type_elims = []);
|
|
764 |
\end{ttbox}
|
|
765 |
The Isabelle theory proceeds to prove facts about $\acc(\prec)$. For
|
|
766 |
instance, $\prec$ is well-founded if and only if its field is contained in
|
|
767 |
$\acc(\prec)$.
|
|
768 |
|
|
769 |
As mentioned in~\S\ref{basic-ind-sec}, a premise of the form $t\in M(R)$
|
|
770 |
gives rise to an unusual induction hypothesis. Let us examine the
|
|
771 |
induction rule, {\tt Acc.induct}:
|
|
772 |
\[ \infer{P(x)}{x\in\acc(r) &
|
|
773 |
\infer*{P(a)}{[r^{-}``\{a\}\in\pow(\{z\in\acc(r).P(z)\}) &
|
|
774 |
a\in\field(r)]_a}}
|
|
775 |
\]
|
|
776 |
The strange induction hypothesis is equivalent to
|
|
777 |
$\forall y. \pair{y,a}\in r\imp y\in\acc(r)\conj P(y)$.
|
|
778 |
Therefore the rule expresses well-founded induction on the accessible part
|
|
779 |
of~$\prec$.
|
|
780 |
|
|
781 |
The use of inverse image is not essential. The Isabelle package can accept
|
|
782 |
introduction rules with arbitrary premises of the form $\forall
|
|
783 |
\vec{y}.P(\vec{y})\imp f(\vec{y})\in R$. The premise can be expressed
|
|
784 |
equivalently as
|
130
|
785 |
\[ \{z\in D. P(\vec{y}) \conj z=f(\vec{y})\} \in \pow(R) \]
|
103
|
786 |
provided $f(\vec{y})\in D$ for all $\vec{y}$ such that~$P(\vec{y})$. The
|
|
787 |
following section demonstrates another use of the premise $t\in M(R)$,
|
|
788 |
where $M=\lst$.
|
|
789 |
|
|
790 |
\subsection{The primitive recursive functions}\label{primrec-sec}
|
|
791 |
The primitive recursive functions are traditionally defined inductively, as
|
|
792 |
a subset of the functions over the natural numbers. One difficulty is that
|
|
793 |
functions of all arities are taken together, but this is easily
|
|
794 |
circumvented by regarding them as functions on lists. Another difficulty,
|
|
795 |
the notion of composition, is less easily circumvented.
|
|
796 |
|
|
797 |
Here is a more precise definition. Letting $\vec{x}$ abbreviate
|
|
798 |
$x_0,\ldots,x_{n-1}$, we can write lists such as $[\vec{x}]$,
|
|
799 |
$[y+1,\vec{x}]$, etc. A function is {\bf primitive recursive} if it
|
|
800 |
belongs to the least set of functions in $\lst(\nat)\to\nat$ containing
|
|
801 |
\begin{itemize}
|
|
802 |
\item The {\bf successor} function $\SC$, such that $\SC[y,\vec{x}]=y+1$.
|
|
803 |
\item All {\bf constant} functions $\CONST(k)$, such that
|
|
804 |
$\CONST(k)[\vec{x}]=k$.
|
|
805 |
\item All {\bf projection} functions $\PROJ(i)$, such that
|
|
806 |
$\PROJ(i)[\vec{x}]=x_i$ if $0\leq i<n$.
|
|
807 |
\item All {\bf compositions} $\COMP(g,[f_0,\ldots,f_{m-1}])$,
|
|
808 |
where $g$ and $f_0$, \ldots, $f_{m-1}$ are primitive recursive,
|
|
809 |
such that
|
|
810 |
\begin{eqnarray*}
|
|
811 |
\COMP(g,[f_0,\ldots,f_{m-1}])[\vec{x}] & = &
|
|
812 |
g[f_0[\vec{x}],\ldots,f_{m-1}[\vec{x}]].
|
|
813 |
\end{eqnarray*}
|
|
814 |
|
|
815 |
\item All {\bf recursions} $\PREC(f,g)$, where $f$ and $g$ are primitive
|
|
816 |
recursive, such that
|
|
817 |
\begin{eqnarray*}
|
|
818 |
\PREC(f,g)[0,\vec{x}] & = & f[\vec{x}] \\
|
|
819 |
\PREC(f,g)[y+1,\vec{x}] & = & g[\PREC(f,g)[y,\vec{x}],\, y,\, \vec{x}].
|
|
820 |
\end{eqnarray*}
|
|
821 |
\end{itemize}
|
|
822 |
Composition is awkward because it combines not two functions, as is usual,
|
|
823 |
but $m+1$ functions. In her proof that Ackermann's function is not
|
|
824 |
primitive recursive, Nora Szasz was unable to formalize this definition
|
|
825 |
directly~\cite{szasz93}. So she generalized primitive recursion to
|
|
826 |
tuple-valued functions. This modified the inductive definition such that
|
|
827 |
each operation on primitive recursive functions combined just two functions.
|
|
828 |
|
|
829 |
\begin{figure}
|
355
|
830 |
\begin{ttbox}
|
103
|
831 |
structure Primrec = Inductive_Fun
|
355
|
832 |
(val thy = Primrec0.thy
|
|
833 |
val rec_doms = [("primrec", "list(nat)->nat")]
|
|
834 |
val sintrs =
|
|
835 |
["SC : primrec",
|
|
836 |
"k: nat ==> CONST(k) : primrec",
|
|
837 |
"i: nat ==> PROJ(i) : primrec",
|
|
838 |
"[| g: primrec; fs: list(primrec) |] ==> COMP(g,fs): primrec",
|
|
839 |
"[| f: primrec; g: primrec |] ==> PREC(f,g): primrec"]
|
|
840 |
val monos = [list_mono]
|
|
841 |
val con_defs = [SC_def,CONST_def,PROJ_def,COMP_def,PREC_def]
|
|
842 |
val type_intrs = pr0_typechecks
|
103
|
843 |
val type_elims = []);
|
355
|
844 |
\end{ttbox}
|
103
|
845 |
\hrule
|
|
846 |
\caption{Inductive definition of the primitive recursive functions}
|
|
847 |
\label{primrec-fig}
|
|
848 |
\end{figure}
|
|
849 |
\def\fs{{\it fs}}
|
|
850 |
Szasz was using ALF, but Coq and HOL would also have problems accepting
|
|
851 |
this definition. Isabelle's package accepts it easily since
|
|
852 |
$[f_0,\ldots,f_{m-1}]$ is a list of primitive recursive functions and
|
|
853 |
$\lst$ is monotonic. There are five introduction rules, one for each of
|
355
|
854 |
the five forms of primitive recursive function. Let us examine the one for
|
|
855 |
$\COMP$:
|
103
|
856 |
\[ \infer{\COMP(g,\fs)\in\primrec}{g\in\primrec & \fs\in\lst(\primrec)} \]
|
|
857 |
The induction rule for $\primrec$ has one case for each introduction rule.
|
|
858 |
Due to the use of $\lst$ as a monotone operator, the composition case has
|
|
859 |
an unusual induction hypothesis:
|
|
860 |
\[ \infer*{P(\COMP(g,\fs))}
|
130
|
861 |
{[g\in\primrec & \fs\in\lst(\{z\in\primrec.P(z)\})]_{\fs,g}} \]
|
103
|
862 |
The hypothesis states that $\fs$ is a list of primitive recursive functions
|
|
863 |
satisfying the induction formula. Proving the $\COMP$ case typically requires
|
|
864 |
structural induction on lists, yielding two subcases: either $\fs=\Nil$ or
|
|
865 |
else $\fs=\Cons(f,\fs')$, where $f\in\primrec$, $P(f)$, and $\fs'$ is
|
|
866 |
another list of primitive recursive functions satisfying~$P$.
|
|
867 |
|
|
868 |
Figure~\ref{primrec-fig} presents the ML invocation. Theory {\tt
|
355
|
869 |
Primrec0.thy} defines the constants $\SC$, $\CONST$, etc. These are not
|
|
870 |
constructors of a new datatype, but functions over lists of numbers. Their
|
|
871 |
definitions, which are omitted, consist of routine list programming. In
|
|
872 |
Isabelle/ZF, the primitive recursive functions are defined as a subset of
|
|
873 |
the function set $\lst(\nat)\to\nat$.
|
103
|
874 |
|
355
|
875 |
The Isabelle theory goes on to formalize Ackermann's function and prove
|
|
876 |
that it is not primitive recursive, using the induction rule {\tt
|
|
877 |
Primrec.induct}. The proof follows Szasz's excellent account.
|
103
|
878 |
|
|
879 |
|
130
|
880 |
\section{Datatypes and codatatypes}\label{data-sec}
|
|
881 |
A (co)datatype definition is a (co)inductive definition with automatically
|
355
|
882 |
defined constructors and a case analysis operator. The package proves that
|
|
883 |
the case operator inverts the constructors and can prove freeness theorems
|
103
|
884 |
involving any pair of constructors.
|
|
885 |
|
|
886 |
|
130
|
887 |
\subsection{Constructors and their domain}\label{univ-sec}
|
355
|
888 |
Conceptually, our two forms of definition are distinct. A (co)inductive
|
|
889 |
definition selects a subset of an existing set; a (co)datatype definition
|
|
890 |
creates a new set. But the package reduces the latter to the former. A
|
|
891 |
set having strong closure properties must serve as the domain of the
|
|
892 |
(co)inductive definition. Constructing this set requires some theoretical
|
|
893 |
effort, which must be done anyway to show that (co)datatypes exist. It is
|
|
894 |
not obvious that standard set theory is suitable for defining codatatypes.
|
103
|
895 |
|
|
896 |
Isabelle/ZF defines the standard notion of Cartesian product $A\times B$,
|
|
897 |
containing ordered pairs $\pair{a,b}$. Now the $m$-tuple
|
355
|
898 |
$\pair{x_1,\ldots,x_m}$ is the empty set~$\emptyset$ if $m=0$, simply
|
|
899 |
$x_1$ if $m=1$ and $\pair{x_1,\pair{x_2,\ldots,x_m}}$ if $m\geq2$.
|
103
|
900 |
Isabelle/ZF also defines the disjoint sum $A+B$, containing injections
|
|
901 |
$\Inl(a)\equiv\pair{0,a}$ and $\Inr(b)\equiv\pair{1,b}$.
|
|
902 |
|
355
|
903 |
A datatype constructor $\Con(x_1,\ldots,x_m)$ is defined to be
|
|
904 |
$h(\pair{x_1,\ldots,x_m})$, where $h$ is composed of $\Inl$ and~$\Inr$.
|
103
|
905 |
In a mutually recursive definition, all constructors for the set~$R_i$ have
|
130
|
906 |
the outer form~$h_{in}$, where $h_{in}$ is the injection described
|
103
|
907 |
in~\S\ref{mutual-sec}. Further nested injections ensure that the
|
|
908 |
constructors for~$R_i$ are pairwise distinct.
|
|
909 |
|
|
910 |
Isabelle/ZF defines the set $\univ(A)$, which contains~$A$ and
|
|
911 |
furthermore contains $\pair{a,b}$, $\Inl(a)$ and $\Inr(b)$ for $a$,
|
|
912 |
$b\in\univ(A)$. In a typical datatype definition with set parameters
|
|
913 |
$A_1$, \ldots, $A_k$, a suitable domain for all the recursive sets is
|
|
914 |
$\univ(A_1\un\cdots\un A_k)$. This solves the problem for
|
|
915 |
datatypes~\cite[\S4.2]{paulson-set-II}.
|
|
916 |
|
|
917 |
The standard pairs and injections can only yield well-founded
|
|
918 |
constructions. This eases the (manual!) definition of recursive functions
|
130
|
919 |
over datatypes. But they are unsuitable for codatatypes, which typically
|
103
|
920 |
contain non-well-founded objects.
|
|
921 |
|
130
|
922 |
To support codatatypes, Isabelle/ZF defines a variant notion of ordered
|
103
|
923 |
pair, written~$\pair{a;b}$. It also defines the corresponding variant
|
|
924 |
notion of Cartesian product $A\otimes B$, variant injections $\QInl(a)$
|
355
|
925 |
and~$\QInr(b)$ and variant disjoint sum $A\oplus B$. Finally it defines
|
103
|
926 |
the set $\quniv(A)$, which contains~$A$ and furthermore contains
|
|
927 |
$\pair{a;b}$, $\QInl(a)$ and $\QInr(b)$ for $a$, $b\in\quniv(A)$. In a
|
130
|
928 |
typical codatatype definition with set parameters $A_1$, \ldots, $A_k$, a
|
|
929 |
suitable domain is $\quniv(A_1\un\cdots\un A_k)$. This approach using
|
355
|
930 |
standard ZF set theory~\cite{paulson-final} is an alternative to adopting
|
|
931 |
Aczel's Anti-Foundation Axiom~\cite{aczel88}.
|
103
|
932 |
|
|
933 |
\subsection{The case analysis operator}
|
130
|
934 |
The (co)datatype package automatically defines a case analysis operator,
|
179
|
935 |
called {\tt$R$\_case}. A mutually recursive definition still has only one
|
|
936 |
operator, whose name combines those of the recursive sets: it is called
|
|
937 |
{\tt$R_1$\_\ldots\_$R_n$\_case}. The case operator is analogous to those
|
|
938 |
for products and sums.
|
103
|
939 |
|
|
940 |
Datatype definitions employ standard products and sums, whose operators are
|
|
941 |
$\split$ and $\case$ and satisfy the equations
|
|
942 |
\begin{eqnarray*}
|
|
943 |
\split(f,\pair{x,y}) & = & f(x,y) \\
|
|
944 |
\case(f,g,\Inl(x)) & = & f(x) \\
|
|
945 |
\case(f,g,\Inr(y)) & = & g(y)
|
|
946 |
\end{eqnarray*}
|
|
947 |
Suppose the datatype has $k$ constructors $\Con_1$, \ldots,~$\Con_k$. Then
|
|
948 |
its case operator takes $k+1$ arguments and satisfies an equation for each
|
|
949 |
constructor:
|
|
950 |
\begin{eqnarray*}
|
|
951 |
R\hbox{\_case}(f_1,\ldots,f_k, {\tt Con}_i(\vec{x})) & = & f_i(\vec{x}),
|
|
952 |
\qquad i = 1, \ldots, k
|
|
953 |
\end{eqnarray*}
|
130
|
954 |
The case operator's definition takes advantage of Isabelle's representation
|
|
955 |
of syntax in the typed $\lambda$-calculus; it could readily be adapted to a
|
|
956 |
theorem prover for higher-order logic. If $f$ and~$g$ have meta-type
|
|
957 |
$i\To i$ then so do $\split(f)$ and
|
|
958 |
$\case(f,g)$. This works because $\split$ and $\case$ operate on their last
|
|
959 |
argument. They are easily combined to make complex case analysis
|
103
|
960 |
operators. Here are two examples:
|
|
961 |
\begin{itemize}
|
|
962 |
\item $\split(\lambda x.\split(f(x)))$ performs case analysis for
|
|
963 |
$A\times (B\times C)$, as is easily verified:
|
|
964 |
\begin{eqnarray*}
|
|
965 |
\split(\lambda x.\split(f(x)), \pair{a,b,c})
|
|
966 |
& = & (\lambda x.\split(f(x))(a,\pair{b,c}) \\
|
|
967 |
& = & \split(f(a), \pair{b,c}) \\
|
|
968 |
& = & f(a,b,c)
|
|
969 |
\end{eqnarray*}
|
|
970 |
|
|
971 |
\item $\case(f,\case(g,h))$ performs case analysis for $A+(B+C)$; let us
|
|
972 |
verify one of the three equations:
|
|
973 |
\begin{eqnarray*}
|
|
974 |
\case(f,\case(g,h), \Inr(\Inl(b)))
|
|
975 |
& = & \case(g,h,\Inl(b)) \\
|
|
976 |
& = & g(b)
|
|
977 |
\end{eqnarray*}
|
|
978 |
\end{itemize}
|
130
|
979 |
Codatatype definitions are treated in precisely the same way. They express
|
103
|
980 |
case operators using those for the variant products and sums, namely
|
|
981 |
$\qsplit$ and~$\qcase$.
|
|
982 |
|
355
|
983 |
\medskip
|
103
|
984 |
|
355
|
985 |
\ifCADE The package has processed all the datatypes discussed in
|
|
986 |
my earlier paper~\cite{paulson-set-II} and the codatatype of lazy lists.
|
|
987 |
Space limitations preclude discussing these examples here, but they are
|
|
988 |
distributed with Isabelle. \typeout{****Omitting datatype examples from
|
|
989 |
CADE version!} \else
|
103
|
990 |
|
|
991 |
To see how constructors and the case analysis operator are defined, let us
|
|
992 |
examine some examples. These include lists and trees/forests, which I have
|
|
993 |
discussed extensively in another paper~\cite{paulson-set-II}.
|
|
994 |
|
|
995 |
\begin{figure}
|
|
996 |
\begin{ttbox}
|
|
997 |
structure List = Datatype_Fun
|
355
|
998 |
(val thy = Univ.thy
|
|
999 |
val rec_specs = [("list", "univ(A)",
|
|
1000 |
[(["Nil"], "i"),
|
|
1001 |
(["Cons"], "[i,i]=>i")])]
|
|
1002 |
val rec_styp = "i=>i"
|
|
1003 |
val ext = None
|
|
1004 |
val sintrs = ["Nil : list(A)",
|
|
1005 |
"[| a: A; l: list(A) |] ==> Cons(a,l) : list(A)"]
|
|
1006 |
val monos = []
|
|
1007 |
val type_intrs = datatype_intrs
|
103
|
1008 |
val type_elims = datatype_elims);
|
|
1009 |
\end{ttbox}
|
|
1010 |
\hrule
|
|
1011 |
\caption{Defining the datatype of lists} \label{list-fig}
|
|
1012 |
|
|
1013 |
\medskip
|
|
1014 |
\begin{ttbox}
|
130
|
1015 |
structure LList = CoDatatype_Fun
|
355
|
1016 |
(val thy = QUniv.thy
|
|
1017 |
val rec_specs = [("llist", "quniv(A)",
|
|
1018 |
[(["LNil"], "i"),
|
|
1019 |
(["LCons"], "[i,i]=>i")])]
|
|
1020 |
val rec_styp = "i=>i"
|
|
1021 |
val ext = None
|
|
1022 |
val sintrs = ["LNil : llist(A)",
|
|
1023 |
"[| a: A; l: llist(A) |] ==> LCons(a,l) : llist(A)"]
|
|
1024 |
val monos = []
|
|
1025 |
val type_intrs = codatatype_intrs
|
130
|
1026 |
val type_elims = codatatype_elims);
|
103
|
1027 |
\end{ttbox}
|
|
1028 |
\hrule
|
130
|
1029 |
\caption{Defining the codatatype of lazy lists} \label{llist-fig}
|
103
|
1030 |
\end{figure}
|
|
1031 |
|
|
1032 |
\subsection{Example: lists and lazy lists}
|
|
1033 |
Figures \ref{list-fig} and~\ref{llist-fig} present the ML definitions of
|
|
1034 |
lists and lazy lists, respectively. They highlight the (many) similarities
|
130
|
1035 |
and (few) differences between datatype and codatatype definitions.
|
103
|
1036 |
|
|
1037 |
Each form of list has two constructors, one for the empty list and one for
|
|
1038 |
adding an element to a list. Each takes a parameter, defining the set of
|
|
1039 |
lists over a given set~$A$. Each uses the appropriate domain from a
|
|
1040 |
Isabelle/ZF theory:
|
|
1041 |
\begin{itemize}
|
|
1042 |
\item $\lst(A)$ specifies domain $\univ(A)$ and parent theory {\tt Univ.thy}.
|
|
1043 |
|
|
1044 |
\item $\llist(A)$ specifies domain $\quniv(A)$ and parent theory {\tt
|
|
1045 |
QUniv.thy}.
|
|
1046 |
\end{itemize}
|
|
1047 |
|
130
|
1048 |
Since $\lst(A)$ is a datatype, it enjoys a structural induction rule, {\tt
|
|
1049 |
List.induct}:
|
103
|
1050 |
\[ \infer{P(x)}{x\in\lst(A) & P(\Nil)
|
|
1051 |
& \infer*{P(\Cons(a,l))}{[a\in A & l\in\lst(A) & P(l)]_{a,l}} }
|
|
1052 |
\]
|
|
1053 |
Induction and freeness yield the law $l\not=\Cons(a,l)$. To strengthen this,
|
|
1054 |
Isabelle/ZF defines the rank of a set and proves that the standard pairs and
|
|
1055 |
injections have greater rank than their components. An immediate consequence,
|
|
1056 |
which justifies structural recursion on lists \cite[\S4.3]{paulson-set-II},
|
|
1057 |
is
|
|
1058 |
\[ \rank(l) < \rank(\Cons(a,l)). \]
|
|
1059 |
|
130
|
1060 |
Since $\llist(A)$ is a codatatype, it has no induction rule. Instead it has
|
|
1061 |
the coinduction rule shown in \S\ref{coind-sec}. Since variant pairs and
|
103
|
1062 |
injections are monotonic and need not have greater rank than their
|
|
1063 |
components, fixedpoint operators can create cyclic constructions. For
|
|
1064 |
example, the definition
|
|
1065 |
\begin{eqnarray*}
|
|
1066 |
\lconst(a) & \equiv & \lfp(\univ(a), \lambda l. \LCons(a,l))
|
|
1067 |
\end{eqnarray*}
|
|
1068 |
yields $\lconst(a) = \LCons(a,\lconst(a))$.
|
|
1069 |
|
|
1070 |
\medskip
|
|
1071 |
It may be instructive to examine the definitions of the constructors and
|
|
1072 |
case operator for $\lst(A)$. The definitions for $\llist(A)$ are similar.
|
|
1073 |
The list constructors are defined as follows:
|
|
1074 |
\begin{eqnarray*}
|
|
1075 |
\Nil & = & \Inl(\emptyset) \\
|
|
1076 |
\Cons(a,l) & = & \Inr(\pair{a,l})
|
|
1077 |
\end{eqnarray*}
|
|
1078 |
The operator $\lstcase$ performs case analysis on these two alternatives:
|
|
1079 |
\begin{eqnarray*}
|
|
1080 |
\lstcase(c,h) & \equiv & \case(\lambda u.c, \split(h))
|
|
1081 |
\end{eqnarray*}
|
|
1082 |
Let us verify the two equations:
|
|
1083 |
\begin{eqnarray*}
|
|
1084 |
\lstcase(c, h, \Nil) & = &
|
|
1085 |
\case(\lambda u.c, \split(h), \Inl(\emptyset)) \\
|
|
1086 |
& = & (\lambda u.c)(\emptyset) \\
|
130
|
1087 |
& = & c\\[1ex]
|
103
|
1088 |
\lstcase(c, h, \Cons(x,y)) & = &
|
|
1089 |
\case(\lambda u.c, \split(h), \Inr(\pair{x,y})) \\
|
|
1090 |
& = & \split(h, \pair{x,y}) \\
|
130
|
1091 |
& = & h(x,y)
|
103
|
1092 |
\end{eqnarray*}
|
|
1093 |
|
|
1094 |
\begin{figure}
|
355
|
1095 |
\begin{ttbox}
|
103
|
1096 |
structure TF = Datatype_Fun
|
355
|
1097 |
(val thy = Univ.thy
|
|
1098 |
val rec_specs = [("tree", "univ(A)",
|
|
1099 |
[(["Tcons"], "[i,i]=>i")]),
|
|
1100 |
("forest", "univ(A)",
|
|
1101 |
[(["Fnil"], "i"),
|
|
1102 |
(["Fcons"], "[i,i]=>i")])]
|
|
1103 |
val rec_styp = "i=>i"
|
|
1104 |
val ext = None
|
|
1105 |
val sintrs =
|
|
1106 |
["[| a:A; f: forest(A) |] ==> Tcons(a,f) : tree(A)",
|
|
1107 |
"Fnil : forest(A)",
|
|
1108 |
"[| t: tree(A); f: forest(A) |] ==> Fcons(t,f) : forest(A)"]
|
|
1109 |
val monos = []
|
|
1110 |
val type_intrs = datatype_intrs
|
103
|
1111 |
val type_elims = datatype_elims);
|
355
|
1112 |
\end{ttbox}
|
103
|
1113 |
\hrule
|
|
1114 |
\caption{Defining the datatype of trees and forests} \label{tf-fig}
|
|
1115 |
\end{figure}
|
|
1116 |
|
|
1117 |
|
|
1118 |
\subsection{Example: mutual recursion}
|
130
|
1119 |
In mutually recursive trees and forests~\cite[\S4.5]{paulson-set-II}, trees
|
103
|
1120 |
have the one constructor $\Tcons$, while forests have the two constructors
|
|
1121 |
$\Fnil$ and~$\Fcons$. Figure~\ref{tf-fig} presents the ML
|
|
1122 |
definition. It has much in common with that of $\lst(A)$, including its
|
|
1123 |
use of $\univ(A)$ for the domain and {\tt Univ.thy} for the parent theory.
|
|
1124 |
The three introduction rules define the mutual recursion. The
|
|
1125 |
distinguishing feature of this example is its two induction rules.
|
|
1126 |
|
|
1127 |
The basic induction rule is called {\tt TF.induct}:
|
|
1128 |
\[ \infer{P(x)}{x\in\TF(A) &
|
|
1129 |
\infer*{P(\Tcons(a,f))}
|
|
1130 |
{\left[\begin{array}{l} a\in A \\
|
|
1131 |
f\in\forest(A) \\ P(f)
|
|
1132 |
\end{array}
|
|
1133 |
\right]_{a,f}}
|
|
1134 |
& P(\Fnil)
|
130
|
1135 |
& \infer*{P(\Fcons(t,f))}
|
103
|
1136 |
{\left[\begin{array}{l} t\in\tree(A) \\ P(t) \\
|
|
1137 |
f\in\forest(A) \\ P(f)
|
|
1138 |
\end{array}
|
|
1139 |
\right]_{t,f}} }
|
|
1140 |
\]
|
|
1141 |
This rule establishes a single predicate for $\TF(A)$, the union of the
|
|
1142 |
recursive sets.
|
|
1143 |
|
|
1144 |
Although such reasoning is sometimes useful
|
|
1145 |
\cite[\S4.5]{paulson-set-II}, a proper mutual induction rule should establish
|
|
1146 |
separate predicates for $\tree(A)$ and $\forest(A)$. The package calls this
|
|
1147 |
rule {\tt TF.mutual\_induct}. Observe the usage of $P$ and $Q$ in the
|
|
1148 |
induction hypotheses:
|
|
1149 |
\[ \infer{(\forall z. z\in\tree(A)\imp P(z)) \conj
|
|
1150 |
(\forall z. z\in\forest(A)\imp Q(z))}
|
|
1151 |
{\infer*{P(\Tcons(a,f))}
|
|
1152 |
{\left[\begin{array}{l} a\in A \\
|
|
1153 |
f\in\forest(A) \\ Q(f)
|
|
1154 |
\end{array}
|
|
1155 |
\right]_{a,f}}
|
|
1156 |
& Q(\Fnil)
|
130
|
1157 |
& \infer*{Q(\Fcons(t,f))}
|
103
|
1158 |
{\left[\begin{array}{l} t\in\tree(A) \\ P(t) \\
|
|
1159 |
f\in\forest(A) \\ Q(f)
|
|
1160 |
\end{array}
|
|
1161 |
\right]_{t,f}} }
|
|
1162 |
\]
|
|
1163 |
As mentioned above, the package does not define a structural recursion
|
|
1164 |
operator. I have described elsewhere how this is done
|
|
1165 |
\cite[\S4.5]{paulson-set-II}.
|
|
1166 |
|
|
1167 |
Both forest constructors have the form $\Inr(\cdots)$,
|
|
1168 |
while the tree constructor has the form $\Inl(\cdots)$. This pattern would
|
|
1169 |
hold regardless of how many tree or forest constructors there were.
|
|
1170 |
\begin{eqnarray*}
|
|
1171 |
\Tcons(a,l) & = & \Inl(\pair{a,l}) \\
|
|
1172 |
\Fnil & = & \Inr(\Inl(\emptyset)) \\
|
|
1173 |
\Fcons(a,l) & = & \Inr(\Inr(\pair{a,l}))
|
|
1174 |
\end{eqnarray*}
|
|
1175 |
There is only one case operator; it works on the union of the trees and
|
|
1176 |
forests:
|
|
1177 |
\begin{eqnarray*}
|
|
1178 |
{\tt tree\_forest\_case}(f,c,g) & \equiv &
|
|
1179 |
\case(\split(f),\, \case(\lambda u.c, \split(g)))
|
|
1180 |
\end{eqnarray*}
|
|
1181 |
|
|
1182 |
\begin{figure}
|
355
|
1183 |
\begin{ttbox}
|
103
|
1184 |
structure Data = Datatype_Fun
|
355
|
1185 |
(val thy = Univ.thy
|
|
1186 |
val rec_specs = [("data", "univ(A Un B)",
|
|
1187 |
[(["Con0"], "i"),
|
|
1188 |
(["Con1"], "i=>i"),
|
|
1189 |
(["Con2"], "[i,i]=>i"),
|
|
1190 |
(["Con3"], "[i,i,i]=>i")])]
|
|
1191 |
val rec_styp = "[i,i]=>i"
|
|
1192 |
val ext = None
|
|
1193 |
val sintrs =
|
|
1194 |
["Con0 : data(A,B)",
|
|
1195 |
"[| a: A |] ==> Con1(a) : data(A,B)",
|
|
1196 |
"[| a: A; b: B |] ==> Con2(a,b) : data(A,B)",
|
|
1197 |
"[| a: A; b: B; d: data(A,B) |] ==> Con3(a,b,d) : data(A,B)"]
|
|
1198 |
val monos = []
|
|
1199 |
val type_intrs = datatype_intrs
|
103
|
1200 |
val type_elims = datatype_elims);
|
355
|
1201 |
\end{ttbox}
|
103
|
1202 |
\hrule
|
|
1203 |
\caption{Defining the four-constructor sample datatype} \label{data-fig}
|
|
1204 |
\end{figure}
|
|
1205 |
|
|
1206 |
\subsection{A four-constructor datatype}
|
|
1207 |
Finally let us consider a fairly general datatype. It has four
|
130
|
1208 |
constructors $\Con_0$, \ldots, $\Con_3$, with the
|
103
|
1209 |
corresponding arities. Figure~\ref{data-fig} presents the ML definition.
|
|
1210 |
Because this datatype has two set parameters, $A$ and~$B$, it specifies
|
|
1211 |
$\univ(A\un B)$ as its domain. The structural induction rule has four
|
|
1212 |
minor premises, one per constructor:
|
|
1213 |
\[ \infer{P(x)}{x\in\data(A,B) &
|
|
1214 |
P(\Con_0) &
|
|
1215 |
\infer*{P(\Con_1(a))}{[a\in A]_a} &
|
|
1216 |
\infer*{P(\Con_2(a,b))}
|
|
1217 |
{\left[\begin{array}{l} a\in A \\ b\in B \end{array}
|
|
1218 |
\right]_{a,b}} &
|
|
1219 |
\infer*{P(\Con_3(a,b,d))}
|
|
1220 |
{\left[\begin{array}{l} a\in A \\ b\in B \\
|
|
1221 |
d\in\data(A,B) \\ P(d)
|
|
1222 |
\end{array}
|
|
1223 |
\right]_{a,b,d}} }
|
|
1224 |
\]
|
|
1225 |
|
|
1226 |
The constructor definitions are
|
|
1227 |
\begin{eqnarray*}
|
|
1228 |
\Con_0 & = & \Inl(\Inl(\emptyset)) \\
|
|
1229 |
\Con_1(a) & = & \Inl(\Inr(a)) \\
|
|
1230 |
\Con_2(a,b) & = & \Inr(\Inl(\pair{a,b})) \\
|
|
1231 |
\Con_3(a,b,c) & = & \Inr(\Inr(\pair{a,b,c})).
|
|
1232 |
\end{eqnarray*}
|
|
1233 |
The case operator is
|
|
1234 |
\begin{eqnarray*}
|
|
1235 |
{\tt data\_case}(f_0,f_1,f_2,f_3) & \equiv &
|
|
1236 |
\case(\begin{array}[t]{@{}l}
|
|
1237 |
\case(\lambda u.f_0,\; f_1),\, \\
|
|
1238 |
\case(\split(f_2),\; \split(\lambda v.\split(f_3(v)))) )
|
|
1239 |
\end{array}
|
|
1240 |
\end{eqnarray*}
|
|
1241 |
This may look cryptic, but the case equations are trivial to verify.
|
|
1242 |
|
|
1243 |
In the constructor definitions, the injections are balanced. A more naive
|
|
1244 |
approach is to define $\Con_3(a,b,c)$ as
|
|
1245 |
$\Inr(\Inr(\Inr(\pair{a,b,c})))$; instead, each constructor has two
|
|
1246 |
injections. The difference here is small. But the ZF examples include a
|
|
1247 |
60-element enumeration type, where each constructor has 5 or~6 injections.
|
|
1248 |
The naive approach would require 1 to~59 injections; the definitions would be
|
|
1249 |
quadratic in size. It is like the difference between the binary and unary
|
|
1250 |
numeral systems.
|
|
1251 |
|
130
|
1252 |
The result structure contains the case operator and constructor definitions as
|
|
1253 |
the theorem list \verb|con_defs|. It contains the case equations, such as
|
103
|
1254 |
\begin{eqnarray*}
|
|
1255 |
{\tt data\_case}(f_0,f_1,f_2,f_3,\Con_3(a,b,c)) & = & f_3(a,b,c),
|
|
1256 |
\end{eqnarray*}
|
|
1257 |
as the theorem list \verb|case_eqns|. There is one equation per constructor.
|
|
1258 |
|
|
1259 |
\subsection{Proving freeness theorems}
|
|
1260 |
There are two kinds of freeness theorems:
|
|
1261 |
\begin{itemize}
|
|
1262 |
\item {\bf injectiveness} theorems, such as
|
|
1263 |
\[ \Con_2(a,b) = \Con_2(a',b') \bimp a=a' \conj b=b' \]
|
|
1264 |
|
|
1265 |
\item {\bf distinctness} theorems, such as
|
|
1266 |
\[ \Con_1(a) \not= \Con_2(a',b') \]
|
|
1267 |
\end{itemize}
|
|
1268 |
Since the number of such theorems is quadratic in the number of constructors,
|
|
1269 |
the package does not attempt to prove them all. Instead it returns tools for
|
|
1270 |
proving desired theorems --- either explicitly or `on the fly' during
|
|
1271 |
simplification or classical reasoning.
|
|
1272 |
|
|
1273 |
The theorem list \verb|free_iffs| enables the simplifier to perform freeness
|
|
1274 |
reasoning. This works by incremental unfolding of constructors that appear in
|
|
1275 |
equations. The theorem list contains logical equivalences such as
|
|
1276 |
\begin{eqnarray*}
|
|
1277 |
\Con_0=c & \bimp & c=\Inl(\Inl(\emptyset)) \\
|
|
1278 |
\Con_1(a)=c & \bimp & c=\Inl(\Inr(a)) \\
|
|
1279 |
& \vdots & \\
|
|
1280 |
\Inl(a)=\Inl(b) & \bimp & a=b \\
|
130
|
1281 |
\Inl(a)=\Inr(b) & \bimp & {\tt False} \\
|
103
|
1282 |
\pair{a,b} = \pair{a',b'} & \bimp & a=a' \conj b=b'
|
|
1283 |
\end{eqnarray*}
|
|
1284 |
For example, these rewrite $\Con_1(a)=\Con_1(b)$ to $a=b$ in four steps.
|
|
1285 |
|
|
1286 |
The theorem list \verb|free_SEs| enables the classical
|
|
1287 |
reasoner to perform similar replacements. It consists of elimination rules
|
355
|
1288 |
to replace $\Con_0=c$ by $c=\Inl(\Inl(\emptyset))$ and so forth, in the
|
103
|
1289 |
assumptions.
|
|
1290 |
|
|
1291 |
Such incremental unfolding combines freeness reasoning with other proof
|
|
1292 |
steps. It has the unfortunate side-effect of unfolding definitions of
|
|
1293 |
constructors in contexts such as $\exists x.\Con_1(a)=x$, where they should
|
|
1294 |
be left alone. Calling the Isabelle tactic {\tt fold\_tac con\_defs}
|
|
1295 |
restores the defined constants.
|
|
1296 |
\fi %CADE
|
|
1297 |
|
355
|
1298 |
\section{Related work}\label{related}
|
|
1299 |
The use of least fixedpoints to express inductive definitions seems
|
|
1300 |
obvious. Why, then, has this technique so seldom been implemented?
|
|
1301 |
|
|
1302 |
Most automated logics can only express inductive definitions by asserting
|
|
1303 |
new axioms. Little would be left of Boyer and Moore's logic~\cite{bm79} if
|
|
1304 |
their shell principle were removed. With ALF the situation is more
|
|
1305 |
complex; earlier versions of Martin-L\"of's type theory could (using
|
|
1306 |
wellordering types) express datatype definitions, but the version
|
|
1307 |
underlying ALF requires new rules for each definition~\cite{dybjer91}.
|
|
1308 |
With Coq the situation is subtler still; its underlying Calculus of
|
|
1309 |
Constructions can express inductive definitions~\cite{huet88}, but cannot
|
|
1310 |
quite handle datatype definitions~\cite{paulin92}. It seems that
|
|
1311 |
researchers tried hard to circumvent these problems before finally
|
|
1312 |
extending the Calculus with rule schemes for strictly positive operators.
|
|
1313 |
|
|
1314 |
Higher-order logic can express inductive definitions through quantification
|
|
1315 |
over unary predicates. The following formula expresses that~$i$ belongs to the
|
|
1316 |
least set containing~0 and closed under~$\succ$:
|
|
1317 |
\[ \forall P. P(0)\conj (\forall x.P(x)\imp P(\succ(x))) \imp P(i) \]
|
|
1318 |
This technique can be used to prove the Knaster-Tarski Theorem, but it is
|
|
1319 |
little used in the HOL system. Melham~\cite{melham89} clearly describes
|
|
1320 |
the development. The natural numbers are defined as shown above, but lists
|
|
1321 |
are defined as functions over the natural numbers. Unlabelled
|
|
1322 |
trees are defined using G\"odel numbering; a labelled tree consists of an
|
|
1323 |
unlabelled tree paired with a list of labels. Melham's datatype package
|
|
1324 |
expresses the user's datatypes in terms of labelled trees. It has been
|
|
1325 |
highly successful, but a fixedpoint approach would have yielded greater
|
|
1326 |
functionality with less effort.
|
|
1327 |
|
|
1328 |
Melham's inductive definition package~\cite{camilleri92} uses
|
|
1329 |
quantification over predicates, which is implicitly a fixedpoint approach.
|
|
1330 |
Instead of formalizing the notion of monotone function, it requires
|
|
1331 |
definitions to consist of finitary rules, a syntactic form that excludes
|
|
1332 |
many monotone inductive definitions.
|
|
1333 |
|
|
1334 |
The earliest use of least fixedpoints is probably Robin Milner's datatype
|
|
1335 |
package for Edinburgh LCF~\cite{milner-ind}. Brian Monahan extended this
|
|
1336 |
package considerably~\cite{monahan84}, as did I in unpublished
|
|
1337 |
work.\footnote{The datatype package described in my LCF
|
|
1338 |
book~\cite{paulson87} does {\it not\/} make definitions, but merely
|
|
1339 |
asserts axioms. I justified this shortcut on grounds of efficiency:
|
|
1340 |
existing packages took tens of minutes to run. Such an explanation would
|
|
1341 |
not do today.}
|
|
1342 |
LCF is a first-order logic of domain theory; the relevant fixedpoint
|
|
1343 |
theorem is not Knaster-Tarski but concerns fixedpoints of continuous
|
|
1344 |
functions over domains. LCF is too weak to express recursive predicates.
|
|
1345 |
Thus it would appear that the Isabelle/ZF package is the first to be based
|
|
1346 |
on the Knaster-Tarski Theorem.
|
|
1347 |
|
|
1348 |
|
103
|
1349 |
\section{Conclusions and future work}
|
355
|
1350 |
Higher-order logic and set theory are both powerful enough to express
|
|
1351 |
inductive definitions. A growing number of theorem provers implement one
|
|
1352 |
of these~\cite{IMPS,saaltink-fme}. The easiest sort of inductive
|
|
1353 |
definition package to write is one that asserts new axioms, not one that
|
|
1354 |
makes definitions and proves theorems about them. But asserting axioms
|
|
1355 |
could introduce unsoundness.
|
|
1356 |
|
|
1357 |
The fixedpoint approach makes it fairly easy to implement a package for
|
|
1358 |
(co)inductive definitions that does not assert axioms. It is efficient: it
|
103
|
1359 |
processes most definitions in seconds and even a 60-constructor datatype
|
|
1360 |
requires only two minutes. It is also simple: the package consists of
|
|
1361 |
under 1100 lines (35K bytes) of Standard ML code. The first working
|
|
1362 |
version took under a week to code.
|
|
1363 |
|
355
|
1364 |
In set theory, care is required to ensure that the inductive definition
|
|
1365 |
yields a set (rather than a proper class). This problem is inherent to set
|
|
1366 |
theory, whether or not the Knaster-Tarski Theorem is employed. We must
|
|
1367 |
exhibit a bounding set (called a domain above). For inductive definitions,
|
|
1368 |
this is often trivial. For datatype definitions, I have had to formalize
|
|
1369 |
much set theory. I intend to formalize cardinal arithmetic and the
|
|
1370 |
$\aleph$-sequence to handle datatype definitions that have infinite
|
|
1371 |
branching. The need for such efforts is not a drawback of the fixedpoint
|
|
1372 |
approach, for the alternative is to take such definitions on faith.
|
103
|
1373 |
|
355
|
1374 |
The approach is not restricted to set theory. It should be suitable for
|
|
1375 |
any logic that has some notion of set and the Knaster-Tarski Theorem. I
|
|
1376 |
intend to use the Isabelle/ZF package as the basis for a higher-order logic
|
|
1377 |
one, using Isabelle/HOL\@. The necessary theory is already
|
130
|
1378 |
mechanized~\cite{paulson-coind}. HOL represents sets by unary predicates;
|
355
|
1379 |
defining the corresponding types may cause complications.
|
103
|
1380 |
|
|
1381 |
|
355
|
1382 |
\bibliographystyle{springer}
|
|
1383 |
\bibliography{string-abbrv,atp,theory,funprog,isabelle}
|
103
|
1384 |
%%%%%\doendnotes
|
|
1385 |
|
|
1386 |
\ifCADE\typeout{****Omitting appendices from CADE version!}
|
|
1387 |
\else
|
|
1388 |
\newpage
|
|
1389 |
\appendix
|
130
|
1390 |
\section{Inductive and coinductive definitions: users guide}
|
|
1391 |
The ML functors \verb|Inductive_Fun| and \verb|CoInductive_Fun| build
|
|
1392 |
inductive and coinductive definitions, respectively. This section describes
|
103
|
1393 |
how to invoke them.
|
|
1394 |
|
|
1395 |
\subsection{The result structure}
|
|
1396 |
Many of the result structure's components have been discussed
|
|
1397 |
in~\S\ref{basic-sec}; others are self-explanatory.
|
|
1398 |
\begin{description}
|
|
1399 |
\item[\tt thy] is the new theory containing the recursive sets.
|
|
1400 |
|
|
1401 |
\item[\tt defs] is the list of definitions of the recursive sets.
|
|
1402 |
|
|
1403 |
\item[\tt bnd\_mono] is a monotonicity theorem for the fixedpoint operator.
|
|
1404 |
|
|
1405 |
\item[\tt unfold] is a fixedpoint equation for the recursive set (the union of
|
|
1406 |
the recursive sets, in the case of mutual recursion).
|
|
1407 |
|
|
1408 |
\item[\tt dom\_subset] is a theorem stating inclusion in the domain.
|
|
1409 |
|
|
1410 |
\item[\tt intrs] is the list of introduction rules, now proved as theorems, for
|
|
1411 |
the recursive sets.
|
|
1412 |
|
|
1413 |
\item[\tt elim] is the elimination rule.
|
|
1414 |
|
|
1415 |
\item[\tt mk\_cases] is a function to create simplified instances of {\tt
|
|
1416 |
elim}, using freeness reasoning on some underlying datatype.
|
|
1417 |
\end{description}
|
|
1418 |
|
|
1419 |
For an inductive definition, the result structure contains two induction rules,
|
130
|
1420 |
{\tt induct} and \verb|mutual_induct|. For a coinductive definition, it
|
|
1421 |
contains the rule \verb|coinduct|.
|
|
1422 |
|
|
1423 |
Figure~\ref{def-result-fig} summarizes the two result signatures,
|
|
1424 |
specifying the types of all these components.
|
103
|
1425 |
|
|
1426 |
\begin{figure}
|
|
1427 |
\begin{ttbox}
|
|
1428 |
sig
|
|
1429 |
val thy : theory
|
|
1430 |
val defs : thm list
|
|
1431 |
val bnd_mono : thm
|
|
1432 |
val unfold : thm
|
|
1433 |
val dom_subset : thm
|
|
1434 |
val intrs : thm list
|
|
1435 |
val elim : thm
|
|
1436 |
val mk_cases : thm list -> string -> thm
|
|
1437 |
{\it(Inductive definitions only)}
|
|
1438 |
val induct : thm
|
|
1439 |
val mutual_induct: thm
|
130
|
1440 |
{\it(Coinductive definitions only)}
|
|
1441 |
val coinduct : thm
|
103
|
1442 |
end
|
|
1443 |
\end{ttbox}
|
|
1444 |
\hrule
|
130
|
1445 |
\caption{The result of a (co)inductive definition} \label{def-result-fig}
|
103
|
1446 |
|
130
|
1447 |
\medskip
|
103
|
1448 |
\begin{ttbox}
|
|
1449 |
sig
|
|
1450 |
val thy : theory
|
|
1451 |
val rec_doms : (string*string) list
|
|
1452 |
val sintrs : string list
|
|
1453 |
val monos : thm list
|
|
1454 |
val con_defs : thm list
|
|
1455 |
val type_intrs : thm list
|
|
1456 |
val type_elims : thm list
|
|
1457 |
end
|
|
1458 |
\end{ttbox}
|
|
1459 |
\hrule
|
130
|
1460 |
\caption{The argument of a (co)inductive definition} \label{def-arg-fig}
|
103
|
1461 |
\end{figure}
|
|
1462 |
|
|
1463 |
\subsection{The argument structure}
|
130
|
1464 |
Both \verb|Inductive_Fun| and \verb|CoInductive_Fun| take the same argument
|
103
|
1465 |
structure (Figure~\ref{def-arg-fig}). Its components are as follows:
|
|
1466 |
\begin{description}
|
|
1467 |
\item[\tt thy] is the definition's parent theory, which {\it must\/}
|
|
1468 |
declare constants for the recursive sets.
|
|
1469 |
|
|
1470 |
\item[\tt rec\_doms] is a list of pairs, associating the name of each recursive
|
|
1471 |
set with its domain.
|
|
1472 |
|
|
1473 |
\item[\tt sintrs] specifies the desired introduction rules as strings.
|
|
1474 |
|
|
1475 |
\item[\tt monos] consists of monotonicity theorems for each operator applied
|
|
1476 |
to a recursive set in the introduction rules.
|
|
1477 |
|
|
1478 |
\item[\tt con\_defs] contains definitions of constants appearing in the
|
130
|
1479 |
introduction rules. The (co)datatype package supplies the constructors'
|
103
|
1480 |
definitions here. Most direct calls of \verb|Inductive_Fun| or
|
130
|
1481 |
\verb|CoInductive_Fun| pass the empty list; one exception is the primitive
|
103
|
1482 |
recursive functions example (\S\ref{primrec-sec}).
|
|
1483 |
|
|
1484 |
\item[\tt type\_intrs] consists of introduction rules for type-checking the
|
|
1485 |
definition, as discussed in~\S\ref{basic-sec}. They are applied using
|
|
1486 |
depth-first search; you can trace the proof by setting
|
|
1487 |
\verb|trace_DEPTH_FIRST := true|.
|
|
1488 |
|
|
1489 |
\item[\tt type\_elims] consists of elimination rules for type-checking the
|
|
1490 |
definition. They are presumed to be `safe' and are applied as much as
|
|
1491 |
possible, prior to the {\tt type\_intrs} search.
|
|
1492 |
\end{description}
|
|
1493 |
The package has a few notable restrictions:
|
|
1494 |
\begin{itemize}
|
|
1495 |
\item The parent theory, {\tt thy}, must declare the recursive sets as
|
|
1496 |
constants. You can extend a theory with new constants using {\tt
|
|
1497 |
addconsts}, as illustrated in~\S\ref{ind-eg-sec}. If the inductive
|
|
1498 |
definition also requires new concrete syntax, then it is simpler to
|
|
1499 |
express the parent theory using a theory file. It is often convenient to
|
|
1500 |
define an infix syntax for relations, say $a\prec b$ for $\pair{a,b}\in
|
|
1501 |
R$.
|
|
1502 |
|
|
1503 |
\item The names of the recursive sets must be identifiers, not infix
|
|
1504 |
operators.
|
|
1505 |
|
|
1506 |
\item Side-conditions must not be conjunctions. However, an introduction rule
|
|
1507 |
may contain any number of side-conditions.
|
|
1508 |
\end{itemize}
|
|
1509 |
|
|
1510 |
|
130
|
1511 |
\section{Datatype and codatatype definitions: users guide}
|
|
1512 |
The ML functors \verb|Datatype_Fun| and \verb|CoDatatype_Fun| define datatypes
|
|
1513 |
and codatatypes, invoking \verb|Datatype_Fun| and
|
|
1514 |
\verb|CoDatatype_Fun| to make the underlying (co)inductive definitions.
|
103
|
1515 |
|
|
1516 |
|
|
1517 |
\subsection{The result structure}
|
130
|
1518 |
The result structure extends that of (co)inductive definitions
|
103
|
1519 |
(Figure~\ref{def-result-fig}) with several additional items:
|
|
1520 |
\begin{ttbox}
|
|
1521 |
val con_thy : theory
|
|
1522 |
val con_defs : thm list
|
|
1523 |
val case_eqns : thm list
|
|
1524 |
val free_iffs : thm list
|
|
1525 |
val free_SEs : thm list
|
|
1526 |
val mk_free : string -> thm
|
|
1527 |
\end{ttbox}
|
|
1528 |
Most of these have been discussed in~\S\ref{data-sec}. Here is a summary:
|
|
1529 |
\begin{description}
|
|
1530 |
\item[\tt con\_thy] is a new theory containing definitions of the
|
130
|
1531 |
(co)datatype's constructors and case operator. It also declares the
|
103
|
1532 |
recursive sets as constants, so that it may serve as the parent
|
130
|
1533 |
theory for the (co)inductive definition.
|
103
|
1534 |
|
|
1535 |
\item[\tt con\_defs] is a list of definitions: the case operator followed by
|
|
1536 |
the constructors. This theorem list can be supplied to \verb|mk_cases|, for
|
|
1537 |
example.
|
|
1538 |
|
|
1539 |
\item[\tt case\_eqns] is a list of equations, stating that the case operator
|
|
1540 |
inverts each constructor.
|
|
1541 |
|
|
1542 |
\item[\tt free\_iffs] is a list of logical equivalences to perform freeness
|
|
1543 |
reasoning by rewriting. A typical application has the form
|
|
1544 |
\begin{ttbox}
|
|
1545 |
by (asm_simp_tac (ZF_ss addsimps free_iffs) 1);
|
|
1546 |
\end{ttbox}
|
|
1547 |
|
|
1548 |
\item[\tt free\_SEs] is a list of `safe' elimination rules to perform freeness
|
|
1549 |
reasoning. It can be supplied to \verb|eresolve_tac| or to the classical
|
|
1550 |
reasoner:
|
|
1551 |
\begin{ttbox}
|
|
1552 |
by (fast_tac (ZF_cs addSEs free_SEs) 1);
|
|
1553 |
\end{ttbox}
|
|
1554 |
|
|
1555 |
\item[\tt mk\_free] is a function to prove freeness properties, specified as
|
|
1556 |
strings. The theorems can be expressed in various forms, such as logical
|
|
1557 |
equivalences or elimination rules.
|
|
1558 |
\end{description}
|
|
1559 |
|
|
1560 |
The result structure also inherits everything from the underlying
|
130
|
1561 |
(co)inductive definition, such as the introduction rules, elimination rule,
|
179
|
1562 |
and (co)induction rule.
|
103
|
1563 |
|
|
1564 |
|
|
1565 |
\begin{figure}
|
|
1566 |
\begin{ttbox}
|
|
1567 |
sig
|
|
1568 |
val thy : theory
|
|
1569 |
val rec_specs : (string * string * (string list*string)list) list
|
|
1570 |
val rec_styp : string
|
|
1571 |
val ext : Syntax.sext option
|
|
1572 |
val sintrs : string list
|
|
1573 |
val monos : thm list
|
|
1574 |
val type_intrs: thm list
|
|
1575 |
val type_elims: thm list
|
|
1576 |
end
|
|
1577 |
\end{ttbox}
|
|
1578 |
\hrule
|
130
|
1579 |
\caption{The argument of a (co)datatype definition} \label{data-arg-fig}
|
103
|
1580 |
\end{figure}
|
|
1581 |
|
|
1582 |
\subsection{The argument structure}
|
130
|
1583 |
Both (co)datatype functors take the same argument structure
|
|
1584 |
(Figure~\ref{data-arg-fig}). It does not extend that for (co)inductive
|
103
|
1585 |
definitions, but shares several components and passes them uninterpreted to
|
|
1586 |
\verb|Datatype_Fun| or
|
130
|
1587 |
\verb|CoDatatype_Fun|. The new components are as follows:
|
103
|
1588 |
\begin{description}
|
130
|
1589 |
\item[\tt thy] is the (co)datatype's parent theory. It {\it must not\/}
|
|
1590 |
declare constants for the recursive sets. Recall that (co)inductive
|
103
|
1591 |
definitions have the opposite restriction.
|
|
1592 |
|
|
1593 |
\item[\tt rec\_specs] is a list of triples of the form ({\it recursive set\/},
|
|
1594 |
{\it domain\/}, {\it constructors\/}) for each mutually recursive set. {\it
|
|
1595 |
Constructors\/} is a list of the form (names, type). See the discussion and
|
|
1596 |
examples in~\S\ref{data-sec}.
|
|
1597 |
|
|
1598 |
\item[\tt rec\_styp] is the common meta-type of the mutually recursive sets,
|
|
1599 |
specified as a string. They must all have the same type because all must
|
|
1600 |
take the same parameters.
|
|
1601 |
|
|
1602 |
\item[\tt ext] is an optional syntax extension, usually omitted by writing
|
|
1603 |
{\tt None}. You can supply mixfix syntax for the constructors by supplying
|
|
1604 |
\begin{ttbox}
|
|
1605 |
Some (Syntax.simple_sext [{\it mixfix declarations\/}])
|
|
1606 |
\end{ttbox}
|
|
1607 |
\end{description}
|
|
1608 |
The choice of domain is usually simple. Isabelle/ZF defines the set
|
|
1609 |
$\univ(A)$, which contains~$A$ and is closed under the standard Cartesian
|
|
1610 |
products and disjoint sums \cite[\S4.2]{paulson-set-II}. In a typical
|
|
1611 |
datatype definition with set parameters $A_1$, \ldots, $A_k$, a suitable
|
|
1612 |
domain for all the recursive sets is $\univ(A_1\un\cdots\un A_k)$. For a
|
130
|
1613 |
codatatype definition, the set
|
103
|
1614 |
$\quniv(A)$ contains~$A$ and is closed under the variant Cartesian products
|
130
|
1615 |
and disjoint sums; the appropriate domain is
|
103
|
1616 |
$\quniv(A_1\un\cdots\un A_k)$.
|
|
1617 |
|
|
1618 |
The {\tt sintrs} specify the introduction rules, which govern the recursive
|
179
|
1619 |
structure of the datatype. Introduction rules may involve monotone
|
|
1620 |
operators and side-conditions to express things that go beyond the usual
|
|
1621 |
notion of datatype. The theorem lists {\tt monos}, {\tt type\_intrs} and
|
|
1622 |
{\tt type\_elims} should contain precisely what is needed for the
|
|
1623 |
underlying (co)inductive definition. Isabelle/ZF defines lists of
|
|
1624 |
type-checking rules that can be supplied for the latter two components:
|
103
|
1625 |
\begin{itemize}
|
179
|
1626 |
\item {\tt datatype\_intrs} and {\tt datatype\_elims} are rules
|
103
|
1627 |
for $\univ(A)$.
|
355
|
1628 |
\item {\tt codatatype\_intrs} and {\tt codatatype\_elims} are
|
103
|
1629 |
rules for $\quniv(A)$.
|
|
1630 |
\end{itemize}
|
|
1631 |
In typical definitions, these theorem lists need not be supplemented with
|
|
1632 |
other theorems.
|
|
1633 |
|
|
1634 |
The constructor definitions' right-hand sides can overlap. A
|
|
1635 |
simple example is the datatype for the combinators, whose constructors are
|
|
1636 |
\begin{eqnarray*}
|
|
1637 |
{\tt K} & \equiv & \Inl(\emptyset) \\
|
|
1638 |
{\tt S} & \equiv & \Inr(\Inl(\emptyset)) \\
|
|
1639 |
p{\tt\#}q & \equiv & \Inr(\Inl(\pair{p,q}))
|
|
1640 |
\end{eqnarray*}
|
|
1641 |
Unlike in previous versions of Isabelle, \verb|fold_tac| now ensures that the
|
|
1642 |
longest right-hand sides are folded first.
|
|
1643 |
|
|
1644 |
\fi
|
|
1645 |
\end{document}
|