Major Section: TUTORIAL-EXAMPLES
The other tutorial examples are rather small and entirely self contained. The present example is rather more elaborate, and makes use of a feature that really adds great power and versatility to ACL2, namely: the use of previously defined collections of lemmas, in the form of ``books.''
This example was written almost entirely by Bill Young of Computational Logic, Inc.
This example is based on one developed by Ricky Butler and Sally Johnson of NASA Langley for the PVS system, and subsequently revised by Judy Crow, et al, at SRI. It is a simple phone book specification. We will not bother to follow their versions closely, but will instead present a style of specification natural for ACL2.
The idea is to model an electronic phone book with the following properties.
Our phone book will store the phone numbers of a city.
It must be possible to retrieve a phone number, given a name.
It must be possible to add and delete entries.
Of course, there are numerous ways to construct such a model. A
natural approach within the Lisp/ACL2 context is to use
``association lists'' or ``alists.'' Briefly, an alist is a list of
pairs (key . value) associating a value with a key. A phone
book could be an alist of pairs (name . pnum). To find the phone
number associated with a given name, we merely search the alist
until we find the appropriate pair. For a large city, such a linear
list would not be efficient, but at this point we are interested
only in modeling the problem, not in deriving an efficient
implementation. We could address that question later by proving our
alist model equivalent, in some desired sense, to a more efficient
data structure.
We could build a theory of alists from scratch, or we can use a
previously constructed theory (book) of alist definitions and facts.
By using an existing book, we build upon the work of others, start
our specification and proof effort from a much richer foundation,
and hopefully devote more of our time to the problem at hand.
Unfortunately, it is not completely simple for the new user to know
what books are available and what they contain. We hope later
to improve the documentation of the growing collection of books
available with the ACL2 distribution; for now, the reader is
encouraged to look in the README file in the books subdirectory.
For present purposes, the beginning user can simply take our word
that a book exists containing useful alist definitions and facts.
On our local machine, these definitions and lemmas can be introduced
into the current theory using the command:
This book has been ``certified,'' which means that the definitions and lemmas have been mechanically checked and stored in a safe manner. (See books and see include-book for details.)(include-book "/slocal/src/acl2/v1-9/books/public/alist-defthms")
Including this book makes available a collection of functions including the following:
Along with these function definitions, the book also provides a number of proved lemmas that aid in simplifying expressions involving these functions. (See rule-classes for the way in which lemmas are used in simplification and rewriting.) For example,(ALISTP A) ; is A an alist (actually a primitive ACL2 function)
(BIND X V A) ; associate the key X with value V in alist A
(BINDING X A) ; return the value associated with key X in alist A
(BOUND? X A) ; is key X associated with any value in alist A
(DOMAIN A) ; return the list of keys bound in alist A
(RANGE A) ; return the list of values bound to keys in alist A
(REMBIND X A) ; remove the binding of key X in alist A
asserts that(defthm bound?-bind (equal (bound? x (bind y v a)) (or (equal x y) (bound? x a))))
x will be bound in (bind y v a) if and only if:
either x = y or x was already bound in a. Also,
asserts that the resulting binding will be(defthm binding-bind (equal (binding x (bind y v a)) (if (equal x y) v (binding x a))))
v, if x = y, or the
binding that x had in a already, if not.Thus, the inclusion of this book essentially extends our specification and reasoning capabilities by the addition of new operations and facts about these operations that allow us to build further specifications on a richer and possibly more intuitive foundation.
However, it must be admitted that the use of a book such as this has two potential limitations:
For example, what is the value ofthe definitions available in a book may not be ideal for your particular problem;
it is (extremely) likely that some useful facts (especially, rewrite rules) are not available in the book and will have to be proved.
binding when given a key that
is not bound in the alist? We can find out by examining the
function definition. Look at the definition of the binding
function (or any other defined function), using the :pe command:
ACL2 !>:pe binding d 33 (INCLUDE-BOOK "/slocal/src/acl2/v1-9/books/public/alist-defthms") >V d (DEFUN BINDING (X A) "The value bound to X in alist A." (DECLARE (XARGS :GUARD (ALISTP A))) (CDR (ASSOC-EQUAL X A)))
This tells us that binding was introduced by the given
include-book form, is currently disabled in the current
theory, and has the definition given by the displayed defun form.
We see that binding is actually defined in terms of the primitive
assoc-equal function. If we look at the definition of
assoc-equal:
ACL2 !>:pe assoc-equal V -489 (DEFUN ASSOC-EQUAL (X ALIST) (DECLARE (XARGS :GUARD (ALISTP ALIST))) (COND ((ENDP ALIST) NIL) ((EQUAL X (CAR (CAR ALIST))) (CAR ALIST)) (T (ASSOC-EQUAL X (CDR ALIST)))))
we can see that assoc-equal returns nil upon reaching the end
of an unsuccessful search down the alist. So binding returns
(cdr nil) in that case, which is nil. Notice that we could also
have investigated this question by trying some simple examples.
ACL2 !>(binding 'a nil) NIL
ACL2 !>(binding 'a (list (cons 'b 2))) NIL
These definitions aren't ideal for all purposes. For one thing,
there's nothing that keeps us from having nil as a value bound to
some key in the alist. Thus, if binding returns nil we don't
always know if that is the value associated with the key in the
alist, or if that key is not bound. We'll have to keep that
ambiguity in mind whenever we use binding in our specification.
Suppose instead that we wanted binding to return some error
string on unbound keys. Well, then we'd just have to write our own
version of binding. But then we'd lose much of the value of
using a previously defined book. As with any specification
technique, certain tradeoffs are necessary.
Why not take a look at the definitions of other alist functions and see how they work together to provide the ability to construct and search alists? We'll be using them rather heavily in what follows so it will be good if you understand basically how they work. Simply start up ACL2 and execute the form shown earlier, but substituting our directory name for the top-level ACL2 directory with yours. Alternatively, the following should work if you start up ACL2 in the directory of the ACL2 sources:
Then, you can use(include-book "books/public/alist-defthms")
:pe to look at function definitions.
You'll soon discover that almost all of the definitions are built on
definitions of other, more primitive functions, as binding is
built on assoc-equal. You can look at those as well, of course,
or in many cases visit their documentation.
The other problem with using a predefined book is that it will
seldom be ``sufficiently complete,'' in the sense that the
collection of rewrite rules supplied won't be adequate to prove
everything we'd like to know about the interactions of the various
functions. If it were, there'd be no real reason to know that
binding is built on top of assoc-equal, because everything
we'd need to know about binding would be nicely expressed in the
collection of theorems supplied with the book. However, that's very
seldom the case. Developing such a collection of rules is currently
more art than science and requires considerable experience. We'll
encounter examples later of ``missing'' facts about binding and
our other alist functions. So, let's get on with the example.
Notice that alists are mappings of keys to values; but, there is no
notion of a ``type'' associated with the keys or with the values.
Our phone book example, however, does have such a notion of types;
we map names to phone numbers. We can introduce these ``types'' by
explicitly defining them, e.g., names are strings and phone numbers
are integers. Alternatively, we can partially define or
axiomatize a recognizer for names without giving a full definition.
A way to safely introduce such ``constrained'' function symbols in
ACL2 is with the encapsulate form. For example, consider the
following form.
(encapsulate ;; Introduce a recognizer for names and give a ``type'' lemma. (((namep *) => *)) ;; (local (defun namep (x) ;; This declare is needed to tell ;; ACL2 that we're aware that the ;; argument x is not used in the body ;; of the function. (declare (ignore x)) t)) ;; (defthm namep-booleanp (booleanp (namep x))))
This encapsulate form introduces the new function namep of one
argument and one result and constrains (namep x) to be Boolean,
for all inputs x. More generally, an encapsulation establishes
an environment in which functions can be defined and theorems and
rules added without necessarily introducing those functions,
theorems, and rules into the environment outside the encapsulation.
To be admissible, all the events in the body of an encapsulate must be
admissible. But the effect of an encapsulate is to assume only the
non-local events.
The first ``argument'' to encapsulate, ((namep (x) t)) above,
declares the intended signatures of new function symbols that
will be ``exported'' from the encapsulation without definition. The
local defun of name defines name within the encapsulation
always to return t. The defthm event establishes that
namep is Boolean. By making the defun local but the defthm
non-local this encapsulate constrains the undefined function
namep to be Boolean; the admissibility of the encapsulation
establishes that there exists a Boolean function (namely the
constant function returning t).
We can subsequently use namep as we use any other Boolean
function, with the proviso that we know nothing about it except that
it always returns either t or nil. We use namep to
``recognize'' legal keys for our phonebook alist.
We wish to do something similar to define what it means to be a legal phone number. We submit the following form to ACL2:
This introduces a Boolean-valued recognizer(encapsulate ;; Introduce a recognizer for phone numbers. (((pnump *) => *)) ;; (local (defun pnump (x) (not (equal x nil)))) ;; (defthm pnump-booleanp (booleanp (pnump x))) ;; (defthm nil-not-pnump (not (pnump nil)))).
pnump, with the
additional proviso that the constant nil is not a pnump. We
impose this restriction to guarantee that we'll never bind a name to
nil in our phone book and thereby introduce the kind of ambiguity
described above regarding the use of binding.
Now a legal phone book is an alist mapping from nameps to
pnumps. We can define this as follows:
Thus, a phone book is really a list of pairs(defun name-phonenum-pairp (x) ;; Recognizes a pair of (name . pnum). (and (consp x) (namep (car x)) (pnump (cdr x))))
(defun phonebookp (l) ;; Recognizes a list of such pairs. (if (not (consp l)) (null l) (and (name-phonenum-pairp (car l)) (phonebookp (cdr l)))))
(name . pnum).
Notice that we have not assumed that the keys of the phone book are
distinct. We'll worry about that question later. (It is not always
desirable to insist that the keys of an alist be distinct. But it
may be a useful requirement for our specific example.)Now we are ready to define some of the functions necessary for our phonebook example. The functions we need are:
(IN-BOOK? NM BK) ; does NM have a phone number in BK
(FIND-PHONE NM BK) ; find NM's phone number in phonebook BK
(ADD-PHONE NM PNUM BK) ; give NM the phone number PNUM in BK
(CHANGE-PHONE NM PNUM BK) ; change NM's phone number to PNUM in BK
(DEL-PHONE NM PNUM) ; remove NM's phone number from BK
Given our underlying theory of alists, it is easy to write these functions. But we must take care to specify appropriate ``boundary'' behavior. Thus, what behavior do we want when, say, we try to change the phone number of a client who is not currently in the book? As usual, there are numerous possibilities; here we'll assume that we return the phone book unchanged if we try anything ``illegal.''
Possible definitions of our phone book functions are as follows.
(Remember, an include-book form such as the ones shown earlier
must be executed in order to provide definitions for functions such
as bound?.)
Notice that we don't have to check whether a name is in the book before deleting, because(defun in-book? (nm bk) (bound? nm bk))
(defun find-phone (nm bk) (binding nm bk))
(defun add-phone (nm pnum bk) ;; If nm already in-book?, make no change. (if (in-book? nm bk) bk (bind nm pnum bk)))
(defun change-phone (nm pnum bk) ;; Make a change only if nm already has a phone number. (if (in-book? nm bk) (bind nm pnum bk) bk))
(defun del-phone (nm bk) ;; Remove the binding from bk, if there is one. (rembind nm bk))
rembind is essentially a no-op if nm
is not bound in bk.In some sense, this completes our specification. But we can't have any real confidence in its correctness without validating our specification in some way. One way to do so is by proving some properties of our specification. Some candidate properties are:
1. A name will be in the book after we add it.
2. We will find the most recently added phone number for a client.
3. If we change a number, we'll find the change.
4. Changing and then deleting a number is the same as just deleting.
5. A name will not be in the book after we delete it.
Let's formulate some of these properties. The first one, for example, is:
You may wonder why we didn't need any hypotheses about the ``types'' of the arguments. In fact,(defthm add-in-book (in-book? nm (add-phone nm pnum bk))).
add-in-book is really expressing a
property that is true of alists in general, not just of the
particular variety of alists we are dealing with. Of course, we
could have added some extraneous hypotheses and proved:
but that would have yielded a weaker and less useful lemma because it would apply to fewer situations. In general, it is best to state lemmas in the most general form possible and to eliminate unnecessary hypotheses whenever possible. The reason for that is simple: lemmas are usually stored as rules and used in later proofs. For a lemma to be used, its hypotheses must be relieved (proved to hold in that instance); extra hypotheses require extra work. So we avoid them whenever possible.(defthm add-in-book (implies (and (namep nm) (pnump pnum) (phonebookp bk)) (in-book? nm (add-phone nm pnum bk)))),
There is another, more important observation to make about our
lemma. Even in its simpler form (without the extraneous
hypotheses), the lemma add-in-book may be useless as a
rewrite rule. Notice that it is stated in terms of the
non-recursive functions in-book? and add-phone. If such
functions appear in the left hand side of the conclusion of a lemma,
the lemma may not ever be used. Suppose in a later proof, the
theorem prover encountered a term of the form:
Since we've already proved(in-book? nm (add-phone nm pnum bk)).
add-in-book, you'd expect that this
would be immediately reduced to true. However, the theorem prover
will often ``expand'' the non-recursive definitions of in-book?
and add-phone using their definitions before it attempts
rewriting with lemmas. After this expansion, lemma add-in-book
won't ``match'' the term and so won't be applied. Look back at
the proof script for add-in-proof and you'll notice that at the
very end the prover warned you of this potential difficulty when it
printed:
The ``Warnings'' line notifies you that there are non-recursive function calls in the left hand side of the conclusion and that this problem might arise. Of course, it may be that you don't ever plan to use the lemma for rewriting or that your intention is to disable these functions. Disabled functions are not expanded and the lemma should apply. However, you should always take note of such warnings and consider an appropriate response. By the way, we noted above thatWarnings: Non-rec Time: 0.18 seconds (prove: 0.05, print: 0.00, other: 0.13) ADD-IN-BOOK
binding is disabled. If it
were not, none of the lemmas about binding in the book we
included would likely be of much use for exactly the reason we just
gave.
For our current example, let's assume that we're just investigating
the properties of our specifications and not concerned about using
our lemmas for rewriting. So let's go on. If we really want to
avoid the warnings, we can add :rule-classes nil to each
defthm event; see rule-classes.
Property 2 is: we always find the most recently added phone number for a client. Try the following formalization:
and you'll find that the proof attempt fails. Examining the proof attempt and our function definitions, we see that the lemma is false if(defthm find-add-first-cut (equal (find-phone nm (add-phone nm pnum bk)) pnum))
nm is already in the book. We can remedy this situation by
reformulating our lemma in at least two different ways:
For technical reasons, lemmas such as(defthm find-add1 (implies (not (in-book? nm bk)) (equal (find-phone nm (add-phone nm pnum bk)) pnum)))
(defthm find-add2 (equal (find-phone nm (add-phone nm pnum bk)) (if (in-book? nm bk) (find-phone nm bk) pnum)))
find-add2, i.e., which do
not have hypotheses, are usually slightly preferable. This lemma is
stored as an ``unconditional'' rewrite rule (i.e., has no
hypotheses) and, therefore, will apply more often than find-add1.
However, for our current purposes either version is all right.Property 3 says: If we change a number, we'll find the change. This is very similar to the previous example. The formalization is as follows.
Property 4 says: changing and then deleting a number is the same as just deleting. We can model this as follows.(defthm find-change (equal (find-phone nm (change-phone nm pnum bk)) (if (in-book? nm bk) pnum (find-phone nm bk))))
Unfortunately, when we try to prove this, we encounter subgoals that seem to be true, but for which the prover is stumped. For example, consider the following goal. (Note:(defthm del-change (equal (del-phone nm (change-phone nm pnum bk)) (del-phone nm bk)))
endp holds of lists that
are empty.)
Our intuition aboutSubgoal *1/4 (IMPLIES (AND (NOT (ENDP BK)) (NOT (EQUAL NM (CAAR BK))) (NOT (BOUND? NM (CDR BK))) (BOUND? NM BK)) (EQUAL (REMBIND NM (BIND NM PNUM BK)) (REMBIND NM BK))).
rembind and bind tells us that this goal
should be true even without the hypotheses. We attempt to prove the
following lemma.
The prover proves this by induction, and stores it as a rewrite rule. After that, the prover has no difficulty in proving(defthm rembind-bind (equal (rembind nm (bind nm pnum bk)) (rembind nm bk)))
del-change.
The need to prove lemma rembind-bind illustrates a point we made
early in this example: the collection of rewrite rules
supplied by a previously certified book will almost never be
everything you'll need. It would be nice if we could operate purely
in the realm of names, phone numbers, and phone books without ever
having to prove any new facts about alists. Unfortunately, we
needed a fact about the relation between rembind and bind that
wasn't supplied with the alists theory. Hopefully, such omissions
will be rare.
Finally, let's consider our property 5 above: a name will not be in the book after we delete it. We formalize this as follows:
This proves easily. But notice that it's only true because(defthm in-book-del (not (in-book? nm (del-phone nm bk))))
del-phone (actually rembind) removes all occurrences of a
name from the phone book. If it only removed, say, the first one it
encountered, we'd need a hypothesis that said that nm occurs at
most once in bk. Ah, maybe that's a property you hadn't
considered. Maybe you want to ensure that any name occurs at
most once in any valid phonebook.To complete this example, let's consider adding an invariant to our specification. In particular, suppose we want to assure that no client has more than one associated phone number. One way to ensure this is to require that the domain of the alist is a ``set'' (has no duplicates).
Now, we want to show under what conditions our operations preserve the property of being a(defun setp (l) (if (atom l) (null l) (and (not (member-equal (car l) (cdr l))) (setp (cdr l)))))
(defun valid-phonebookp (bk) (and (phonebookp bk) (setp (domain bk))))
valid-phonebookp. The operations
in-book? and find-phone don't return a phone book, so we
don't really need to worry about them. Since we're really
interested in the ``types'' of values preserved by our phonebook
functions, let's look at the types of those operations as well.
Note the ``(defthm in-book-booleanp (booleanp (in-book? nm bk)))
(defthm in-book-namep (implies (and (phonebookp bk) (in-book? nm bk)) (namep nm)) :hints (("Goal" :in-theory (enable bound?))))
(defthm find-phone-pnump (implies (and (phonebookp bk) (in-book? nm bk)) (pnump (find-phone nm bk))) :hints (("Goal" :in-theory (enable bound? binding))))
:hints'' on the last two lemmas. Neither of these
would prove without these hints, because once again there are
some facts about bound? and binding not available in our
current context. Now, we could figure out what those facts are and
try to prove them. Alternatively, we can enable bound? and
binding and hope that by opening up these functions, the
conjectures will reduce to versions that the prover does know enough
about or can prove by induction. In this case, this strategy works.
The hints tell the prover to enable the functions in question
when considering the designated goal.
Below we develop the theorems showing that add-phone,
change-phone, and del-phone preserve our proposed invariant.
Notice that along the way we have to prove some subsidiary facts,
some of which are pretty ugly. It would be a good idea for you to
try, say, add-phone-preserves-invariant without introducing the
following four lemmas first. See if you can develop the proof and
only add these lemmas as you need assistance. Then try
change-phone-preserves-invariant and del-phone-preserves-invariant.
They will be easier. It is illuminating to think about why
del-phone-preserves-invariant does not need any ``type''
hypotheses.
(defthm bind-preserves-phonebookp (implies (and (phonebookp bk) (namep nm) (pnump num)) (phonebookp (bind nm num bk)))) (defthm member-equal-strip-cars-bind (implies (and (not (equal x y)) (not (member-equal x (strip-cars a)))) (not (member-equal x (strip-cars (bind y z a)))))) (defthm bind-preserves-domain-setp (implies (and (alistp bk) (setp (domain bk))) (setp (domain (bind nm num bk)))) :hints (("Goal" :in-theory (enable domain)))) (defthm phonebookp-alistp (implies (phonebookp bk) (alistp bk))) (defthm ADD-PHONE-PRESERVES-INVARIANT (implies (and (valid-phonebookp bk) (namep nm) (pnump num)) (valid-phonebookp (add-phone nm num bk))) :hints (("Goal" :in-theory (disable domain-bind)))) (defthm CHANGE-PHONE-PRESERVES-INVARIANT (implies (and (valid-phonebookp bk) (namep nm) (pnump num)) (valid-phonebookp (change-phone nm num bk))) :hints (("Goal" :in-theory (disable domain-bind)))) (defthm remove-equal-preserves-setp (implies (setp l) (setp (remove-equal x l)))) (defthm rembind-preserves-phonebookp (implies (phonebookp bk) (phonebookp (rembind nm bk)))) (defthm DEL-PHONE-PRESERVES-INVARIANT (implies (valid-phonebookp bk) (valid-phonebookp (del-phone nm bk))))
As a final test of your understanding, try to formulate and prove an invariant that says that no phone number is assigned to more than one name. The following hints may help.
1. Define the appropriate invariant. (Hint: remember the function
range.)2. Do our current definitions of
add-phoneandchange-phonenecessarily preserve this property? If not, consider what hypotheses are necessary in order to guarantee that they do preserve this property.3. Study the definition of the function
rangeand notice that it is defined in terms of the functionstrip-cdrs. Understand how this defines the range of an alist.4. Formulate the correctness theorems and attempt to prove them. You'll probably benefit from studying the invariant proof above. In particular, you may need some fact about the function
strip-cdrsanalogous to the lemmamember-equal-strip-cars-bindabove.
Below is one solution to this exercise. Don't look at the solution,
however, until you've struggled a bit with it. Notice that we
didn't actually change the definitions of add-phone and
change-phone, but added a hypothesis saying that the number is
``new.'' We could have changed the definitions to check this and
return the phonebook unchanged if the number was already in use.
(defun pnums-in-use (bk) (range bk)) (defun phonenums-unique (bk) (setp (pnums-in-use bk))) (defun new-pnump (pnum bk) (not (member-equal pnum (pnums-in-use bk)))) (defthm member-equal-strip-cdrs-rembind (implies (not (member-equal x (strip-cdrs y))) (not (member-equal x (strip-cdrs (rembind z y)))))) (defthm DEL-PHONE-PRESERVES-PHONENUMS-UNIQUE (implies (phonenums-unique bk) (phonenums-unique (del-phone nm bk))) :hints (("Goal" :in-theory (enable range)))) (defthm strip-cdrs-bind-non-member (implies (and (not (bound? x a)) (alistp a)) (equal (strip-cdrs (bind x y a)) (append (strip-cdrs a) (list y)))) :hints (("Goal" :in-theory (enable bound?)))) (defthm setp-append-list (implies (setp l) (equal (setp (append l (list x))) (not (member-equal x l))))) (defthm ADD-PHONE-PRESERVES-PHONENUMS-UNIQUE (implies (and (phonenums-unique bk) (new-pnump pnum bk) (alistp bk)) (phonenums-unique (add-phone nm pnum bk))) :hints (("Goal" :in-theory (enable range)))) (defthm member-equal-strip-cdrs-bind (implies (and (not (member-equal z (strip-cdrs a))) (not (equal z y))) (not (member-equal z (strip-cdrs (bind x y a)))))) (defthm CHANGE-PHONE-PRESERVES-PHONENUMS-UNIQUE (implies (and (phonenums-unique bk) (new-pnump pnum bk) (alistp bk)) (phonenums-unique (change-phone nm pnum bk))) :hints (("Goal" :in-theory (enable range))))
