VERIFY-GUARDS

verify the guards of a function 
Major Section:  EVENTS

See guard for a general discussion of guards.

Before discussing the verify-guards event, we first discuss guard verification, which can take place at definition time or, later, using verify-guards. Typically, guard verification takes place at definition time if a guard (or type, or stobjs) has been supplied explicitly unless :verify-guards nil has been specified; see defun and see xargs, and see set-verify-guards-eagerness for how to change this default. The point of guard verification is to ensure that during evaluation of an expression without free variables, no guard violation takes place.

Guard verification is intended to guarantee that for any call of a given function, if its guard holds for that call then the guard will hold for every function call in the body of that function. Moreover, in order to avoid guard violations during evaluation of the function's guard itself, guard verification also is intended to guarantee that the guards are satisfied for all calls in the guard itself. Consider the following simple example. 

(defun f (x)
  (declare (xargs :guard (and (consp x)
                              (integerp (car x)))))
  (if (rationalp (cdr x))
      (+ (car x) (cdr x))
    17))
If you evaluate (f t), for example, in the top-level loop, you will (by default) get a guard error. The point of guard verification is to guarantee the absence of guard errors, and we start by using this example to illustrate the proof obligations that guarantee such absence.

The body of the above definition has the following function calls, where the first is the entire body. 

  (if (rationalp (cdr x))
      (< (car x) (cdr x))
    17)
  (rationalp (cdr x)) ; the test of the top-level IF call
  (cdr x)             ; from (rationalp (cdr x))
  (< (car x) (cdr x)) ; the true branch of the top-level IF call
  (car x)             ; from (< (car x) (cdr x))
  (cdr x)             ; from (< (car x) (cdr x))
We thus see potentially six conditions to prove, one for each call. The guards of the function symbols of those calls are t for if and rationalp, (or (consp x) (equal x nil)) for both (car x) and (cdr x), and finally that both arguments are rationals for <. Moreover, we can take advantage of ``contextual assumptions'': the if-test conditions and the top-level :guard. Thus, for verify-guards the proof obligation from the body of f is as follows. 
(implies 
 (and (consp x) (integerp (car x))) ; from the :guard
 (and t ; from the top-level IF call
      t ; from (rationalp (cdr x))
      (or (consp x) (equal x nil)) ; from the first (cdr x)
      (implies
       (rationalp (cdr x)) ; IF-test for calls in the true branch
       (and (or (consp x) (equal x nil)) ; from (car x)
            (or (consp x) (equal x nil)) ; from the second (cdr x)
            (and (rationalp (car x)) (rationalp (cdr x))) ; from the < call
            ))))
But the :guard itself generates a similar sort of proof obligation. Note that the guard (and (consp x) (integerp (car x))) is really an abbreviation (i.e. via the macro AND) for the term (if (consp x) (integerp (car x)) nil). The guard proof obligation for the guard itself is thus as follows. 
(and t ; from (consp x)
     (implies (consp x)
              (and t         ; from (integerp (car x)) ;
                   (consp x) ; from (car x) ;
                   )))
All of the above proof obligations are indeed theorems, and guard verification succeeds for the above definition of f.

The example above illustrates the general procedure for generating the guard proof obligation. Each function call is considered in the body or guard of the function, and it is required that the guard is met for that call, under certain ``contextual assumptions'', which are as follows. In the case of the body of the named function, it is assumed that the guard holds for that function on its formal parameters. And in both cases -- the body of the named function and also its guard -- the governing tests from superior calls of IF are also assumed.

As mentioned above, if the guard on a function is not t, then guard verification requires not only consideration of the body under the assumption that the guard is true, but also consideration of the guard itself. Thus, for example, guard verification fails in the following example, even though there are no proof obligations arising from the body, because the guard itself can cause a guard violation when evaluated for an arbitrary value of x: 

(defun foo (x)
  (declare (xargs :guard (car x)))
  x)

We turn now to the verify-guards event as a way of verifying the guards for a function or theorem. 

Examples:
(verify-guards flatten)
(verify-guards flatten
               :hints (("Goal" :use (:instance assoc-of-app)))
               :otf-flg t
               :guard-debug t ; default = nil
               :doc "string")

General Form:
(verify-guards name
        :hints        hints
        :otf-flg      otf-flg
        :guard-debug  t ; typically t, but any value is legal
        :doc          doc-string)
In the General Form above, name is the name of a :logic function (see defun-mode) or of a theorem or axiom. In the most common case name is the name of a function that has not yet had its guards verified, each subroutine of which has had its guards verified. The values hints, otf-flg, and guard-debug are as described in the corresponding documentation entries; and doc-string, if supplied, is a string not beginning with ``:Doc-Section''. The four keyword arguments above are all optional. To admit this event, the conjunction of the guard proof obligations must be proved. If that proof is successful, name is considered to have had its guards verified.

See verify-guards-formula for a utility that lets you view the formula to be proved by verify-guards, but without creating an event.

If name is one of several functions in a mutually recursive clique, verify-guards will attempt to verify the guards of all of the functions.

If name is a theorem or axiom name, verify-guards verifies the guards of the associated formula. When a theorem has had its guards verified then you know that the theorem will evaluate to non-nil in all Common Lisps, without causing a runtime error (other than possibly a resource error). In particular, you know that the theorem's validity does not depend upon ACL2's arbitrary completion of the domains of partial Common Lisp functions.

For example, if app is defined as 

(defun app (x y)
  (declare (xargs :guard (true-listp x)))
  (if (endp x)
      y
      (cons (car x) (app (cdr x) y))))
then we can verify the guards of app and we can prove the theorem: 
(defthm assoc-of-app
  (equal (app (app a b) c) (app a (app b c))))
However, if you go into almost any Common Lisp in which app is defined as shown and evaluate 
(equal (app (app 1 2) 3) (app 1 (app 2 3)))
we get an error or, perhaps, something worse like nil! How can this happen since the formula is an instance of a theorem? It is supposed to be true!

It happens because the theorem exploits the fact that ACL2 has completed the domains of the partially defined Common Lisp functions like car and cdr, defining them to be nil on all non-conses. The formula above violates the guards on app. It is therefore ``unreasonable'' to expect it to be valid in Common Lisp.

But the following formula is valid in Common Lisp: 

(if (and (true-listp a)
         (true-listp b))
    (equal (app (app a b) c) (app a (app b c)))
    t)
That is, no matter what the values of a, b and c the formula above evaluates to t in all Common Lisps (unless the Lisp engine runs out of memory or stack computing it). Furthermore the above formula is a theorem:

(defthm guarded-assoc-of-app
  (if (and (true-listp a)
           (true-listp b))
      (equal (app (app a b) c) (app a (app b c)))
      t))
This formula, guarded-assoc-of-app, is very easy to prove from assoc-of-app. So why prove it? The interesting thing about guarded-assoc-of-app is that we can verify the guards of the formula. That is, (verify-guards guarded-assoc-of-app) succeeds. Note that it has to prove that if a and b are true lists then so is (app a b) to establish that the guard on the outermost app on the left is satisfied. By verifying the guards of the theorem we know it will evaluate to true in all Common Lisps. Put another way, we know that the validity of the formula does not depend on ACL2's completion of the partial functions or that the formula is ``well-typed.''

One last complication: The careful reader might have thought we could state guarded-assoc-of-app as 

(implies (and (true-listp a)
              (true-listp b))
         (equal (app (app a b) c)
                (app a (app b c))))
rather than using the if form of the theorem. We cannot! The reason is technical: implies is defined as a function in ACL2. When it is called, both arguments are evaluated and then the obvious truth table is checked. That is, implies is not ``lazy.'' Hence, when we write the guarded theorem in the implies form we have to prove the guards on the conclusion without knowing that the hypothesis is true. It would have been better had we defined implies as a macro that expanded to the if form, making it lazy. But we did not and after we introduced guards we did not want to make such a basic change.

Recall however that verify-guards is almost always used to verify the guards on a function definition rather than a theorem. We now return to that discussion.

Because name is not uniquely associated with the verify-guards event (it necessarily names a previously defined function) the documentation string, doc-string, is not stored in the documentation database. Thus, we actually prohibit doc-string from having the form of an ACL2 documentation string; see doc-string.

Verify-guards must often be used when the value of a recursive call of a defined function is given as an argument to a subroutine that is guarded. An example of such a situation is given below. Suppose app (read ``append'') has a guard requiring its first argument to be a true-listp. Consider 

(defun rev (x)
  (declare (xargs :guard (true-listp x)))
  (cond ((endp x) nil)
        (t (app (rev (cdr x)) (list (car x))))))
Observe that the value of a recursive call of rev is being passed into a guarded subroutine, app. In order to verify the guards of this definition we must show that (rev (cdr x)) produces a true-listp, since that is what the guard of app requires. How do we know that (rev (cdr x)) is a true-listp? The most elegant argument is a two-step one, appealing to the following two lemmas: (1) When x is a true-listp, (cdr x) is a true-listp. (2) When z is a true-listp, (rev z) is a true-listp. But the second lemma is a generalized property of rev, the function we are defining. This property could not be stated before rev is defined and so is not known to the theorem prover when rev is defined.

Therefore, we might break the admission of rev into three steps: define rev without addressing its guard verification, prove some general properties about rev, and then verify the guards. This can be done as follows: 

(defun rev (x)
  (declare (xargs :guard (true-listp x)
                  :verify-guards nil))    ; Note this additional xarg.
  (cond ((endp x) nil)
        (t (app (rev (cdr x)) (list (car x))))))

(defthm true-listp-rev
  (implies (true-listp x2)
           (true-listp (rev x2))))

(verify-guards rev)
The ACL2 system can actually admit the original definition of rev, verifying the guards as part of the defun event. The reason is that, in this particular case, the system's heuristics just happen to hit upon the lemma true-listp-rev. But in many more complicated functions it is necessary for the user to formulate the inductively provable properties before guard verification is attempted.

Remark on computation of guard conjectures and evaluation. When ACL2 computes the guard conjecture for the body of a function, it evaluates any ground subexpressions (those with no free variables), for calls of functions whose :executable-counterpart runes are enabled. Note that here, ``enabled'' refers to the current global theory, not to any :hints given to the guard verification process; after all, the guard conjecture is computed even before its initial goal is produced. Also note that this evaluation is done in an environment as though :set-guard-checking :all had been executed, so that we can trust that this evaluation takes place without guard violations; see set-guard-checking.

If you want to verify the guards on functions that are built into ACL2, you will first need to put them into :logic mode. See verify-termination, specifically the ``Remark on system functions'' in that documentation.