NON-LINEAR-ARITHMETIC

Non-linear Arithmetic 
Major Section:  MISCELLANEOUS

This documentation topic is divided into two parts. We first discuss the practical aspect of how to use the non-linear arithmetic extension to ACL2, and then the theory behind it. We assume that the reader is familiar with the material in linear-arithmetic and that on :linear rules.

We begin our discussion of how to use non-linear arithmetic with a simple example. Assume that we wish to prove: 

(thm
 (implies (and (rationalp x)
               (rationalp y)
               (rationalp z)
               (< 0 y)
               (< x (* y z)))
          (< (floor x y) z)))

Note that (floor x y) <= (/ x y). Note also that if we divide both sides of x < (* y z) by y, since 0 < y, we obtain (/ x y) < z. By chaining these two inequalities together, we get the inequality we desired to prove.

We now proceed with our example session:

(skip-proofs
 (progn

; Since the truth of this theorem depends on the linear properties
; of floor, we will need the linear lemma:

   (defthm floor-bounds-1
       (implies (and (rationalp x)
                     (rationalp y))
                (and (< (+ (/ x y) -1)
                        (floor x y))
                     (<= (floor x y)
                         (/ x y))))
       :rule-classes ((:linear :trigger-terms ((floor x y)))))

; We now disable floor, so that the linear lemma will be used.

   (in-theory (disable floor))

; We create five rewrite rules which we will use during non-linear
; arithmetic.  The necessity for these is due to one of the differences in
; ACL2's behaviour when non-linear arithmetic is turned on.  Although
; the conclusions of linear lemmas have always been rewritten before
; they are used, now, when non-linear arithmetic is turned on, the
; conclusions are rewritten under a different theory than under ``normal''
; rewriting.  This theory is also used in other, similar, circumstances
; described below.

   (defthm |arith (* -1 x)|
       (equal (* -1 x)
              (- x)))

   (defthm |arith (* 1 x)|
       (equal (* 1 x)
              (fix x)))

   (defthm |arith (* x (/ x) y)|
       (equal (* x (/ x) y)
              (if (equal (fix x) 0)
                  0
                  (fix y))))

   (defthm |arith (* y x)|
       (equal (* y x)
              (* x y)))

   (defthm |arith (fix x)|
       (implies (acl2-numberp x)
                (equal (fix x)
                       x))))
 )  ; End skip-proofs.

; We disable the above rewrite rules from normal use.

(in-theory (disable |arith (* -1 x)|
                    |arith (* 1 x)|
                    |arith (* x (/ x) y)|
                    |arith (* y x)|
                    |arith (fix x)|))

; We create an arithmetic-theory.  Note that we must give a quoted
; constant for the theory -- none of the normal theory-functions
; are applicable to in-arithmetic-theory.

(in-arithmetic-theory '(|arith (* -1 x)|
                        |arith (* 1 x)|
                        |arith (* x (/ x) y)|
                        |arith (* y x)|
                        |arith (fix x)|))

; We turn non-linear arithmetic on.

(set-non-linearp t)

; We can now go ahead and prove our theorem.

(thm
 (implies (and (rationalp x)
               (rationalp y)
               (rationalp z)
               (< 0 y)
               (< x (* y z)))
          (< (floor x y) z)))

The above example illustrates the two practical requirements for using non-linear arithmetic in ACL2. First, one must set up an arithmetic-theory. Usually, one would not set up an arithmetic-theory on one's own but would instead load a library book or books which do so. Second, one must turn the non-linear arithmetic extension on. This one must do explicitly -- no book can do this for you.

For a brief discussion of why this is so, even though (set-non-linearp t) is an embeddable event, see acl2-defaults-table (in particular, the final paragraph). (Note that (set-non-linearp t) modifies the acl2-defaults-table.) Also see set-non-linearp, see embedded-event-form, and see events.

You can also enable non-linear arithmetic with the hint :nonlinearp t. See hints. We, in fact, recommend the use of a hint which will enable nonlinear arithmetic only when the goal has stabilized under rewriting. Using default-hints can make this easier.

(defun nonlinearp-default-hint (stable-under-simplificationp hist pspv)
  (cond (stable-under-simplificationp
         (if (not (access rewrite-constant
                          (access prove-spec-var pspv :rewrite-constant)
                          :nonlinearp))
             '(:computed-hint-replacement t :nonlinearp t)
           nil))
        ((access rewrite-constant
                 (access prove-spec-var pspv :rewrite-constant)
                 :nonlinearp)
         (if (not (equal (caar hist) 'SETTLED-DOWN-CLAUSE))
             '(:computed-hint-replacement t :nonlinearp nil)
           nil))
        (t nil)))

(set-default-hints '((nonlinearp-default-hint stable-under-simplificationp
                                              hist pspv)))

This has proven to be a helpful strategy which allows faster proof times.

We now proceed to briefly describe the theory behind the non-linear extension to ACL2. In linear-arithmetic it was stated that, ``[L]inear polynomial inequalities can be combined by cross-multiplication and addition to permit the deduction of a third inequality....'' That is, if 

0 < poly1,
0 < poly2,
and c and d are positive rational constants, then 
0 < c*poly1 + d*poly2.

Similarly, given the above, 

0 < poly1*poly2.

In the linear arithmetic case, we are taking advantage of the facts that multiplication by a positive rational constant does not change the sign of a polynomial and that the sum of two positive polynomials is itself positive. In the non-linear arithmetic case, we are using the fact that the product of two positive polynomials is itself positive.

For example, suppose we have the three assumptions: 

p1:  3*x*y + 7*a < 4
p2:            3 < 2*x  or p2': 0 < -3 + 2*x
p3:            1 < y    or p3': 0 < -1 + y,
and we wish to prove that a < 0. As described elsewhere (see linear-arithmetic), we proceed by assuming the negation of our goal: 
p4:            0 <= a,
and looking for a contradiction.

There are no cancellations which can be performed by linear arithmetic in the above situation. (Recall that two polynomials are cancelled against each other only when they have the same largest unknown.) However, p1 has a product as its largest unknown, and for each of the factors of that product there is a poly that has that factor as a largest unknown. When non-linear arithmetic is enabled, ACL2 will therefore multiply p1' and p2' obtaining 

p5:            0 < 3 + -2*x + -3*y + 2*x*y.
The addition of this polynomial will allow cancelation to continue and, in this case, we will prove our goal. Thus, just as ACL2 adds two polynomials together when they have the same largest unknown of opposite signs in order to create a new ``smaller'' polynomial; so ACL2 multiplies polynomials together when the product of their largest unknowns is itself the largest unknown of another polynomial. As the use of :linear lemmas to further seed the arithmetic database may allow cancellation to proceed, so may the use of non-linear arithmetic.

This multiplication of polynomials is the motivation for an arithmetic-theory distinct from than the normal one. Because this may be done so often, and because the individual factors have presumably already been rewritten, it is important that this be done in an efficient way. The use of a small, specialized, theory helps avoid the repeated application of rewrite rules to already stabilized terms.