Computability and Logic

298 DECIDABILITY OF ARITHMETIC WITHOUT MULTIPLICATION
contain the variable x.) Repeat these three steps over and over until no disjunct has
more than one lower inequality among its conjuncts.
(15) Carry out an analogous process for upper inequalities, until no disjunct has more
than one lower or upper inequality among its conjuncts.
(16) Replace each formula of form
by
D m (kx), D m (---kx), D m (kx+t), or D m (---kx+t)
D m (kx --- 0), D m (kx --- 0), D m (kx --- (---t)), or D m (kx --- t)
as the case may be. This step is justified because for any number a, m divides a if
and only if m divides −a.
(17) At this point all atomic formulas with D m and involving x have the form
D m (kx --- t), where k is a positive integer. Replace any formula of this form by the
disjunction of all conjunctions
D m (kx --- i)&D m (t --- i)
for 0 ≤ i < m. To see that this step is justified, note that m divides the difference of
two numbers a and b if and only if a and b leave the same remainder on division
by m, and that the remainder on dividing a (respectively, b) bym is the unique i
with 0 ≤ i < m such that m divides a − i (respectively, b − i).
(18) Put the result back into disjunctive normal form.
(19) At this point all atomic formulas with D m and involving x have the form
D m (kx --- i), where k is a positive integer and 0 ≤ i < m. Replace any formula of
this form with k >1 by the disjunction of the formulas D m (x --- j) for all j with
0 ≤ j < m such that m divides kj − i. This step is justified because for any
number a, ka leaves a remainder of i on division by m if and only if kj does, where
j is the remainder on dividing a by m.
(20) Put the result back into disjunctive normal form.
(21) At this point all atomic formulas with D m and involving x have the form
D m (x --- i), where i is a nonnegative integer. In any such case consider the prime
decomposition of m; that is, write
m = p e 1
1 ···pe k
k
where p 1 < p 2 < ··· < pk and all ps are primes.
If k >1, then let m 1 = p e 1
1 ,...,m k = p e k
k , and replace D m(x --- i) by
D m1 (x --- i)& ... & D mk (x --- i).
This step is justified because the product of two given numbers having no common
factor (such as powers of distinct primes) divides a given number if and only if
each of the two given numbers does.
(22) At this point all atomic formulas with Ds and involving x have the form
D m (x − i), where i is a nonnegative integer, and m a power of a prime. If in a
given disjunct there are two conjuncts D m (x − i) and D n (x − j) where m and n
are powers of the same prime, say m = p d , n = p e , d ≤ e, then drop D m (x --- i) in
favor of D n (i − j), which does not involve x. This step is justified because, since