Computability and Logic

DECIDABILITY OF ARITHMETIC WITHOUT MULTIPLICATION 299
m divides n, for any number a,ifa leaves remainder j on division by n, a will
leave remainder i on division by m if and only if j does.
(23) Repeat the preceding step until for any two conjuncts D m (x --- i) and D n (x --- j)ina
single disjunct, m and n are powers of distinct primes, and therefore have no
common factors.
(24) Replace each D m (x --- i)byD m (x --- i*), where i* is the remainder on dividing i
by m.
(25) Rewrite each disjunct so that all atomic formulas with with Ds and involving x are
on the left.
(26) At this point each disjunct has the form
D m1 (x --- i 1 )& ... & D mk (x --- i k ) & (other conjuncts)
where 0 ≤ i 1 < m 1 ,...,0 ≤ i k < m k . Let m = m 1 ·····m k . According to the
Chinese remainder theorem (see Lemma 15.5), there exists a (unique) i with
0 ≤ i < m such that i leaves remainder i 1 on division by m 1 ,...,i leaves
remainder i k on division by m k . Replace the conjuncts involving Ds by the single
formula D m (x --- i).
(27) At this point we have a disjunction F 1 ∨···∨F k each of whose disjuncts is a
conjunction containing at most one lower inequality, at most one upper inequality,
and at most one formula of form D m (x − i). Rewrite ∃x(F 1 ∨···∨F k )as
∃xF 1 ∨···∨∃xF k .
(28) Within each disjunct ∃xF, rewrite the conjunction F so that any and all conjuncts
involving x occur on the left, and confine the quantifier to these conjuncts, of
which there are at most three; if there are none, simply omit the quantifier.
(29) At this point, the only occurrences of x are in sentences of one of the seven types
listed in Table 24-1. Replace these by the sentences listed on the right.
Table 24-1. Elimination of quantifiers
∃x s< jx 0