Computability and Logic

DECIDABILITY OF ARITHMETIC WITHOUT MULTIPLICATION 297
(2) Replace each formula of form r = s by (r < s + 1 & s < r + 1).
(3) Replace each formula of form r ≠ s by (r < s ∨ s < r).
(4) Put the result back into disjunctive normal form.
(5) Replace each formula of form ∼r < s by s < r + 1.
(6) Replace each formula of form ∼D m s by the disjunction of D m (s + i) for all i with
0 < i < m. The result is coextensive with the original, because for any number
a, m divides exactly one of a, a + 1, a + 2,...,a + m − 1.
(7) Put the result back into disjunctive normal form.
(8) At this point we have a disjunction of conjunctions of atomic formulas of the
forms r < s and D m s. Replace each formula of form r < s by 0