(VAR x y zs ys) (RULES isprime(x) -> mem(x, primes(x)) mem(x, nil) -> FALSE mem(x, cons(y, zs)) -> TRUE :|: x = y mem(x, cons(y, zs)) -> mem(x, zs) :|: y > x mem(x, cons(y, zs)) -> mem(x, zs) :|: x > y primes(x) -> sieve(nats(2, x)) nats(x, y) -> nil :|: x > y nats(x, y) -> cons(x, nil) :|: x = y nats(x, y) -> cons(x, nats(x + 1, y)) :|: y > x sieve(nil) -> nil sieve(cons(x, ys)) -> cons(x, sieve(filter(x, ys))) filter(x, nil) -> nil filter(x, cons(y, zs)) -> if_1(isdiv(x, y), x, y, zs) if_1(TRUE, x, y, zs) -> filter(x, zs) filter(x, cons(y, zs)) -> if_2(isdiv(x, y), x, y, zs) if_2(FALSE, x, y, zs) -> cons(x, filter(x, zs)) isdiv(x, 0) -> TRUE :|: x > 0 isdiv(x, y) -> FALSE :|: x > y && y > 0 isdiv(x, y) -> isdiv(x, (-x) + y) :|: y >= x && x > 0 isdiv(x, y) -> isdiv(x, -y) :|: 0 > y isdiv(x, y) -> isdiv(-x, y) :|: 0 > x )
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