075-is_multiply_prime.dfy

predicate Prime(p: nat)
    {
        p > 1 &&
        forall k :: 1 < k < p ==> p % k != 0
    }

    method is_multiply_prime(x: nat) returns (ans : bool)
        // pre-conditions-start
        requires x > 1
        // pre-conditions-end
        // post-conditions-start
        ensures ans <==> exists a: nat, b: nat, c: nat :: Prime(a) && Prime(b) && Prime(c) && x == a * b * c
        // post-conditions-end
        {
            // impl-start
            for a := 2 to x
                // invariants-start
                invariant forall i: nat, j: nat, k: nat :: (Prime(i) && Prime(j) && Prime(k) && i < a) ==> x != i * j * k
                // invariants-end
            {
                if Prime(a) {
                    for b := 2 to x
                        // invariants-start
                        invariant forall j: nat, k: nat :: (Prime(j) && Prime(k) && j < b) ==> x != a * j * k
                        // invariants-end
                    {
                        if Prime(b) {
                            for c := 2 to x
                                // invariants-start
                                invariant forall k: nat :: (Prime(k) && k < c) ==> x != a * b * k
                                // invariants-end
                            {
                                if Prime(c) && x == a * b * c {
                                    return true;
                                }
                            }
                        }
                    }
                }
            }
            return false;
            // impl-end
        }