Definition R_prime (x y : Z) := 1 < x /\ 1 < y /\ x < y.Definition f_Z (x : Z) := Zabs_nat x.

Lemma not_prime_gen : forall a b : Z, 1 < a -> 1 < b -> b < a -> ~ prime a -> (forall c : Z, b < c < a -> rel_prime c a) -> exists q : Z, exists b : Z, a = q * b /\ 1 < q /\ 1 < b.