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

(* Specifications: *)
Lemma R_prime_wf : well_founded R_prime.
Lemma ind_prime : forall P : Z -> Prop, (forall x : Z, (forall y : Z, (R_prime y x -> P y)) -> P x) -> forall x : Z, P x.
Lemma prime_dec_gen : forall a b : Z, 1 < b -> b < a -> (forall c : Z, b < c < a -> rel_prime c a) -> prime a \/ ~ prime a.
Lemma prime_dec : forall a : Z, prime a \/ ~ prime a.
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.
Lemma not_prime : forall a : Z, 1 < a -> ~ prime a -> exists q : Z, exists b : Z, a = q * b /\ 1 < q /\ 1 < b.