(** * Primality on nat *)

Definition Prime (n : nat) : Prop :=
  n > 1 /\ (forall q : nat, Divides q n -> q = 1 \/ q = n).(** Prime with bounded all quantifier. *)

Definition bPrime (n : nat) :=
  n > 1 /\ (forall q : nat, q < S n -> Divides q n -> q = 1 \/ q = n).

Theorem primepropdiv : forall n : nat, n > 1 -> (forall q : nat, Prime q -> Divides q n -> q * q > n) -> Prime n.