Definition prod : forall (n : nat) (x : nat -> nat), nat.
intros.
induction n as [| n Hrecn].
exact 1.
exact (x n * Hrecn).
Defined.
Definition factorial (n : nat) : nat := prod n S.

Lemma primeDiv :
forall a : nat, 1 < a -> exists p : nat, Prime p /\ Divides p a.