intros.
induction n as [| n Hrecn].
exact 0.
exact (max (x n) Hrecn).
Defined.

(* Specifications: *)
Lemma maxBetaLe :
forall (n : nat) (x : nat -> nat) (i : nat), i < n -> x i <= maxBeta n x.
Theorem divProd2 :
forall (n : nat) (x : nat -> nat) (i : nat),
i <= n -> Divides (prod i x) (prod n x).
Theorem betaTheorem1 :
forall (n : nat) (y : nat -> nat),
{a : nat * nat |
forall z : nat,
z < n ->
y z =
snd (proj1_sig (modulo (coPrimeBeta z (snd a)) (gtBeta z (snd a)) (fst a)))}.