Fixpoint zip {X} (f1 f2 : nat -> X) n :=
  match n with
  | O => f1 O
  | S O => f2 0
  | S (S n) => zip (fun n => f1 (S n)) (fun n => f2 (S n)) n
  end.

Lemma finite_product: forall {A B} {PA : A -> Prop} {PB : B -> Prop}, finite PA -> finite PB -> finite (fun ab => PA (fst ab) /\ PB (snd ab)).