Fact fin_reif n : (forall i, i < n -> exists x, R i x)
                 -> exists s, forall i (Hi : i < n), R i (s i Hi).Fact fin_reif_nat (R : nat -> nat -> Prop) n :
         (forall i, i < n -> ex (R i)) -> exists s, forall i, i < n -> R i (s i).

Lemma first_non_zero (f : nat -> nat) n : f 0 = 0 -> f n <> 0 -> exists i, i < n /\ (forall k, k <= i -> f k = 0) /\ f (i+1) <> 0.