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 bounded_min (P : nat -> Prop) : (forall x, P x \/ ~ P x) -> forall n, (exists k, k < n /\ P k /\ forall i, i < k -> ~ P i) \/ forall k, k < n -> ~ P k.