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