(** bounded quantifiers preserve decidability *)

(* Specifications: *)
Theorem alldec :
forall (P : nat -> Prop) (N : nat),
(forall n : nat, P n \/ ~ P n) ->
(forall x : nat, x < N -> P x) \/ ~ (forall x : nat, x < N -> P x).
Theorem exdec :
forall (P : nat -> Prop) (N : nat),
(forall n : nat, P n \/ ~ P n) ->
(exists x : nat, x < N /\ P x) \/ ~ (exists x : nat, x < N /\ P x).