(** [X] is the translation of [Y] as a predicate *)

Definition X P n := exists l, length l = n /\ Y P (true::l).

Lemma inductively_barred_at_decidable : forall P, (forall l, P l \/ ~ P l) -> forall n l, inductively_barred_at P n l \/ ~ inductively_barred_at P n l.