(** * Emptiness is decidable *)
Definition Empty (E : R -> Prop) :=
Lub_Rbar (fun x => x = 0 \/ E x) = Glb_Rbar (fun x => x = 0 \/ E x)
/\ Lub_Rbar (fun x => x = 1 \/ E x) = Glb_Rbar (fun x => x = 1 \/ E x).

Lemma not_Empty_dec (E : R -> Prop) : (Decidable.decidable (exists x, E x)) ->
{(exists x, E x)} + {(forall x, ~ E x)}.