(** * Emptyness *)
Definition empty (i: interv) : Prop := fst i >= snd i.

(* Specifications: *)
Lemma empty_dec:
forall i, {empty i} + {~empty i}.
Lemma is_notempty:
forall i, fst i < snd i -> ~empty i.
Lemma empty_notin:
forall x i, empty i -> ~In x i.
Lemma in_notempty:
forall x i, In x i -> ~empty i.