Definition eqopt (a b : option key) :=
 match a, b with
 | Some x, Some y => K.eq x y
 | _ , _ => False
 end.

Lemma searchtree_expand_right: forall k t lo hi, searchtree lo (Some k) t -> (eqopt (Some k) hi \/ ltopt (Some k) hi) -> searchtree lo hi t.