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

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