Inductive interp: tree -> (key -> Prop) :=
| member_here: forall x y c tl tr, K.eq x y -> interp (T c tl y tr) x
| member_left: forall x y c tl tr, interp tl x -> interp (T c tl y tr) x
| member_right: forall x y c tl tr, interp tr x -> interp (T c tl y tr) x.

Lemma interp_range:
forall x tr lo hi, searchtree lo hi tr -> interp tr x -> ltopt lo (Some x) /\ ltopt (Some x) hi.