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.

(* Specifications: *)
Lemma interp_empty: forall x, ~ interp EE x.
Hint Constructors interp.
Lemma generalize_bounds:
forall t, searchtree None None t -> exists lo, exists hi, searchtree lo hi t.
Lemma interp_range:
forall x tr lo hi, searchtree lo hi tr -> interp tr x -> ltopt lo (Some x) /\ ltopt (Some x) hi.