(* Inductive Abs : tree -> total_map V -> Prop :=
    Abs_E : Abs E (t_empty default)
  | Abs_T : forall (a b: total_map V) (l: tree) (k: key) (v: V) (r: tree),
            Abs l a ->
            Abs r b ->
	    Abs (T l k v r) (t_update (combine k a b) (Id k) v)
*)
(** Because the [combine] function is chosen very carefully, it turns out
  that this abstraction relation even works on bogus trees! *)

Remark abstraction_of_bogus_tree: 
 forall v2 v3,
   Abs (T (T E 3 v3 E) 2 v2 E) (t_update (t_empty default) 2 v2).

Lemma can_relate: forall t, SearchTree t -> exists cts, Abs t cts.