(* backward induction on valid natmaps *)

Lemma valid_indb (P : natmap A -> Prop) :
P Unit ->
(forall u, P (1 \-> u)) ->
(forall t u h, P h -> last_key h < t ->
valid (t \-> u \+ h) -> P (t \-> u \+ h)) ->
forall h, valid h -> P h.