(* induction over union maps, expressed with pts and \+ *)

(* backward progressing over keys *)
Lemma um_indb (P : U -> Prop) :
P um_undef -> P Unit ->
(forall k v f, P f -> valid (pts k v \+ f) ->
(forall x, x \in dom f -> ord x k) ->
P (pts k v \+ f)) ->
forall f, P f.