Section UmpleqLemmas.
Variables (K : ordType) (A : Type) (U : union_map_class K A).
Implicit Types (x y a b : U) (p : pred K).
(* monotonicity lemmas *)
Hint Resolve umpleq_filtI : core.
(* x <= y and subdom *)
(* some/none lemmas *)
(* um_filter *)
(* um_all *)
(* others *)
(* 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.