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 *)

Lemma um_eta k f :
k \in dom f -> exists v, find k f = Some v /\ f = pts k v \+ free k f.