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 \+ *)

(* validity holds pairwise *)
Lemma um_valid3 f1 f2 f3 :
valid (f1 \+ f2 \+ f3) =
[&& valid (f1 \+ f2), valid (f2 \+ f3) & valid (f1 \+ f3)].