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.

(* in valid case, we can define the order by means of filter *)
Lemma umpleq_filtE a x :
valid x -> [pcm a <= x] <-> um_filter (mem (dom a)) x = a.