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 *)

Lemma umpleq_some x1 x2 t s :
valid x2 -> [pcm x1 <= x2] -> find t x1 = Some s -> find t x2 = Some s.