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 *)

(* meet is the greatest lower bound *)
Lemma umpleq_meet a x y1 y2 :
valid (x \+ y1 \+ y2) ->
[pcm a <= x \+ y1] -> [pcm a <= x \+ y2] -> [pcm a <= x].