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 freePt2 k1 k2 v :
cond U k2 ->
free k1 (pts k2 v : U) = if k1 == k2 then Unit else pts k2 v.