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 \+ *)

(* points-to is a prime element of the monoid *)
Lemma um_prime f1 f2 k v :
cond U k ->
f1 \+ f2 = pts k v ->
f1 = pts k v /\ f2 = Unit \/
f1 = Unit /\ f2 = pts k v.