Instance eq_bigcap T U : Proper (set_eq ==> pointwise_relation T (@set_eq U) ==> set_eq) bigcap.

Lemma union_lift_subset_compat {A} (s1 s2 : set (option A)) (s3 s4 : set A) :
s1 \subset lift s3 ->
s2 \subset lift s4 ->
(s1 :|: s2) \subset lift (s3 :|: s4).