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

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