Coercion seq_In T : seq T -> set T := fun s x => List.In x s.Coercion list_In T : list T -> set T := fun s x => List.In x s.

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).