Add Relation t Subset
 reflexivity proved by Subset_refl
 transitivity proved by Subset_trans
 as SubsetSetoid.#[global]
Instance In_s_m : Morphisms.Proper (E.eq ==> Subset ++> Basics.impl) In | 1.Add Morphism Empty with signature Subset --> Basics.impl as Empty_s_m.Add Morphism add with signature E.eq ==> Subset ++> Subset as add_s_m.Add Morphism remove with signature E.eq ==> Subset ++> Subset as remove_s_m.Add Morphism union with signature Subset ++> Subset ++> Subset as union_s_m.Add Morphism inter with signature Subset ++> Subset ++> Subset as inter_s_m.Add Morphism diff with signature Subset ++> Subset --> Subset as diff_s_m.

(* Specifications: *)
Lemma filter_equal : forall f, compat_bool E.eq f -> forall s s', s[=]s' -> filter f s [=] filter f s'.
Lemma filter_ext : forall f f', compat_bool E.eq f -> (forall x, f x = f' x) -> forall s s', s[=]s' -> filter f s [=] filter f' s'.
Lemma filter_subset : forall f, compat_bool E.eq f -> forall s s', s[<=]s' -> filter f s [<=] filter f s'.