Notation empty_1 := empty_spec (only parsing).Notation fold_1 := fold_spec (only parsing).Notation cardinal_1 := cardinal_spec (only parsing).Notation partition_1 := partition_spec1 (only parsing).Notation partition_2 := partition_spec2 (only parsing).Notation choose_1 := choose_spec1 (only parsing).Notation choose_2 := choose_spec2 (only parsing).Notation elements_3w := elements_spec2w (only parsing).

(* Specifications: *)
Lemma In_eq_iff : E.eq x y -> (In x s <-> In y s).
Lemma mem_iff : In x s <-> mem x s = true.
Lemma not_mem_iff : ~In x s <-> mem x s = false.
Lemma equal_iff : s[=]s' <-> equal s s' = true.
Lemma subset_iff : s[<=]s' <-> subset s s' = true.
Lemma empty_iff : In x empty <-> False.
Lemma is_empty_iff : Empty s <-> is_empty s = true.
Lemma singleton_iff : In y (singleton x) <-> E.eq x y.
Lemma add_iff : In y (add x s) <-> E.eq x y \/ In y s.
Lemma add_neq_iff : ~ E.eq x y -> (In y (add x s) <-> In y s).
Lemma remove_iff : In y (remove x s) <-> In y s /\ ~E.eq x y.
Lemma remove_neq_iff : ~ E.eq x y -> (In y (remove x s) <-> In y s).
Lemma for_all_iff : Proper (E.eq==>Logic.eq) f -> (For_all (fun x => f x = true) s <-> for_all f s = true).
Lemma exists_iff : Proper (E.eq==>Logic.eq) f -> (Exists (fun x => f x = true) s <-> exists_ f s = true).
Lemma elements_iff : In x s <-> InA E.eq x (elements s).