Definition elements_spec2w : NoDupA E.eq (elements s)
   := @M.elements_3w s.Definition choose_spec1 : choose s = Some x -> In x s
   := @M.choose_1 s x.Definition choose_spec2 : choose s = None -> Empty s
   := @M.choose_2 s.Definition filter_spec : forall f, Proper (E.eq==>Logic.eq) f ->
   (In x (filter f s) <-> In x s /\ f x = true)
   := @MF.filter_iff s x.Definition partition_spec1 : forall f, Proper (E.eq==>Logic.eq) f ->
   Equal (fst (partition f s)) (filter f s)
   := @M.partition_1 s.Definition partition_spec2 : forall f, Proper (E.eq==>Logic.eq) f ->
   Equal (snd (partition f s)) (filter (fun x => negb (f x)) s)
   := @M.partition_2 s.

(* Specifications: *)
Lemma for_all_spec : forall f, Proper (E.eq==>Logic.eq) f -> (for_all f s = true <-> For_all (fun x => f x = true) s).
Lemma exists_spec : forall f, Proper (E.eq==>Logic.eq) f -> (exists_ f s = true <-> Exists (fun x => f x = true) s).