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).

Lemma remove_iff : In y (remove x s) <-> In y s /\ ~E.eq x y.