Definition Subset (s s': t) := Subset' (proj1_sig s) (proj1_sig s').
Definition Empty (s: t) := Empty' (proj1_sig s).
Definition For_all (P : elt -> Prop) (s: t) := For_all' P (proj1_sig s).
Definition Exists (P : elt -> Prop) (s: t) := Exists' P  (proj1_sig s).
Definition eq : t -> t -> Prop := Equal.

Lemma partition_aux_filter:
forall f s n1 l1 n2 l2, partition_aux f s  (n1,l1) (n2,l2) =
(filter_aux f s n1 l1, filter_aux (compose negb f) s n2 l2).