intros.
exists (cast_map_to_subset H).
auto with algebra.
Qed.
Hint Resolve subset_seq_castable: algebra.

Lemma cast_seq_nice :
forall (A B : Setoid) (v : MAP B A) (W : part_set A)
(H : forall i : B, in_part (v i) W) (P : Predicate (MAP B A)),
Pred_fun P v -> Pred_fun P (Map_embed (cast_map_to_subset H)).