Proof with auto.
unfold vec.nb_nats.
intros.
apply vec.perm_sym.
apply vec.NoDup_incl_Permutation.
rewrite <- vec.List_map.
apply NoDup_map.
intros.
apply natBelow_unique...
apply vec.NoDup_nats.
do 2 intro.
rewrite <- vec.List_map in H0.
destruct (In_map_inv H0). clear H0.
destruct H1. subst.
destruct (vec.In_nats_inv _ _ _ H1). clear H1.
rewrite <- vec.List_map.
apply H...
rewrite vec.length.
omega.
Qed.

(* Specifications: *)
Lemma nats_Permutation_IndexSeq' b n (l: Vector.t Index n):
vec.Permutation (vec.map nb_val l) (vec.nb_nats b n) -> IndexSeq b l.