Definition plusnb (y x: nat) (nb: natBelow x): natBelow (x + y) :=
  match nb in natBelow x' return natBelow (x' + y) with
  | mkNatBelow a b => eq_rec_r natBelow (mkNatBelow a (b + y)) (eq_S _ _ (plus_assoc_reverse a b y))
  end.

Lemma Permutation_mapping (X: Set) n (a b: Vector.t X n): Permutation a b -> exists l: Vector.t (natBelow n) n, (forall x, List.In x l) /\ map (nth b) l = a.