(* Turn a vector in a list of scalar to be used for multiple product *)
Fixpoint v2l (n: nat) : vect n -> list K :=
  match n return vect n -> list K with
    O => fun k => k :: nil
  | S n1 => fun x => let (x1, x2) := x in
                      (v2l n1 x1 ++ v2l n1 x2)%list
  end.

Lemma all_prods_free n: free _ (all_prods _ (base n 1)).