(* Base of k vector *)
Fixpoint base (n: nat) k:list (vect n) :=
match n return list (vect n) with
| O =>  match k with | O => 1%f::nil | _ => nil end
| S n1 =>
match k with | O =>
map (lift n1) (base n1 k) | S k1 =>
(map (dlift n1) (base n1 k1) ++
map (lift n1) (base n1 k))%list
end
end.

Lemma proj_base_length n (x: vect n) k :
length (proj n k x) = length (base n k).