(* Hodge Duality *)
Fixpoint dual n : vect n -> vect n :=
match n return vect n ->  vect n with
|    0 => fun a => a
| S n1 => fun v => let (x,y) := v in (dual n1 (y ^_'f), dual n1 x)
end.
Notation "'@ x " := (dual _ x) (at level 9).

Lemma dual_scal n k (v: vect n) : '@(k .* v) = k .* '@v.