(* 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_hom n k v : hom n k v -> hom n (n - k) '@v.
Hint Resolve dual_hom.