(* Ad-hoc conjugate function in order to define the product *)
Fixpoint conj (n : nat) (b: bool) {struct n}: vect n -> vect n :=
match n return (vect n -> vect n) with
| 0%nat => fun a => if b then (- a)%f else a
| S n1 =>
fun l1 =>
let (l2, l3) := l1 in (conj n1 (negb b) l2, conj n1 b l3)
end.
Notation "x ^_ b" := (conj _ b x)  (at level 30).
Notation "x ^_'t" := (conj _ true x)  (at level 30).
Notation "x ^_'f" := (conj _ false x)  (at level 30).

(* We can swap add-hoc conjugate *)
Lemma conj_swap n b1 b2 (x: vect n) : x ^_ b2 ^_ b1 =  x ^_ b1 ^_ b2.