(* Ad-hoc conjugate function in order to define the join *)
Fixpoint dconj (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 (dconj n1 b l2, dconj n1 (negb b) l3)
end.
Notation "x ^d_ b" := (dconj _ b x)  (at level 29, left associativity).
Notation "x ^d_'t" := (dconj _ true x)  (at level 29, left associativity).
Notation "x ^d_'f" := (dconj _ false x)  (at level 29, left associativity).

Lemma conj_dual n b (x: vect n): '@(x ^_ b) = '@x ^d_ b.