Definition pos_div_2 (eps : posreal) := mkposreal _ (is_pos_div_2 eps).(** * The sign function *)

Definition sign (x : R) :=
  match total_order_T 0 x with
  | inleft (left _) => 1
  | inleft (right _) => 0
  | inright _ => -1
  end.

Lemma pairmap_rcons {T T0 : Type} (f : T -> T -> T0) (s : seq T) h0 h x0 : pairmap f x0 (rcons (rcons s h0) h) = rcons (pairmap f x0 (rcons s h0)) (f h0 h).