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 interval_finite_subdiv_between (a b : R) (eps : posreal) (Hab : a <= b) : let l := proj1_sig (interval_finite_subdiv a b eps Hab) in forall i, (i < size l)%nat -> a <= nth 0 l i <= b.