(** * 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.