Definition cond_positivity (x:R) : bool :=
match Rle_dec 0 x with
| left _ => true
| right _ => false
end.

(* Specifications: *)
(** Sequential characterisation of continuity *)
Lemma continuity_seq :
forall (f:R -> R) (Un:nat -> R) (l:R),
continuity_pt f l -> Un_cv Un l -> Un_cv (fun i:nat => f (Un i)) (f l).
Lemma dicho_lb_car :
forall (x y:R) (P:R -> bool) (n:nat),
P x = false -> P (dicho_lb x y P n) = false.
Lemma dicho_up_car :
forall (x y:R) (P:R -> bool) (n:nat),
P y = true -> P (dicho_up x y P n) = true.
(* A general purpose corollary. *)
Lemma cv_pow_half : forall a, Un_cv (fun n => a/2^n) 0.
intros a; unfold Rdiv; replace 0 with (a * 0) by ring.