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

(* Specifications: *)
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.
Lemma cv_pow_half : forall a, Un_cv (fun n => a/2^n) 0.
Lemma IVT : forall (f:R -> R) (x y:R), continuity f -> x < y -> f x < 0 -> 0 < f y -> { z:R | x <= z <= y /\ f z = 0 }.
Lemma IVT_cor : forall (f:R -> R) (x y:R), continuity f -> x <= y -> f x * f y <= 0 -> { z:R | x <= z <= y /\ f z = 0 }.
Lemma Rsqrt_exists : forall y:R, 0 <= y -> { z:R | 0 <= z /\ y = Rsqr z }.