(** [sqrt_up] is a monotone function (but not a strict one) *)

(** And we can't find better approximations in general. - The upper bound is exact for squares - Concerning the lower bound, for any c>0, take [a=b=c²+1], then [√°(a*b) = c²+1] while [P √°a = P √°b = c] *)
(** [sqrt_up] and successor : - the [sqrt_up] function climbs by at most 1 at a time - otherwise it stays at the same value - the +1 steps occur after squares *)
Lemma sqrt_up_succ_le : forall a, 0<=a -> √°(S a) <= S (√°a).