(** Sqrt is exact on squares *)

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