Section Double_round_sqrt.
Definition double_round_sqrt_hyp fexp1 fexp2 :=
(forall ex, (2 * fexp1 ex <= fexp1 (2 * ex))%Z)
/\ (forall ex, (2 * fexp1 ex <= fexp1 (2 * ex - 1))%Z)
/\ (forall ex, (fexp1 (2 * ex) < 2 * ex)%Z ->
(fexp2 ex + ex <= 2 * fexp1 ex - 2)%Z).

Lemma ln_beta_sqrt_disj :
forall x,
0 < x ->
(ln_beta x = 2 * ln_beta (sqrt x) - 1 :> Z)%Z
\/ (ln_beta x = 2 * ln_beta (sqrt x) :> Z)%Z.