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.