Definition double_round_sqrt_beta_ge_4_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 - 1)%Z).

Lemma ln_beta_div_disj : forall x y : R, 0 < x -> 0 < y -> ((ln_beta (x / y) = ln_beta x - ln_beta y :> Z)%Z \/ (ln_beta (x / y) = ln_beta x - ln_beta y + 1 :> Z)%Z).