End Double_round_mult_FTZ.
End Double_round_mult.
Section Double_round_plus.

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