End Double_round_mult_FTZ.
End Double_round_mult.
Section Double_round_plus.

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