Definition double_round_mult_hyp fexp1 fexp2 :=
  (forall ex ey, (fexp2 (ex + ey) <= fexp1 ex + fexp1 ey)%Z)
  /\ (forall ex ey, (fexp2 (ex + ey - 1) <= fexp1 ex + fexp1 ey)%Z).

Lemma ln_beta_minus_separated : forall fexp : Z -> Z, Valid_exp fexp -> forall x y, 0 < x -> 0 < y -> y < x -> bpow (ln_beta x - 1) < x -> generic_format beta fexp x -> (ln_beta y <= fexp (ln_beta x))%Z -> (ln_beta (x - y) = ln_beta x :> Z).