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

Lemma double_round_plus_aux0_aux :
forall (fexp1 fexp2 : Z -> Z),
forall x y,
(fexp2 (ln_beta (x + y))%Z <= fexp1 (ln_beta x))%Z ->
(fexp2 (ln_beta (x + y))%Z <= fexp1 (ln_beta y))%Z ->
generic_format beta fexp1 x -> generic_format beta fexp1 y ->
generic_format beta fexp2 (x + y).