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_minus_aux2_aux :
forall (fexp : Z -> Z),
Valid_exp fexp ->
forall x y,
0 < y -> y < x ->
(ln_beta y <= fexp (ln_beta x) - 1)%Z ->
generic_format beta fexp x ->
generic_format beta fexp y ->
round beta fexp Zceil (x - y) - (x - y) <= y.