Definition double_round_plus_beta_ge_3_hyp fexp1 fexp2 :=
  (forall ex ey, (fexp1 (ex + 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 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Z)
  /\ (forall ex ey, (ex - 1 <= ey)%Z -> (fexp2 ex <= fexp1 ey)%Z).

Lemma double_round_minus_beta_ge_3_aux1 : forall (fexp1 fexp2 : Z -> Z), Valid_exp fexp1 -> Valid_exp fexp2 -> double_round_plus_beta_ge_3_hyp fexp1 fexp2 -> forall x y, 0 < y -> y < x -> (ln_beta y <= fexp1 (ln_beta x) - 1)%Z -> (fexp1 (ln_beta (x - y)) <= ln_beta y)%Z -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> generic_format beta fexp2 (x - y).