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

Lemma double_round_div_aux : forall fexp1 fexp2 : Z -> Z, Valid_exp fexp1 -> Valid_exp fexp2 -> forall (choice1 choice2 : Z -> bool), (exists n, (beta = 2 * n :> Z)%Z) -> double_round_div_hyp fexp1 fexp2 -> forall x y, 0 < x -> 0 < y -> generic_format beta fexp1 x -> generic_format beta fexp1 y -> double_round_eq fexp1 fexp2 choice1 choice2 (x / y).