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).

Theorem double_round_minus_beta_ge_3_FTZ : (3 <= beta)%Z -> forall choice1 choice2, (emin' + prec' <= emin + 1)%Z -> (2 * prec <= prec')%Z -> forall x y, FTZ_format beta emin prec x -> FTZ_format beta emin prec y -> double_round_eq (FTZ_exp emin prec) (FTZ_exp emin' prec') choice1 choice2 (x - y).