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).

Theorem double_round_plus_FTZ : forall choice1 choice2, (emin' + prec' <= emin + 1)%Z -> (2 * prec + 1 <= 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).