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_FLX : forall choice1 choice2, (2 * prec + 1 <= prec')%Z -> forall x y, FLX_format beta prec x -> FLX_format beta prec y -> double_round_eq (FLX_exp prec) (FLX_exp prec') choice1 choice2 (x + y).