Definition binary_normalize mode m e szero :=
  match m with
  | Z0 => B754_zero szero
  | Zpos m => FF2B _ (proj1 (binary_round_correct mode false m e))
  | Zneg m => FF2B _ (proj1 (binary_round_correct mode true m e))
  end.

Theorem binary_normalize_correct : forall m mx ex szero, if Rlt_bool (Rabs (round radix2 fexp (round_mode m) (F2R (Float radix2 mx ex)))) (bpow radix2 emax) then B2R (binary_normalize m mx ex szero) = round radix2 fexp (round_mode m) (F2R (Float radix2 mx ex)) /\ is_finite (binary_normalize m mx ex szero) = true /\ Bsign (binary_normalize m mx ex szero) = match Rcompare (F2R (Float radix2 mx ex)) 0 with | Eq => szero | Lt => true | Gt => false end else B2FF (binary_normalize m mx ex szero) = binary_overflow m (Rlt_bool (F2R (Float radix2 mx ex)) 0).