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).