Definition Bmult_FF mult_nan m x y :=
let f pl := F754_nan (fst pl) (snd pl) in
match x, y with
| F754_nan _ _, _ | _, F754_nan _ _ => f (mult_nan x y)
| F754_infinity sx, F754_infinity sy => F754_infinity (xorb sx sy)
| F754_infinity sx, F754_finite sy _ _ => F754_infinity (xorb sx sy)
| F754_finite sx _ _, F754_infinity sy => F754_infinity (xorb sx sy)
| F754_infinity _, F754_zero _ => f (mult_nan x y)
| F754_zero _, F754_infinity _ => f (mult_nan x y)
| F754_finite sx _ _, F754_zero sy => F754_zero (xorb sx sy)
| F754_zero sx, F754_finite sy _ _ => F754_zero (xorb sx sy)
| F754_zero sx, F754_zero sy => F754_zero (xorb sx sy)
| F754_finite sx mx ex, F754_finite sy my ey =>
binary_round_aux m (xorb sx sy) (mx * my) (ex + ey) loc_Exact
end.

Theorem B2FF_Bmult :
forall mult_nan mult_nan_ff,
forall m x y,
mult_nan_ff (B2FF x) (B2FF y) = (let '(sr, exist plr _) := mult_nan x y in (sr, plr)) ->
B2FF (Bmult mult_nan m x y) = Bmult_FF mult_nan_ff m (B2FF x) (B2FF y).