(** Rounding modes *)
Inductive mode := mode_NE | mode_ZR | mode_DN | mode_UP | mode_NA.
Definition round_mode m :=
match m with
| mode_NE => ZnearestE
| mode_ZR => Ztrunc
| mode_DN => Zfloor
| mode_UP => Zceil
| mode_NA => ZnearestA
end.
Definition choice_mode m sx mx lx :=
match m with
| mode_NE => cond_incr (round_N (negb (Zeven mx)) lx) mx
| mode_ZR => mx
| mode_DN => cond_incr (round_sign_DN sx lx) mx
| mode_UP => cond_incr (round_sign_UP sx lx) mx
| mode_NA => cond_incr (round_N true lx) mx
end.
Global Instance valid_rnd_round_mode : forall m, Valid_rnd (round_mode m).
Definition overflow_to_inf m s :=
match m with
| mode_NE => true
| mode_NA => true
| mode_ZR => false
| mode_UP => negb s
| mode_DN => s
end.
Definition binary_overflow m s :=
if overflow_to_inf m s then F754_infinity s
else F754_finite s (match (Zpower 2 prec - 1)%Z with Zpos p => p | _ => xH end) (emax - prec).
Definition binary_round_aux mode sx mx ex lx :=
let '(mrs', e') := shr_fexp (Zpos mx) ex lx in
let '(mrs'', e'') := shr_fexp (choice_mode mode sx (shr_m mrs') (loc_of_shr_record mrs')) e' loc_Exact in
match shr_m mrs'' with
| Z0 => F754_zero sx
| Zpos m => if Zle_bool e'' (emax - prec) then F754_finite sx m e'' else binary_overflow mode sx
| _ => F754_nan false xH (* dummy *)
end.

Theorem binary_round_aux_correct :
forall mode x mx ex lx,
inbetween_float radix2 (Zpos mx) ex (Rabs x) lx ->
(ex <= fexp (Zdigits radix2 (Zpos mx) + ex))%Z ->
let z := binary_round_aux mode (Rlt_bool x 0) mx ex lx in
valid_binary z = true /\
if Rlt_bool (Rabs (round radix2 fexp (round_mode mode) x)) (bpow radix2 emax) then
FF2R radix2 z = round radix2 fexp (round_mode mode) x /\
is_finite_FF z = true /\ sign_FF z = Rlt_bool x 0
else
z = binary_overflow mode (Rlt_bool x 0).