Proof with auto with typeclass_instances.
intros m minus_nan x y Fx Fy.
replace (negb (Bsign y)) with (Bsign (Bopp pair y)).
unfold Rminus.
erewrite <- B2R_Bopp.
apply Bplus_correct.
exact Fx.
rewrite is_finite_Bopp. auto. now destruct y as [ | | | ].
Qed.
(** Division *)
Definition Fdiv_core_binary m1 e1 m2 e2 :=
let d1 := Zdigits2 m1 in
let d2 := Zdigits2 m2 in
let e := (e1 - e2)%Z in
let (m, e') :=
match (d2 + prec - d1)%Z with
| Zpos p => (Z.shiftl m1 (Zpos p), e + Zneg p)%Z
| _ => (m1, e)
end in
let '(q, r) :=  Zfast_div_eucl m m2 in
(q, e', new_location m2 r loc_Exact).

(* Specifications: *)
Lemma Bdiv_correct_aux :
forall m sx mx ex sy my ey,
let x := F2R (Float radix2 (cond_Zopp sx (Zpos mx)) ex) in
let y := F2R (Float radix2 (cond_Zopp sy (Zpos my)) ey) in
let z :=
let '(mz, ez, lz) := Fdiv_core_binary (Zpos mx) ex (Zpos my) ey in
match mz with
| Zpos mz => binary_round_aux m (xorb sx sy) mz ez lz
| _ => F754_nan false xH (* dummy *)
end in
valid_binary z = true /\
if Rlt_bool (Rabs (round radix2 fexp (round_mode m) (x / y))) (bpow radix2 emax) then
FF2R radix2 z = round radix2 fexp (round_mode m) (x / y) /\
is_finite_FF z = true /\ sign_FF z = xorb sx sy
else
z = binary_overflow m (xorb sx sy).
Lemma Bdiv_correct_aux :
forall m sx mx ex sy my ey,
let x := F2R (Float radix2 (cond_Zopp sx (Zpos mx)) ex) in
let y := F2R (Float radix2 (cond_Zopp sy (Zpos my)) ey) in
let z :=
let '(mz, ez, lz) := Fdiv_core_binary (Zpos mx) ex (Zpos my) ey in
match mz with
| Zpos mz => binary_round_aux m (xorb sx sy) mz ez lz
| _ => F754_nan false xH (* dummy *)
end in
valid_binary z = true /\
if Rlt_bool (Rabs (round radix2 fexp (round_mode m) (x / y))) (bpow radix2 emax) then
FF2R radix2 z = round radix2 fexp (round_mode m) (x / y) /\
is_finite_FF z = true /\ sign_FF z = xorb sx sy
else
z = binary_overflow m (xorb sx sy).