Proof with auto with typeclass_instances.
intros m x mx ex lx Bx Ex z.
unfold binary_round_aux in z.
revert z.
rewrite shr_truncate. 2: easy.
refine (_ (round_trunc_sign_any_correct _ _ (round_mode m) (choice_mode m) _ x (Zpos mx) ex lx Bx (or_introl _ Ex))).
refine (_ (truncate_correct_partial _ _ _ _ _ _ _ Bx Ex)).
destruct (truncate radix2 fexp (Zpos mx, ex, lx)) as ((m1, e1), l1).
rewrite loc_of_shr_record_of_loc, shr_m_shr_record_of_loc.
set (m1' := choice_mode m (Rlt_bool x 0) m1 l1).
intros (H1a,H1b) H1c.
rewrite H1c.
assert (Hm: (m1 <= m1')%Z).
(* . *)
unfold m1', choice_mode, cond_incr.
case m ;
try apply Zle_refl ;
match goal with |- (m1 <= if ?b then _ else _)%Z =>
case b ; [ apply Zle_succ | apply Zle_refl ] end.
assert (Hr: Rabs (round radix2 fexp (round_mode m) x) = F2R (Float radix2 m1' e1)).
(* . *)
rewrite <- (Zabs_eq m1').
replace (Zabs m1') with (Zabs (cond_Zopp (Rlt_bool x 0) m1')).
rewrite F2R_Zabs.
now apply f_equal.
apply abs_cond_Zopp.
apply Zle_trans with (2 := Hm).
apply Zlt_succ_le.
apply F2R_gt_0_reg with radix2 e1.
apply Rle_lt_trans with (1 := Rabs_pos x).
exact (proj2 (inbetween_float_bounds _ _ _ _ _ H1a)).
(* . *)
assert (Br: inbetween_float radix2 m1' e1 (Rabs (round radix2 fexp (round_mode m) x)) loc_Exact).
now apply inbetween_Exact.
destruct m1' as [|m1'|m1'].
(* . m1' = 0 *)
rewrite shr_truncate. 2: apply Zle_refl.
generalize (truncate_0 radix2 fexp e1 loc_Exact).
destruct (truncate radix2 fexp (Z0, e1, loc_Exact)) as ((m2, e2), l2).
rewrite shr_m_shr_record_of_loc.
intros Hm2.
rewrite Hm2.
repeat split.
rewrite Rlt_bool_true.
repeat split.
apply sym_eq.
case Rlt_bool ; apply F2R_0.
rewrite <- F2R_Zabs, abs_cond_Zopp, F2R_0.
apply bpow_gt_0.
(* . 0 < m1' *)
assert (He: (e1 <= fexp (Zdigits radix2 (Zpos m1') + e1))%Z).
rewrite <- ln_beta_F2R_Zdigits, <- Hr, ln_beta_abs.
2: discriminate.
rewrite H1b.
rewrite canonic_exp_abs.
fold (canonic_exp radix2 fexp (round radix2 fexp (round_mode m) x)).
apply canonic_exp_round_ge...
rewrite H1c.
case (Rlt_bool x 0).
apply Rlt_not_eq.
now apply F2R_lt_0_compat.
apply Rgt_not_eq.
now apply F2R_gt_0_compat.
refine (_ (truncate_correct_partial _ _ _ _ _ _ _ Br He)).
2: now rewrite Hr ; apply F2R_gt_0_compat.
refine (_ (truncate_correct_format radix2 fexp (Zpos m1') e1 _ _ He)).
2: discriminate.
rewrite shr_truncate. 2: easy.
destruct (truncate radix2 fexp (Zpos m1', e1, loc_Exact)) as ((m2, e2), l2).
rewrite shr_m_shr_record_of_loc.
intros (H3,H4) (H2,_).
destruct m2 as [|m2|m2].
elim Rgt_not_eq with (2 := H3).
rewrite F2R_0.
now apply F2R_gt_0_compat.
rewrite F2R_cond_Zopp, H3, abs_cond_Ropp, <- F2R_abs.
simpl Zabs.
case_eq (Zle_bool e2 (emax - prec)) ; intros He2.
assert (bounded m2 e2 = true).
apply andb_true_intro.
split.
unfold canonic_mantissa.
apply Zeq_bool_true.
rewrite Zpos_digits2_pos.
rewrite <- ln_beta_F2R_Zdigits.
apply sym_eq.
now rewrite H3 in H4.
discriminate.
exact He2.
apply (conj H).
rewrite Rlt_bool_true.
repeat split.
apply F2R_cond_Zopp.
now apply bounded_lt_emax.
rewrite (Rlt_bool_false _ (bpow radix2 emax)).
refine (conj _ (refl_equal _)).
unfold binary_overflow.
case overflow_to_inf.
apply refl_equal.
unfold valid_binary, bounded.
rewrite Zle_bool_refl.
rewrite Bool.andb_true_r.
apply Zeq_bool_true.
rewrite Zpos_digits2_pos.
replace (Zdigits radix2 (Zpos (match (Zpower 2 prec - 1)%Z with Zpos p => p | _ => xH end))) with prec.
unfold fexp, FLT_exp, emin.
generalize (prec_gt_0 prec).
clear -Hmax ; zify ; omega.
change 2%Z with (radix_val radix2).
case_eq (Zpower radix2 prec - 1)%Z.
simpl Zdigits.
generalize (Zpower_gt_1 radix2 prec (prec_gt_0 prec)).
clear ; omega.
intros p Hp.
apply Zle_antisym.
cut (prec - 1 < Zdigits radix2 (Zpos p))%Z. clear ; omega.
apply Zdigits_gt_Zpower.
simpl Zabs. rewrite <- Hp.
cut (Zpower radix2 (prec - 1) < Zpower radix2 prec)%Z. clear ; omega.
apply lt_Z2R.
rewrite 2!Z2R_Zpower. 2: now apply Zlt_le_weak.
apply bpow_lt.
apply Zlt_pred.
now apply Zlt_0_le_0_pred.
apply Zdigits_le_Zpower.
simpl Zabs. rewrite <- Hp.
apply Zlt_pred.
intros p Hp.
generalize (Zpower_gt_1 radix2 _ (prec_gt_0 prec)).
clear -Hp ; zify ; omega.
apply Rnot_lt_le.
intros Hx.
generalize (refl_equal (bounded m2 e2)).
unfold bounded at 2.
rewrite He2.
rewrite Bool.andb_false_r.
rewrite bounded_canonic_lt_emax with (2 := Hx).
discriminate.
unfold canonic.
now rewrite <- H3.
elim Rgt_not_eq with (2 := H3).
apply Rlt_trans with R0.
now apply F2R_lt_0_compat.
now apply F2R_gt_0_compat.
rewrite <- Hr.
apply generic_format_abs...
apply generic_format_round...
(* . not m1' < 0 *)
elim Rgt_not_eq with (2 := Hr).
apply Rlt_le_trans with R0.
now apply F2R_lt_0_compat.
apply Rabs_pos.
(* *)
apply Rlt_le_trans with (2 := proj1 (inbetween_float_bounds _ _ _ _ _ Bx)).
now apply F2R_gt_0_compat.
(* all the modes are valid *)
clear. case m.
exact inbetween_int_NE_sign.
exact inbetween_int_ZR_sign.
exact inbetween_int_DN_sign.
exact inbetween_int_UP_sign.
exact inbetween_int_NA_sign.
Qed.
(** Multiplication *)

Lemma Bmult_correct_aux :
forall m sx mx ex (Hx : bounded mx ex = true) sy my ey (Hy : bounded my ey = true),
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 := binary_round_aux m (xorb sx sy) (mx * my) (ex + ey) loc_Exact 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).