Section FRoundP.
Variable b : Fbound.
Variable radix : Z.
Variable precision : nat.
Let FtoRradix := FtoR radix.
Coercion FtoRradix : float >-> R.
Hypothesis radixMoreThanOne : (1 < radix)%Z.
Let radixMoreThanZERO := Zlt_1_O _ (Zlt_le_weak _ _ radixMoreThanOne).
Hint Resolve radixMoreThanZERO: zarith.
Hypothesis precisionGreaterThanOne : 1 < precision.
Hypothesis pGivesBound : Zpos (vNum b) = Zpower_nat radix precision.
intros p q fmin Fp Fq H' H'0 H'1 H'2.
cut (0 <= Fnum p * Fnum q)%Z;
[ intros multPos
| apply Zle_mult_gen; apply (LeR0Fnum radix); auto with arith ].
cut (ex (fun m : Z => FtoRradix fmin = Float m (Fexp (Fmult p q)))).
2: unfold FtoRradix in |- *;
apply
RoundedModeRep
with (b := b) (precision := precision) (P := isMin b radix);
auto.
2: apply MinRoundedModeP with (precision := precision); auto.
2: rewrite (Fmult_correct radix); auto with zarith.
intros H'3; elim H'3; intros m E; clear H'3.
exists (Fminus radix (Fmult p q) (Float m (Fexp (Fmult p q)))).
split.
rewrite E; unfold FtoRradix in |- *; repeat rewrite Fminus_correct;
repeat rewrite Fmult_correct; auto with zarith.
split.
cut (fmin <= Fmult p q)%R;
[ idtac
| unfold FtoRradix in |- *; rewrite Fmult_correct; auto; case H'2;
auto with real zarith; (intros H1 H2; case H2; auto with zarith) ].
rewrite E; unfold Fmult, Fminus, Fopp, Fplus in |- *; simpl in |- *; auto.
repeat rewrite Zmin_n_n; repeat rewrite <- Zminus_diag_reverse; auto.
simpl in |- *; repeat rewrite Zpower_nat_O; repeat rewrite Zmult_1_r.
unfold FtoRradix, FtoR in |- *; simpl in |- *.
intros H'3;
(cut (m <= Fnum p * Fnum q)%Z;
[ idtac
| apply le_IZR;
apply Rmult_le_reg_l with (r := powerRZ radix (Fexp p + Fexp q));
auto with real zarith;
repeat rewrite (Rmult_comm (powerRZ radix (Fexp p + Fexp q)));
auto with zarith ]); intros H'4.
repeat split; simpl in |- *; auto.
case (ZquotientProp (Fnum p * Fnum q) (Zpower_nat radix precision));
auto with zarith.
intros x (H'5, (H'6, H'7)).
cut
(Zquotient (Fnum p * Fnum q) (Zpower_nat radix precision) *
powerRZ radix (precision + (Fexp p + Fexp q)) <= fmin)%R;
[ rewrite E; intros H'8 | idtac ].
cut
(Zquotient (Fnum p * Fnum q) (Zpower_nat radix precision) *
powerRZ radix precision <= m)%R; [ intros H'9 | idtac ].
rewrite Z.abs_eq; auto with zarith.
apply Z.le_lt_trans with x; auto.
replace x with
(Fnum p * Fnum q +
-
(Zquotient (Fnum p * Fnum q) (Zpower_nat radix precision) *
Zpower_nat radix precision))%Z.
apply Zplus_le_compat_l; auto.
apply Zle_Zopp.
apply le_IZR; auto.
rewrite Rmult_IZR.
rewrite Zpower_nat_Z_powerRZ; auto with zarith.
pattern (Fnum p * Fnum q)%Z at 1 in |- *; rewrite H'5; ring.
rewrite pGivesBound.
rewrite <- (Z.abs_eq (Zpower_nat radix precision)); auto with zarith.
apply Zlt_Zabs_inv2; auto.
apply Rmult_le_reg_l with (r := powerRZ radix (Fexp p + Fexp q));
auto with real zarith.
repeat rewrite (Rmult_comm (powerRZ radix (Fexp p + Fexp q))); auto.
rewrite Rmult_assoc; rewrite <- powerRZ_add; auto with real zarith.
case
(FboundedMbound _ radixMoreThanOne b precision)
with
(z := (precision + (Fexp p + Fexp q))%Z)
(m := Zquotient (Fnum p * Fnum q) (Zpower_nat radix precision));
auto with zarith.
apply Zmult_le_reg_r with (p := Zpower_nat radix precision); auto with zarith.
apply Z.lt_gt; auto with zarith.
pattern (Zpower_nat radix precision) at 2 in |- *;
rewrite <- (fun x => Z.abs_eq (Zpower_nat radix x)).
rewrite <- Zabs_Zmult.
apply Z.le_trans with (1 := H'6); auto with zarith.
rewrite Zabs_Zmult.
apply Z.le_trans with (Zpower_nat radix precision * Z.abs (Fnum q))%Z.
apply Zle_Zmult_comp_r; auto with zarith.
apply Zlt_le_weak; rewrite <- pGivesBound; case Fp; auto with float.
apply Zle_Zmult_comp_l; auto with zarith.
apply Zlt_le_weak; rewrite <- pGivesBound; case Fq; auto with float.
auto with zarith.
intros x0 (H'8, H'9); rewrite <- H'9.
case H'2.
intros H'10 (H'11, H'12); apply H'12; auto.
rewrite H'9; auto.
rewrite powerRZ_add; auto with real zarith.
rewrite <- Rmult_assoc.
unfold FtoRradix in |- *; rewrite <- Fmult_correct; auto with zarith.
unfold Fmult, FtoR in |- *; simpl in |- *.
repeat rewrite (fun x => Rmult_comm x (powerRZ radix (Fexp p + Fexp q))).
apply Rmult_le_compat_l; auto with real zarith.
rewrite <- Zpower_nat_Z_powerRZ; auto with zarith.
pattern (Fnum p * Fnum q)%Z at 2 in |- *;
rewrite <- (Z.abs_eq (Fnum p * Fnum q)); auto.
rewrite <- Rmult_IZR; apply Rle_IZR; apply Zle_Zabs_inv2; auto.
simpl in |- *; auto.
apply Zmin_n_n; auto.
Qed.
intros p q fmax Fp Fq H' H'0 H'1 H'2.
cut (0 <= Fnum p * Fnum q)%Z;
[ intros multPos
| apply Zle_mult_gen; apply (LeR0Fnum radix); auto with arith ].
case (ZquotientProp (Fnum p * Fnum q) (Zpower_nat radix precision));
auto with zarith.
intros r; intros (H'3, (H'4, H'5)).
cut (0 <= Zquotient (Fnum p * Fnum q) (Zpower_nat radix precision))%Z;
[ intros Z2 | apply ZquotientPos; auto with zarith ].
cut (0 <= r)%Z;
[ intros Z3
| replace r with
(Fnum p * Fnum q -
Zquotient (Fnum p * Fnum q) (Zpower_nat radix precision) *
Zpower_nat radix precision)%Z;
[ idtac | pattern (Fnum p * Fnum q)%Z at 1 in |- *; rewrite H'3; ring ];
auto ].
2: apply Zle_Zminus_ZERO; rewrite Z.abs_eq in H'4; auto with zarith;
rewrite Z.abs_eq in H'4; auto with zarith.
case (Z.eq_dec r 0); intros Z4.
exists (Fzero (Fexp p + Fexp q)); repeat (split; auto with float).
replace (FtoRradix (Fzero (Fexp p + Fexp q))) with 0%R;
[ idtac | unfold Fzero, FtoRradix, FtoR in |- *; simpl in |- *; ring ].
apply Rplus_eq_reg_l with (r := FtoRradix fmax).
replace (fmax + 0)%R with (FtoRradix fmax); [ idtac | ring ].
replace (fmax + (p * q - fmax))%R with (p * q)%R; [ idtac | ring ].
unfold FtoRradix in |- *; rewrite <- (Fmult_correct radix); auto with zarith.
case
(FboundedMbound _ radixMoreThanOne b precision)
with
(z := (precision + (Fexp p + Fexp q))%Z)
(m := Zquotient (Fnum p * Fnum q) (Zpower_nat radix precision));
auto with zarith.
apply Zmult_le_reg_r with (p := Zpower_nat radix precision); auto with zarith.
apply Z.lt_gt; auto with zarith.
pattern (Zpower_nat radix precision) at 2 in |- *;
rewrite <- (fun x => Z.abs_eq (Zpower_nat radix x)).
rewrite <- Zabs_Zmult.
apply Z.le_trans with (1 := H'4); auto with zarith.
rewrite Zabs_Zmult.
apply Z.le_trans with (Zpower_nat radix precision * Z.abs (Fnum q))%Z.
apply Zle_Zmult_comp_r; auto with zarith.
apply Zlt_le_weak; rewrite <- pGivesBound; case Fp; auto with float.
apply Zle_Zmult_comp_l; auto with zarith.
apply Zlt_le_weak; rewrite <- pGivesBound; case Fq; auto with float.
auto with zarith.
intros x (H'6, H'7).
cut (FtoR radix (Fmult p q) = FtoR radix x).
intros H'8; rewrite H'8.
apply sym_eq; apply (ProjectMax b radix); auto.
rewrite <- H'8; auto.
rewrite Fmult_correct; auto with zarith.
rewrite H'7.
unfold FtoRradix, FtoR in |- *; simpl in |- *.
rewrite powerRZ_add with (n := Z_of_nat precision); auto with real zarith.
pattern (Fnum p * Fnum q)%Z at 1 in |- *; rewrite H'3.
rewrite plus_IZR; rewrite Rmult_IZR.
repeat rewrite Zpower_nat_Z_powerRZ; auto with real zarith.
rewrite Z4; simpl;ring.
cut (ex (fun m : Z => FtoRradix fmax = Float m (Fexp (Fmult p q))));
[ intros Z5 | idtac ].
2: unfold FtoRradix in |- *;
apply
RoundedModeRep
with (b := b) (precision := precision) (P := isMax b radix);
auto.
2: apply MaxRoundedModeP with (precision := precision); auto.
2: rewrite (Fmult_correct radix); auto with zarith.
elim Z5; intros m E; clear Z5.
exists (Fopp (Fminus radix (Float m (Fexp (Fmult p q))) (Fmult p q))).
split.
rewrite E; unfold FtoRradix in |- *; repeat rewrite Fopp_correct;
repeat rewrite Fminus_correct; repeat rewrite Fmult_correct;
auto with zarith; ring.
cut
(Fexp (Fopp (Fminus radix (Float m (Fexp (Fmult p q))) (Fmult p q))) =
(Fexp p + Fexp q)%Z); [ intros Z5 | idtac ].
split; auto.
split; [ idtac | rewrite Z5; auto ].
cut (Fmult p q <= fmax)%R;
[ idtac
| unfold FtoRradix in |- *; rewrite Fmult_correct; auto; case H'2;
auto with real zarith; (intros H1 H2; case H2; auto) ].
cut
(fmax <=
Z.succ (Zquotient (Fnum p * Fnum q) (Zpower_nat radix precision)) *
powerRZ radix (precision + (Fexp p + Fexp q)))%R.
rewrite E; repeat rewrite Zmin_n_n; repeat rewrite <- Zminus_diag_reverse;
repeat rewrite Zpower_nat_O; repeat rewrite Zmult_1_r;
auto.
unfold Fmult, Fminus, Fplus, Fopp in |- *; simpl in |- *.
repeat rewrite Zmin_n_n; repeat rewrite <- Zminus_diag_reverse;
repeat rewrite Zpower_nat_O; repeat rewrite Zmult_1_r;
auto.
intros H1 H2; rewrite Zabs_Zopp; apply Zlt_Zabs_intro.
apply Z.lt_le_trans with 0%Z; auto with zarith.
cut (Fnum p * Fnum q <= m)%Z; auto with zarith.
apply le_IZR;
apply (Rle_monotony_contra_exp radix) with (z := (Fexp p + Fexp q)%Z);
auto with zarith.
cut
(m <=
Z.succ (Zquotient (Fnum p * Fnum q) (Zpower_nat radix precision)) *
Zpower_nat radix precision)%Z; [ intros H'9 | idtac ].
apply Z.le_lt_trans with (Zpower_nat radix precision - r)%Z;
[ idtac | rewrite pGivesBound; auto with zarith ].
replace r with
(Fnum p * Fnum q -
Zquotient (Fnum p * Fnum q) (Zpower_nat radix precision) *
Zpower_nat radix precision)%Z.
replace
(Zpower_nat radix precision -
(Fnum p * Fnum q -
Zquotient (Fnum p * Fnum q) (Zpower_nat radix precision) *
Zpower_nat radix precision))%Z with
(Z.succ (Zquotient (Fnum p * Fnum q) (Zpower_nat radix precision)) *
Zpower_nat radix precision - Fnum p * Fnum q)%Z;
auto with zarith.
unfold Z.succ in |- *; simpl in |- *; ring.
pattern (Fnum p * Fnum q)%Z at 1 in |- *; rewrite H'3; ring.
apply le_IZR;
apply (Rle_monotony_contra_exp radix) with (z := (Fexp p + Fexp q)%Z);
auto with zarith.
replace
(IZR
(Z.succ (Zquotient (Fnum p * Fnum q) (Zpower_nat radix precision)) *
Zpower_nat radix precision) * powerRZ radix (Fexp p + Fexp q))%R with
(Z.succ (Zquotient (Fnum p * Fnum q) (Zpower_nat radix precision)) *
powerRZ radix (precision + (Fexp p + Fexp q)))%R;
[ auto | idtac ].
rewrite powerRZ_add; auto with real zarith.
repeat rewrite Rmult_IZR; repeat rewrite Zpower_nat_Z_powerRZ; auto with zarith.
ring.
case
(FboundedMbound _ radixMoreThanOne b precision)
with
(z := (precision + (Fexp p + Fexp q))%Z)
(m := Z.succ (Zquotient (Fnum p * Fnum q) (Zpower_nat radix precision)));
auto with arith.
rewrite Z.abs_eq; auto with zarith.
apply Zlt_le_succ.
case (Zle_lt_or_eq _ _ multPos); intros Eq1.
cut (0 < Z.abs (Fnum p))%Z; [ intros Eq2 | idtac ].
cut (0 < Z.abs (Fnum q))%Z; [ intros Eq3 | idtac ].
apply Zlt_mult_simpl_l with (c := Zpower_nat radix precision);
auto with zarith.
rewrite (fun x y z => Zmult_comm x (Zquotient y z)).
apply Z.le_lt_trans with (Fnum p * Fnum q)%Z.
rewrite Z.abs_eq in H'4; auto with zarith; rewrite Z.abs_eq in H'4;
auto with zarith.
rewrite <- (Z.abs_eq (Fnum p * Fnum q)); auto with zarith; rewrite Zabs_Zmult.
apply Z.lt_trans with (Z.abs (Fnum p) * Zpower_nat radix precision)%Z.
cut (Z.abs (Fnum q) < Zpower_nat radix precision)%Z;
[ intros Eq4; apply Zmult_gt_0_lt_compat_l
| rewrite <- pGivesBound; case Fq ]; auto with zarith.
cut (Z.abs (Fnum p) < Zpower_nat radix precision)%Z;
[ intros Eq4; apply Zmult_gt_0_lt_compat_r
| rewrite <- pGivesBound; case Fp ]; auto with zarith.
case (Zle_lt_or_eq _ _ (Zle_ZERO_Zabs (Fnum q))); auto.
intros Eq3; Contradict Eq1; replace (Fnum q) with 0%Z; auto with zarith.
generalize Eq3; case (Fnum q); simpl in |- *; auto; intros; discriminate.
case (Zle_lt_or_eq _ _ (Zle_ZERO_Zabs (Fnum p))); auto.
intros Eq3; Contradict Eq1; replace (Fnum p) with 0%Z; auto with zarith.
generalize Eq3; case (Fnum p); simpl in |- *; auto; intros; discriminate.
rewrite <- Eq1; replace (Zquotient 0 (Zpower_nat radix precision)) with 0%Z;
auto with zarith.
apply Z.le_trans with (1 := H'1); auto with zarith.
intros f1 (Hf1, Hf2); rewrite <- Hf2.
case H'2; intros L1 (L2, L3); apply L3; auto.
rewrite Hf2; unfold Fmult, FtoRradix, FtoR in |- *.
replace
(Fnum p * powerRZ radix (Fexp p) * (Fnum q * powerRZ radix (Fexp q)))%R with
(Fnum p * Fnum q * powerRZ radix (Fexp p + Fexp q))%R.
replace
(Z.succ (Zquotient (Fnum p * Fnum q) (Zpower_nat radix precision)) *
powerRZ radix (precision + (Fexp p + Fexp q)))%R with
((Zquotient (Fnum p * Fnum q) (Zpower_nat radix precision) *
Zpower_nat radix precision + Zpower_nat radix precision)%Z *
powerRZ radix (Fexp p + Fexp q))%R.
apply Rle_monotone_exp; auto with real zarith.
rewrite <- Rmult_IZR; apply Rle_IZR.
pattern (Fnum p * Fnum q)%Z at 1 in |- *; rewrite H'3;
cut (r < Zpower_nat radix precision)%Z; auto with zarith.
rewrite Z.abs_eq in H'5; auto with zarith; rewrite Z.abs_eq in H'5;
auto with zarith.
unfold Z.succ in |- *; repeat rewrite Rmult_IZR || rewrite plus_IZR;
simpl in |- *.
rewrite (powerRZ_add radix precision); auto with real zarith;
rewrite <- (Zpower_nat_Z_powerRZ radix precision); auto with real zarith;
ring.
rewrite powerRZ_add; auto with real zarith; ring.
unfold Fopp, Fminus, Fmult in |- *; simpl in |- *; auto.
apply Zmin_n_n.
Qed.
intros P H' p q pq H'0.
case
(RoundedModeRep b radix precision) with (p := Fmult p q) (q := pq) (P := P);
auto with zarith.
rewrite Fmult_correct; auto with zarith.
simpl in |- *; intros x H'1.
case
(eqExpLess _ radixMoreThanOne b)
with (p := pq) (q := Float x (Fexp (Fmult p q)));
auto.
apply RoundedModeBounded with (radix := radix) (P := P) (r := (p * q)%R);
auto.
simpl in |- *; intros x0 H'2; elim H'2; intros H'3 H'4; elim H'4;
intros H'5 H'6; clear H'4 H'2.
exists x0; split; [ idtac | split ]; auto.
unfold FtoRradix in |- *; rewrite H'5; auto.
apply le_IZR; auto.
Qed.
intros P H' p q pq H'0 H'1 H'2 H'3.
replace (precision + (Fexp p + Fexp q))%Z with
(Fexp (Float (pPred (vNum b)) (precision + (Fexp p + Fexp q))));
[ idtac | simpl in |- *; auto ].
unfold FtoRradix in |- *; apply eqExpMax; auto.
apply RoundedModeBounded with (radix := radix) (P := P) (r := (p * q)%R);
auto; auto.
unfold pPred in |- *; apply maxFbounded; auto.
apply Z.le_trans with (1 := H'3); auto with zarith.
replace (FtoR radix (Float (pPred (vNum b)) (precision + (Fexp p + Fexp q))))
with (radix * Float (pPred (vNum b)) (pred precision + (Fexp p + Fexp q)))%R.
rewrite Fabs_correct; auto with zarith.
unfold FtoRradix in |- *;
apply
RoundedModeMultAbs
with (b := b) (precision := precision) (P := P) (r := (p * q)%R);
auto.
unfold pPred in |- *; apply maxFbounded; auto with zarith.
rewrite Rabs_mult; auto.
apply
Rle_trans
with
(FtoR radix
(Fmult (Float (pPred (vNum b)) (Fexp p))
(Float (pPred (vNum b)) (Fexp q)))).
rewrite Fmult_correct; auto with arith.
apply Rmult_le_compat; auto with real.
rewrite <- (Fabs_correct radix); try apply maxMax1; auto with zarith.
rewrite <- (Fabs_correct radix); try apply maxMax1; auto with zarith.
unfold Fmult, FtoR in |- *; simpl in |- *; auto.
rewrite powerRZ_add with (n := Z_of_nat (pred precision));
auto with real arith.
repeat rewrite <- Rmult_assoc.
repeat rewrite (fun (z : Z) (x : R) => Rmult_comm x (powerRZ radix z));
auto.
apply Rmult_le_compat_l; auto with real arith.
rewrite <- Rmult_assoc.
rewrite (fun x : R => Rmult_comm x radix).
rewrite <- powerRZ_Zs; auto with real arith.
replace (Z.succ (pred precision)) with (Z_of_nat precision).
rewrite Rmult_IZR; auto.
apply Rmult_le_compat; auto with real arith.
replace 0%R with (IZR 0); unfold pPred in |- *; try apply Rle_IZR;
auto with real zarith.
replace 0%R with (IZR 0); unfold pPred in |- *; try apply Rle_IZR;
auto with real zarith.
unfold pPred in |- *; rewrite pGivesBound; rewrite <- Zpower_nat_Z_powerRZ;
auto with real zarith.
rewrite inj_pred; auto with arith zarith.
auto with real zarith.
auto with real zarith.
unfold FtoRradix, FtoR in |- *; simpl in |- *.
repeat rewrite (Rmult_comm (pPred (vNum b))).
rewrite <- Rmult_assoc.
rewrite <- powerRZ_Zs; auto with real zarith.
rewrite inj_pred; auto with arith zarith.
replace (Z.succ (Z.pred precision + (Fexp p + Fexp q))) with
(precision + (Fexp p + Fexp q))%Z; auto; unfold Z.succ, Z.pred in |- *;
ring.
Qed.

Theorem errorBoundedMultPos :
forall P,
RoundedModeP b radix P ->
forall p q f : float,
Fbounded b p ->
Fbounded b q ->
(0 <= p)%R ->
(0 <= q)%R ->
(- dExp b <= Fexp p + Fexp q)%Z ->
P (p * q)%R f ->
exists r : float,
r = (p * q - f)%R :>R /\ Fbounded b r /\ Fexp r = (Fexp p + Fexp q)%Z.