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.
Definition Fulp (p : float) :=
powerRZ radix (Fexp (Fnormalize radix b precision p)).
intros p q H' H'0 H'1; unfold Fulp in |- *.
rewrite
FcanonicUnique
with
(p := Fnormalize radix b precision p)
(q := Fnormalize radix b precision q)
(3 := pGivesBound); auto with float arith.
apply trans_eq with (FtoR radix p).
apply FnormalizeCorrect; auto.
apply trans_eq with (FtoR radix q); auto.
apply sym_eq; apply FnormalizeCorrect; auto.
Qed.
intros p H'; unfold Fulp, FtoRradix, FtoR, Fnormalize in |- *; simpl in |- *;
rewrite Rmult_1_l.
case (Z_zerop (Fnum p)); simpl in |- *; auto.
intros H'0; apply (Rle_powerRZ radix (- dExp b) (Fexp p));
auto with float real zarith.
intros H'0; apply Rle_powerRZ; auto with real zarith arith.
Qed.
intros x; unfold is_Fzero, Fulp, Fnormalize in |- *; case (Z_zerop (Fnum x));
simpl in |- *; auto.
intros H' H'0; Contradict H'; auto.
Qed.
intros p H1 H2; unfold Fulp in |- *.
replace (FtoRradix p) with (FtoRradix (Fnormalize radix b precision p));
[ idtac | unfold FtoRradix in |- *; apply FnormalizeCorrect; auto ].
apply Rmult_le_reg_l with (powerRZ radix (Z.pred precision)).
apply powerRZ_lt; auto with real arith.
replace
(powerRZ radix (Z.pred precision) *
(Rabs (Fnormalize radix b precision p) *
powerRZ radix (Z.succ (- precision))))%R with
(Rabs (Fnormalize radix b precision p)).
unfold FtoRradix in |- *; rewrite <- Fabs_correct; auto with arith real.
unfold Fabs, FtoR in |- *; simpl in |- *.
apply Rmult_le_compat_r; [ apply powerRZ_le | rewrite <- inj_pred ];
auto with real arith zarith.
rewrite <- Zpower_nat_Z_powerRZ.
replace (Zpower_nat radix (pred precision)) with (nNormMin radix precision);
auto; apply Rle_IZR.
apply pNormal_absolu_min with b; auto with arith zarith real.
apply
trans_eq
with
(Rabs (Fnormalize radix b precision p) *
(powerRZ radix (Z.pred precision) * powerRZ radix (Z.succ (- precision))))%R;
[ idtac | ring ].
rewrite <- powerRZ_add; auto with zarith real.
replace (Z.pred precision + Z.succ (- precision))%Z with 0%Z;
[ simpl in |- *; ring | unfold Z.succ, Z.pred in |- *; ring ];
auto with real zarith.
Qed.
intros p H.
replace (FtoRradix p) with (FtoRradix (Fnormalize radix b precision p));
[ idtac | unfold FtoRradix in |- *; apply FnormalizeCorrect; auto ].
unfold FtoRradix in |- *; rewrite <- Fabs_correct; auto with arith real.
unfold FtoR in |- *; simpl in |- *; unfold Fulp in |- *.
apply Rmult_le_compat_r; [ apply powerRZ_le | idtac ];
auto with real arith zarith.
apply Rle_trans with (IZR (Z.pred (Zpos (vNum b))));
[ apply Rle_IZR; auto with float zarith | idtac ].
unfold Z.pred in |- *; right; rewrite pGivesBound; replace 1%R with (IZR 1);
auto with real.
rewrite <- Zpower_nat_Z_powerRZ; rewrite Z_R_minus;auto.
Qed.
intros x y Hx Hy H1 H2; unfold Fulp in |- *.
apply Rle_powerRZ; auto with real zarith.
apply Fcanonic_Rle_Zle with radix b precision; auto with zarith arith.
apply FnormalizeCanonic; auto with zarith arith.
apply FnormalizeCanonic; auto with zarith arith.
repeat rewrite FnormalizeCorrect; auto with zarith arith real.
repeat rewrite Rabs_right; auto with zarith arith real.
apply Rge_trans with (FtoRradix x); auto with real.
Qed.
intros p H'.
replace (Fulp p) with (powerRZ radix (Fexp p)).
2: unfold Fulp in |- *; replace (Fnormalize radix b precision p) with p; auto;
apply sym_equal; apply FcanonicUnique with (3 := pGivesBound);
auto with arith; apply FnormalizeCanonic || apply FnormalizeCorrect;
auto with float zarith.
2: apply FcanonicBound with (1 := H'); auto with float zarith.
unfold FtoRradix in |- *; rewrite <- Fminus_correct; auto with zarith.
case (Z.eq_dec (Fnum p) (- nNormMin radix precision)); intros H1'.
case (Z.eq_dec (Fexp p) (- dExp b)); intros H2'.
rewrite FSuccDiff2; auto with arith.
unfold FtoR in |- *; simpl in |- *; rewrite Rmult_1_l; auto with real.
rewrite FSuccDiff3; auto with arith.
unfold FtoR in |- *; simpl in |- *; rewrite Rmult_1_l.
apply Rlt_le; apply Rlt_powerRZ; auto with real zarith.
unfold Z.pred in |- *; auto with zarith.
rewrite FSuccDiff1; auto with arith.
unfold FtoR in |- *; simpl in |- *; rewrite Rmult_1_l; auto with real.
Qed.
intros p H'.
replace (Fulp p) with (Fulp (Fnormalize radix b precision p)).
replace (FtoRradix p) with (FtoRradix (Fnormalize radix b precision p)).
unfold FNSucc in |- *; apply FulpSucCan; auto with float arith.
unfold FtoRradix in |- *; apply FnormalizeCorrect; auto.
apply FulpComp; auto with float arith.
unfold FtoRradix in |- *; apply FnormalizeCorrect; auto.
Qed.
intros p H'.
replace (Fulp p) with (powerRZ radix (Fexp p)).
2: unfold Fulp in |- *; replace (Fnormalize radix b precision p) with p; auto;
apply sym_equal; apply FcanonicUnique with (3 := pGivesBound);
auto with arith; apply FnormalizeCanonic || apply FnormalizeCorrect;
auto with arith.
unfold FtoRradix in |- *; rewrite <- Fminus_correct; auto with arith.
case (Z.eq_dec (Fnum p) (nNormMin radix precision)); intros H1'.
case (Z.eq_dec (Fexp p) (- dExp b)); intros H2'.
rewrite FPredDiff2; auto with arith.
unfold FtoR in |- *; simpl in |- *; rewrite Rmult_1_l; auto with real.
rewrite FPredDiff3; auto with arith.
unfold FtoR in |- *; simpl in |- *; rewrite Rmult_1_l; auto with real.
apply Rlt_le; apply Rlt_powerRZ; auto with real zarith.
replace 1%R with (INR 1); auto with real arith.
unfold Z.pred in |- *; auto with zarith.
rewrite FPredDiff1; auto with arith.
unfold FtoR in |- *; simpl in |- *; rewrite Rmult_1_l; auto with real.
apply FcanonicBound with (1 := H').
Qed.
intros p H'.
replace (Fulp p) with (Fulp (Fnormalize radix b precision p)).
replace (FtoRradix p) with (FtoRradix (Fnormalize radix b precision p)).
unfold FNPred in |- *; apply FulpPredCan; auto with float arith.
unfold FtoRradix in |- *; apply FnormalizeCorrect; auto.
apply FulpComp; auto with float arith.
unfold FtoRradix in |- *; apply FnormalizeCorrect; auto.
Qed.
intros x H.
unfold FtoRradix in |- *; apply FSuccDiff1; auto with arith.
Contradict H; unfold FtoRradix, FtoR in |- *; simpl in |- *; rewrite H.
apply Rlt_not_le.
replace 0%R with (0 * powerRZ radix (Fexp x))%R; [ idtac | ring ].
apply Rlt_monotony_exp; auto with real arith.
generalize (nNormPos _ radixMoreThanOne precision);
replace 0%R with (IZR (- 0%nat)); auto with real zarith arith.
Qed.
intros f Hf1 Hf2 H.
apply LeFulpPos; auto with zarith float; unfold FtoRradix in |- *.
apply R0RltRlePred; auto with arith.
apply Rlt_le; apply FPredLt; auto with arith.
Qed.
intros p H; unfold Fulp in |- *.
rewrite FcanonicFnormalizeEq; auto with arith.
unfold FtoRradix, FtoR in |- *; simpl in |- *; ring.
Qed.
intros x H H0; rewrite CanonicFulp; auto.
apply FSuccDiffPos; auto.
Qed.
intros x H H0.
unfold FNSucc in |- *.
rewrite FcanonicFnormalizeEq; auto with arith.
apply FSuccUlpPos; auto.
Qed.
intros f; unfold Fulp in |- *; case (Rle_or_lt 0 f); intros H'.
replace (Fabs f) with f; auto; unfold Fabs in |- *; apply floatEq;
simpl in |- *; auto with zarith real.
apply sym_eq; apply Z.abs_eq; apply LeR0Fnum with radix; auto with zarith real.
replace (Fabs f) with (Fopp f);
[ rewrite Fnormalize_Fopp | apply floatEq; simpl in |- * ];
auto with arith.
apply sym_eq; apply Zabs_eq_opp; apply R0LeFnum with radix;
auto with zarith real.
Qed.
intros P H' p q H'0.
case (Req_dec p q); intros Eq1.
rewrite <- Eq1.
replace (p - p)%R with 0%R; [ idtac | ring ].
rewrite Rabs_R0; auto.
unfold Fulp, FtoRradix, FtoR in |- *; simpl in |- *; auto with real arith.
case H'.
intros H'1 H'2; elim H'2; intros H'3 H'4; elim H'4; intros H'5 H'6;
case H'5 with (1 := H'0); clear H'5 H'4 H'2; intros H'5.
rewrite Rabs_right; auto.
cut (Fbounded b q); [ intros B0 | case H'5; auto ].
apply Rlt_le_trans with (2 := FulpSuc q B0).
apply Rplus_lt_reg_l with (r := FtoR radix q).
repeat rewrite Rplus_minus; auto.
case (Rle_or_lt (FNSucc b radix precision q) p); auto.
intros H'2; absurd (FNSucc b radix precision q <= q)%R; auto.
apply Rgt_not_le; red in |- *; unfold FtoRradix in |- *;
auto with real float arith.
case H'5; auto.
intros H'4 H'7; elim H'7; intros H'8 H'9; apply H'9; clear H'7; auto.
apply (FcanonicBound radix b); auto with float arith.
apply Rle_ge; apply Rplus_le_reg_l with (r := FtoR radix q).
repeat rewrite Rplus_minus; auto.
rewrite Rplus_0_r; apply isMin_inv1 with (1 := H'5); auto.
rewrite Faux.Rabsolu_left1; auto.
rewrite Ropp_minus_distr; auto.
cut (Fbounded b q); [ intros B0 | case H'5; auto ].
apply Rlt_le_trans with (2 := FulpPred q B0).
apply Ropp_lt_cancel; repeat rewrite Rminus_0_l.
apply Rplus_lt_reg_l with (r := FtoR radix q).
repeat rewrite Rplus_minus; auto.
case (Rle_or_lt p (FNPred b radix precision q)); auto.
intros H'2; absurd (q <= FNPred b radix precision q)%R; auto.
apply Rgt_not_le; red in |- *; unfold FtoRradix in |- *;
auto with real float arith.
case H'5; auto.
intros H'4 H'7; elim H'7; intros H'8 H'9; apply H'9; clear H'7; auto.
apply (FcanonicBound radix b); auto with float arith.
intros H1; apply Rplus_lt_compat_l; auto with real; apply Ropp_lt_contravar;
unfold Rminus in |- *; auto with real.
apply Rplus_le_reg_l with (r := FtoR radix q).
repeat rewrite Rplus_minus; auto.
rewrite Rplus_0_r; apply isMax_inv1 with (1 := H'5).
Qed.
intros P H p q x H0 H1 H2 H3 H4.
apply Zlt_powerRZ with (e := IZR radix); auto with real zarith.
apply Rle_lt_trans with (FtoRradix (Fabs q)).
replace (powerRZ radix (Fexp q)) with (FtoRradix (Float 1%nat (Fexp q)));
unfold FtoRradix in |- *;
[ apply Fop.RleFexpFabs; auto with arith
| unfold FtoR in |- *; simpl in |- *; ring ].
rewrite (Fabs_correct radix); auto with zarith.
fold FtoRradix in |- *; rewrite H3.
apply Rlt_le_trans with (Fulp p);
[ apply RoundedModeUlp with P; auto | unfold Fulp in |- * ].
apply Rle_powerRZ; auto with real zarith.
apply FcanonicLeastExp with radix b precision; auto with real arith.
apply sym_eq; apply FnormalizeCorrect; auto.
apply FnormalizeCanonic; auto with zarith.
Qed.
intros P p H' H.
elim H'; intros H'0 H'1; elim H'1; intros H'2 H'3; elim H'3; intros H'4 H'5;
clear H'3 H'1.
case (H'0 p).
intros x H'6.
apply (H'2 p p x); auto.
apply sym_eq; apply (RoundedProjector _ _ P H'); auto.
Qed.
intros P r q H' H'0.
case H'.
intros H'1 H'2; elim H'2; intros H'3 H'4; elim H'4; intros H'5 H'6;
case H'5 with (1 := H'0); clear H'4 H'2; intros H;
case H; auto.
Qed.
intros P p q H' H'0 H'1.
cut (MinOrMaxP b radix P);
[ intros Mn; case (Mn p q); auto; intros Mn1 | auto with inv ].
apply sym_eq; apply MinEq with (1 := Mn1); auto.
apply (RoundedModeProjectorIdem (isMin b radix)); auto.
apply MinRoundedModeP with (precision := precision); auto.
apply sym_eq; apply MaxEq with (1 := Mn1); auto.
apply (RoundedModeProjectorIdem (isMax b radix)); auto.
apply MaxRoundedModeP with (precision := precision); auto.
Qed.
intros P H' r q q' H'0 H'1.
replace (radix * q')%R with (FtoRradix (Float (Fnum q') (Fexp q' + 1%nat))).
intros H'2; case H'2.
intros H'3; case H'; intros H'4 H'5; elim H'5; intros H'6 H'7; elim H'7;
intros H'8 H'9; apply H'9 with (1 := H'3); clear H'7 H'5;
auto.
apply RoundedModeProjectorIdem; auto.
apply FBoundedScale; auto.
intros H'3.
replace (FtoRradix q) with (FtoRradix (Float (Fnum q') (Fexp q' + 1%nat)));
auto with real.
apply (RoundedProjector _ _ P H'); auto.
apply FBoundedScale; auto.
case H'.
intros H'4 H'5; elim H'5; intros H'6 H'7; clear H'5.
apply (H'6 r (Float (Fnum q') (Fexp q' + 1%nat)) q); auto.
apply RoundedModeBounded with (P := P) (r := r); auto.
rewrite (FvalScale _ radixMoreThanOne b).
rewrite powerRZ_1; auto.
Qed.
intros P H' r q q' H'0 H'1.
replace (radix * q')%R with (FtoRradix (Float (Fnum q') (Fexp q' + 1%nat))).
intros H'2; case H'2.
intros H'3; case H'; intros H'4 H'5; elim H'5; intros H'6 H'7; elim H'7;
intros H'8 H'9; apply H'9 with (1 := H'3); clear H'7 H'5;
auto.
apply RoundedModeProjectorIdem; auto.
apply FBoundedScale; auto.
intros H'3.
replace (FtoRradix q) with (FtoRradix (Float (Fnum q') (Fexp q' + 1%nat)));
auto with real.
apply (RoundedProjector _ _ P H'); auto.
apply FBoundedScale; auto.
unfold FtoRradix in H'3; rewrite H'3; auto.
case H'.
intros H'4 H'5; elim H'5; intros H'6 H'7; clear H'5.
unfold FtoRradix in |- *; rewrite FvalScale; auto.
rewrite powerRZ_1; auto.
Qed.
intros r min H' H'0.
rewrite <- (FzeroisZero radix b).
case H'; intros H'1.
apply (MonotoneMin b radix) with (p := FtoRradix (Fzero (- dExp b))) (q := r);
auto.
unfold FtoRradix in |- *; rewrite (FzeroisZero radix b); auto.
apply (RoundedModeProjectorIdem (isMin b radix)); auto.
apply MinRoundedModeP with (precision := precision); auto with float.
apply FboundedFzero; auto.
replace (FtoR radix (Fzero (- dExp b))) with (FtoRradix min); auto with real.
apply sym_eq; apply (ProjectMin b radix).
apply FboundedFzero; auto.
rewrite <- H'1 in H'0; rewrite <- (FzeroisZero radix b) in H'0; auto.
Qed.
intros P p r H' H'0 H'1.
case H'.
intros H'2 H'3; Elimc H'3; intros H'3 H'4; Elimc H'4; intros H'4 H'5;
case (H'4 r p); auto; intros H'6.
apply RleMinR0 with (r := r); auto.
apply Rle_trans with r; auto; apply isMax_inv1 with (1 := H'6).
Qed.
intros r max H' H'0.
rewrite <- (FzeroisZero radix b).
case H'; intros H'1.
apply (MonotoneMax b radix) with (q := FtoRradix (Fzero (- dExp b))) (p := r);
auto.
unfold FtoRradix in |- *; rewrite FzeroisZero; auto.
apply (RoundedModeProjectorIdem (isMax b radix)); auto.
apply MaxRoundedModeP with (precision := precision); auto.
apply FboundedFzero; auto.
replace (FtoR radix (Fzero (- dExp b))) with (FtoRradix max); auto with real.
apply sym_eq; apply (ProjectMax b radix).
apply FboundedFzero; auto.
rewrite H'1 in H'0; rewrite <- (FzeroisZero radix b) in H'0; auto.
Qed.
intros P p r H' H'0 H'1.
case H'.
intros H'2 H'3; Elimc H'3; intros H'3 H'4; Elimc H'4; intros H'4 H'5;
case (H'4 r p); auto; intros H'6.
apply Rle_trans with r; auto; apply isMin_inv1 with (1 := H'6).
apply RleMaxR0 with (r := r); auto.
Qed.
intros p min H' H'0 H'1.
cut (min <= / 2%nat * p)%R;
[ intros Rl1; Casec Rl1; intros Rl1
| apply isMin_inv1 with (1 := H'1); auto ].
case (eqExpMax _ radixMoreThanOne b min p); auto.
case H'1; auto.
rewrite Fabs_correct; auto with arith.
rewrite Rabs_right; auto.
apply Rle_trans with (/ 2%nat * p)%R; auto.
apply Rlt_le; auto.
apply Rmult_le_reg_l with (r := INR 2); auto with real.
rewrite <- Rmult_assoc; rewrite Rinv_r; auto with real; rewrite Rmult_ne_r;
auto with real.
apply Rle_ge; apply RleMinR0 with (r := (/ 2%nat * p)%R); auto.
apply Rmult_le_reg_l with (r := INR 2); auto with real.
rewrite <- Rmult_assoc; rewrite Rinv_r; auto with real; rewrite Rmult_1_l;
rewrite Rmult_0_r; auto with real.
intros min' H'2; elim H'2; intros H'3 H'4; elim H'4; intros H'5 H'6;
clear H'4 H'2.
case (FboundNext _ radixMoreThanOne b precision) with (p := min');
auto with arith; fold FtoRradix in |- *.
intros Smin H'2; elim H'2; intros H'4 H'7; clear H'2.
exists Smin; split; auto.
rewrite H'7; auto.
unfold FtoRradix in |- *.
rewrite <- H'5; auto.
replace (Float (Z.succ (Fnum min')) (Fexp min')) with
(Float (Fnum (Fshift radix (Z.abs_nat (Fexp p - Fexp min')) p) - Fnum min')
(Fexp min')); auto.
unfold FtoRradix, FtoR in |- *; simpl in |- *.
rewrite <- Z_R_minus.
rewrite (fun x y z : R => Rmult_comm (x - y) z); rewrite Rmult_minus_distr_l;
repeat rewrite (fun x : Z => Rmult_comm (powerRZ radix x)).
rewrite Rmult_IZR.
rewrite Zpower_nat_powerRZ_absolu; auto with zarith.
rewrite Rmult_assoc.
rewrite <- (powerRZ_add radix); auto with real zarith.
replace (Fexp p - Fexp min' + Fexp min')%Z with (Fexp p); [ auto | ring ].
apply floatEq; auto; simpl in |- *.
apply Zle_antisym.
apply Zlt_succ_le.
apply Zplus_lt_reg_l with (p := Fnum min'); auto.
cut (forall x y : Z, (x + (y - x))%Z = y);
[ intros tmp; rewrite tmp; clear tmp | intros; ring ].
replace (Fnum min' + Z.succ (Z.succ (Fnum min')))%Z with
(2%nat * Z.succ (Fnum min'))%Z.
apply (Rlt_Float_Zlt radix) with (r := Fexp min'); auto;
fold FtoRradix in |- *.
replace (FtoRradix (Float (2%nat * Z.succ (Fnum min')) (Fexp min'))) with
(2%nat * Float (Z.succ (Fnum min')) (Fexp min'))%R.
rewrite <- H'7.
replace
(Float (Fnum p * Zpower_nat radix (Z.abs_nat (Fexp p - Fexp min')))
(Fexp min')) with (Fshift radix (Z.abs_nat (Fexp p - Fexp min')) p).
unfold FtoRradix in |- *; rewrite FshiftCorrect; auto.
apply Rmult_lt_reg_l with (r := (/ 2%nat)%R); auto with real.
rewrite <- Rmult_assoc; rewrite Rinv_l; auto with real; rewrite Rmult_1_l;
auto with real.
case (Rle_or_lt Smin (/ 2%nat * FtoR radix p)); auto.
intros H'2; absurd (min < Smin)%R.
apply Rle_not_lt.
case H'1; auto.
intros H'8 H'9; elim H'9; intros H'10 H'11; apply H'11; clear H'9; auto.
rewrite H'7; unfold FtoRradix in |- *; rewrite <- H'5; auto.
unfold FtoR in |- *; simpl in |- *; apply Rlt_monotony_exp;
auto with real zarith.
unfold Fshift in |- *; simpl in |- *.
replace (Fexp p - Z.abs_nat (Fexp p - Fexp min'))%Z with (Fexp min'); auto.
rewrite inj_abs; auto.
ring.
auto with zarith.
replace (FtoRradix (Float (2%nat * Z.succ (Fnum min')) (Fexp min'))) with
((2%nat * Z.succ (Fnum min'))%Z * powerRZ radix (Fexp min'))%R.
rewrite Rmult_IZR; auto.
unfold FtoRradix, FtoR in |- *; simpl in |- *; ring.
simpl in |- *; auto.
replace (Z_of_nat 2) with (Z.succ (Z.succ 0)).
repeat rewrite <- Zmult_succ_l_reverse; unfold Z.succ in |- *; ring.
simpl in |- *; auto.
apply Zlt_le_succ; auto.
apply Zplus_lt_reg_l with (p := Fnum min'); auto.
cut (forall x y : Z, (x + (y - x))%Z = y);
[ intros tmp; rewrite tmp; clear tmp | intros; ring ].
replace (Fnum min' + Fnum min')%Z with (2%nat * Fnum min')%Z.
apply (Rlt_Float_Zlt radix) with (r := Fexp min'); auto;
fold FtoRradix in |- *.
replace (FtoRradix (Float (2%nat * Fnum min') (Fexp min'))) with
(2%nat * Float (Fnum min') (Fexp min'))%R.
replace
(Float (Fnum p * Zpower_nat radix (Z.abs_nat (Fexp p - Fexp min')))
(Fexp min')) with (Fshift radix (Z.abs_nat (Fexp p - Fexp min')) p).
unfold FtoRradix in |- *; rewrite FshiftCorrect; auto.
apply Rmult_lt_reg_l with (r := (/ 2%nat)%R); auto with real.
rewrite <- Rmult_assoc; rewrite Rinv_l; auto with real; rewrite Rmult_1_l;
auto with real.
replace (FtoR radix (Float (Fnum min') (Fexp min'))) with (FtoR radix min);
auto.
unfold Fshift in |- *; simpl in |- *.
replace (Fexp p - Z.abs_nat (Fexp p - Fexp min'))%Z with (Fexp min'); auto.
rewrite inj_abs; auto.
ring.
auto with zarith.
replace (FtoRradix (Float (2%nat * Fnum min') (Fexp min'))) with
((2%nat * Fnum min')%Z * powerRZ radix (Fexp min'))%R.
rewrite Rmult_IZR; auto.
unfold FtoRradix, FtoR in |- *; simpl in |- *; ring.
simpl in |- *; auto.
replace (Z_of_nat 2) with (Z.succ (Z.succ 0)).
repeat rewrite <- Zmult_succ_l_reverse; unfold Z.succ in |- *; ring.
simpl in |- *; auto.
exists min; split; auto.
case H'1; auto.
rewrite Rl1.
pattern (FtoRradix p) at 2 in |- *;
replace (FtoRradix p) with (2%nat * (/ 2%nat * p))%R.
simpl in |- *; ring.
rewrite <- Rmult_assoc; rewrite Rinv_r; auto with real; rewrite Rmult_ne_r;
auto with real.
Qed.
intros p min max P H' H'0 H'1; apply Rle_antisym.
apply Rplus_le_reg_l with (r := (- max)%R).
replace (- max + p)%R with (p - max)%R; [ idtac | ring ].
replace (- max + (min + max))%R with (FtoRradix min); [ idtac | ring ].
rewrite <- (Fminus_correct radix); auto with arith.
case H'0.
intros H'2 H'3; elim H'3; intros H'4 H'5; apply H'5; clear H'3; auto.
apply Sterbenz; auto.
case H'1; auto.
apply Rle_trans with (FtoRradix max); auto.
apply Rmult_le_reg_l with (r := INR 2); auto with real.
rewrite <- Rmult_assoc; rewrite Rinv_r; auto with real; rewrite Rmult_1_l;
auto with real.
apply Rledouble; auto.
apply Rle_trans with (FtoRradix min); auto.
apply RleMinR0 with (r := (/ 2%nat * p)%R); auto.
apply Rmult_le_reg_l with (r := INR 2); auto with real.
rewrite <- Rmult_assoc; rewrite Rinv_r; auto with real; rewrite Rmult_1_l;
rewrite Rmult_0_r; auto with real.
apply Rle_trans with (/ 2%nat * p)%R; auto; apply isMax_inv1 with (1 := H'1).
case H'1.
intros H'3 H'6; elim H'6; intros H'7 H'8; apply H'8; clear H'6; auto.
apply Rmult_le_reg_l with (r := INR 2); auto with real.
rewrite <- Rmult_assoc; rewrite Rinv_r; auto with real; rewrite Rmult_1_l;
auto with real.
apply Rmult_le_reg_l with (r := (/ 2%nat)%R); auto with real.
rewrite <- Rmult_assoc; rewrite Rinv_l; auto with real; rewrite Rmult_1_l;
auto with real.
apply isMax_inv1 with (1 := H'1).
rewrite Fminus_correct; auto with arith.
apply Rplus_le_reg_l with (r := FtoR radix max).
replace (FtoR radix max + (FtoR radix p - FtoR radix max))%R with
(FtoR radix p); [ idtac | ring ].
apply Rplus_le_reg_l with (r := (- (/ 2%nat * p))%R).
replace (- (/ 2%nat * p) + FtoR radix p)%R with (/ 2%nat * p)%R.
replace (- (/ 2%nat * p) + (FtoR radix max + / 2%nat * p))%R with
(FtoR radix max); [ apply isMax_inv1 with (1 := H'1) | ring ].
replace (FtoR radix p) with (2%nat * (/ 2%nat * p))%R.
simpl in |- *; ring.
rewrite <- Rmult_assoc; rewrite Rinv_r; auto with real.
apply Rplus_le_reg_l with (r := (- min)%R).
replace (- min + p)%R with (p - min)%R; [ idtac | ring ].
replace (- min + (min + max))%R with (FtoRradix max); [ idtac | ring ].
case (PminPos p min); auto.
intros x H'2; elim H'2; intros H'3 H'4; elim H'4; clear H'2.
case H'1.
intros H'2 H'5; elim H'5; intros H'6 H'7; apply H'7; clear H'5; auto.
unfold FtoRradix in H'4; rewrite H'4; auto.
fold FtoRradix in |- *; apply Rplus_le_reg_l with (r := FtoRradix min).
replace (min + (p - min))%R with (FtoRradix p); [ idtac | ring ].
apply Rplus_le_reg_l with (r := (- (/ 2%nat * p))%R).
replace (- (/ 2%nat * p) + p)%R with (/ 2%nat * p)%R.
replace (- (/ 2%nat * p) + (min + / 2%nat * p))%R with (FtoRradix min);
[ apply isMin_inv1 with (1 := H'0) | ring ].
pattern (FtoRradix p) at 3 in |- *;
replace (FtoRradix p) with (2%nat * (/ 2%nat * p))%R.
simpl in |- *; ring.
rewrite <- Rmult_assoc; rewrite Rinv_r; auto with real.
Qed.
intros p min max H' H'0 H'1; case (Rle_or_lt 0 p); intros H'2.
apply div2IsBetweenPos; auto.
cut (forall x y : R, (- x)%R = (- y)%R -> x = y);
[ intros H'3; apply H'3; clear H'3 | idtac ].
replace (- (min + max))%R with (- max + - min)%R; [ idtac | ring ].
repeat rewrite <- (Fopp_correct radix); auto with float.
apply div2IsBetweenPos; auto with float.
rewrite (Fopp_correct radix); auto.
replace 0%R with (-0)%R; try apply Rlt_le; auto with real.
replace (/ 2%nat * Fopp p)%R with (- (/ 2%nat * p))%R; auto with float.
rewrite (Fopp_correct radix); auto; fold FtoRradix; ring.
replace (/ 2%nat * Fopp p)%R with (- (/ 2%nat * p))%R; auto with float.
rewrite (Fopp_correct radix); auto; fold FtoRradix;ring.
intros x y H'3; rewrite <- (Ropp_involutive x);
rewrite <- (Ropp_involutive y); rewrite H'3; auto.
Qed.
intros P H' r q q' H'0 H'1 H'2.
case (Rle_or_lt 0 r); intros Rl0.
rewrite Rabs_right; auto.
apply RoundedModeMult with (P := P) (r := r); auto.
rewrite <- (Rabs_right r); auto with real.
apply Rle_ge; apply RleRoundedR0 with (P := P) (r := r); auto.
rewrite Faux.Rabsolu_left1; auto.
replace (radix * q')%R with (- (radix * - q'))%R;
[ apply Ropp_le_contravar | ring ].
rewrite <- (Fopp_correct radix).
apply RoundedModeMultLess with (P := P) (r := r); auto.
apply oppBounded; auto.
unfold FtoRradix in |- *; rewrite Fopp_correct.
rewrite <- (Ropp_involutive r).
replace (radix * - FtoR radix q')%R with (- (radix * q'))%R;
[ apply Ropp_le_contravar | fold FtoRradix;ring ]; auto.
rewrite <- (Faux.Rabsolu_left1 r); auto.
apply Rlt_le; auto.
apply RleRoundedLessR0 with (P := P) (r := r); auto.
apply Rlt_le; auto.
Qed.
intros r1 r2 min max H' H'0 H'1 H'2; split.
case H'; auto.
split.
apply Rlt_le; auto.
intros f H'3 H'4.
case H'; auto.
intros H'5 H'6; elim H'6; intros H'7 H'8; apply H'8; clear H'6; auto.
case (Rle_or_lt (FtoR radix f) r1); auto; intros C1.
absurd (FtoR radix f < max)%R.
apply Rle_not_lt.
case H'0.
intros H'6 H'9; elim H'9; intros H'10 H'11; apply H'11; clear H'9; auto.
apply Rlt_le; auto.
apply Rle_lt_trans with (2 := H'2); auto.
Qed.
intros r1 r2 min max H' H'0 H'1 H'2; split.
case H'0; auto.
split.
apply Rlt_le; auto.
intros f H'3 H'4.
case H'0; auto.
intros H'5 H'6; elim H'6; intros H'7 H'8; apply H'8; clear H'6; auto.
case (Rle_or_lt r1 (FtoR radix f)); auto; intros C1.
absurd (min < FtoR radix f)%R.
apply Rle_not_lt.
case H'.
intros H'6 H'9; elim H'9; intros H'10 H'11; apply H'11; clear H'9; auto.
apply Rlt_le; auto.
apply Rlt_le_trans with (1 := H'1); auto.
Qed.
intros P p r H' H'0 H'1.
cut (0 < Float 1%nat (- dExp b))%R;
[ intros Rl0
| unfold FtoRradix, FtoR in |- *; simpl in |- *; rewrite Rmult_1_l;
auto with real arith ].
cut (0 < r)%R; [ intros Rl1 | apply Rlt_le_trans with (1 := Rl0) ]; auto.
cut (0 <= r)%R; [ intros Rl2 | apply Rlt_le; auto ].
case H'.
intros H'2 H'3; Elimc H'3; intros H'3 H'4; Elimc H'4; intros H'4 H'5;
case (H'4 r p); auto; intros H'6.
apply Rle_trans with r; auto with real.
apply isMin_inv1 with (1 := H'6).
rewrite Rmult_comm; pattern r at 1 in |- *; replace r with (r * 1%nat)%R;
[ apply Rmult_le_compat_l | simpl; ring ]; auto with real arith.
case (MinEx b radix precision) with (r := r); auto with arith;
intros min Hmin.
cut (Fbounded b (Float (Fnum min) (Z.succ (Fexp min)))); [ intros F2 | idtac ].
cut (FtoRradix (Float (Fnum min) (Z.succ (Fexp min))) = (radix * min)%R :>R);
[ intros F2Eq | idtac ].
apply Rle_trans with (FtoRradix (Float (Fnum min) (Z.succ (Fexp min)))).
case H'6.
intros H'7 H'8; elim H'8; intros H'9 H'10; apply H'10; clear H'8; auto.
case (Rle_or_lt r (Float (Fnum min) (Z.succ (Fexp min)))); auto; intros Rlt0.
absurd (Float (Fnum min) (Z.succ (Fexp min)) <= min)%R.
apply Rgt_not_le.
rewrite F2Eq; auto with real.
rewrite Rmult_comm.
pattern (FtoRradix min) at 2 in |- *;
replace (FtoRradix min) with (min * 1%nat)%R; auto with real.
red in |- *; apply Rmult_lt_compat_l; auto with real arith.
case (RleMinR0 r min); auto.
intros H'8; case H'1.
intros H'11; absurd (Float 1%nat (- dExp b) <= min)%R.
apply Rgt_not_le; auto.
rewrite <- H'8; auto.
apply
(MonotoneMin b radix)
with (p := FtoRradix (Float 1%nat (- dExp b))) (q := r);
auto.
apply (RoundedModeProjectorIdem (isMin b radix)); auto.
apply MinRoundedModeP with (precision := precision); auto.
repeat split.
simpl in |- *; auto with zarith.
apply (vNumbMoreThanOne radix) with (precision := precision);
auto with zarith.
simpl in |- *; auto with zarith.
intros H'11; absurd (min = Float 1%nat (- dExp b) :>R).
rewrite <- H'8.
apply Rlt_dichotomy_converse; left; auto.
apply (MinUniqueP b radix r); auto.
rewrite <- H'11.
apply (RoundedModeProjectorIdem (isMin b radix)); auto.
apply MinRoundedModeP with (precision := precision); auto.
repeat split.
simpl in |- *; auto with zarith.
apply (vNumbMoreThanOne radix) with (precision := precision);
auto with zarith.
simpl in |- *; auto with zarith.
case Hmin.
intros H'8 H'11; elim H'11; intros H'12 H'13; apply H'13; clear H'11; auto.
apply Rlt_le; auto.
rewrite F2Eq.
apply Rmult_le_compat_l; auto with real arith.
replace 0%R with (INR 0); auto with real arith.
apply isMin_inv1 with (1 := Hmin).
unfold FtoRradix, FtoR in |- *; simpl in |- *.
rewrite powerRZ_Zs; auto with real zarith; ring.
cut (Fbounded b min);
[ unfold Fbounded in |- *; intros Fb0 | case Hmin; auto ].
elim Fb0; intros H H0; auto.
repeat (split; simpl in |- *); auto.
apply Z.le_trans with (Fexp min); auto with zarith.
Qed.
intros P p r H' H'0 H'1.
cut (Float (- 1%nat) (- dExp b) < 0)%R;
[ intros Rl0
| unfold FtoRradix, FtoR in |- *; simpl in |- *; unfold IZR at 1; rewrite Ropp_mult_distr_l_reverse;
rewrite Rmult_1_l;
auto with real arith ].
2: replace 0%R with (-0)%R; auto with real arith; ring.
cut (r < 0)%R; [ intros Rl1 | apply Rle_lt_trans with (2 := Rl0) ]; auto.
cut (r <= 0)%R; [ intros Rl2 | apply Rlt_le; auto ].
case H'.
intros H'2 H'3; Elimc H'3; intros H'3 H'4; Elimc H'4; intros H'4 H'5;
case (H'4 r p); auto; intros H'6.
case (MaxEx b radix precision) with (r := r); auto with arith;
intros max Hmax.
cut (Fbounded b (Float (Fnum max) (Z.succ (Fexp max)))); [ intros F2 | idtac ].
cut (FtoRradix (Float (Fnum max) (Z.succ (Fexp max))) = (radix * max)%R :>R);
[ intros F2Eq | idtac ].
apply Rle_trans with (FtoRradix (Float (Fnum max) (Z.succ (Fexp max)))).
rewrite F2Eq; auto with real.
apply Rmult_le_compat_l; auto with real arith.
replace 0%R with (INR 0); auto with real arith.
apply isMax_inv1 with (1 := Hmax); auto.
case H'6.
intros H'7 H'8; elim H'8; intros H'9 H'10; apply H'10; clear H'8; auto.
case (Rle_or_lt (Float (Fnum max) (Z.succ (Fexp max))) r); auto; intros Rlt0.
absurd (max <= Float (Fnum max) (Z.succ (Fexp max)))%R.
apply Rgt_not_le.
rewrite F2Eq.
replace (radix * max)%R with (- (- max * radix))%R; [ idtac | ring ].
pattern (FtoRradix max) at 1 in |- *;
replace (FtoRradix max) with (- (- max * 1%nat))%R;
[ idtac | simpl in |- *; ring ].
apply Ropp_lt_gt_contravar; apply Rmult_lt_compat_l; auto with real.
replace 0%R with (-0)%R; [ apply Ropp_lt_contravar | ring ].
case (RleMaxR0 r max); auto.
intros H'8; case H'1.
intros H'11; absurd (max <= Float (- 1%nat) (- dExp b))%R.
apply Rgt_not_le; auto.
rewrite H'8; auto.
apply
(MonotoneMax b radix)
with (q := FtoRradix (Float (- 1%nat) (- dExp b))) (p := r);
auto.
apply (RoundedModeProjectorIdem (isMax b radix)); auto.
apply MaxRoundedModeP with (precision := precision); auto.
repeat split.
simpl in |- *; auto with zarith.
apply (vNumbMoreThanOne radix) with (precision := precision);
auto with zarith.
simpl in |- *; auto with zarith.
intros H'11; absurd (Float (- 1%nat) (- dExp b) = max :>R).
rewrite H'8; auto.
apply Rlt_dichotomy_converse; left; auto.
apply (MaxUniqueP b radix r); auto.
rewrite H'11.
apply (RoundedModeProjectorIdem (isMax b radix)); auto.
apply MaxRoundedModeP with (precision := precision); auto.
repeat split.
simpl in |- *; auto with zarith.
apply (vNumbMoreThanOne radix) with (precision := precision);
auto with zarith.
simpl in |- *; auto with zarith.
case Hmax.
intros H'8 H'11; elim H'11; intros H'12 H'13; apply H'13; clear H'11; auto.
apply Rlt_le; auto.
unfold FtoRradix, FtoR in |- *; simpl in |- *.
rewrite powerRZ_Zs; auto with real zarith; ring.
cut (Fbounded b max);
[ unfold Fbounded in |- *; intros Fb0 | case Hmax; auto ].
elim Fb0; intros H H0; repeat (split; simpl in |- *); auto.
apply Z.le_trans with (Fexp max); auto with zarith.
apply Rle_trans with r; auto with real.
pattern r at 2 in |- *; replace r with (- (- r * 1%nat))%R; [ idtac | simpl; ring ].
replace (radix * r)%R with (- (- r * radix))%R; [ idtac | ring ].
apply Ropp_le_contravar; apply Rmult_le_compat_l; auto with real arith.
replace 0%R with (-0)%R; auto with real arith.
apply isMax_inv1 with (1 := H'6).
Qed.
intros P p r H' H'0 H'1; case (Rle_or_lt 0 r); intros H'2.
repeat rewrite Rabs_right; auto with real.
apply roundedModeLessMult with (P := P); auto.
rewrite <- (Rabs_right r); auto with real.
apply Rle_ge; apply (RleRoundedR0 P) with (r := r); auto.
repeat rewrite Faux.Rabsolu_left1; auto.
replace (radix * - r)%R with (- (radix * r))%R;
[ apply Ropp_le_contravar | ring ].
apply roundedModeMoreMult with (P := P); auto.
rewrite <- (Ropp_involutive r); rewrite <- (Faux.Rabsolu_left1 r); auto.
replace (Float (- 1%nat) (- dExp b)) with (Fopp (Float 1%nat (- dExp b))).
unfold FtoRradix in |- *; rewrite Fopp_correct; auto with real.
simpl in |- *; auto.
apply Rlt_le; auto.
apply Rlt_le; auto.
apply (RleRoundedLessR0 P) with (r := r); auto.
apply Rlt_le; auto.
Qed.
intros P H' p q r H'0 H'1 H'2; case H'1; intros H'3.
cut (MonotoneP radix P);
[ intros Mn | apply RoundedModeP_inv4 with (1 := H'); auto ].
apply (Mn p r); auto.
apply RoundedModeProjectorIdem with (P := P); auto.
rewrite RoundedModeProjectorIdemEq with (P := P) (p := p) (q := q);
auto with real.
cut (CompatibleP b radix P);
[ intros Cp | apply RoundedModeP_inv2 with (1 := H'); auto ].
apply (Cp r p q); auto.
apply RoundedModeBounded with (P := P) (r := r); auto.
Qed.
intros P H' p q r H'0 H'1 H'2; case H'1; intros H'3.
cut (MonotoneP radix P);
[ intros Mn | apply RoundedModeP_inv4 with (1 := H'); auto ].
apply (Mn r p); auto.
apply RoundedModeProjectorIdem with (P := P); auto.
rewrite RoundedModeProjectorIdemEq with (P := P) (p := p) (q := q);
auto with real.
cut (CompatibleP b radix P);
[ intros Cp | apply RoundedModeP_inv2 with (1 := H'); auto ].
apply (Cp r p q); auto.
apply RoundedModeBounded with (P := P) (r := r); auto.
Qed.
intros P p q r H' H'0 H'1 H'2.
case (Rle_or_lt 0 p); intros Rl1.
rewrite Rabs_right; auto with real.
apply RleBoundRoundr with (P := P) (r := p); auto with real.
rewrite <- (Rabs_right p); auto with real.
apply Rle_ge; apply RleRoundedR0 with (P := P) (r := p); auto.
rewrite Faux.Rabsolu_left1; auto.
rewrite <- (Ropp_involutive r); apply Ropp_le_contravar.
rewrite <- (Fopp_correct radix); auto.
apply RleBoundRoundl with (P := P) (r := p); auto with float.
rewrite (Fopp_correct radix); rewrite <- (Ropp_involutive p);
rewrite <- (Faux.Rabsolu_left1 p); auto with real;
apply Rlt_le; auto.
apply RleRoundedLessR0 with (P := P) (r := p); auto; apply Rlt_le; auto.
Qed.
intros P p q r H' H'0 H'1 H'2.
case (Rle_or_lt 0 p); intros Rl1.
rewrite Rabs_right; auto.
apply RleBoundRoundl with (P := P) (r := p); auto.
rewrite <- (Rabs_right p); auto with real.
apply Rle_ge; apply RleRoundedR0 with (P := P) (r := p); auto.
rewrite Faux.Rabsolu_left1; auto.
rewrite <- (Ropp_involutive r); apply Ropp_le_contravar.
rewrite <- (Fopp_correct radix); auto.
apply RleBoundRoundr with (P := P) (r := p); auto with float.
rewrite (Fopp_correct radix); rewrite <- (Ropp_involutive p);
rewrite <- (Faux.Rabsolu_left1 p); auto with real;
apply Rlt_le; auto.
apply RleRoundedLessR0 with (P := P) (r := p); auto; apply Rlt_le; auto.
Qed.
(* Rounded of natural numbers are natural *)
intros P z p HP H'.
case
(RoundedModeRep b radix precision)
with (P := P) (p := Float z 0%nat) (q := p); auto.
cut (CompatibleP b radix P);
[ intros Cp | apply RoundedModeP_inv2 with (1 := HP); auto ];
auto.
apply Cp with (1 := H'); auto.
unfold FtoRradix, FtoR in |- *; simpl in |- *.
rewrite Rmult_1_r; auto.
apply RoundedModeBounded with (P := P) (r := IZR z); auto.
intros x H'0; exists x; auto.
unfold FtoRradix in |- *; rewrite H'0.
unfold FtoRradix, FtoR in |- *; simpl in |- *.
rewrite Rmult_1_r; auto.
Qed.

Theorem NroundN :
forall (P : R -> float -> Prop) (n : nat) (p : float),
RoundedModeP b radix P -> P n p -> exists n' : nat, p = n' :>R.