Section ClosestP.
Variable b : Fbound.
Variable radix : Z.
Variable precision : nat.
Let FtoRradix := FtoR radix.
Local 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 pq H' H'0 H'1 H'2.
cut (ex (fun m : Z => pq = Float m (Fexp (Fplus radix p q)) :>R)).
2: unfold FtoRradix in |- *;
apply
RoundedModeRep
with (b := b) (precision := precision) (P := Closest b radix);
auto.
2: apply ClosestRoundedModeP with (precision := precision); auto.
2: rewrite (Fplus_correct radix); auto with arith.
intros H'3; elim H'3; intros m E; clear H'3.
exists
(Fabs (Fminus radix q (Fminus radix (Float m (Fexp (Fplus radix p q))) p))).
cut (forall A B : Prop, A -> (A -> B) -> A /\ B);
[ intros tmp; apply tmp; clear tmp | auto ].
unfold FtoRradix in |- *; rewrite Fabs_correct; auto with arith.
cut (forall p q : R, p = q -> Rabs p = Rabs q);
[ intros tmp; apply tmp; clear tmp | intros p' q' H; rewrite H; auto ].
unfold FtoRradix in |- *; repeat rewrite Fminus_correct; auto with arith.
unfold FtoRradix in E; rewrite E; auto.
ring.
intros H'4.
cut (Rabs (pq - (p + q)) <= Rabs (q - (p + q)))%R.
2: elim H'2; auto.
replace (q - (p + q))%R with (- FtoRradix p)%R.
2: ring.
rewrite Rabs_Ropp.
unfold FtoRradix in |- *; rewrite <- Fabs_correct; auto with arith.
rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr.
unfold FtoRradix in H'4; rewrite <- H'4.
simpl in |- *.
rewrite Zmin_le1; auto.
generalize H'1 H'; case p; case q; unfold Fabs, Fminus, Fopp, Fplus in |- *;
simpl in |- *; clear H'1 H'.
intros Fnum1 Fexp1 Fnum2 Fexp2 H'5 H'6.
repeat rewrite Zmin_n_n; auto.
repeat rewrite (Zmin_le2 _ _ H'5); auto with zarith.
replace (Z.abs_nat (Fexp2 - Fexp2)) with 0.
rewrite Zpower_nat_O.
cut (forall z : Z, (z * 1%nat)%Z = z);
[ intros tmp; repeat rewrite tmp; clear tmp | auto with zarith ].
unfold FtoRradix, FtoR in |- *; simpl in |- *.
intros H'.
repeat split; simpl in |- *.
rewrite (fun x => Z.abs_eq (Z.abs x)); auto with zarith.
apply Z.le_lt_trans with (Z.abs Fnum2); auto.
apply le_IZR.
apply (Rle_monotony_contra_exp radix) with (z := Fexp2); auto.
case H'6; auto.
case H'6; auto.
intros; simpl in |- *; ring.
replace (Fexp2 - Fexp2)%Z with 0%Z; simpl in |- *; auto with zarith.
Qed.
intros p q pq H' H'0 H'1.
case (Zle_or_lt (Fexp p) (Fexp q)); intros H'2.
apply errorBoundedPlusLe; auto.
replace (p + q)%R with (q + p)%R; [ idtac | ring ].
replace (Z.min (Fexp p) (Fexp q)) with (Z.min (Fexp q) (Fexp p));
[ idtac | apply Zmin_sym ].
apply errorBoundedPlusLe; auto.
auto with zarith.
apply (ClosestCompatible b radix (p + q)%R (q + p)%R pq); auto.
ring.
case H'1; auto.
Qed.
intros p q pq H' H'0 H'1.
case (errorBoundedPlusAbs p q pq); auto.
intros x H'2; elim H'2; intros H'3 H'4; elim H'4; intros H'5 H'6;
clear H'4 H'2.
generalize H'3; clear H'3.
unfold Rabs in |- *; case (Rcase_abs (p + q - pq)).
intros H'2 H'3; exists (Fopp x); split; auto.
unfold FtoRradix in |- *; rewrite Fopp_correct; auto.
unfold FtoRradix in H'3; rewrite H'3; ring.
split.
apply oppBounded; auto.
rewrite <- H'6; auto.
intros H'2 H'3; exists x; split; auto.
Qed.
intros p q r H' H'0 H'1 H'2.
cut
(2%nat * Rabs (FtoR radix (Fplus radix p q) - FtoR radix r) <=
Float 1%nat (Fexp r))%R;
[ rewrite Fplus_correct; auto with zarith; intros Rl1 | idtac ].
case errorBoundedPlus with (p := p) (q := q) (pq := r); auto.
intros x H'3; elim H'3; intros H'4 H'5; elim H'5; intros H'6 H'7;
clear H'5 H'3.
unfold FtoRradix in H'4; rewrite <- H'4 in Rl1.
2: apply Rle_trans with (Fulp b radix precision r); auto.
2: apply (ClosestUlp b radix precision); auto.
2: rewrite Fplus_correct; auto with zarith.
2: unfold FtoRradix in |- *; apply FulpLe; auto.
2: apply
RoundedModeBounded
with (radix := radix) (P := Closest b radix) (r := (p + q)%R);
auto.
2: apply ClosestRoundedModeP with (precision := precision); auto.
cut (x = 0%R :>R); [ unfold FtoRradix in |- *; intros Eq1 | idtac ].
replace (FtoR radix r) with (FtoR radix r + 0)%R; [ idtac | ring ].
rewrite <- Eq1.
rewrite H'4; ring.
apply (is_Fzero_rep1 radix).
case (Z_zerop (Fnum x)); simpl in |- *; auto.
intros H'3; Contradict Rl1.
apply Rgt_not_le.
red in |- *; apply Rle_lt_trans with (Rabs (FtoR radix x)).
unfold FtoRradix, FtoR in |- *; simpl in |- *; auto.
rewrite Rabs_mult.
apply Rmult_le_compat; auto with real arith.
generalize H'3; case (Fnum x); simpl in |- *; auto with real zarith.
intros H'5; case H'5; auto.
intros p0 H'5; rewrite Rabs_right; auto with real.
replace 1%R with (INR 1); unfold IZR; repeat rewrite <- INR_IPR; auto with real arith.
intros p0 H'5; rewrite Faux.Rabsolu_left1; auto.
unfold IZR; rewrite Ropp_involutive.
repeat rewrite <- INR_IPR; simpl; replace 1%R with (INR 1); auto with real arith.
unfold IZR; repeat rewrite <- INR_IPR; replace 0%R with (- 0%nat)%R; auto with real.
rewrite Rabs_right; auto with real arith.
apply Rle_powerRZ; auto with real arith.
auto with zarith.
apply Rle_ge; cut (1 < radix)%Z; auto with float real zarith.
cut (forall r : R, (2%nat * r)%R = (r + r)%R);
[ intros tmp; rewrite tmp; clear tmp | intros f; simpl in |- *; ring ].
pattern (Rabs (FtoR radix x)) at 1 in |- *;
replace (Rabs (FtoR radix x)) with (Rabs (FtoR radix x) + 0)%R;
[ idtac | ring ].
apply Rplus_lt_compat_l; auto.
case (Rabs_pos (FtoR radix x)); auto.
rewrite <- Fabs_correct; auto with arith.
intros H'5; Contradict H'3.
cut (Fnum (Fabs x) = 0%Z).
unfold Fabs in |- *; simpl in |- *; case (Fnum x); simpl in |- *; auto;
intros; discriminate.
change (is_Fzero (Fabs x)) in |- *.
apply (is_Fzero_rep2 radix); auto with arith.
Qed.
intros p0 q0 r0 H' H'0 H'1 H'2;
case (Zle_or_lt (Fexp r0) (Z.min (Fexp p0) (Fexp q0)));
auto.
intros H'3; Contradict H'2.
apply plusExact1; auto.
Qed.
intros p q r H' H'0 H'1 H'2 H'3.
apply plusExact1; auto.
apply FcanonicBound with (1 := H'0); auto.
case (Zle_or_lt (Fexp p) (Fexp q)); intros Zl1.
rewrite Zmin_le1; auto with zarith.
apply Z.le_trans with (Z.pred (Fexp p)); auto with zarith.
unfold Z.pred in |- *; auto with zarith.
rewrite Zmin_le2; auto with zarith.
case (Zlt_next _ _ Zl1); intros Zl2.
rewrite Zl2 in H'3.
replace (Fexp q) with (Z.pred (Z.succ (Fexp q))); auto with zarith;
unfold Z.pred, Z.succ in |- *; ring.
case H'0; clear H'0; intros H'0.
absurd (r < Float (nNormMin radix precision) (Z.pred (Fexp p)))%R.
apply Rle_not_lt; auto.
unfold FtoRradix in |- *;
apply
(ClosestMonotone b radix
(Float (nNormMin radix precision) (Z.pred (Fexp p))) (
p + q)%R); auto; auto.
cut (Float (nNormMin radix precision) (Fexp p) <= p)%R;
[ intros Eq1 | idtac ].
case (Rle_or_lt 0 q); intros Rl1.
apply Rlt_le_trans with (FtoRradix p).
apply
Rlt_le_trans with (FtoRradix (Float (nNormMin radix precision) (Fexp p)));
auto.
unfold FtoRradix, FtoR in |- *; simpl in |- *; auto.
apply Rmult_lt_compat_l; auto with real arith.
replace 0%R with (IZR 0%nat); auto with real; auto with real float arith.
apply Rlt_IZR; apply nNormPos; auto with zarith.
unfold Z.pred in |- *; auto with real float zarith arith.
pattern (FtoRradix p) at 1 in |- *; replace (FtoRradix p) with (p + 0)%R;
auto with real.
apply Rplus_lt_reg_l with (r := (- q)%R); auto.
replace (- q + (p + q))%R with (FtoRradix p); [ idtac | ring ].
apply
Rlt_le_trans with (FtoRradix (Float (nNormMin radix precision) (Fexp p)));
auto.
apply
Rlt_le_trans
with (2%nat * Float (nNormMin radix precision) (Z.pred (Fexp p)))%R;
auto.
cut (forall r : R, (2%nat * r)%R = (r + r)%R);
[ intros tmp; rewrite tmp; clear tmp | intros; simpl in |- *; ring ].
rewrite (Rplus_comm (- q)).
apply Rplus_lt_compat_l.
rewrite <- Faux.Rabsolu_left1; auto.
rewrite <- (Fabs_correct radix); auto with arith.
unfold FtoRradix in |- *; apply maxMaxBis with (b := b); auto with zarith.
apply Rlt_le; auto.
apply
Rle_trans with (radix * Float (nNormMin radix precision) (Z.pred (Fexp p)))%R.
apply Rmult_le_compat_r; auto.
apply (LeFnumZERO radix); simpl in |- *; auto with arith.
apply Zlt_le_weak; apply nNormPos; auto with zarith.
rewrite INR_IZR_INZ; apply Rle_IZR; simpl in |- *; cut (1 < radix)%Z;
auto with real zarith.
pattern (Fexp p) at 2 in |- *; replace (Fexp p) with (Z.succ (Z.pred (Fexp p)));
[ idtac | unfold Z.succ, Z.pred in |- *; ring ].
unfold FtoRradix, FtoR in |- *; simpl in |- *.
rewrite powerRZ_Zs; auto with real zarith.
repeat rewrite <- Rmult_assoc.
rewrite (Rmult_comm radix); auto with real.
unfold FtoRradix, FtoR in |- *; simpl in |- *; auto.
apply Rmult_le_compat_r; auto with real zarith.
apply Rle_IZR.
rewrite <- (Z.abs_eq (Fnum p)); auto with zarith.
apply pNormal_absolu_min with (b := b); auto with arith.
unfold FtoRradix, FtoR in |- *; simpl in |- *; auto.
apply (LeR0Fnum radix); auto with arith.
apply (RoundedModeProjectorIdem b radix (Closest b radix)); auto.
apply ClosestRoundedModeP with (precision := precision); auto.
repeat split; simpl in |- *.
rewrite Z.abs_eq; auto with zarith.
apply ZltNormMinVnum; auto with arith.
apply Zlt_le_weak; apply nNormPos; auto with zarith.
apply Z.le_trans with (Fexp q); auto with float zarith.
case (Rle_or_lt 0 r); intros Rl1.
rewrite <- (Rabs_right r); auto with real.
rewrite <- (Fabs_correct radix); auto with arith.
unfold FtoRradix in |- *; apply maxMaxBis with (b := b); auto with zarith.
apply
RoundedModeBounded
with (radix := radix) (P := Closest b radix) (r := (p + q)%R);
auto.
apply ClosestRoundedModeP with (precision := precision); auto with real.
apply Rlt_le_trans with 0%R; auto.
apply (LeFnumZERO radix); simpl in |- *; auto with arith.
apply Zlt_le_weak; apply nNormPos; auto with zarith.
absurd (- dExp b <= Fexp q)%Z; auto with float.
apply Zlt_not_le.
case H'0; intros Z1 (Z2, Z3); rewrite <- Z2; auto with zarith.
Qed.
intros p q r H' H'0 H'1 H'2.
case (Rle_or_lt 0 p); intros Rl1.
apply plusExact2Aux; auto.
replace (p + q)%R with (- (Fopp p + Fopp q))%R.
rewrite <- (plusExact2Aux (Fopp p) (Fopp q) (Fopp r)); auto.
unfold FtoRradix in |- *; rewrite Fopp_correct; ring.
unfold FtoRradix in |- *; rewrite Fopp_correct.
apply Rlt_le; replace 0%R with (-0)%R; auto with real.
apply FcanonicFopp; auto with arith.
apply oppBounded; auto.
replace (Fopp p + Fopp q)%R with (- (p + q))%R.
apply ClosestOpp; auto.
unfold FtoRradix in |- *; repeat rewrite Fopp_correct; ring.
unfold FtoRradix in |- *; repeat rewrite Fopp_correct; ring.
Qed.
intros p q r H' H'0 H'1 H'2.
cut (r = FtoRradix (Fzero (- dExp b)) :>R);
[ intros Eq1; rewrite Eq1
| rewrite H'2; apply sym_eq; unfold FtoRradix in |- *; apply FzeroisZero ].
apply plusExact1; auto.
apply (ClosestCompatible b radix (p + q)%R (p + q)%R r); auto.
apply FboundedFzero; auto.
simpl in |- *; auto.
unfold Z.min in |- *; case (Fexp p ?= Fexp q)%Z; auto with float.
Qed.

Theorem plusErrorBound1 :
forall p q r : float,
Fbounded b p ->
Fbounded b q ->
Closest b radix (p + q) r ->
~ is_Fzero r ->
(Rabs (r - (p + q)) < Rabs r * / 2%nat * (radix * / pPred (vNum b)))%R.