Section Fnormalized_Def.
Variable radix : Z.
Hypothesis radixMoreThanOne : (1 < radix)%Z.
Let radixMoreThanZERO := Zlt_1_O _ (Zlt_le_weak _ _ radixMoreThanOne).
Hint Resolve radixMoreThanZERO: zarith.
Let FtoRradix := FtoR radix.
Coercion FtoRradix : float >-> R.
Variable b : Fbound.
Definition Fnormal (p : float) :=
Fbounded b p /\ (Zpos (vNum b) <= Z.abs (radix * Fnum p))%Z.
intros p H; case H; auto.
Qed.
intros p H; case H; auto.
Qed.
Hint Resolve FnormalBounded FnormalBound: float.
unfold is_Fzero in |- *; intros p H; red in |- *; intros H1.
case H; rewrite H1.
replace (Z.abs (radix * 0)) with 0%Z; auto with zarith.
rewrite Zmult_comm; simpl in |- *; auto.
Qed.
intros p H; split; auto with float.
replace (Z.abs (radix * Fnum (Fopp p))) with (Z.abs (radix * Fnum p));
auto with float.
case p; simpl in |- *; auto with zarith.
intros Fnum1 Fexp1; rewrite <- Zopp_mult_distr_r; apply sym_equal;
apply Zabs_Zopp.
Qed.
intros p; case p; intros a e H; split; auto with float.
simpl in |- *; case H; intros H1 H2; simpl in |- *; auto.
rewrite <- (Z.abs_eq radix); auto with zarith.
rewrite <- Zabs_Zmult.
rewrite (fun x => Z.abs_eq (Z.abs x)); auto with float zarith.
Qed.
Definition pPred x := Z.pred (Zpos x).
intros p z H H0; unfold FtoRradix in |- *.
rewrite <-
(FshiftCorrect _ radixMoreThanOne (Z.abs_nat (z - Fexp p))
(Float (pPred (vNum b)) z)).
unfold FtoR, Fabs in |- *; simpl in |- *; auto with zarith.
rewrite Rmult_IZR; rewrite Zpower_nat_Z_powerRZ; auto with zarith.
repeat rewrite inj_abs; auto with zarith.
replace (z - (z - Fexp p))%Z with (Fexp p); [ idtac | ring ].
rewrite Rmult_assoc; rewrite <- powerRZ_add; auto with real zarith.
replace (z - Fexp p + Fexp p)%Z with z; [ idtac | ring ].
apply Rle_trans with (pPred (vNum b) * powerRZ radix (Fexp p))%R.
apply Rle_monotone_exp; auto with zarith; repeat rewrite Rmult_IZR;
apply Rle_IZR; unfold pPred in |- *; apply Zle_Zpred;
auto with float real zarith.
apply Rmult_le_compat_l; auto with real zarith.
replace 0%R with (IZR 0); auto with real; apply Rle_IZR; unfold pPred in |- *;
apply Zle_Zpred; auto with float zarith.
apply Rle_powerRZ; auto with float real zarith.
Qed.
intros p H'; unfold FtoRradix, FtoR in |- *; simpl in |- *.
pattern (Fexp p) at 2 in |- *; replace (Fexp p) with (Z.succ (Z.pred (Fexp p)));
[ rewrite powerRZ_Zs; auto with real zarith
| unfold Z.succ, Z.pred in |- *; ring ].
repeat rewrite <- Rmult_assoc.
apply Rmult_lt_compat_r; auto with real arith.
rewrite <- Rmult_IZR; apply Rlt_IZR.
unfold pPred in |- *; cut (Zpos (vNum b) <= Z.abs (Fnum p) * radix)%Z;
auto with zarith.
rewrite <- (Z.abs_eq radix); auto with float zarith; rewrite <- Zabs_Zmult;
rewrite Zmult_comm; auto with float real zarith.
Qed.
Definition Fsubnormal (p : float) :=
Fbounded b p /\
Fexp p = (- dExp b)%Z /\ (Z.abs (radix * Fnum p) < Zpos (vNum b))%Z.
intros p H; case H; auto.
Qed.
intros p H; case H; auto.
intros H1 H2; case H2; auto.
Qed.
intros p H; case H; auto.
intros H1 H2; case H2; auto.
Qed.
Hint Resolve FsubnormalFbounded FsubnormalBound FsubnormalFexp: float.
intros p H'; repeat split; simpl in |- *; auto with zarith float.
rewrite Zabs_Zopp; auto with float.
rewrite <- Zopp_mult_distr_r; rewrite Zabs_Zopp; auto with float.
Qed.
intros p; case p; intros a e H; split; auto with float.
simpl in |- *; split; auto with float.
case H; intros H1 (H2, H3); auto.
rewrite <- (Z.abs_eq radix); auto with zarith.
rewrite <- Zabs_Zmult.
rewrite (fun x => Z.abs_eq (Z.abs x)); auto with float zarith.
case H; intros H1 (H2, H3); auto.
Qed.
intros p q H' H'0 H'1.
apply FtoREqInv2 with (radix := radix); auto.
generalize H' H'0; unfold Fsubnormal in |- *; auto with zarith.
Qed.
intros p q H' H'0 H'1.
apply Rlt_Fexp_eq_Zlt with (radix := radix); auto with zarith.
apply trans_equal with (- dExp b)%Z.
case H'; auto.
intros H1 H2; case H2; auto.
apply sym_equal; case H'0; auto.
intros H1 H2; case H2; auto.
Qed.
intros p q H' H'0 H'1.
case (Rtotal_order p q); auto; intros Test; case Test; clear Test;
intros Test; Contradict H'1.
unfold FtoRradix in Test; rewrite sameExpEq with (2 := Test); auto.
auto with zarith.
apply trans_equal with (- dExp b)%Z.
case H'; auto.
intros H1 H2; case H2; auto.
apply sym_equal; case H'0.
intros H1 H2; case H2; auto.
apply Zle_not_lt.
apply Zlt_le_weak.
apply FsubnormalLt; auto.
Qed.
Definition Fcanonic (a : float) := Fnormal a \/ Fsubnormal a.
intros p H; case H; auto with float.
Qed.
Hint Resolve FcanonicBound: float.
auto with float.
Qed.
intros p H'; case H'; intros H'1.
left; apply FnormalFop; auto.
right; apply FsubnormFopp; auto.
Qed.
intros p H'; case H'; clear H'; auto with float.
intros H; left; auto with float.
apply FnormalFabs; auto.
intros H; right; auto with float.
apply FsubnormFabs; auto.
Qed.
intros p; red in |- *; intros H; elim H; intros H0 H1; clear H.
absurd (Z.abs (radix * Fnum p) < Zpos (vNum b))%Z;
auto with float zarith.
Qed.
intros.
replace (Rabs x) with (FtoR radix (Fabs x)).
unfold FtoRradix in |- *.
apply maxMax with (b := b); auto with *.
unfold FtoRradix in |- *.
apply Fabs_correct; auto with *.
Qed.
(* What depends of the precision *)
Variable precision : nat.
Hypothesis precisionNotZero : precision <> 0.
Hypothesis pGivesBound : Zpos (vNum b) = Zpower_nat radix precision.
intros p H'.
case (Zle_lt_or_eq (Z.succ (Fnum p)) (Zpos (vNum b))); auto with float.
case (Zle_or_lt 0 (Fnum p)); intros H1.
rewrite <- (Z.abs_eq (Fnum p)); auto with float zarith.
apply Z.le_trans with 0%Z; auto with zarith.
intros H'0; exists (Float (Z.succ (Fnum p)) (Fexp p)); split; auto with float.
repeat split; simpl in |- *; auto with float.
case (Zle_or_lt 0 (Fnum p)); intros H1; auto with zarith.
rewrite Z.abs_eq; auto with zarith.
apply Z.lt_trans with (Z.abs (Fnum p)); auto with float zarith.
repeat rewrite Zabs_eq_opp; auto with zarith.
intros H'0;
exists (Float (Zpower_nat radix (pred precision)) (Z.succ (Fexp p)));
split; auto.
repeat split; simpl in |- *; auto with zarith arith float.
rewrite pGivesBound.
rewrite Z.abs_eq; auto with zarith.
rewrite H'0; rewrite pGivesBound.
pattern precision at 2 in |- *; replace precision with (1 + pred precision).
rewrite Zpower_nat_is_exp.
rewrite Zpower_nat_1.
unfold FtoRradix, FtoR in |- *; simpl in |- *.
rewrite powerRZ_Zs; auto with real zarith.
rewrite Rmult_IZR; ring.
generalize precisionNotZero; case precision; simpl in |- *; auto with arith.
intros H'1; case H'1; auto.
Qed.
apply digitInv; auto.
rewrite pGivesBound.
rewrite Z.abs_eq; auto with zarith.
rewrite Z.abs_eq; auto with zarith.
Qed.
apply digitInv; auto.
rewrite pGivesBound.
rewrite Z.abs_eq; auto with zarith.
rewrite Z.abs_eq; auto with zarith.
rewrite pGivesBound; auto with zarith.
Qed.
intros n H'.
rewrite <- (Z.abs_eq (Zpos (vNum b))); auto with zarith.
apply digit_anti_monotone_lt with (n := radix); auto.
rewrite digitVNumiSPrecision; auto with arith.
Qed.
intros p H; unfold Fdigit in |- *.
rewrite <- digitPredVNumiSPrecision.
apply digit_monotone; auto with zarith.
rewrite (fun x => Z.abs_eq (Z.pred x)); auto with float zarith.
Qed.
intros p H; apply vNumPrecision; auto.
Qed.
intros z H'; repeat (split; simpl in |- *; auto with zarith).
rewrite pGivesBound; auto.
apply Z.lt_le_trans with (Zpower_nat radix 1); auto with zarith.
rewrite Zpower_nat_1; auto with zarith.
Qed.
intros z m H' H'0 H'1; case (Zle_lt_or_eq _ _ H'0); intros H'2.
exists (Float m z); split; auto with zarith.
repeat split; simpl in |- *; auto with zarith.
rewrite Z.abs_eq; auto; rewrite pGivesBound; auto.
case (FboundNext (Float (Z.pred (Zpos (vNum b))) z)); auto with float.
intros f' (H1, H2); exists f'; split; auto.
rewrite H2; rewrite pGivesBound.
unfold FtoRradix, FtoR in |- *; simpl in |- *; auto.
rewrite <- Zsucc_pred; rewrite <- H'2; auto; ring.
Qed.
intros z m H H0.
case (Zle_or_lt 0 m); intros H1.
case (FboundedMboundPos z (Z.abs m)); auto; try rewrite Z.abs_eq; auto.
intros f (H2, H3); exists f; split; auto.
case (FboundedMboundPos z (Z.abs m)); auto; try rewrite Zabs_eq_opp;
auto with zarith.
intros f (H2, H3); exists (Fopp f); split; auto with float.
rewrite (Fopp_correct radix); auto with arith; fold FtoRradix in |- *;
rewrite H3.
rewrite Ropp_Ropp_IZR; ring.
Qed.
intros p H; apply le_antisym; auto with float.
apply pGivesDigit; auto with float.
apply le_S_n.
rewrite <- digitVNumiSPrecision.
unfold Fdigit in |- *.
replace (S (digit radix (Fnum p))) with (digit radix (Fnum p) + 1).
rewrite <- digitAdd; auto with zarith.
apply digit_monotone; auto with float.
rewrite (fun x => Z.abs_eq (Zpos x)); auto with float zarith.
rewrite Zmult_comm; rewrite Zpower_nat_1; auto with float zarith.
red in |- *; intros H1; case H.
intros H0 H2; Contradict H2; rewrite H1.
replace (Z.abs (radix * 0)) with 0%Z; auto with zarith.
rewrite Zmult_comm; simpl in |- *; auto.
rewrite plus_comm; simpl in |- *; auto.
Qed.
Hint Resolve FnormalPrecision: float.
intros p q H' H'0 H'1.
apply (FdigitEq radix); auto.
apply FnormalNotZero; auto.
apply trans_equal with (y := precision); auto with float.
apply sym_equal; auto with float.
Qed.
intros p q H' H'0 H'1 H'2.
case (Zle_or_lt (Fexp q) (Fexp p)); auto.
intros H'3; right.
case (Zle_lt_or_eq _ _ H'3); intros H'4.
2: split; auto.
2: apply Rlt_Fexp_eq_Zlt with (radix := radix); auto with zarith.
absurd (Fnum (Fshift radix (Z.abs_nat (Fexp p - Fexp q)) p) < Fnum q)%Z; auto.
2: apply Rlt_Fexp_eq_Zlt with (radix := radix); auto with zarith.
2: unfold FtoRradix in |- *; rewrite FshiftCorrect; auto.
2: unfold Fshift in |- *; simpl in |- *; auto with zarith.
2: replace (Z_of_nat (Z.abs_nat (Fexp p - Fexp q))) with (Fexp p - Fexp q)%Z;
auto with zarith.
2: cut (0 < Fexp p - Fexp q)%Z; auto with zarith.
2: case (Fexp p - Fexp q)%Z; simpl in |- *; auto with zarith.
2: intros p0; rewrite (inject_nat_convert (Zpos p0)); auto with arith.
2: intros p0 H'5; discriminate.
red in |- *; intros H'5.
absurd
(Fdigit radix (Fshift radix (Z.abs_nat (Fexp p - Fexp q)) p) <=
Fdigit radix q); auto with arith.
rewrite FshiftFdigit; auto with arith.
replace (Fdigit radix p) with precision.
replace (Fdigit radix q) with precision; auto with zarith.
cut (0 < Fexp p - Fexp q)%Z; auto with zarith.
case (Fexp p - Fexp q)%Z; simpl in |- *; auto with zarith.
intros p0 H'6; generalize (convert_not_O p0); auto with zarith.
intros p0 H'6; discriminate.
apply sym_equal; auto with float.
apply sym_equal; auto with float.
apply FnormalNotZero; auto with arith.
unfold Fdigit in |- *; apply digit_monotone; auto with arith.
repeat rewrite Z.abs_eq; auto with zarith.
apply LeR0Fnum with (radix := radix); auto with zarith.
apply Rle_trans with (r2 := FtoRradix p); auto with real.
apply LeR0Fnum with (radix := radix); auto with zarith.
unfold FtoRradix in |- *; rewrite FshiftCorrect; auto.
Qed.
intros p q H' H'0 H'1 H'2.
cut
((Fexp (Fopp q) < Fexp (Fopp p))%Z \/
Fexp (Fopp q) = Fexp (Fopp p) /\ (Fnum (Fopp q) < Fnum (Fopp p))%Z).
simpl in |- *.
intros H'3; elim H'3; clear H'3; intros H'3;
[ idtac | elim H'3; clear H'3; intros H'3 H'4 ]; auto;
right; split; auto with zarith.
apply FnormalLtPos; try apply FnormalFop; auto; unfold FtoRradix in |- *;
repeat rewrite Fopp_correct; replace 0%R with (-0)%R;
auto with real.
Qed.
Definition nNormMin := Zpower_nat radix (pred precision).
unfold nNormMin in |- *; auto with zarith.
Qed.
unfold nNormMin, Fdigit in |- *; simpl in |- *; apply digitInv;
auto with zarith arith.
rewrite Z.abs_eq; auto with zarith.
Qed.
rewrite pGivesBound.
unfold nNormMin in |- *; simpl in |- *; auto with zarith arith.
Qed.
Hint Resolve nNrMMimLevNum: arith.
Definition firstNormalPos := Float nNormMin (- dExp b).
repeat split; unfold firstNormalPos in |- *; simpl in |- *; auto with zarith.
rewrite pGivesBound.
rewrite Z.abs_eq; auto with zarith.
unfold nNormMin in |- *; simpl in |- *; auto with zarith arith.
apply Zlt_le_weak; auto with zarith.
apply nNormPos.
rewrite pGivesBound.
replace precision with (pred precision + 1).
rewrite Zpower_nat_is_exp; auto with zarith.
rewrite Zpower_nat_1; auto with zarith.
rewrite (fun x => Zmult_comm x radix); unfold nNormMin in |- *;
auto with zarith.
unfold nNormMin in |- *; auto with zarith.
Qed.
intros p H; apply Zmult_le_reg_r with (p := radix); auto with zarith.
unfold nNormMin in |- *.
pattern radix at 2 in |- *; rewrite <- (Zpower_nat_1 radix).
rewrite <- Zpower_nat_is_exp; auto with zarith.
replace (pred precision + 1) with precision.
rewrite <- pGivesBound; auto with float.
rewrite <- (Z.abs_eq radix); auto with zarith.
rewrite <- Zabs_Zmult; rewrite Zmult_comm; auto with float.
generalize precisionNotZero; case precision; simpl in |- *;
try (intros tmp; Contradict tmp; auto; fail); intros;
rewrite plus_comm; simpl in |- *; auto.
Qed.
intros p z H' H'0;
apply
Rlt_le_trans with (FtoR radix (Float (Zpos (vNum b)) (Z.pred z))).
unfold FtoRradix in |- *; apply maxMax; auto with zarith;
unfold Z.pred in |- *; auto with zarith.
unfold FtoRradix, FtoR, nNormMin in |- *; simpl in |- *.
pattern z at 2 in |- *; replace z with (Z.succ (Z.pred z));
[ rewrite powerRZ_Zs; auto with real zarith
| unfold Z.succ, Z.pred in |- *; ring ].
rewrite <- Rmult_assoc.
apply Rmult_le_compat_r; auto with real arith.
pattern radix at 2 in |- *; rewrite <- (Zpower_nat_1 radix).
rewrite <- Rmult_IZR.
rewrite <- Zpower_nat_is_exp.
replace (pred precision + 1) with precision.
replace (INR (nat_of_P (vNum b))) with (IZR (Zpos (vNum b))).
rewrite pGivesBound; auto with real.
simpl; unfold IZR; rewrite <- INR_IPR; auto.
generalize precisionNotZero; case precision; simpl in |- *; auto with arith.
intros H'1; Contradict H'1; auto.
intros; rewrite plus_comm; simpl in |- *; auto.
Qed.
intros p H' H'0.
case (Rle_or_lt firstNormalPos p); intros Lt0; auto with real.
case (FnormalLtPos p firstNormalPos); auto.
apply firstNormalPosNormal.
intros H'1; Contradict H'1; unfold firstNormalPos in |- *; simpl in |- *.
apply Zle_not_lt; auto with float.
intros H'1; elim H'1; intros H'2 H'3; Contradict H'3.
unfold firstNormalPos in |- *; simpl in |- *.
apply Zle_not_lt.
rewrite <- (Z.abs_eq (Fnum p)); auto with float zarith.
apply pNormal_absolu_min; auto.
apply LeR0Fnum with (radix := radix); auto with arith.
Qed.
intros p H' H'0.
rewrite <- (Ropp_involutive p); unfold FtoRradix in |- *;
repeat rewrite Fopp_correct.
apply Ropp_le_contravar; rewrite <- Fopp_correct.
apply FnormalLtFirstNormalPos.
apply FnormalFop; auto.
replace 0%R with (-0)%R; unfold FtoRradix in |- *; try rewrite Fopp_correct;
auto with real.
Qed.

Theorem FsubnormalDigit :
forall p : float, Fsubnormal p -> Fdigit radix p < precision.