Commit 3aac99c3 authored by BOLDO Sylvie's avatar BOLDO Sylvie

FTS, TS, presque tout underflow ErrFMA

parent 603d5937
......@@ -367,6 +367,67 @@ rewrite inj_abs; simpl; omega.
apply pff_round_NE_is_round.
Qed.
Lemma FloatFexp_gt: forall e f, Fbounded b f ->
(bpow beta (e+p) <= Rabs (FtoR beta f))%R ->
(e < Float.Fexp f)%Z.
Proof.
intros e f Ff H1.
apply lt_bpow with beta.
apply Rmult_lt_reg_r with (bpow beta p).
apply bpow_gt_0.
rewrite <- bpow_plus.
apply Rle_lt_trans with (1:=H1).
rewrite <- Fabs_correct.
2: apply radix_gt_0.
unfold Fabs, FtoR; simpl; rewrite Rmult_comm.
rewrite bpow_powerRZ, Z2R_IZR.
apply Rmult_lt_compat_l.
apply powerRZ_lt.
replace 0%R with (IZR 0) by reflexivity.
apply IZR_lt, radix_gt_0.
destruct Ff as (T1,T2).
rewrite bpow_powerRZ.
rewrite Z2R_IZR.
replace p with (Z.of_nat (Zabs_nat p)).
rewrite <- Zpower_nat_Z_powerRZ.
apply IZR_lt.
now rewrite <- pGivesBound.
rewrite inj_abs; omega.
Qed.
Lemma CanonicGeNormal: forall f, Fcanonic beta b f ->
(bpow beta (-dExp b+p-1) <= Rabs (FtoR beta f))%R ->
(Fnormal beta b f)%Z.
Proof.
intros f Cf H1.
case Cf; trivial.
intros (H2,(H3,H4)).
contradict H1; apply Rlt_not_le.
rewrite <- Fabs_correct.
2: apply radix_gt_0.
unfold FtoR, Fabs; simpl.
rewrite H3, <- (Z2R_IZR beta), <- bpow_powerRZ.
apply Rlt_le_trans with (bpow beta (p-1)*bpow beta (-dExp b))%R.
2: rewrite <- bpow_plus.
2: right; f_equal; ring.
apply Rmult_lt_compat_r.
apply bpow_gt_0.
apply Rmult_lt_reg_l with (Z2R beta).
change 0%R with (Z2R 0); apply Z2R_lt, radix_gt_0.
rewrite Z2R_IZR, <- mult_IZR.
apply Rlt_le_trans with (IZR (Z.pos (vNum b))).
apply IZR_lt.
rewrite <- (Zabs_eq beta).
now rewrite <- Zabs_Zmult.
apply Zlt_le_weak, radix_gt_0.
rewrite pGivesBound.
rewrite <- Z2R_IZR, Z2R_Zpower_nat.
rewrite <- Z2R_IZR, <- bpow_1.
rewrite <- bpow_plus.
apply bpow_le.
rewrite inj_abs; omega.
Qed.
End Equiv.
Section Equiv_instanc.
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