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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * Copyright INRIA, CNRS and contributors *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
Require Import ZArith Int63 SpecFloat PrimFloat FloatOps FloatAxioms.
Require Import Psatz.
(** * Support results involving frexp and ldexp *)
Lemma shift_value : shift = (2*emax + prec)%Z.
reflexivity.
Qed.
Theorem frexp_spec : forall f, let (m,e) := frexp f in (Prim2SF m, e) = SFfrexp prec emax (Prim2SF f).
intro.
unfold frexp.
case_eq (frshiftexp f).
intros.
assert (H' := frshiftexp_spec f).
now rewrite H in H'.
Qed.
Theorem ldexp_spec : forall f e, Prim2SF (ldexp f e) = SFldexp prec emax (Prim2SF f) e.
intros.
unfold ldexp.
rewrite (ldshiftexp_spec f _).
assert (Hv := Prim2SF_valid f).
destruct (Prim2SF f); auto.
unfold SFldexp.
unfold binary_round.
assert (Hmod_elim : forall e, (φ (of_Z (Z.max (Z.min e (emax - emin)) (emin - emax - 1) + shift))%int63 - shift = Z.max (Z.min e (emax - emin)) (emin - emax - 1))%Z).
{
intro e1.
rewrite of_Z_spec, shift_value.
unfold wB, size; simpl.
unfold Z.pow_pos; simpl.
set (n := Z.max (Z.min _ _) _).
set (wB := 9223372036854775808%Z). (* Z.pow_pos 2 63 *)
assert (-2099 <= n <= 2098)%Z by (unfold n; lia).
rewrite Z.mod_small by (unfold wB; lia).
now rewrite Z.add_simpl_r.
}
rewrite Hmod_elim.
clear Hmod_elim.
revert Hv.
unfold valid_binary, bounded, canonical_mantissa.
unfold fexp.
rewrite Bool.andb_true_iff.
intro H'.
destruct H' as (H1,H2).
apply Zeq_bool_eq in H1.
apply Z.max_case_strong.
apply Z.min_case_strong.
- reflexivity.
- intros He _.
destruct (Z.max_spec (Z.pos (digits2_pos m) + e0 - prec) emin) as [ (H, Hm) | (H, Hm) ].
+ rewrite Hm in H1.
rewrite <- H1.
rewrite !Z.max_l by (revert He; unfold emax, emin, prec; lia).
replace (emin + _)%Z with emax by ring.
unfold shl_align.
rewrite <- H1 in H.
replace (Z.pos _ + _ - _ - _)%Z with (Z.pos (digits2_pos m) - prec)%Z by ring.
remember (Zpos _ - _)%Z as z'.
destruct z' ; [ lia | lia | ].
unfold binary_round_aux.
unfold shr_fexp.
unfold fexp.
unfold Zdigits2.
unfold shr_record_of_loc, shr.
rewrite !Z.max_l by (revert H He; unfold emax, emin, prec; lia).
replace (_ - _)%Z with (Z.pos (digits2_pos (shift_pos p m)) - prec)%Z by ring.
assert (Hs : (Z.pos (digits2_pos (shift_pos p m)) <= prec)%Z).
{
assert (H' : forall p p', digits2_pos (shift_pos p p') = (digits2_pos p' + p)%positive).
{
induction p0.
intro p'.
simpl.
rewrite IHp0.
rewrite IHp0.
lia.
intro p'.
simpl.
rewrite IHp0.
rewrite IHp0.
lia.
intro p'.
simpl.
lia.
}
rewrite H'.
lia.
}
replace (Z.pos (digits2_pos m) + (emin + e) - prec - (emin + e))%Z with (Z.neg p) by lia.
unfold shr_m, loc_of_shr_record.
unfold round_nearest_even.
remember (Z.pos (digits2_pos (shift_pos p m)) - prec)%Z as ds.
destruct ds.
* rewrite Z.max_l by (revert He; unfold emax, emin, prec; lia).
replace (_ - _)%Z with Z0 by lia.
replace (_ <=? _)%Z with false by (symmetry; rewrite Z.leb_gt; lia).
rewrite Z.max_l by (revert He; unfold emax, emin, prec; lia).
replace (_ - _)%Z with Z0 by lia.
rewrite Z.max_l by (revert He; unfold emax, emin, prec; lia).
replace (_ - _)%Z with Z0 by lia.
replace (_ <=? _)%Z with false by (symmetry; rewrite Z.leb_gt; lia).
reflexivity.
* exfalso; lia.
* rewrite Z.max_l by (revert He; unfold emax, emin, prec; lia).
replace (_ - _)%Z with (Zneg p0) by lia.
replace (_ <=? _)%Z with false by (symmetry; rewrite Z.leb_gt; lia).
rewrite Z.max_l by (revert He; unfold emax, emin, prec; lia).
replace (_ - _)%Z with (Zneg p0) by lia.
rewrite Z.max_l by (revert He; unfold emax, emin, prec; lia).
replace (_ - _)%Z with (Zneg p0) by lia.
replace (_ <=? _)%Z with false by (symmetry; rewrite Z.leb_gt; lia).
reflexivity.
+ rewrite !Z.max_l by (revert H He; unfold emax, emin, prec; lia).
rewrite Hm in H1.
clear Hm.
replace (Zpos _ + _ - _)%Z with (e0 + (emax - emin))%Z by (rewrite <- H1 at 1; ring).
replace (Zpos _ + _ - _)%Z with (e0 + e)%Z by (rewrite <- H1 at 1; ring).
unfold shl_align.
replace (_ - _)%Z with Z0 by ring.
replace (e0 + e - _)%Z with Z0 by ring.
unfold binary_round_aux.
unfold shr_fexp.
unfold fexp.
unfold Zdigits2.
rewrite !Z.max_l by (revert H He; unfold emax, emin, prec; lia).
unfold shr_record_of_loc.
unfold shr.
unfold Zdigits2.
replace (Zpos _ + _ - _ - _)%Z with Z0 by lia.
unfold shr_m.
unfold loc_of_shr_record.
unfold round_nearest_even.
rewrite Z.max_l by (revert H He; unfold emax, emin, prec; lia).
replace (Zpos _ + _ - _ - _)%Z with Z0 by lia.
replace (_ <=? _)%Z with false by (symmetry; rewrite Z.leb_gt; lia).
replace (Zpos _ + _ - _ - _)%Z with Z0 by lia.
rewrite Z.max_l by (revert H He; unfold emax, emin, prec; lia).
replace (Zpos _ + _ - _ - _)%Z with Z0 by lia.
replace (_ <=? _)%Z with false by (symmetry; rewrite Z.leb_gt; lia).
reflexivity.
- rewrite Z.min_le_iff.
intro H.
destruct H as [ He | Habs ]; [ | revert Habs; now unfold emin, emax ].
unfold shl_align.
assert (Hprec : (Z.pos (digits2_pos m) <= prec)%Z).
{
destruct (Z.max_spec (Z.pos (digits2_pos m) + e0 - prec) emin) as [ (Hpi, Hpe) | (Hpi, Hpe) ]; rewrite Hpe in H1; lia.
}
assert (Hshr : forall p s, Zdigits2 (shr_m (iter_pos shr_1 p s)) = Z.max Z0 (Zdigits2 (shr_m s) - Z.pos p)%Z).
{
assert (Hshr1 : forall s, Zdigits2 (shr_m (shr_1 s)) = Z.max 0 (Zdigits2 (shr_m s) - 1)%Z).
{
intro s0.
destruct s0.
unfold shr_1.
destruct shr_m; try (simpl; lia).
- destruct p; unfold Zdigits2, shr_m, digits2_pos; lia.
- destruct p; unfold Zdigits2, shr_m, digits2_pos; lia.
}
induction p.
simpl.
intro s0.
do 2 rewrite IHp.
rewrite Hshr1.
lia.
intros.
simpl.
do 2 rewrite IHp.
lia.
apply Hshr1.
}
assert (Hd0 : forall z, Zdigits2 z = 0%Z -> z = 0%Z).
{
intro z.
unfold Zdigits2.
now destruct z.
}
assert (Hshr_p0 : forall p0, (prec < Z.pos p0)%Z -> shr_m (iter_pos shr_1 p0 {| shr_m := Z.pos m; shr_r := false; shr_s := false |}) = Z0).
{
intros p0 Hp0.
apply Hd0.
rewrite Hshr.
rewrite Z.max_l; [ reflexivity | ].
unfold shr_m.
unfold Zdigits2.
lia.
}
assert (Hshr_p0_r : forall p0, (prec < Z.pos p0)%Z -> shr_r (iter_pos shr_1 p0 {| shr_m := Z.pos m; shr_r := false; shr_s := false |}) = false).
{
intros p0 Hp0.
assert (Hshr_p0m1 : shr_m (iter_pos shr_1 (p0-1) {| shr_m := Z.pos m; shr_r := false; shr_s := false |}) = Z0).
{
apply Hd0.
rewrite Hshr.
rewrite Z.max_l; [ reflexivity | ].
unfold shr_m.
unfold Zdigits2.
lia.
}
assert (Hiter_pos : forall A (f : A -> A) p e, iter_pos f (p + 1) e = f (iter_pos f p e)).
{
assert (Hiter_pos' : forall A (f : A -> A) p e, iter_pos f p (f e) = f (iter_pos f p e)).
{
intros A f'.
induction p.
intro e'.
simpl.
now do 2 rewrite IHp.
intro e'.
simpl.
now do 2 rewrite IHp.
intro e'.
now simpl.
}
intros A f'.
induction p.
intros.
simpl.
rewrite <- Pos.add_1_r.
do 2 rewrite IHp.
now do 3 rewrite Hiter_pos'.
intros.
simpl.
now do 2 rewrite Hiter_pos'.
intros.
now simpl.
}
replace p0 with (p0 - 1 + 1)%positive.
rewrite Hiter_pos.
unfold shr_1 at 1.
remember (iter_pos _ _ _) as shr_p0m1.
destruct shr_p0m1.
unfold SpecFloat.shr_m in Hshr_p0m1.
now rewrite Hshr_p0m1.
rewrite Pos.add_1_r.
rewrite Pos.sub_1_r.
apply Pos.succ_pred.
lia.
}
rewrite Z.leb_le in H2.
destruct (Z.max_spec (Z.pos (digits2_pos m) + (e0 + (emin - emax - 1)) - prec) emin) as [ (H, Hm) | (H, Hm) ].
+ rewrite Hm.
replace (_ - _)%Z with (emax - e0 + 1)%Z by ring.
remember (emax - e0 + 1)%Z as z'.
destruct z'; [ exfalso; lia | | exfalso; lia ].
unfold binary_round_aux.
unfold shr_fexp, fexp.
unfold shr, shr_record_of_loc.
unfold Zdigits2.
rewrite Hm.
replace (_ - _)%Z with (Z.pos p) by (rewrite Heqz'; ring).
set (rne := round_nearest_even _ _).
assert (rne = 0%Z).
{
unfold rne.
unfold round_nearest_even.
assert (Hp0 : (prec < Z.pos p)%Z) by lia.
unfold loc_of_shr_record.
specialize (Hshr_p0_r _ Hp0).
specialize (Hshr_p0 _ Hp0).
revert Hshr_p0_r Hshr_p0.
set (shr_p0 := iter_pos shr_1 _ _).
destruct shr_p0.
unfold SpecFloat.shr_r, SpecFloat.shr_m.
intros Hshr_r Hshr_m.
rewrite Hshr_r, Hshr_m.
now destruct shr_s.
}
rewrite H0.
rewrite Z.max_r by (rewrite Heqz'; unfold prec; lia).
replace (_ - _)%Z with 0%Z by lia.
unfold shr_m.
rewrite Z.max_r by lia.
remember (emin - (e0 + e))%Z as eminmze.
destruct eminmze; [ exfalso; lia | | exfalso; lia ].
rewrite Z.max_r by lia.
rewrite <- Heqeminmze.
set (rne' := round_nearest_even _ _).
assert (Hrne'0 : rne' = 0%Z).
{
unfold rne'.
unfold round_nearest_even.
assert (Hp1 : (prec < Z.pos p0)%Z) by lia.
unfold loc_of_shr_record.
specialize (Hshr_p0_r _ Hp1).
specialize (Hshr_p0 _ Hp1).
revert Hshr_p0_r Hshr_p0.
set (shr_p1 := iter_pos shr_1 _ _).
destruct shr_p1.
unfold SpecFloat.shr_r, SpecFloat.shr_m.
intros Hshr_r Hshr_m.
rewrite Hshr_r, Hshr_m.
now destruct shr_s.
}
rewrite Hrne'0.
rewrite Z.max_r by (rewrite Heqeminmze; unfold prec; lia).
replace (_ - _)%Z with 0%Z by lia.
reflexivity.
+ exfalso; lia.
Qed.
|
