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Diffstat (limited to 'src/common/IntegerExtra.v')
-rw-r--r-- | src/common/IntegerExtra.v | 287 |
1 files changed, 287 insertions, 0 deletions
diff --git a/src/common/IntegerExtra.v b/src/common/IntegerExtra.v index c9b5dbd..f44cac2 100644 --- a/src/common/IntegerExtra.v +++ b/src/common/IntegerExtra.v @@ -362,4 +362,291 @@ Module IntExtra. rewrite testbit_repr; auto. Abort. + Lemma div_divs_equiv : + forall x y, + signed x >= 0 -> + signed y >= 0 -> + divs x y = divu x y. + Proof. + unfold divs, divu. + intros. + rewrite !signed_eq_unsigned; + try rewrite Zquot.Zquot_Zdiv_pos; try reflexivity; + lazymatch goal with + | |- unsigned _ <= max_signed => + solve [rewrite <- signed_positive; assumption] + | |- 0 <= unsigned _ => solve [apply unsigned_range_2] + end. + Qed. + + Lemma neg_signed' : + forall x : int, + unsigned x <> 2147483648 -> + signed (neg x) = - signed x. + Proof. + intros x Hhalf. + Transparent repr. + unfold signed. + simpl. + rewrite Z_mod_modulus_eq. + replace modulus with 4294967296; auto. + replace half_modulus with 2147483648; auto. + repeat match goal with | |- context[if ?x then _ else _] => destruct x end. + - destruct (Z.eq_dec (unsigned x) 0). + + rewrite e. auto. + + pose proof (Z.mod_opp_l_nz (unsigned x) 4294967296). + assert (4294967296 <> 0) by lia. + apply H in H0. + rewrite H0 in l. + pose proof (Z.mod_small (unsigned x) 4294967296). + assert (0 <= unsigned x < 4294967296). + pose proof (unsigned_range_2 x). lia. + apply H1 in H2. rewrite H2 in l. lia. + rewrite Z.mod_small. assumption. + pose proof (unsigned_range_2 x). lia. + - destruct (Z.eq_dec (unsigned x) 0). + + lia. + + rewrite Z.mod_opp_l_nz; try lia. + rewrite Z.opp_sub_distr. + rewrite Z.mod_small. lia. + pose proof (unsigned_range_2 x). + simplify; lia. + rewrite Z.mod_small. assumption. + pose proof (unsigned_range_2 x). + simplify; lia. + - destruct (Z.eq_dec (unsigned x) 0). + + rewrite e in *. rewrite Z.opp_0 in *. rewrite Zmod_0_l in g. lia. + + rewrite Z.mod_opp_l_nz; try lia. + rewrite Z.mod_small. lia. + pose proof (unsigned_range_2 x). lia. + rewrite Z.mod_small. assumption. + pose proof (unsigned_range_2 x). lia. + - destruct (Z.eq_dec (unsigned x) 0). + + lia. + + rewrite Z.mod_opp_l_nz in g; try lia. + rewrite Z.mod_small in g. + assert (unsigned x < 2147483648) by lia. lia. + pose proof (unsigned_range_2 x). + replace max_unsigned with 4294967295 in * by auto. lia. + rewrite Z.mod_small. assumption. + pose proof (unsigned_range_2 x). + replace max_unsigned with 4294967295 in * by auto. lia. + Qed. + + Lemma neg_divs_distr_l : + forall x y, + unsigned x <> 2147483648 -> + neg (divs x y) = divs (neg x) y. + Proof. + intros x y Hhalf. unfold divs, neg. + set (x' := signed x). set (y' := signed y). + apply eqm_samerepr. + apply eqm_trans with (- (Z.quot x' y')). + auto with ints. + replace (- (Z.quot x' y')) with (Z.quot (- x') y') + by (rewrite Zquot.Zquot_opp_l; auto). + unfold x'. + rewrite <- neg_signed'. + auto with ints. + assumption. + Qed. + + Lemma neg_signed : + forall x : int, + unsigned x <> 2147483648 -> + signed x < 0 -> + signed (neg x) >= 0. + Proof. + intros. + rewrite neg_signed'. lia. + assumption. + Qed. + + Lemma shl_signed_positive : + forall y, + unsigned y <= 30 -> + signed (shl one y) >= 0. + Proof. + intros. + unfold signed, shl. + destruct (zlt (unsigned (repr (Z.shiftl (unsigned one) (unsigned y)))) half_modulus). + - rewrite unsigned_repr. + + rewrite Z.shiftl_1_l. + apply Z.le_ge. apply Z.pow_nonneg. lia. + + rewrite Z.shiftl_1_l. split. + apply Z.pow_nonneg. lia. + simplify. + replace (4294967295) with (2 ^ 32 - 1); try lia. + transitivity (2 ^ 31); try lia. + apply Z.pow_le_mono_r; lia. + - simplify. rewrite Z.shiftl_1_l in g. + unfold half_modulus, modulus, wordsize, + Wordsize_32.wordsize in *. unfold two_power_nat in *. simplify. + unfold Z_mod_modulus in *. + destruct (2 ^ unsigned y) eqn:?. + apply Z.ge_le in g. exfalso. + replace (4294967296 / 2) with (2147483648) in g; auto. + rewrite Z.shiftl_1_l. rewrite Heqz. + unfold wordsize in *. unfold Wordsize_32.wordsize in *. + rewrite Zbits.P_mod_two_p_eq in *. + replace (4294967296 / 2) with (2147483648) in g; auto. + rewrite <- Heqz in g. + rewrite Z.mod_small in g. + replace (2147483648) with (2 ^ 31) in g. + pose proof (Z.pow_le_mono_r 2 (unsigned y) 30). + apply Z.ge_le in g. + assert (0 < 2) by lia. apply H0 in H1. lia. assumption. lia. + split. lia. rewrite two_power_nat_equiv. + apply Z.pow_lt_mono_r; lia. + + pose proof (Zlt_neg_0 p). + pose proof (pow2_nonneg (unsigned y)). rewrite <- Heqz in H0. + lia. + Qed. + + Lemma is_power2_shl : + forall y, + unsigned y <= 30 -> + is_power2 (shl one y) = Some y. + Proof. + intros. + unfold is_power2, shl. + destruct (Zbits.Z_is_power2 (unsigned (repr (Z.shiftl (unsigned one) (unsigned y))))) eqn:?. + - simplify. + rewrite Z_mod_modulus_eq in Heqo. + rewrite Z.mod_small in Heqo. rewrite Z.shiftl_1_l in Heqo. + rewrite <- two_p_correct in Heqo. + rewrite Zbits.Z_is_power2_complete in Heqo. inv Heqo. + rewrite repr_unsigned. auto. + pose proof (unsigned_range_2 y). lia. + rewrite Z.shiftl_1_l. unfold modulus, wordsize, Wordsize_32.wordsize. + rewrite two_power_nat_equiv. + split. apply pow2_nonneg. + apply Z.pow_lt_mono_r; lia. + - simplify. + rewrite Z_mod_modulus_eq in Heqo. + rewrite Z.mod_small in Heqo. rewrite Z.shiftl_1_l in Heqo. + rewrite <- two_p_correct in Heqo. + rewrite Zbits.Z_is_power2_complete in Heqo. discriminate. + pose proof (unsigned_range_2 y). lia. + rewrite Z.shiftl_1_l. unfold modulus, wordsize, Wordsize_32.wordsize. + rewrite two_power_nat_equiv. + split. apply pow2_nonneg. + apply Z.pow_lt_mono_r; lia. + Qed. + + Definition shrx_alt (x y : int) : int := + if zlt (signed x) 0 + then neg (shru (neg x) y) + else shru x y. + + Lemma shrx_shrx_alt_equiv_ne : + forall x y, + unsigned x <> 2147483648 -> + unsigned y <= 30 -> + shrx x y = shrx_alt x y. + Proof. + intros x y Hhalf H. + unfold shrx, shrx_alt, lt. + destruct (Z_ge_lt_dec (signed x) 0); + [rewrite zlt_false | rewrite zlt_true]; + + repeat lazymatch goal with + | |- is_power2 _ = Some _ => apply is_power2_shl + | |- signed (shl one _) >= 0 => apply shl_signed_positive + | |- signed (neg _) >= 0 => apply neg_signed + | |- divs _ _ = divu _ _ => apply div_divs_equiv + | |- divs ?x (shl one ?y) = neg (shru (neg ?x) ?y) => + rewrite <- neg_involutive at 1; rewrite neg_divs_distr_l; + try assumption; f_equal + | |- divs ?x (shl one ?y) = shru ?x ?y => + let H := fresh "H" in + pose proof (divu_pow2 x (shl one y) y) as H; + rewrite <- H + end; try assumption. + Qed. + + Lemma shrx_shrx_alt_equiv_eq : + forall x y, + unsigned x = 2147483648 -> + unsigned y <= 30 -> + shrx x y = shrx_alt x y. + Proof. + intros. + repeat unfold shrx, shrx_alt, signed, divs, neg. + replace half_modulus with 2147483648 by auto. + replace modulus with 4294967296 by auto. + simplify. + rewrite !Z_mod_modulus_eq. + rewrite !H. + simplify. + assert (Hshl: Z.shiftl 1 (unsigned y) mod 4294967296 = Z.shiftl 1 (unsigned y)). + { apply Z.mod_small. + rewrite Z.shiftl_1_l. + split. + apply pow2_nonneg. + replace 4294967296 with (2^32) by auto. + apply Z.le_lt_trans with (m := 2 ^ 31); try lia. + apply Z.pow_le_mono_r; lia. + } + rewrite !Hshl. + f_equal. + assert ((Z.shiftl 1 (unsigned y)) < 2147483648). + rewrite Z.shiftl_1_l. + replace 2147483648 with (2^31) by auto. + apply Z.le_lt_trans with (m := 2 ^ 30); try lia. + apply Z.pow_le_mono_r; lia. + destruct (zlt (Z.shiftl 1 (unsigned y)) 2147483648); try lia. + replace (-2147483648 mod 4294967296) with 2147483648 by auto. + assert (Hmodeq : Z.shiftr 2147483648 (unsigned y) mod 4294967296 + = Z.shiftr 2147483648 (unsigned y)). + { apply Z.mod_small. split. + apply Z.shiftr_nonneg. lia. + rewrite Z.shiftr_div_pow2. + replace 4294967296 with (Z.succ 4294967295); auto. + apply Zle_lt_succ. + replace 4294967295 with (4294967295 * (2 ^ unsigned y) / (2 ^ unsigned y)). + 2: { + apply Z.div_mul. + pose proof (Z.pow_pos_nonneg 2 (unsigned y)). + apply not_eq_sym. + apply Z.le_neq. apply H2; try lia. + apply unsigned_range_2. + } + + apply Z.div_le_mono. + apply Z.pow_pos_nonneg. lia. + apply unsigned_range_2. + transitivity 4294967295; try lia. + apply Z.le_mul_diag_r; try lia. + replace 1 with (Z.succ 0) by auto. + apply Z.le_succ_l. + apply Z.pow_pos_nonneg; try lia. + apply unsigned_range_2. apply unsigned_range_2. + } + rewrite !Hmodeq. + replace (-2147483648) with (Z.opp 2147483648) by auto. + rewrite Zquot.Zquot_opp_l. + f_equal. + rewrite Zquot.Zquot_Zdiv_pos. + rewrite Z.shiftr_div_pow2. + rewrite Z.shiftl_1_l. auto. + apply unsigned_range_2. + lia. + rewrite Z.shiftl_1_l. + apply Z.lt_le_incl. + apply Z.pow_pos_nonneg; try lia. + apply unsigned_range_2. + Qed. + + Theorem shrx_shrx_alt_equiv : + forall x y, + unsigned y <= 30 -> + shrx x y = shrx_alt x y. + Proof. + intros. + destruct (Z.eq_dec (unsigned x) 2147483648); + [ apply shrx_shrx_alt_equiv_eq | apply shrx_shrx_alt_equiv_ne]; auto. + Qed. + End IntExtra. |