diff options
| -rw-r--r-- | src/BaseSystem.v | 2 | ||||
| -rw-r--r-- | src/BaseSystemProofs.v | 2 | ||||
| -rw-r--r-- | src/Encoding/ModularWordEncodingTheorems.v | 6 | ||||
| -rw-r--r-- | src/ModularArithmetic/ModularBaseSystemOpt.v | 8 | ||||
| -rw-r--r-- | src/ModularArithmetic/ModularBaseSystemProofs.v | 28 | ||||
| -rw-r--r-- | src/ModularArithmetic/PrimeFieldTheorems.v | 4 | ||||
| -rw-r--r-- | src/ModularArithmetic/PseudoMersenneBaseParamProofs.v | 12 | ||||
| -rw-r--r-- | src/Testbit.v | 10 | ||||
| -rw-r--r-- | src/Util/NumTheoryUtil.v | 16 | ||||
| -rw-r--r-- | src/Util/ZUtil.v | 1419 |
10 files changed, 747 insertions, 760 deletions
diff --git a/src/BaseSystem.v b/src/BaseSystem.v index 743cdfde8..c22af95ca 100644 --- a/src/BaseSystem.v +++ b/src/BaseSystem.v @@ -111,7 +111,7 @@ Section PolynomialBaseCoefs. rewrite in_map_iff in *. destruct H; destruct H. subst. - apply pos_pow_nat_pos. + apply Z.pos_pow_nat_pos. Qed. Lemma poly_base_defn : forall i, (i < length poly_base)%nat -> diff --git a/src/BaseSystemProofs.v b/src/BaseSystemProofs.v index 85835aabe..5746cb5f1 100644 --- a/src/BaseSystemProofs.v +++ b/src/BaseSystemProofs.v @@ -177,7 +177,7 @@ Section BaseSystemProofs. Lemma nth_error_base_nonzero : forall n x, nth_error base n = Some x -> x <> 0. Proof. - eauto using (@nth_error_value_In Z), Zgt0_neq0, base_positive. + eauto using (@nth_error_value_In Z), Z.gt0_neq0, base_positive. Qed. Hint Rewrite plus_0_r. diff --git a/src/Encoding/ModularWordEncodingTheorems.v b/src/Encoding/ModularWordEncodingTheorems.v index 41b75e216..033e99665 100644 --- a/src/Encoding/ModularWordEncodingTheorems.v +++ b/src/Encoding/ModularWordEncodingTheorems.v @@ -43,12 +43,12 @@ Section SignBit. pose proof (F_opp_spec x) as opp_spec_x. apply F_eq in opp_spec_x. rewrite FieldToZ_add in opp_spec_x. - rewrite <-opp_spec_x, Z_odd_mod in sign_zero by (pose proof prime_ge_2 m prime_m; omega). - replace (Z.odd m) with true in sign_zero by (destruct (ZUtil.prime_odd_or_2 m prime_m); auto || omega). + rewrite <-opp_spec_x, Z.odd_mod in sign_zero by (pose proof prime_ge_2 m prime_m; omega). + replace (Z.odd m) with true in sign_zero by (destruct (Z.prime_odd_or_2 m prime_m); auto || omega). rewrite Z.odd_add, F_FieldToZ_add_opp, Z.div_same, Bool.xorb_true_r in sign_zero by assumption || omega. apply Bool.xorb_eq. rewrite <-Bool.negb_xorb_l. assumption. Qed. -End SignBit.
\ No newline at end of file +End SignBit. diff --git a/src/ModularArithmetic/ModularBaseSystemOpt.v b/src/ModularArithmetic/ModularBaseSystemOpt.v index 116fe10e5..9bb3128ad 100644 --- a/src/ModularArithmetic/ModularBaseSystemOpt.v +++ b/src/ModularArithmetic/ModularBaseSystemOpt.v @@ -22,7 +22,7 @@ Definition Z_div_opt := Eval compute in Z.div. Definition Z_pow_opt := Eval compute in Z.pow. Definition Z_opp_opt := Eval compute in Z.opp. Definition Z_shiftl_opt := Eval compute in Z.shiftl. -Definition Z_shiftl_by_opt := Eval compute in Z_shiftl_by. +Definition Z_shiftl_by_opt := Eval compute in Z.shiftl_by. Definition nth_default_opt {A} := Eval compute in @nth_default A. Definition set_nth_opt {A} := Eval compute in @set_nth A. @@ -499,11 +499,11 @@ Section Multiplication. cbv [BaseSystem.mul mul mul_each mul_bi mul_bi' zeros ext_base reduce]. rewrite <- mul'_opt_correct. change @base with base_opt. - rewrite map_shiftl by apply k_nonneg. + rewrite Z.map_shiftl by apply k_nonneg. rewrite c_subst. rewrite k_subst. change @map with @map_opt. - change @Z_shiftl_by with @Z_shiftl_by_opt. + change @Z.shiftl_by with @Z_shiftl_by_opt. reflexivity. Defined. @@ -668,4 +668,4 @@ Section Canonicalization. auto using freeze_opt_preserves_rep. Qed. -End Canonicalization.
\ No newline at end of file +End Canonicalization. diff --git a/src/ModularArithmetic/ModularBaseSystemProofs.v b/src/ModularArithmetic/ModularBaseSystemProofs.v index 6f82a8950..a8bd93097 100644 --- a/src/ModularArithmetic/ModularBaseSystemProofs.v +++ b/src/ModularArithmetic/ModularBaseSystemProofs.v @@ -110,7 +110,7 @@ Section PseudoMersenneProofs. rewrite Z.sub_sub_distr, Z.sub_diag. simpl. rewrite Z.mul_comm. - rewrite mod_mult_plus; auto using modulus_nonzero. + rewrite Z.mod_add_l; auto using modulus_nonzero. rewrite <- Zplus_mod; auto. Qed. @@ -304,8 +304,8 @@ Section CarryProofs. rewrite nth_default_base_succ by omega. rewrite Z.mul_assoc. rewrite (Z.mul_comm _ (2 ^ log_cap i)). - rewrite mul_div_eq; try ring. - apply gt_lt_symmetry. + rewrite Z.mul_div_eq; try ring. + apply Z.gt_lt_iff. apply Z.pow_pos_nonneg; omega || apply log_cap_nonneg. Qed. @@ -337,7 +337,7 @@ Section CarryProofs. rewrite <- Z.add_opp_l, <- Z.opp_sub_distr. unfold pow2_mod. rewrite Z.land_ones by apply log_cap_nonneg. - rewrite <- mul_div_eq by (apply gt_lt_symmetry; apply Z.pow_pos_nonneg; omega || apply log_cap_nonneg). + rewrite <- Z.mul_div_eq by (apply Z.gt_lt_iff; apply Z.pow_pos_nonneg; omega || apply log_cap_nonneg). rewrite Z.shiftr_div_pow2 by apply log_cap_nonneg. rewrite Zopp_mult_distr_r. rewrite Z.mul_comm. @@ -485,7 +485,7 @@ Section CanonicalizationProofs. Lemma max_bound_pos : forall i, (i < length base)%nat -> 0 < max_bound i. Proof. - unfold max_bound, log_cap; intros; apply Z_ones_pos_pos. + unfold max_bound, log_cap; intros; apply Z.ones_pos_pos. apply limb_widths_pos. rewrite nth_default_eq. apply nth_In. @@ -495,7 +495,7 @@ Section CanonicalizationProofs. Lemma max_bound_nonneg : forall i, 0 <= max_bound i. Proof. - unfold max_bound; intros; auto using Z_ones_nonneg. + unfold max_bound; intros; auto using Z.ones_nonneg. Qed. Local Hint Resolve max_bound_nonneg. @@ -874,7 +874,7 @@ Section CanonicalizationProofs. apply Z.add_le_mono. + apply carry_bounds_0_upper; auto; omega. + apply Z.mul_le_mono_pos_l; auto. - apply Z_shiftr_ones; auto; + apply Z.shiftr_ones; auto; [ | pose proof (B_compat_log_cap (pred (length base))); omega ]. split. - apply carry_bounds_lower; auto; omega. @@ -913,7 +913,7 @@ Section CanonicalizationProofs. + rewrite <-max_bound_log_cap, <-Z.add_1_l. apply Z.add_le_mono. - rewrite Z.shiftr_div_pow2 by apply log_cap_nonneg. - apply Z_div_floor; auto. + apply Z.div_floor; auto. destruct i. * simpl. eapply Z.le_lt_trans; [ apply carry_full_bounds_0; auto | ]. @@ -996,7 +996,7 @@ Section CanonicalizationProofs. + rewrite <-max_bound_log_cap, <-Z.add_1_l. rewrite Z.shiftr_div_pow2 by apply log_cap_nonneg. apply Z.add_le_mono. - - apply Z_div_floor; auto. + - apply Z.div_floor; auto. eapply Z.le_lt_trans; [ eapply carry_full_2_bounds_0; eauto | ]. replace (Z.succ 1) with (2 ^ 1) by ring. rewrite <-max_bound_log_cap. @@ -1202,7 +1202,7 @@ Section CanonicalizationProofs. Lemma max_ones_nonneg : 0 <= max_ones. Proof. unfold max_ones. - apply Z_ones_nonneg. + apply Z.ones_nonneg. pose proof limb_widths_nonneg. induction limb_widths. cbv; congruence. @@ -1217,19 +1217,19 @@ Section CanonicalizationProofs. unfold max_ones. intros ? ? x_range. rewrite Z.land_comm. - rewrite Z.land_ones by apply Z_le_fold_right_max_initial. + rewrite Z.land_ones by apply Z.le_fold_right_max_initial. apply Z.mod_small. split; try omega. eapply Z.lt_le_trans; try eapply x_range. apply Z.pow_le_mono_r; try omega. rewrite log_cap_eq. destruct (lt_dec i (length limb_widths)). - + apply Z_le_fold_right_max. + + apply Z.le_fold_right_max. - apply limb_widths_nonneg. - rewrite nth_default_eq. auto using nth_In. + rewrite nth_default_out_of_bounds by omega. - apply Z_le_fold_right_max_initial. + apply Z.le_fold_right_max_initial. Qed. Lemma full_isFull'_true : forall j us, (length us = length base) -> @@ -1817,7 +1817,7 @@ Section CanonicalizationProofs. + match goal with |- (?a ?= ?b) = (?c ?= ?d) => rewrite (Z.compare_antisym b a); rewrite (Z.compare_antisym d c) end. apply CompOpp_inj; rewrite !CompOpp_involutive. - apply gt_lt_symmetry in Hgt. + apply Z.gt_lt_iff in Hgt. etransitivity; try apply Z_compare_decode_step_lt; auto; omega. Qed. diff --git a/src/ModularArithmetic/PrimeFieldTheorems.v b/src/ModularArithmetic/PrimeFieldTheorems.v index 2021e8514..a2f606f30 100644 --- a/src/ModularArithmetic/PrimeFieldTheorems.v +++ b/src/ModularArithmetic/PrimeFieldTheorems.v @@ -460,8 +460,8 @@ Section SquareRootsPrime5Mod8. apply Z2N.inj_iff; try zero_bounds. rewrite <- Z.mul_cancel_l with (p := 2) by omega. ring_simplify. - rewrite mul_div_eq by omega. - rewrite mul_div_eq by omega. + rewrite Z.mul_div_eq by omega. + rewrite Z.mul_div_eq by omega. rewrite (Zmod_div_mod 2 8 q) by (try omega; apply Zmod_divide; omega || auto). rewrite q_5mod8. diff --git a/src/ModularArithmetic/PseudoMersenneBaseParamProofs.v b/src/ModularArithmetic/PseudoMersenneBaseParamProofs.v index 49b1875ce..9e4e4b3ba 100644 --- a/src/ModularArithmetic/PseudoMersenneBaseParamProofs.v +++ b/src/ModularArithmetic/PseudoMersenneBaseParamProofs.v @@ -163,7 +163,7 @@ Section PseudoMersenneBaseParamProofs. rewrite (Z.mul_comm r). subst r. assert (i + j - length base < length base)%nat by omega. - rewrite mul_div_eq by (apply gt_lt_symmetry; apply Z.mul_pos_pos; + rewrite Z.mul_div_eq by (apply Z.gt_lt_iff; apply Z.mul_pos_pos; [ | subst b; rewrite nth_default_base; try assumption ]; apply Z.pow_pos_nonneg; omega || apply k_nonneg || apply sum_firstn_limb_widths_nonneg). rewrite (Zminus_0_l_reverse (b i * b j)) at 1. @@ -172,7 +172,7 @@ Section PseudoMersenneBaseParamProofs. repeat rewrite nth_default_base by assumption. do 2 rewrite <- Z.pow_add_r by (apply sum_firstn_limb_widths_nonneg || apply k_nonneg). symmetry. - apply mod_same_pow. + apply Z.mod_same_pow. split. + apply Z.add_nonneg_nonneg; apply sum_firstn_limb_widths_nonneg || apply k_nonneg. + rewrite base_length in *; apply limb_widths_match_modulus; assumption. @@ -183,7 +183,7 @@ Section PseudoMersenneBaseParamProofs. Proof. intros. repeat rewrite nth_default_base by omega. - apply mod_same_pow. + apply Z.mod_same_pow. split; [apply sum_firstn_limb_widths_nonneg | ]. destruct (NPeano.Nat.eq_dec i 0); subst. + case_eq limb_widths; intro; unfold sum_firstn; simpl; try omega; intros l' lw_eq. @@ -218,7 +218,7 @@ Section PseudoMersenneBaseParamProofs. destruct In_b_base as [i nth_err_b]. apply nth_error_subst in nth_err_b. rewrite nth_err_b. - apply gt_lt_symmetry. + apply Z.gt_lt_iff. apply Z.pow_pos_nonneg; omega || apply sum_firstn_limb_widths_nonneg. Qed. @@ -236,9 +236,9 @@ Section PseudoMersenneBaseParamProofs. intros; subst b r. repeat rewrite nth_default_base by omega. rewrite (Z.mul_comm _ (2 ^ (sum_firstn limb_widths (i+j)))). - rewrite mul_div_eq by (apply gt_lt_symmetry; apply Z.pow_pos_nonneg; omega || apply sum_firstn_limb_widths_nonneg). + rewrite Z.mul_div_eq by (apply Z.gt_lt_iff; apply Z.pow_pos_nonneg; omega || apply sum_firstn_limb_widths_nonneg). rewrite <- Z.pow_add_r by apply sum_firstn_limb_widths_nonneg. - rewrite mod_same_pow; try ring. + rewrite Z.mod_same_pow; try ring. split; [ apply sum_firstn_limb_widths_nonneg | ]. apply limb_widths_good. rewrite <- base_length; assumption. diff --git a/src/Testbit.v b/src/Testbit.v index 2bfcc3df6..d735bbe21 100644 --- a/src/Testbit.v +++ b/src/Testbit.v @@ -107,7 +107,7 @@ Proof. rewrite <- nth_default_eq in uniform. erewrite nth_error_value_eq_nth_default in uniform; eauto. subst. - destruct r; [ | apply pos_pow_nat_pos | pose proof (Zlt_neg_0 p) ] ; omega. + destruct r; [ | apply Z.pos_pow_nat_pos | pose proof (Zlt_neg_0 p) ] ; omega. + intros. rewrite nth_default_eq. rewrite uniform; auto. @@ -151,7 +151,7 @@ Proof. induction us; boring. rewrite <- (IHus base) by (omega || eauto using no_overflow_tail). rewrite decode_cons by (eapply uniform_base_BaseVector; eauto; - rewrite gt_lt_symmetry; apply Z_pow_gt0; omega). + rewrite Z.gt_lt_iff; apply Z.pow_pos_nonneg; omega). simpl. f_equal. + symmetry. eapply no_overflow_cons; eauto. @@ -174,12 +174,12 @@ Proof. auto using Z.land_0_l. + destruct i; simpl. - rewrite nth_default_cons. - rewrite Z.shiftr_0_r, Z_land_add_land by omega. + rewrite Z.shiftr_0_r, Z.land_add_land by omega. symmetry; eapply no_overflow_cons; eauto. - rewrite nth_default_cons_S. erewrite IHus; eauto using no_overflow_tail. remember (i * limb_width)%nat as k. - rewrite Z_shiftr_add_land by omega. + rewrite Z.shiftr_add_land by omega. replace (limb_width + k - limb_width)%nat with k by omega. reflexivity. Qed. @@ -190,7 +190,7 @@ Lemma unfold_bits_testbit : forall limb_width us n, (0 < limb_width)%nat -> Proof. unfold testbit; intros. erewrite unfold_bits_indexing; eauto. - rewrite <- Z_testbit_low by + rewrite <- Z.testbit_low by (split; try apply Nat2Z.inj_lt; pose proof (mod_bound_pos n limb_width); omega). rewrite Z.shiftr_spec by apply Nat2Z.is_nonneg. f_equal. diff --git a/src/Util/NumTheoryUtil.v b/src/Util/NumTheoryUtil.v index 10ce148b0..c16b87639 100644 --- a/src/Util/NumTheoryUtil.v +++ b/src/Util/NumTheoryUtil.v @@ -66,7 +66,7 @@ Qed. Lemma p_odd : Z.odd p = true. Proof. - pose proof (prime_odd_or_2 p prime_p). + pose proof (Z.prime_odd_or_2 p prime_p). destruct H; auto. Qed. @@ -124,12 +124,12 @@ Proof. assert (b mod p <> 0) as b_nonzero. { intuition. rewrite <- Z.pow_2_r in a_square. - rewrite mod_exp_0 in a_square by prime_bound. + rewrite Z.mod_exp_0 in a_square by prime_bound. rewrite <- a_square in a_nonzero. auto. } pose proof (squared_fermat_little b b_nonzero). - rewrite mod_pow in * by prime_bound. + rewrite Z.mod_pow in * by prime_bound. rewrite <- a_square. rewrite Z.mod_mod; prime_bound. Qed. @@ -172,10 +172,10 @@ Proof. intros. destruct (exists_primitive_root_power) as [y [in_ZPGroup_y [y_order gpow_y]]]; auto. destruct (gpow_y a a_range) as [j [j_range pow_y_j]]; clear gpow_y. - rewrite mod_pow in pow_a_x by prime_bound. + rewrite Z.mod_pow in pow_a_x by prime_bound. replace a with (a mod p) in pow_y_j by (apply Z.mod_small; omega). rewrite <- pow_y_j in pow_a_x. - rewrite <- mod_pow in pow_a_x by prime_bound. + rewrite <- Z.mod_pow in pow_a_x by prime_bound. rewrite <- Z.pow_mul_r in pow_a_x by omega. assert (p - 1 | j * x) as divide_mul_j_x. { rewrite <- phi_is_order in y_order. @@ -193,13 +193,13 @@ Proof. rewrite <- Z_div_plus by omega. rewrite Z.mul_comm. rewrite x_id_inv in divide_mul_j_x; auto. - apply (divide_mul_div _ j 2) in divide_mul_j_x; + apply (Z.divide_mul_div _ j 2) in divide_mul_j_x; try (apply prime_pred_divide2 || prime_bound); auto. rewrite <- Zdivide_Zdiv_eq by (auto || omega). rewrite Zplus_diag_eq_mult_2. replace (a mod p) with a in pow_y_j by (symmetry; apply Z.mod_small; omega). rewrite Z_div_mult by omega; auto. - apply divide2_even_iff. + apply Z.divide2_even_iff. apply prime_pred_even. Qed. @@ -281,7 +281,7 @@ Lemma div2_p_1mod4 : forall (p : Z) (prime_p : prime p) (neq_p_2: p <> 2), (p / 2) * 2 + 1 = p. Proof. intros. - destruct (prime_odd_or_2 p prime_p); intuition. + destruct (Z.prime_odd_or_2 p prime_p); intuition. rewrite <- Zdiv2_div. pose proof (Zdiv2_odd_eqn p); break_if; congruence || omega. Qed. diff --git a/src/Util/ZUtil.v b/src/Util/ZUtil.v index a8b18ffef..ab107b7a2 100644 --- a/src/Util/ZUtil.v +++ b/src/Util/ZUtil.v @@ -17,7 +17,7 @@ Hint Resolve (fun a b H => proj1 (Z.mod_pos_bound a b H)) (fun a b H => proj2 (Z this database. *) Create HintDb zsimplify discriminated. Hint Rewrite Z.div_1_r Z.mul_1_r Z.mul_1_l Z.sub_diag Z.mul_0_r Z.mul_0_l Z.add_0_l Z.add_0_r Z.opp_involutive Z.sub_0_r : zsimplify. -Hint Rewrite Z.div_mul Z.div_1_l Z.div_same Z.mod_same Z.div_small Z.mod_small Z.div_add Z.div_add_l using lia : zsimplify. +Hint Rewrite Z.div_mul Z.div_1_l Z.div_same Z.mod_same Z.div_small Z.mod_small Z.div_add Z.div_add_l Z.mod_add using lia : zsimplify. (** "push" means transform [-f x] to [f (-x)]; "pull" means go the other way *) Create HintDb push_Zopp discriminated. @@ -43,318 +43,299 @@ Hint Rewrite Z.div_small_iff using lia : zstrip_div. We'll put, e.g., [mul_div_eq] into it below. *) Create HintDb zstrip_div. -Lemma gt_lt_symmetry: forall n m, n > m <-> m < n. -Proof. - intros; split; omega. -Qed. - -Lemma positive_is_nonzero : forall x, x > 0 -> x <> 0. -Proof. - intros; omega. -Qed. -Hint Resolve positive_is_nonzero. - -Lemma div_positive_gt_0 : forall a b, a > 0 -> b > 0 -> a mod b = 0 -> - a / b > 0. -Proof. - intros; rewrite gt_lt_symmetry. - apply Z.div_str_pos. - split; intuition. - apply Z.divide_pos_le; try (apply Zmod_divide); omega. -Qed. - -Lemma elim_mod : forall a b m, a = b -> a mod m = b mod m. -Proof. - intros; subst; auto. -Qed. -Hint Resolve elim_mod. - -Lemma mod_mult_plus: forall a b c, (b <> 0) -> (a * b + c) mod b = c mod b. -Proof. - intros. - rewrite Zplus_mod. - rewrite Z.mod_mul; auto; simpl. - rewrite Zmod_mod; auto. -Qed. - -Lemma pos_pow_nat_pos : forall x n, - Z.pos x ^ Z.of_nat n > 0. - do 2 (intros; induction n; subst; simpl in *; auto with zarith). - rewrite <- Pos.add_1_r, Zpower_pos_is_exp. - apply Zmult_gt_0_compat; auto; reflexivity. -Qed. - -Lemma Z_div_mul' : forall a b : Z, b <> 0 -> (b * a) / b = a. - intros. rewrite Z.mul_comm. apply Z.div_mul; auto. -Qed. - -Hint Rewrite Z_div_mul' using lia : zsimplify. - -Lemma Zgt0_neq0 : forall x, x > 0 -> x <> 0. - intuition. -Qed. - -Lemma pow_Z2N_Zpow : forall a n, 0 <= a -> - ((Z.to_nat a) ^ n = Z.to_nat (a ^ Z.of_nat n)%Z)%nat. -Proof. - intros; induction n; try reflexivity. - rewrite Nat2Z.inj_succ. - rewrite pow_succ_r by apply le_0_n. - rewrite Z.pow_succ_r by apply Zle_0_nat. - rewrite IHn. - rewrite Z2Nat.inj_mul; auto using Z.pow_nonneg. -Qed. - -Lemma pow_Zpow : forall a n : nat, Z.of_nat (a ^ n) = Z.of_nat a ^ Z.of_nat n. -Proof with auto using Zle_0_nat, Z.pow_nonneg. - intros; apply Z2Nat.inj... - rewrite <- pow_Z2N_Zpow, !Nat2Z.id... -Qed. - -Lemma mod_exp_0 : forall a x m, x > 0 -> m > 1 -> a mod m = 0 -> - a ^ x mod m = 0. -Proof. - intros. - replace x with (Z.of_nat (Z.to_nat x)) in * by (apply Z2Nat.id; omega). - induction (Z.to_nat x). { - simpl in *; omega. - } { - rewrite Nat2Z.inj_succ in *. - rewrite Z.pow_succ_r by omega. - rewrite Z.mul_mod by omega. - case_eq n; intros. { - subst. simpl. - rewrite Zmod_1_l by omega. - rewrite H1. - apply Zmod_0_l. +Module Z. + Lemma positive_is_nonzero : forall x, x > 0 -> x <> 0. + Proof. intros; omega. Qed. + + Hint Resolve positive_is_nonzero : zarith. + + Lemma div_positive_gt_0 : forall a b, a > 0 -> b > 0 -> a mod b = 0 -> + a / b > 0. + Proof. + intros; rewrite Z.gt_lt_iff. + apply Z.div_str_pos. + split; intuition. + apply Z.divide_pos_le; try (apply Zmod_divide); omega. + Qed. + + Lemma elim_mod : forall a b m, a = b -> a mod m = b mod m. + Proof. intros; subst; auto. Qed. + + Hint Resolve elim_mod : zarith. + + Lemma mod_add_l : forall a b c, b <> 0 -> (a * b + c) mod b = c mod b. + Proof. intros; rewrite (Z.add_comm _ c); autorewrite with zsimplify; reflexivity. Qed. + Hint Rewrite mod_add_l using lia : zsimplify. + + Lemma mod_add' : forall a b c, b <> 0 -> (a + b * c) mod b = a mod b. + Proof. intros; rewrite (Z.mul_comm _ c); autorewrite with zsimplify; reflexivity. Qed. + Lemma mod_add_l' : forall a b c, a <> 0 -> (a * b + c) mod a = c mod a. + Proof. intros; rewrite (Z.mul_comm _ b); autorewrite with zsimplify; reflexivity. Qed. + Hint Rewrite mod_add' mod_add_l' using lia : zsimplify. + + Lemma pos_pow_nat_pos : forall x n, + Z.pos x ^ Z.of_nat n > 0. + Proof. + do 2 (intros; induction n; subst; simpl in *; auto with zarith). + rewrite <- Pos.add_1_r, Zpower_pos_is_exp. + apply Zmult_gt_0_compat; auto; reflexivity. + Qed. + + Lemma div_mul' : forall a b : Z, b <> 0 -> (b * a) / b = a. + Proof. intros. rewrite Z.mul_comm. apply Z.div_mul; auto. Qed. + Hint Rewrite div_mul' using lia : zsimplify. + + (** TODO: Should we get rid of this duplicate? *) + Notation gt0_neq0 := positive_is_nonzero (only parsing). + + Lemma pow_Z2N_Zpow : forall a n, 0 <= a -> + ((Z.to_nat a) ^ n = Z.to_nat (a ^ Z.of_nat n)%Z)%nat. + Proof. + intros; induction n; try reflexivity. + rewrite Nat2Z.inj_succ. + rewrite pow_succ_r by apply le_0_n. + rewrite Z.pow_succ_r by apply Zle_0_nat. + rewrite IHn. + rewrite Z2Nat.inj_mul; auto using Z.pow_nonneg. + Qed. + + Lemma pow_Zpow : forall a n : nat, Z.of_nat (a ^ n) = Z.of_nat a ^ Z.of_nat n. + Proof with auto using Zle_0_nat, Z.pow_nonneg. + intros; apply Z2Nat.inj... + rewrite <- pow_Z2N_Zpow, !Nat2Z.id... + Qed. + + Lemma mod_exp_0 : forall a x m, x > 0 -> m > 1 -> a mod m = 0 -> + a ^ x mod m = 0. + Proof. + intros. + replace x with (Z.of_nat (Z.to_nat x)) in * by (apply Z2Nat.id; omega). + induction (Z.to_nat x). { + simpl in *; omega. + } { + rewrite Nat2Z.inj_succ in *. + rewrite Z.pow_succ_r by omega. + rewrite Z.mul_mod by omega. + case_eq n; intros. { + subst. simpl. + rewrite Zmod_1_l by omega. + rewrite H1. + apply Zmod_0_l. + } { + subst. + rewrite IHn by (rewrite Nat2Z.inj_succ in *; omega). + rewrite H1. + auto. + } + } + Qed. + + Lemma mod_pow : forall (a m b : Z), (0 <= b) -> (m <> 0) -> + a ^ b mod m = (a mod m) ^ b mod m. + Proof. + intros; rewrite <- (Z2Nat.id b) by auto. + induction (Z.to_nat b); auto. + rewrite Nat2Z.inj_succ. + do 2 rewrite Z.pow_succ_r by apply Nat2Z.is_nonneg. + rewrite Z.mul_mod by auto. + rewrite (Z.mul_mod (a mod m) ((a mod m) ^ Z.of_nat n) m) by auto. + rewrite <- IHn by auto. + rewrite Z.mod_mod by auto. + reflexivity. + Qed. + + Ltac divide_exists_mul := let k := fresh "k" in + match goal with + | [ H : (?a | ?b) |- _ ] => apply Z.mod_divide in H; try apply Zmod_divides in H; destruct H as [k H] + | [ |- (?a | ?b) ] => apply Z.mod_divide; try apply Zmod_divides + end; (omega || auto). + + Lemma divide_mul_div: forall a b c (a_nonzero : a <> 0) (c_nonzero : c <> 0), + (a | b * (a / c)) -> (c | a) -> (c | b). + Proof. + intros ? ? ? ? ? divide_a divide_c_a; do 2 divide_exists_mul. + rewrite divide_c_a in divide_a. + rewrite div_mul' in divide_a by auto. + replace (b * k) with (k * b) in divide_a by ring. + replace (c * k * k0) with (k * (k0 * c)) in divide_a by ring. + rewrite Z.mul_cancel_l in divide_a by (intuition; rewrite H in divide_c_a; ring_simplify in divide_a; intuition). + eapply Zdivide_intro; eauto. + Qed. + + Lemma divide2_even_iff : forall n, (2 | n) <-> Z.even n = true. + Proof. + intro; split. { + intro divide2_n. + divide_exists_mul; [ | pose proof (Z.mod_pos_bound n 2); omega]. + rewrite divide2_n. + apply Z.even_mul. } { - subst. - rewrite IHn by (rewrite Nat2Z.inj_succ in *; omega). - rewrite H1. - auto. + intro n_even. + pose proof (Zmod_even n). + rewrite n_even in H. + apply Zmod_divide; omega || auto. } - } -Qed. - -Lemma mod_pow : forall (a m b : Z), (0 <= b) -> (m <> 0) -> - a ^ b mod m = (a mod m) ^ b mod m. -Proof. - intros; rewrite <- (Z2Nat.id b) by auto. - induction (Z.to_nat b); auto. - rewrite Nat2Z.inj_succ. - do 2 rewrite Z.pow_succ_r by apply Nat2Z.is_nonneg. - rewrite Z.mul_mod by auto. - rewrite (Z.mul_mod (a mod m) ((a mod m) ^ Z.of_nat n) m) by auto. - rewrite <- IHn by auto. - rewrite Z.mod_mod by auto. - reflexivity. -Qed. - -Ltac Zdivide_exists_mul := let k := fresh "k" in -match goal with -| [ H : (?a | ?b) |- _ ] => apply Z.mod_divide in H; try apply Zmod_divides in H; destruct H as [k H] -| [ |- (?a | ?b) ] => apply Z.mod_divide; try apply Zmod_divides -end; (omega || auto). - -Lemma divide_mul_div: forall a b c (a_nonzero : a <> 0) (c_nonzero : c <> 0), - (a | b * (a / c)) -> (c | a) -> (c | b). -Proof. - intros ? ? ? ? ? divide_a divide_c_a; do 2 Zdivide_exists_mul. - rewrite divide_c_a in divide_a. - rewrite Z_div_mul' in divide_a by auto. - replace (b * k) with (k * b) in divide_a by ring. - replace (c * k * k0) with (k * (k0 * c)) in divide_a by ring. - rewrite Z.mul_cancel_l in divide_a by (intuition; rewrite H in divide_c_a; ring_simplify in divide_a; intuition). - eapply Zdivide_intro; eauto. -Qed. - -Lemma divide2_even_iff : forall n, (2 | n) <-> Z.even n = true. -Proof. - intro; split. { - intro divide2_n. - Zdivide_exists_mul; [ | pose proof (Z.mod_pos_bound n 2); omega]. - rewrite divide2_n. - apply Z.even_mul. - } { - intro n_even. - pose proof (Zmod_even n). - rewrite n_even in H. - apply Zmod_divide; omega || auto. - } -Qed. - -Lemma prime_odd_or_2 : forall p (prime_p : prime p), p = 2 \/ Z.odd p = true. -Proof. - intros. - apply Decidable.imp_not_l; try apply Z.eq_decidable. - intros p_neq2. - pose proof (Zmod_odd p) as mod_odd. - destruct (Sumbool.sumbool_of_bool (Z.odd p)) as [? | p_not_odd]; auto. - rewrite p_not_odd in mod_odd. - apply Zmod_divides in mod_odd; try omega. - destruct mod_odd as [c c_id]. - rewrite Z.mul_comm in c_id. - apply Zdivide_intro in c_id. - apply prime_divisors in c_id; auto. - destruct c_id; [omega | destruct H; [omega | destruct H; auto]]. - pose proof (prime_ge_2 p prime_p); omega. -Qed. - -Lemma mul_div_eq : (forall a m, m > 0 -> m * (a / m) = (a - a mod m))%Z. -Proof. - intros. - rewrite (Z_div_mod_eq a m) at 2 by auto. - ring. -Qed. - -Lemma mul_div_eq' : (forall a m, m > 0 -> (a / m) * m = (a - a mod m))%Z. -Proof. - intros. - rewrite (Z_div_mod_eq a m) at 2 by auto. - ring. -Qed. - -Hint Rewrite mul_div_eq mul_div_eq' using lia : zdiv_to_mod. -Hint Rewrite <- mul_div_eq' using lia : zmod_to_div. - -Ltac prime_bound := match goal with -| [ H : prime ?p |- _ ] => pose proof (prime_ge_2 p H); try omega -end. - -Lemma Zlt_minus_lt_0 : forall n m, m < n -> 0 < n - m. -Proof. - intros; omega. -Qed. - - -Lemma Z_testbit_low : forall n x i, (0 <= i < n) -> - Z.testbit x i = Z.testbit (Z.land x (Z.ones n)) i. -Proof. - intros. - rewrite Z.land_ones by omega. - symmetry. - apply Z.mod_pow2_bits_low. - omega. -Qed. - - -Lemma Z_testbit_shiftl : forall i, (0 <= i) -> forall a b n, (i < n) -> - Z.testbit (a + Z.shiftl b n) i = Z.testbit a i. -Proof. - intros. - erewrite Z_testbit_low; eauto. - rewrite Z.land_ones, Z.shiftl_mul_pow2 by omega. - rewrite Z.mod_add by (pose proof (Z.pow_pos_nonneg 2 n); omega). - auto using Z.mod_pow2_bits_low. -Qed. - -Lemma Z_mod_div_eq0 : forall a b, 0 < b -> (a mod b) / b = 0. -Proof. - intros. - apply Z.div_small. - auto using Z.mod_pos_bound. -Qed. - -Lemma Z_shiftr_add_land : forall n m a b, (n <= m)%nat -> - Z.shiftr ((Z.land a (Z.ones (Z.of_nat n))) + (Z.shiftl b (Z.of_nat n))) (Z.of_nat m) = Z.shiftr b (Z.of_nat (m - n)). -Proof. - intros. - rewrite Z.land_ones by apply Nat2Z.is_nonneg. - rewrite !Z.shiftr_div_pow2 by apply Nat2Z.is_nonneg. - rewrite Z.shiftl_mul_pow2 by apply Nat2Z.is_nonneg. - rewrite (le_plus_minus n m) at 1 by assumption. - rewrite Nat2Z.inj_add. - rewrite Z.pow_add_r by apply Nat2Z.is_nonneg. - rewrite <- Z.div_div by first - [ pose proof (Z.pow_pos_nonneg 2 (Z.of_nat n)); omega - | apply Z.pow_pos_nonneg; omega ]. - rewrite Z.div_add by (pose proof (Z.pow_pos_nonneg 2 (Z.of_nat n)); omega). - rewrite Z_mod_div_eq0 by (pose proof (Z.pow_pos_nonneg 2 (Z.of_nat n)); omega); auto. -Qed. - -Lemma Z_land_add_land : forall n m a b, (m <= n)%nat -> - Z.land ((Z.land a (Z.ones (Z.of_nat n))) + (Z.shiftl b (Z.of_nat n))) (Z.ones (Z.of_nat m)) = Z.land a (Z.ones (Z.of_nat m)). -Proof. - intros. - rewrite !Z.land_ones by apply Nat2Z.is_nonneg. - rewrite Z.shiftl_mul_pow2 by apply Nat2Z.is_nonneg. - replace (b * 2 ^ Z.of_nat n) with - ((b * 2 ^ Z.of_nat (n - m)) * 2 ^ Z.of_nat m) by - (rewrite (le_plus_minus m n) at 2; try assumption; - rewrite Nat2Z.inj_add, Z.pow_add_r by apply Nat2Z.is_nonneg; ring). - rewrite Z.mod_add by (pose proof (Z.pow_pos_nonneg 2 (Z.of_nat m)); omega). - symmetry. apply Znumtheory.Zmod_div_mod; try (apply Z.pow_pos_nonneg; omega). - rewrite (le_plus_minus m n) by assumption. - rewrite Nat2Z.inj_add, Z.pow_add_r by apply Nat2Z.is_nonneg. - apply Z.divide_factor_l. -Qed. - -Lemma Z_pow_gt0 : forall a, 0 < a -> forall b, 0 <= b -> 0 < a ^ b. -Proof. - intros until 1. - apply natlike_ind; try (simpl; omega). - intros. - rewrite Z.pow_succ_r by assumption. - apply Z.mul_pos_pos; assumption. -Qed. - -Lemma div_pow2succ : forall n x, (0 <= x) -> - n / 2 ^ Z.succ x = Z.div2 (n / 2 ^ x). -Proof. - intros. - rewrite Z.pow_succ_r, Z.mul_comm by auto. - rewrite <- Z.div_div by (try apply Z.pow_nonzero; omega). - rewrite Zdiv2_div. - reflexivity. -Qed. - -Lemma shiftr_succ : forall n x, - Z.shiftr n (Z.succ x) = Z.shiftr (Z.shiftr n x) 1. -Proof. - intros. - rewrite Z.shiftr_shiftr by omega. - reflexivity. -Qed. - - -Definition Z_shiftl_by n a := Z.shiftl a n. - -Lemma Z_shiftl_by_mul_pow2 : forall n a, 0 <= n -> Z.mul (2 ^ n) a = Z_shiftl_by n a. -Proof. - intros. - unfold Z_shiftl_by. - rewrite Z.shiftl_mul_pow2 by assumption. - apply Z.mul_comm. -Qed. - -Lemma map_shiftl : forall n l, 0 <= n -> map (Z.mul (2 ^ n)) l = map (Z_shiftl_by n) l. -Proof. - intros; induction l; auto using Z_shiftl_by_mul_pow2. - simpl. - rewrite IHl. - f_equal. - apply Z_shiftl_by_mul_pow2. - assumption. -Qed. - -Lemma Z_odd_mod : forall a b, (b <> 0)%Z -> - Z.odd (a mod b) = if Z.odd b then xorb (Z.odd a) (Z.odd (a / b)) else Z.odd a. -Proof. - intros. - rewrite Zmod_eq_full by assumption. - rewrite <-Z.add_opp_r, Z.odd_add, Z.odd_opp, Z.odd_mul. - case_eq (Z.odd b); intros; rewrite ?Bool.andb_true_r, ?Bool.andb_false_r; auto using Bool.xorb_false_r. -Qed. - -Lemma mod_same_pow : forall a b c, 0 <= c <= b -> a ^ b mod a ^ c = 0. -Proof. - intros. - replace b with (b - c + c) by ring. - rewrite Z.pow_add_r by omega. - apply Z_mod_mult. -Qed. - - Lemma Z_ones_succ : forall x, (0 <= x) -> + Qed. + + Lemma prime_odd_or_2 : forall p (prime_p : prime p), p = 2 \/ Z.odd p = true. + Proof. + intros. + apply Decidable.imp_not_l; try apply Z.eq_decidable. + intros p_neq2. + pose proof (Zmod_odd p) as mod_odd. + destruct (Sumbool.sumbool_of_bool (Z.odd p)) as [? | p_not_odd]; auto. + rewrite p_not_odd in mod_odd. + apply Zmod_divides in mod_odd; try omega. + destruct mod_odd as [c c_id]. + rewrite Z.mul_comm in c_id. + apply Zdivide_intro in c_id. + apply prime_divisors in c_id; auto. + destruct c_id; [omega | destruct H; [omega | destruct H; auto]]. + pose proof (prime_ge_2 p prime_p); omega. + Qed. + + Lemma mul_div_eq : forall a m, m > 0 -> m * (a / m) = (a - a mod m). + Proof. + intros. + rewrite (Z_div_mod_eq a m) at 2 by auto. + ring. + Qed. + + Lemma mul_div_eq' : (forall a m, m > 0 -> (a / m) * m = (a - a mod m))%Z. + Proof. + intros. + rewrite (Z_div_mod_eq a m) at 2 by auto. + ring. + Qed. + + Hint Rewrite mul_div_eq mul_div_eq' using lia : zdiv_to_mod. + Hint Rewrite <- mul_div_eq' using lia : zmod_to_div. + + Ltac prime_bound := match goal with + | [ H : prime ?p |- _ ] => pose proof (prime_ge_2 p H); try omega + end. + + Lemma testbit_low : forall n x i, (0 <= i < n) -> + Z.testbit x i = Z.testbit (Z.land x (Z.ones n)) i. + Proof. + intros. + rewrite Z.land_ones by omega. + symmetry. + apply Z.mod_pow2_bits_low. + omega. + Qed. + + + Lemma testbit_shiftl : forall i, (0 <= i) -> forall a b n, (i < n) -> + Z.testbit (a + Z.shiftl b n) i = Z.testbit a i. + Proof. + intros. + erewrite Z.testbit_low; eauto. + rewrite Z.land_ones, Z.shiftl_mul_pow2 by omega. + rewrite Z.mod_add by (pose proof (Z.pow_pos_nonneg 2 n); omega). + auto using Z.mod_pow2_bits_low. + Qed. + + Lemma mod_div_eq0 : forall a b, 0 < b -> (a mod b) / b = 0. + Proof. + intros. + apply Z.div_small. + auto using Z.mod_pos_bound. + Qed. + Hint Rewrite mod_div_eq0 using lia : zsimplify. + + Lemma shiftr_add_land : forall n m a b, (n <= m)%nat -> + Z.shiftr ((Z.land a (Z.ones (Z.of_nat n))) + (Z.shiftl b (Z.of_nat n))) (Z.of_nat m) = Z.shiftr b (Z.of_nat (m - n)). + Proof. + intros. + rewrite Z.land_ones by apply Nat2Z.is_nonneg. + rewrite !Z.shiftr_div_pow2 by apply Nat2Z.is_nonneg. + rewrite Z.shiftl_mul_pow2 by apply Nat2Z.is_nonneg. + rewrite (le_plus_minus n m) at 1 by assumption. + rewrite Nat2Z.inj_add. + rewrite Z.pow_add_r by apply Nat2Z.is_nonneg. + rewrite <- Z.div_div by first + [ pose proof (Z.pow_pos_nonneg 2 (Z.of_nat n)); omega + | apply Z.pow_pos_nonneg; omega ]. + rewrite Z.div_add by (pose proof (Z.pow_pos_nonneg 2 (Z.of_nat n)); omega). + rewrite mod_div_eq0 by (pose proof (Z.pow_pos_nonneg 2 (Z.of_nat n)); omega); auto. + Qed. + + Lemma land_add_land : forall n m a b, (m <= n)%nat -> + Z.land ((Z.land a (Z.ones (Z.of_nat n))) + (Z.shiftl b (Z.of_nat n))) (Z.ones (Z.of_nat m)) = Z.land a (Z.ones (Z.of_nat m)). + Proof. + intros. + rewrite !Z.land_ones by apply Nat2Z.is_nonneg. + rewrite Z.shiftl_mul_pow2 by apply Nat2Z.is_nonneg. + replace (b * 2 ^ Z.of_nat n) with + ((b * 2 ^ Z.of_nat (n - m)) * 2 ^ Z.of_nat m) by + (rewrite (le_plus_minus m n) at 2; try assumption; + rewrite Nat2Z.inj_add, Z.pow_add_r by apply Nat2Z.is_nonneg; ring). + rewrite Z.mod_add by (pose proof (Z.pow_pos_nonneg 2 (Z.of_nat m)); omega). + symmetry. apply Znumtheory.Zmod_div_mod; try (apply Z.pow_pos_nonneg; omega). + rewrite (le_plus_minus m n) by assumption. + rewrite Nat2Z.inj_add, Z.pow_add_r by apply Nat2Z.is_nonneg. + apply Z.divide_factor_l. + Qed. + + Lemma div_pow2succ : forall n x, (0 <= x) -> + n / 2 ^ Z.succ x = Z.div2 (n / 2 ^ x). + Proof. + intros. + rewrite Z.pow_succ_r, Z.mul_comm by auto. + rewrite <- Z.div_div by (try apply Z.pow_nonzero; omega). + rewrite Zdiv2_div. + reflexivity. + Qed. + + Lemma shiftr_succ : forall n x, + Z.shiftr n (Z.succ x) = Z.shiftr (Z.shiftr n x) 1. + Proof. + intros. + rewrite Z.shiftr_shiftr by omega. + reflexivity. + Qed. + + + Definition shiftl_by n a := Z.shiftl a n. + + Lemma shiftl_by_mul_pow2 : forall n a, 0 <= n -> Z.mul (2 ^ n) a = Z.shiftl_by n a. + Proof. + intros. + unfold Z.shiftl_by. + rewrite Z.shiftl_mul_pow2 by assumption. + apply Z.mul_comm. + Qed. + + Lemma map_shiftl : forall n l, 0 <= n -> map (Z.mul (2 ^ n)) l = map (Z.shiftl_by n) l. + Proof. + intros; induction l; auto using Z.shiftl_by_mul_pow2. + simpl. + rewrite IHl. + f_equal. + apply Z.shiftl_by_mul_pow2. + assumption. + Qed. + + Lemma odd_mod : forall a b, (b <> 0)%Z -> + Z.odd (a mod b) = if Z.odd b then xorb (Z.odd a) (Z.odd (a / b)) else Z.odd a. + Proof. + intros. + rewrite Zmod_eq_full by assumption. + rewrite <-Z.add_opp_r, Z.odd_add, Z.odd_opp, Z.odd_mul. + case_eq (Z.odd b); intros; rewrite ?Bool.andb_true_r, ?Bool.andb_false_r; auto using Bool.xorb_false_r. + Qed. + + Lemma mod_same_pow : forall a b c, 0 <= c <= b -> a ^ b mod a ^ c = 0. + Proof. + intros. + replace b with (b - c + c) by ring. + rewrite Z.pow_add_r by omega. + apply Z_mod_mult. + Qed. + Hint Rewrite mod_same_pow using lia : zsimplify. + + Lemma ones_succ : forall x, (0 <= x) -> Z.ones (Z.succ x) = 2 ^ x + Z.ones x. Proof. unfold Z.ones; intros. @@ -365,14 +346,14 @@ Qed. rewrite Z.pow_succ_r; omega. Qed. - Lemma Z_div_floor : forall a b c, 0 < b -> a < b * (Z.succ c) -> a / b <= c. + Lemma div_floor : forall a b c, 0 < b -> a < b * (Z.succ c) -> a / b <= c. Proof. intros. apply Z.lt_succ_r. apply Z.div_lt_upper_bound; try omega. Qed. - Lemma Z_shiftr_1_r_le : forall a b, a <= b -> + Lemma shiftr_1_r_le : forall a b, a <= b -> Z.shiftr a 1 <= Z.shiftr b 1. Proof. intros. @@ -380,7 +361,7 @@ Qed. apply Z.div_le_mono; omega. Qed. - Lemma Z_ones_pred : forall i, 0 < i -> Z.ones (Z.pred i) = Z.shiftr (Z.ones i) 1. + Lemma ones_pred : forall i, 0 < i -> Z.ones (Z.pred i) = Z.shiftr (Z.ones i) 1. Proof. induction i; [ | | pose proof (Pos2Z.neg_is_neg p) ]; try omega. intros. @@ -394,7 +375,7 @@ Qed. f_equal. omega. Qed. - Lemma Z_shiftr_ones' : forall a n, 0 <= a < 2 ^ n -> forall i, (0 <= i) -> + Lemma shiftr_ones' : forall a n, 0 <= a < 2 ^ n -> forall i, (0 <= i) -> Z.shiftr a i <= Z.ones (n - i) \/ n <= i. Proof. intros until 1. @@ -408,17 +389,17 @@ Qed. left. rewrite shiftr_succ. replace (n - Z.succ x) with (Z.pred (n - x)) by omega. - rewrite Z_ones_pred by omega. - apply Z_shiftr_1_r_le. + rewrite Z.ones_pred by omega. + apply Z.shiftr_1_r_le. assumption. Qed. - Lemma Z_shiftr_ones : forall a n i, 0 <= a < 2 ^ n -> (0 <= i) -> (i <= n) -> + Lemma shiftr_ones : forall a n i, 0 <= a < 2 ^ n -> (0 <= i) -> (i <= n) -> Z.shiftr a i <= Z.ones (n - i) . Proof. intros a n i G G0 G1. destruct (Z_le_lt_eq_dec i n G1). - + destruct (Z_shiftr_ones' a n G i G0); omega. + + destruct (Z.shiftr_ones' a n G i G0); omega. + subst; rewrite Z.sub_diag. destruct (Z_eq_dec a 0). - subst; rewrite Z.shiftr_0_l; reflexivity. @@ -426,7 +407,7 @@ Qed. apply Z.log2_lt_pow2; omega. Qed. - Lemma Z_shiftr_upper_bound : forall a n, 0 <= n -> 0 <= a <= 2 ^ n -> Z.shiftr a n <= 1. + Lemma shiftr_upper_bound : forall a n, 0 <= n -> 0 <= a <= 2 ^ n -> Z.shiftr a n <= 1. Proof. intros a ? ? [a_nonneg a_upper_bound]. apply Z_le_lt_eq_dec in a_upper_bound. @@ -442,439 +423,445 @@ Qed. omega. Qed. -(* prove that combinations of known positive/nonnegative numbers are positive/nonnegative *) -Ltac zero_bounds' := - repeat match goal with - | [ |- 0 <= _ + _] => apply Z.add_nonneg_nonneg - | [ |- 0 <= _ - _] => apply Z.le_0_sub - | [ |- 0 <= _ * _] => apply Z.mul_nonneg_nonneg - | [ |- 0 <= _ / _] => apply Z.div_pos - | [ |- 0 <= _ ^ _ ] => apply Z.pow_nonneg - | [ |- 0 <= Z.shiftr _ _] => apply Z.shiftr_nonneg - | [ |- 0 < _ + _] => try solve [apply Z.add_pos_nonneg; zero_bounds']; - try solve [apply Z.add_nonneg_pos; zero_bounds'] - | [ |- 0 < _ - _] => apply Z.lt_0_sub - | [ |- 0 < _ * _] => apply Z.lt_0_mul; left; split - | [ |- 0 < _ / _] => apply Z.div_str_pos - | [ |- 0 < _ ^ _ ] => apply Z.pow_pos_nonneg - end; try omega; try prime_bound; auto. - -Ltac zero_bounds := try omega; try prime_bound; zero_bounds'. - -Hint Extern 1 => progress zero_bounds : zero_bounds. - -Lemma Z_ones_nonneg : forall i, (0 <= i) -> 0 <= Z.ones i. -Proof. - apply natlike_ind. - + unfold Z.ones. simpl; omega. - + intros. - rewrite Z_ones_succ by assumption. - zero_bounds. -Qed. - -Lemma Z_ones_pos_pos : forall i, (0 < i) -> 0 < Z.ones i. -Proof. - intros. - unfold Z.ones. - rewrite Z.shiftl_1_l. - apply Z.lt_succ_lt_pred. - apply Z.pow_gt_1; omega. -Qed. - -Lemma N_le_1_l : forall p, (1 <= N.pos p)%N. -Proof. - destruct p; cbv; congruence. -Qed. - -Lemma Pos_land_upper_bound_l : forall a b, (Pos.land a b <= N.pos a)%N. -Proof. - induction a; destruct b; intros; try solve [cbv; congruence]; - simpl; specialize (IHa b); case_eq (Pos.land a b); intro; simpl; - try (apply N_le_1_l || apply N.le_0_l); intro land_eq; - rewrite land_eq in *; unfold N.le, N.compare in *; - rewrite ?Pos.compare_xI_xI, ?Pos.compare_xO_xI, ?Pos.compare_xO_xO; - try assumption. - destruct (p ?=a)%positive; cbv; congruence. -Qed. - -Lemma Z_land_upper_bound_l : forall a b, (0 <= a) -> (0 <= b) -> - Z.land a b <= a. -Proof. - intros. - destruct a, b; try solve [exfalso; auto]; try solve [cbv; congruence]. - cbv [Z.land]. - rewrite <-N2Z.inj_pos, <-N2Z.inj_le. - auto using Pos_land_upper_bound_l. -Qed. - -Lemma Z_land_upper_bound_r : forall a b, (0 <= a) -> (0 <= b) -> - Z.land a b <= b. -Proof. - intros. - rewrite Z.land_comm. - auto using Z_land_upper_bound_l. -Qed. - -Lemma Z_le_fold_right_max : forall low l x, (forall y, In y l -> low <= y) -> - In x l -> x <= fold_right Z.max low l. -Proof. - induction l; intros ? lower_bound In_list; [cbv [In] in *; intuition | ]. - simpl. - destruct (in_inv In_list); subst. - + apply Z.le_max_l. - + etransitivity. - - apply IHl; auto; intuition. - - apply Z.le_max_r. -Qed. - -Lemma Z_le_fold_right_max_initial : forall low l, low <= fold_right Z.max low l. -Proof. - induction l; intros; try reflexivity. - etransitivity; [ apply IHl | apply Z.le_max_r ]. -Qed. - -Ltac Zltb_to_Zlt := - repeat match goal with - | [ H : (?x <? ?y) = ?b |- _ ] - => let H' := fresh in - rename H into H'; - pose proof (Zlt_cases x y) as H; - rewrite H' in H; - clear H' - end. - -Ltac Zcompare_to_sgn := - repeat match goal with - | [ H : _ |- _ ] => progress rewrite <- ?Z.sgn_neg_iff, <- ?Z.sgn_pos_iff, <- ?Z.sgn_null_iff in H - | _ => progress rewrite <- ?Z.sgn_neg_iff, <- ?Z.sgn_pos_iff, <- ?Z.sgn_null_iff - end. - -Local Ltac replace_to_const c := - repeat match goal with - | [ H : ?x = ?x |- _ ] => clear H - | [ H : ?x = c, H' : context[?x] |- _ ] => rewrite H in H' - | [ H : c = ?x, H' : context[?x] |- _ ] => rewrite <- H in H' - | [ H : ?x = c |- context[?x] ] => rewrite H - | [ H : c = ?x |- context[?x] ] => rewrite <- H - end. - -Lemma Zlt_div_0 n m : n / m < 0 <-> ((n < 0 < m \/ m < 0 < n) /\ 0 < -(n / m)). -Proof. - Zcompare_to_sgn; rewrite Z.sgn_opp; simpl. - pose proof (Zdiv_sgn n m) as H. - pose proof (Z.sgn_spec (n / m)) as H'. - repeat first [ progress intuition - | progress simpl in * - | congruence - | lia - | progress replace_to_const (-1) - | progress replace_to_const 0 - | progress replace_to_const 1 - | match goal with - | [ x : Z |- _ ] => destruct x - end ]. -Qed. - -Lemma two_times_x_minus_x x : 2 * x - x = x. -Proof. lia. Qed. - -Lemma Zmul_div_le x y z - (Hx : 0 <= x) (Hy : 0 <= y) (Hz : 0 < z) - (Hyz : y <= z) - : x * y / z <= x. -Proof. - transitivity (x * z / z); [ | rewrite Z.div_mul by lia; lia ]. - apply Z_div_le; nia. -Qed. - -Lemma Zdiv_mul_diff a b c - (Ha : 0 <= a) (Hb : 0 < b) (Hc : 0 <= c) - : c * a / b - c * (a / b) <= c. -Proof. - pose proof (Z.mod_pos_bound a b). - etransitivity; [ | apply (Zmul_div_le c (a mod b) b); lia ]. - rewrite (Z_div_mod_eq a b) at 1 by lia. - rewrite Z.mul_add_distr_l. - replace (c * (b * (a / b))) with ((c * (a / b)) * b) by lia. - rewrite Z.div_add_l by lia. - lia. -Qed. - -Lemma Zdiv_mul_le_le a b c - : 0 <= a -> 0 < b -> 0 <= c -> c * (a / b) <= c * a / b <= c * (a / b) + c. -Proof. - pose proof (Zdiv_mul_diff a b c); split; try apply Z.div_mul_le; lia. -Qed. - -Lemma Zdiv_mul_le_le_offset a b c - : 0 <= a -> 0 < b -> 0 <= c -> c * a / b - c <= c * (a / b). -Proof. - pose proof (Zdiv_mul_le_le a b c); lia. -Qed. - -Hint Resolve Zmult_le_compat_r Zmult_le_compat_l Z_div_le Zdiv_mul_le_le_offset Z.add_le_mono Z.sub_le_mono : zarith. - -(** * [Zsimplify_fractions_le] *) -(** The culmination of this series of tactics, - [Zsimplify_fractions_le], will use the fact that [a * (b / c) <= - (a * b) / c], and do some reasoning modulo associativity and - commutativity in [Z] to perform such a reduction. It may leave - over goals if it cannot prove that some denominators are non-zero. - If the rewrite [a * (b / c)] → [(a * b) / c] is safe to do on the - LHS of the goal, this tactic should not turn a solvable goal into - an unsolvable one. - - After running, the tactic does some basic rewriting to simplify - fractions, e.g., that [a * b / b = a]. *) -Ltac Zsplit_sums_step := - match goal with - | [ |- _ + _ <= _ ] - => etransitivity; [ eapply Z.add_le_mono | ] - | [ |- _ - _ <= _ ] - => etransitivity; [ eapply Z.sub_le_mono | ] - end. -Ltac Zsplit_sums := - try (Zsplit_sums_step; [ Zsplit_sums.. | ]). -Ltac Zpre_reorder_fractions_step := - match goal with - | [ |- context[?x / ?y * ?z] ] - => rewrite (Z.mul_comm (x / y) z) - | _ => let LHS := match goal with |- ?LHS <= ?RHS => LHS end in - match LHS with - | context G[?x * (?y / ?z)] - => let G' := context G[(x * y) / z] in - transitivity G' - end - end. -Ltac Zpre_reorder_fractions := - try first [ Zsplit_sums_step; [ Zpre_reorder_fractions.. | ] - | Zpre_reorder_fractions_step; [ .. | Zpre_reorder_fractions ] ]. -Ltac Zsplit_comparison := - match goal with - | [ |- ?x <= ?x ] => reflexivity - | [ H : _ >= _ |- _ ] - => apply Z.ge_le_iff in H - | [ |- ?x * ?y <= ?z * ?w ] - => lazymatch goal with - | [ H : 0 <= x |- _ ] => idtac - | [ H : x < 0 |- _ ] => fail - | _ => destruct (Z_lt_le_dec x 0) - end; - [ .. - | lazymatch goal with - | [ H : 0 <= y |- _ ] => idtac - | [ H : y < 0 |- _ ] => fail - | _ => destruct (Z_lt_le_dec y 0) + (* prove that combinations of known positive/nonnegative numbers are positive/nonnegative *) + Ltac zero_bounds' := + repeat match goal with + | [ |- 0 <= _ + _] => apply Z.add_nonneg_nonneg + | [ |- 0 <= _ - _] => apply Z.le_0_sub + | [ |- 0 <= _ * _] => apply Z.mul_nonneg_nonneg + | [ |- 0 <= _ / _] => apply Z.div_pos + | [ |- 0 <= _ ^ _ ] => apply Z.pow_nonneg + | [ |- 0 <= Z.shiftr _ _] => apply Z.shiftr_nonneg + | [ |- 0 < _ + _] => try solve [apply Z.add_pos_nonneg; zero_bounds']; + try solve [apply Z.add_nonneg_pos; zero_bounds'] + | [ |- 0 < _ - _] => apply Z.lt_0_sub + | [ |- 0 < _ * _] => apply Z.lt_0_mul; left; split + | [ |- 0 < _ / _] => apply Z.div_str_pos + | [ |- 0 < _ ^ _ ] => apply Z.pow_pos_nonneg + end; try omega; try prime_bound; auto. + + Ltac zero_bounds := try omega; try prime_bound; zero_bounds'. + + Hint Extern 1 => progress zero_bounds : zero_bounds. + + Lemma ones_nonneg : forall i, (0 <= i) -> 0 <= Z.ones i. + Proof. + apply natlike_ind. + + unfold Z.ones. simpl; omega. + + intros. + rewrite Z.ones_succ by assumption. + zero_bounds. + Qed. + + Lemma ones_pos_pos : forall i, (0 < i) -> 0 < Z.ones i. + Proof. + intros. + unfold Z.ones. + rewrite Z.shiftl_1_l. + apply Z.lt_succ_lt_pred. + apply Z.pow_gt_1; omega. + Qed. + + Lemma N_le_1_l : forall p, (1 <= N.pos p)%N. + Proof. + destruct p; cbv; congruence. + Qed. + + Lemma Pos_land_upper_bound_l : forall a b, (Pos.land a b <= N.pos a)%N. + Proof. + induction a; destruct b; intros; try solve [cbv; congruence]; + simpl; specialize (IHa b); case_eq (Pos.land a b); intro; simpl; + try (apply N_le_1_l || apply N.le_0_l); intro land_eq; + rewrite land_eq in *; unfold N.le, N.compare in *; + rewrite ?Pos.compare_xI_xI, ?Pos.compare_xO_xI, ?Pos.compare_xO_xO; + try assumption. + destruct (p ?=a)%positive; cbv; congruence. + Qed. + + Lemma land_upper_bound_l : forall a b, (0 <= a) -> (0 <= b) -> + Z.land a b <= a. + Proof. + intros. + destruct a, b; try solve [exfalso; auto]; try solve [cbv; congruence]. + cbv [Z.land]. + rewrite <-N2Z.inj_pos, <-N2Z.inj_le. + auto using Pos_land_upper_bound_l. + Qed. + + Lemma land_upper_bound_r : forall a b, (0 <= a) -> (0 <= b) -> + Z.land a b <= b. + Proof. + intros. + rewrite Z.land_comm. + auto using Z.land_upper_bound_l. + Qed. + + Lemma le_fold_right_max : forall low l x, (forall y, In y l -> low <= y) -> + In x l -> x <= fold_right Z.max low l. + Proof. + induction l; intros ? lower_bound In_list; [cbv [In] in *; intuition | ]. + simpl. + destruct (in_inv In_list); subst. + + apply Z.le_max_l. + + etransitivity. + - apply IHl; auto; intuition. + - apply Z.le_max_r. + Qed. + + Lemma le_fold_right_max_initial : forall low l, low <= fold_right Z.max low l. + Proof. + induction l; intros; try reflexivity. + etransitivity; [ apply IHl | apply Z.le_max_r ]. + Qed. + + Ltac ltb_to_lt := + repeat match goal with + | [ H : (?x <? ?y) = ?b |- _ ] + => let H' := fresh in + rename H into H'; + pose proof (Zlt_cases x y) as H; + rewrite H' in H; + clear H' + end. + + Ltac compare_to_sgn := + repeat match goal with + | [ H : _ |- _ ] => progress rewrite <- ?Z.sgn_neg_iff, <- ?Z.sgn_pos_iff, <- ?Z.sgn_null_iff in H + | _ => progress rewrite <- ?Z.sgn_neg_iff, <- ?Z.sgn_pos_iff, <- ?Z.sgn_null_iff + end. + + Local Ltac replace_to_const c := + repeat match goal with + | [ H : ?x = ?x |- _ ] => clear H + | [ H : ?x = c, H' : context[?x] |- _ ] => rewrite H in H' + | [ H : c = ?x, H' : context[?x] |- _ ] => rewrite <- H in H' + | [ H : ?x = c |- context[?x] ] => rewrite H + | [ H : c = ?x |- context[?x] ] => rewrite <- H + end. + + Lemma lt_div_0 n m : n / m < 0 <-> ((n < 0 < m \/ m < 0 < n) /\ 0 < -(n / m)). + Proof. + Z.compare_to_sgn; rewrite Z.sgn_opp; simpl. + pose proof (Zdiv_sgn n m) as H. + pose proof (Z.sgn_spec (n / m)) as H'. + repeat first [ progress intuition + | progress simpl in * + | congruence + | lia + | progress replace_to_const (-1) + | progress replace_to_const 0 + | progress replace_to_const 1 + | match goal with + | [ x : Z |- _ ] => destruct x + end ]. + Qed. + + Lemma two_times_x_minus_x x : 2 * x - x = x. + Proof. lia. Qed. + + Lemma mul_div_le x y z + (Hx : 0 <= x) (Hy : 0 <= y) (Hz : 0 < z) + (Hyz : y <= z) + : x * y / z <= x. + Proof. + transitivity (x * z / z); [ | rewrite Z.div_mul by lia; lia ]. + apply Z_div_le; nia. + Qed. + + Lemma div_mul_diff a b c + (Ha : 0 <= a) (Hb : 0 < b) (Hc : 0 <= c) + : c * a / b - c * (a / b) <= c. + Proof. + pose proof (Z.mod_pos_bound a b). + etransitivity; [ | apply (mul_div_le c (a mod b) b); lia ]. + rewrite (Z_div_mod_eq a b) at 1 by lia. + rewrite Z.mul_add_distr_l. + replace (c * (b * (a / b))) with ((c * (a / b)) * b) by lia. + rewrite Z.div_add_l by lia. + lia. + Qed. + + Lemma div_mul_le_le a b c + : 0 <= a -> 0 < b -> 0 <= c -> c * (a / b) <= c * a / b <= c * (a / b) + c. + Proof. + pose proof (Z.div_mul_diff a b c); split; try apply Z.div_mul_le; lia. + Qed. + + Lemma div_mul_le_le_offset a b c + : 0 <= a -> 0 < b -> 0 <= c -> c * a / b - c <= c * (a / b). + Proof. + pose proof (Z.div_mul_le_le a b c); lia. + Qed. + + Hint Resolve Zmult_le_compat_r Zmult_le_compat_l Z_div_le Z.div_mul_le_le_offset Z.add_le_mono Z.sub_le_mono : zarith. + + (** * [Z.simplify_fractions_le] *) + (** The culmination of this series of tactics, + [Z.simplify_fractions_le], will use the fact that [a * (b / c) <= + (a * b) / c], and do some reasoning modulo associativity and + commutativity in [Z] to perform such a reduction. It may leave + over goals if it cannot prove that some denominators are non-zero. + If the rewrite [a * (b / c)] → [(a * b) / c] is safe to do on the + LHS of the goal, this tactic should not turn a solvable goal into + an unsolvable one. + + After running, the tactic does some basic rewriting to simplify + fractions, e.g., that [a * b / b = a]. *) + Ltac split_sums_step := + match goal with + | [ |- _ + _ <= _ ] + => etransitivity; [ eapply Z.add_le_mono | ] + | [ |- _ - _ <= _ ] + => etransitivity; [ eapply Z.sub_le_mono | ] + end. + Ltac split_sums := + try (split_sums_step; [ split_sums.. | ]). + Ltac pre_reorder_fractions_step := + match goal with + | [ |- context[?x / ?y * ?z] ] + => rewrite (Z.mul_comm (x / y) z) + | _ => let LHS := match goal with |- ?LHS <= ?RHS => LHS end in + match LHS with + | context G[?x * (?y / ?z)] + => let G' := context G[(x * y) / z] in + transitivity G' + end + end. + Ltac pre_reorder_fractions := + try first [ split_sums_step; [ pre_reorder_fractions.. | ] + | pre_reorder_fractions_step; [ .. | pre_reorder_fractions ] ]. + Ltac split_comparison := + match goal with + | [ |- ?x <= ?x ] => reflexivity + | [ H : _ >= _ |- _ ] + => apply Z.ge_le_iff in H + | [ |- ?x * ?y <= ?z * ?w ] + => lazymatch goal with + | [ H : 0 <= x |- _ ] => idtac + | [ H : x < 0 |- _ ] => fail + | _ => destruct (Z_lt_le_dec x 0) end; [ .. - | apply Zmult_le_compat; [ | | assumption | assumption ] ] ] - | [ |- ?x / ?y <= ?z / ?y ] - => lazymatch goal with - | [ H : 0 < y |- _ ] => idtac - | [ H : y <= 0 |- _ ] => fail - | _ => destruct (Z_lt_le_dec 0 y) - end; - [ apply Z_div_le; [ apply gt_lt_symmetry; assumption | ] - | .. ] - | [ |- ?x / ?y <= ?x / ?z ] - => lazymatch goal with - | [ H : 0 <= x |- _ ] => idtac - | [ H : x < 0 |- _ ] => fail - | _ => destruct (Z_lt_le_dec x 0) - end; - [ .. - | lazymatch goal with - | [ H : 0 < z |- _ ] => idtac - | [ H : z <= 0 |- _ ] => fail - | _ => destruct (Z_lt_le_dec 0 z) + | lazymatch goal with + | [ H : 0 <= y |- _ ] => idtac + | [ H : y < 0 |- _ ] => fail + | _ => destruct (Z_lt_le_dec y 0) + end; + [ .. + | apply Zmult_le_compat; [ | | assumption | assumption ] ] ] + | [ |- ?x / ?y <= ?z / ?y ] + => lazymatch goal with + | [ H : 0 < y |- _ ] => idtac + | [ H : y <= 0 |- _ ] => fail + | _ => destruct (Z_lt_le_dec 0 y) end; - [ apply Z.div_le_compat_l; [ assumption | split; [ assumption | ] ] - | .. ] ] - | [ |- _ + _ <= _ + _ ] - => apply Z.add_le_mono - | [ |- _ - _ <= _ - _ ] - => apply Z.sub_le_mono - | [ |- ?x * (?y / ?z) <= (?x * ?y) / ?z ] - => apply Z.div_mul_le - end. -Ltac Zsplit_comparison_fin_step := - match goal with - | _ => assumption - | _ => lia - | _ => progress subst - | [ H : ?n * ?m < 0 |- _ ] - => apply (proj1 (Z.lt_mul_0 n m)) in H; destruct H as [[??]|[??]] - | [ H : ?n / ?m < 0 |- _ ] - => apply (proj1 (Zlt_div_0 n m)) in H; destruct H as [[[??]|[??]]?] - | [ H : (?x^?y) <= ?n < _, H' : ?n < 0 |- _ ] - => assert (0 <= x^y) by zero_bounds; lia - | [ H : (?x^?y) < 0 |- _ ] - => assert (0 <= x^y) by zero_bounds; lia - | [ H : (?x^?y) <= 0 |- _ ] - => let H' := fresh in - assert (H' : 0 <= x^y) by zero_bounds; - assert (x^y = 0) by lia; - clear H H' - | [ H : _^_ = 0 |- _ ] - => apply Z.pow_eq_0_iff in H; destruct H as [?|[??]] - | [ H : 0 <= ?x, H' : ?x - 1 < 0 |- _ ] - => assert (x = 0) by lia; clear H H' - | [ |- ?x <= ?y ] => is_evar x; reflexivity - | [ |- ?x <= ?y ] => is_evar y; reflexivity - end. -Ltac Zsplit_comparison_fin := repeat Zsplit_comparison_fin_step. -Ltac Zsimplify_fractions_step := - match goal with - | _ => rewrite Z.div_mul by (try apply Z.pow_nonzero; zero_bounds) - | [ |- context[?x * ?y / ?x] ] - => rewrite (Z.mul_comm x y) - | [ |- ?x <= ?x ] => reflexivity - end. -Ltac Zsimplify_fractions := repeat Zsimplify_fractions_step. -Ltac Zsimplify_fractions_le := - Zpre_reorder_fractions; - [ repeat Zsplit_comparison; Zsplit_comparison_fin; zero_bounds.. - | Zsimplify_fractions ]. - -Lemma Zlog2_nonneg' n a : n <= 0 -> n <= Z.log2 a. -Proof. - intros; transitivity 0; auto with zarith. -Qed. + [ apply Z_div_le; [ apply Z.gt_lt_iff; assumption | ] + | .. ] + | [ |- ?x / ?y <= ?x / ?z ] + => lazymatch goal with + | [ H : 0 <= x |- _ ] => idtac + | [ H : x < 0 |- _ ] => fail + | _ => destruct (Z_lt_le_dec x 0) + end; + [ .. + | lazymatch goal with + | [ H : 0 < z |- _ ] => idtac + | [ H : z <= 0 |- _ ] => fail + | _ => destruct (Z_lt_le_dec 0 z) + end; + [ apply Z.div_le_compat_l; [ assumption | split; [ assumption | ] ] + | .. ] ] + | [ |- _ + _ <= _ + _ ] + => apply Z.add_le_mono + | [ |- _ - _ <= _ - _ ] + => apply Z.sub_le_mono + | [ |- ?x * (?y / ?z) <= (?x * ?y) / ?z ] + => apply Z.div_mul_le + end. + Ltac split_comparison_fin_step := + match goal with + | _ => assumption + | _ => lia + | _ => progress subst + | [ H : ?n * ?m < 0 |- _ ] + => apply (proj1 (Z.lt_mul_0 n m)) in H; destruct H as [[??]|[??]] + | [ H : ?n / ?m < 0 |- _ ] + => apply (proj1 (lt_div_0 n m)) in H; destruct H as [[[??]|[??]]?] + | [ H : (?x^?y) <= ?n < _, H' : ?n < 0 |- _ ] + => assert (0 <= x^y) by zero_bounds; lia + | [ H : (?x^?y) < 0 |- _ ] + => assert (0 <= x^y) by zero_bounds; lia + | [ H : (?x^?y) <= 0 |- _ ] + => let H' := fresh in + assert (H' : 0 <= x^y) by zero_bounds; + assert (x^y = 0) by lia; + clear H H' + | [ H : _^_ = 0 |- _ ] + => apply Z.pow_eq_0_iff in H; destruct H as [?|[??]] + | [ H : 0 <= ?x, H' : ?x - 1 < 0 |- _ ] + => assert (x = 0) by lia; clear H H' + | [ |- ?x <= ?y ] => is_evar x; reflexivity + | [ |- ?x <= ?y ] => is_evar y; reflexivity + end. + Ltac split_comparison_fin := repeat split_comparison_fin_step. + Ltac simplify_fractions_step := + match goal with + | _ => rewrite Z.div_mul by (try apply Z.pow_nonzero; zero_bounds) + | [ |- context[?x * ?y / ?x] ] + => rewrite (Z.mul_comm x y) + | [ |- ?x <= ?x ] => reflexivity + end. + Ltac simplify_fractions := repeat simplify_fractions_step. + Ltac simplify_fractions_le := + pre_reorder_fractions; + [ repeat split_comparison; split_comparison_fin; zero_bounds.. + | simplify_fractions ]. + + Lemma log2_nonneg' n a : n <= 0 -> n <= Z.log2 a. + Proof. + intros; transitivity 0; auto with zarith. + Qed. + + Hint Resolve log2_nonneg' : zarith. -Hint Resolve Zlog2_nonneg' : zarith. + (** We create separate databases for two directions of transformations + involving [Z.log2]; combining them leads to loops. *) + (* for hints that take in hypotheses of type [log2 _], and spit out conclusions of type [_ ^ _] *) + Create HintDb hyp_log2. -(** We create separate databases for two directions of transformations - involving [Z.log2]; combining them leads to loops. *) -(* for hints that take in hypotheses of type [log2 _], and spit out conclusions of type [_ ^ _] *) -Create HintDb hyp_log2. + (* for hints that take in hypotheses of type [_ ^ _], and spit out conclusions of type [log2 _] *) + Create HintDb concl_log2. -(* for hints that take in hypotheses of type [_ ^ _], and spit out conclusions of type [log2 _] *) -Create HintDb concl_log2. + Hint Resolve (fun a b H => proj1 (Z.log2_lt_pow2 a b H)) (fun a b H => proj1 (Z.log2_le_pow2 a b H)) : concl_log2. + Hint Resolve (fun a b H => proj2 (Z.log2_lt_pow2 a b H)) (fun a b H => proj2 (Z.log2_le_pow2 a b H)) : hyp_log2. -Hint Resolve (fun a b H => proj1 (Z.log2_lt_pow2 a b H)) (fun a b H => proj1 (Z.log2_le_pow2 a b H)) : concl_log2. -Hint Resolve (fun a b H => proj2 (Z.log2_lt_pow2 a b H)) (fun a b H => proj2 (Z.log2_le_pow2 a b H)) : hyp_log2. + Lemma le_lt_to_log2 x y z : 0 <= z -> 0 < y -> 2^x <= y < 2^z -> x <= Z.log2 y < z. + Proof. + destruct (Z_le_gt_dec 0 x); auto with concl_log2 lia. + Qed. -Lemma Zle_lt_to_log2 x y z : 0 <= z -> 0 < y -> 2^x <= y < 2^z -> x <= Z.log2 y < z. -Proof. - destruct (Z_le_gt_dec 0 x); auto with concl_log2 lia. -Qed. + Lemma div_x_y_x x y : 0 < x -> 0 < y -> x / y / x = 1 / y. + Proof. + intros; rewrite Z.div_div, (Z.mul_comm y x), <- Z.div_div, Z.div_same by lia. + reflexivity. + Qed. -Lemma Zdiv_x_y_x x y : 0 < x -> 0 < y -> x / y / x = 1 / y. -Proof. - intros; rewrite Z.div_div, (Z.mul_comm y x), <- Z.div_div, Z.div_same by lia. - reflexivity. -Qed. + Hint Rewrite div_x_y_x using lia : zsimplify. -Hint Rewrite Zdiv_x_y_x using lia : zsimplify. + Lemma mod_opp_l_z_iff a b (H : b <> 0) : a mod b = 0 <-> (-a) mod b = 0. + Proof. + split; intro H'; apply Z.mod_opp_l_z in H'; rewrite ?Z.opp_involutive in H'; assumption. + Qed. -Lemma Zmod_opp_l_z_iff a b (H : b <> 0) : a mod b = 0 <-> (-a) mod b = 0. -Proof. - split; intro H'; apply Z.mod_opp_l_z in H'; rewrite ?Z.opp_involutive in H'; assumption. -Qed. + Lemma opp_eq_0_iff a : -a = 0 <-> a = 0. + Proof. lia. Qed. -Lemma Zopp_eq_0_iff a : -a = 0 <-> a = 0. -Proof. lia. Qed. + Hint Rewrite <- mod_opp_l_z_iff using lia : zsimplify. + Hint Rewrite opp_eq_0_iff : zsimplify. -Hint Rewrite <- Zmod_opp_l_z_iff using lia : zsimplify. -Hint Rewrite Zopp_eq_0_iff : zsimplify. + Lemma sub_pos_bound a b X : 0 <= a < X -> 0 <= b < X -> -X < a - b < X. + Proof. lia. Qed. -Lemma Zsub_pos_bound a b X : 0 <= a < X -> 0 <= b < X -> -X < a - b < X. -Proof. lia. Qed. + Lemma div_opp_l_complete a b (Hb : b <> 0) : -a/b = -(a/b) - (if Z_zerop (a mod b) then 0 else 1). + Proof. + destruct (Z_zerop (a mod b)); autorewrite with zsimplify push_Zopp; reflexivity. + Qed. -Lemma Zdiv_opp_l_complete a b (Hb : b <> 0) : -a/b = -(a/b) - (if Z_zerop (a mod b) then 0 else 1). -Proof. - destruct (Z_zerop (a mod b)); autorewrite with zsimplify push_Zopp; reflexivity. -Qed. + Lemma div_opp_l_complete' a b (Hb : b <> 0) : -(a/b) = -a/b + (if Z_zerop (a mod b) then 0 else 1). + Proof. + destruct (Z_zerop (a mod b)); autorewrite with zsimplify pull_Zopp; lia. + Qed. -Lemma Zdiv_opp_l_complete' a b (Hb : b <> 0) : -(a/b) = -a/b + (if Z_zerop (a mod b) then 0 else 1). -Proof. - destruct (Z_zerop (a mod b)); autorewrite with zsimplify pull_Zopp; lia. -Qed. + Hint Rewrite Z.div_opp_l_complete using lia : pull_Zopp. + Hint Rewrite Z.div_opp_l_complete' using lia : push_Zopp. -Hint Rewrite Zdiv_opp_l_complete using lia : pull_Zopp. -Hint Rewrite Zdiv_opp_l_complete' using lia : push_Zopp. + Lemma div_opp a : a <> 0 -> -a / a = -1. + Proof. + intros; autorewrite with pull_Zopp zsimplify; lia. + Qed. -Lemma Zdiv_opp a : a <> 0 -> -a / a = -1. -Proof. - intros; autorewrite with pull_Zopp zsimplify; lia. -Qed. + Hint Rewrite Z.div_opp using lia : zsimplify. -Hint Rewrite Zdiv_opp using lia : zsimplify. + Lemma div_sub_1_0 x : x > 0 -> (x - 1) / x = 0. + Proof. auto with zarith lia. Qed. -Lemma Zdiv_sub_1_0 x : x > 0 -> (x - 1) / x = 0. -Proof. auto with zarith lia. Qed. + Hint Rewrite div_sub_1_0 using lia : zsimplify. -Hint Rewrite Zdiv_sub_1_0 using lia : zsimplify. + Lemma sub_pos_bound_div a b X : 0 <= a < X -> 0 <= b < X -> -1 <= (a - b) / X <= 0. + Proof. + intros H0 H1; pose proof (Z.sub_pos_bound a b X H0 H1). + assert (Hn : -X <= a - b) by lia. + assert (Hp : a - b <= X - 1) by lia. + split; etransitivity; [ | apply Z_div_le, Hn; lia | apply Z_div_le, Hp; lia | ]; + instantiate; autorewrite with zsimplify; try reflexivity. + Qed. -Lemma Zsub_pos_bound_div a b X : 0 <= a < X -> 0 <= b < X -> -1 <= (a - b) / X <= 0. -Proof. - intros H0 H1; pose proof (Zsub_pos_bound a b X H0 H1). - assert (Hn : -X <= a - b) by lia. - assert (Hp : a - b <= X - 1) by lia. - split; etransitivity; [ | apply Z_div_le, Hn; lia | apply Z_div_le, Hp; lia | ]; - instantiate; autorewrite with zsimplify; try reflexivity. -Qed. + Hint Resolve (fun a b X H0 H1 => proj1 (Z.sub_pos_bound_div a b X H0 H1)) + (fun a b X H0 H1 => proj1 (Z.sub_pos_bound_div a b X H0 H1)) : zarith. -Hint Resolve (fun a b X H0 H1 => proj1 (Zsub_pos_bound_div a b X H0 H1)) - (fun a b X H0 H1 => proj1 (Zsub_pos_bound_div a b X H0 H1)) : zarith. + Lemma sub_pos_bound_div_eq a b X : 0 <= a < X -> 0 <= b < X -> (a - b) / X = if a <? b then -1 else 0. + Proof. + intros H0 H1; pose proof (Z.sub_pos_bound_div a b X H0 H1). + destruct (a <? b) eqn:?; Z.ltb_to_lt. + { cut ((a - b) / X <> 0); [ lia | ]. + autorewrite with zstrip_div; auto with zarith lia. } + { autorewrite with zstrip_div; auto with zarith lia. } + Qed. -Lemma Zsub_pos_bound_div_eq a b X : 0 <= a < X -> 0 <= b < X -> (a - b) / X = if a <? b then -1 else 0. -Proof. - intros H0 H1; pose proof (Zsub_pos_bound_div a b X H0 H1). - destruct (a <? b) eqn:?; Zltb_to_Zlt. - { cut ((a - b) / X <> 0); [ lia | ]. - autorewrite with zstrip_div; auto with zarith lia. } - { autorewrite with zstrip_div; auto with zarith lia. } -Qed. + Lemma add_opp_pos_bound_div_eq a b X : 0 <= a < X -> 0 <= b < X -> (-b + a) / X = if a <? b then -1 else 0. + Proof. + rewrite !(Z.add_comm (-_)), !Z.add_opp_r. + apply Z.sub_pos_bound_div_eq. + Qed. -Lemma Zadd_opp_pos_bound_div_eq a b X : 0 <= a < X -> 0 <= b < X -> (-b + a) / X = if a <? b then -1 else 0. -Proof. - rewrite !(Z.add_comm (-_)), !Z.add_opp_r. - apply Zsub_pos_bound_div_eq. -Qed. + Hint Rewrite Z.sub_pos_bound_div_eq Z.add_opp_pos_bound_div_eq using lia : zstrip_div. -Hint Rewrite Zsub_pos_bound_div_eq Zadd_opp_pos_bound_div_eq using lia : zstrip_div. + Lemma div_small_sym a b : 0 <= a < b -> 0 = a / b. + Proof. intros; symmetry; apply Z.div_small; assumption. Qed. -Lemma Zdiv_small_sym a b : 0 <= a < b -> 0 = a / b. -Proof. intros; symmetry; apply Z.div_small; assumption. Qed. + Lemma mod_small_sym a b : 0 <= a < b -> a = a mod b. + Proof. intros; symmetry; apply Z.mod_small; assumption. Qed. -Lemma Zmod_small_sym a b : 0 <= a < b -> a = a mod b. -Proof. intros; symmetry; apply Z.mod_small; assumption. Qed. + Hint Resolve div_small_sym mod_small_sym : zarith. -Hint Resolve Zdiv_small_sym Zmod_small_sym : zarith. + Lemma div_add' a b c : c <> 0 -> (a + c * b) / c = a / c + b. + Proof. intro; rewrite <- Z.div_add, (Z.mul_comm c); try lia. Qed. -Lemma Zdiv_add' a b c : c <> 0 -> (a + c * b) / c = a / c + b. -Proof. intro; rewrite <- Z.div_add, (Z.mul_comm c); try lia. Qed. + Lemma div_add_l' a b c : b <> 0 -> (b * a + c) / b = a + c / b. + Proof. intro; rewrite <- Z.div_add_l, (Z.mul_comm b); lia. Qed. -Lemma Zdiv_add_l' a b c : b <> 0 -> (b * a + c) / b = a + c / b. -Proof. intro; rewrite <- Z.div_add_l, (Z.mul_comm b); lia. Qed. + Hint Rewrite div_add_l' div_add' using lia : zsimplify. -Hint Rewrite Zdiv_add_l' Zdiv_add' using lia : zsimplify. + Lemma div_add_sub_l a b c d : b <> 0 -> (a * b + c - d) / b = a + (c - d) / b. + Proof. rewrite <- Z.add_sub_assoc; apply Z.div_add_l. Qed. -Lemma Zdiv_add_sub_l a b c d : b <> 0 -> (a * b + c - d) / b = a + (c - d) / b. -Proof. rewrite <- Z.add_sub_assoc; apply Z.div_add_l. Qed. + Lemma div_add_sub_l' a b c d : b <> 0 -> (b * a + c - d) / b = a + (c - d) / b. + Proof. rewrite <- Z.add_sub_assoc; apply Z.div_add_l'. Qed. -Lemma Zdiv_add_sub_l' a b c d : b <> 0 -> (b * a + c - d) / b = a + (c - d) / b. -Proof. rewrite <- Z.add_sub_assoc; apply Zdiv_add_l'. Qed. + Lemma div_add_sub a b c d : c <> 0 -> (a + b * c - d) / c = (a - d) / c + b. + Proof. rewrite (Z.add_comm _ (_ * _)), (Z.add_comm (_ / _)); apply Z.div_add_sub_l. Qed. -Lemma Zdiv_add_sub a b c d : c <> 0 -> (a + b * c - d) / c = (a - d) / c + b. -Proof. rewrite (Z.add_comm _ (_ * _)), (Z.add_comm (_ / _)); apply Zdiv_add_sub_l. Qed. + Lemma div_add_sub' a b c d : c <> 0 -> (a + c * b - d) / c = (a - d) / c + b. + Proof. rewrite (Z.add_comm _ (_ * _)), (Z.add_comm (_ / _)); apply Z.div_add_sub_l'. Qed. -Lemma Zdiv_add_sub' a b c d : c <> 0 -> (a + c * b - d) / c = (a - d) / c + b. -Proof. rewrite (Z.add_comm _ (_ * _)), (Z.add_comm (_ / _)); apply Zdiv_add_sub_l'. Qed. + Hint Rewrite Z.div_add_sub Z.div_add_sub' Z.div_add_sub_l Z.div_add_sub_l' using lia : zsimplify. -Hint Rewrite Zdiv_add_sub Zdiv_add_sub' Zdiv_add_sub_l Zdiv_add_sub_l' using lia : zsimplify. + Lemma div_mul_skip a b k : 0 < b -> 0 < k -> a * b / k / b = a / k. + Proof. + intros; rewrite Z.div_div, (Z.mul_comm k), <- Z.div_div by lia. + autorewrite with zsimplify; reflexivity. + Qed. -Lemma Zdiv_mul_skip a b k : 0 < b -> 0 < k -> a * b / k / b = a / k. -Proof. - intros; rewrite Z.div_div, (Z.mul_comm k), <- Z.div_div by lia. - autorewrite with zsimplify; reflexivity. -Qed. + Lemma div_mul_skip' a b k : 0 < b -> 0 < k -> b * a / k / b = a / k. + Proof. + intros; rewrite Z.div_div, (Z.mul_comm k), <- Z.div_div by lia. + autorewrite with zsimplify; reflexivity. + Qed. -Lemma Zdiv_mul_skip' a b k : 0 < b -> 0 < k -> b * a / k / b = a / k. -Proof. - intros; rewrite Z.div_div, (Z.mul_comm k), <- Z.div_div by lia. - autorewrite with zsimplify; reflexivity. -Qed. + Hint Rewrite Z.div_mul_skip Z.div_mul_skip' using lia : zsimplify. +End Z. -Hint Rewrite Zdiv_mul_skip Zdiv_mul_skip' using lia : zsimplify. +Module Export BoundsTactics. + Ltac prime_bound := Z.prime_bound. + Ltac zero_bounds := Z.zero_bounds. +End BoundsTactics. |