From e4cd2e81a5eacb131fee96e0acdfac9422b176bc Mon Sep 17 00:00:00 2001 From: Jason Gross Date: Fri, 29 Jul 2016 13:37:08 -0700 Subject: Add some lemmas and hints about Zmod After | File Name | Before || Change ---------------------------------------------------------------------------------- 1m42.02s | Total | 1m42.15s || -0m00.13s ---------------------------------------------------------------------------------- 0m32.64s | Specific/GF25519 | 0m32.66s || -0m00.01s 0m15.18s | ModularArithmetic/ModularBaseSystemProofs | 0m15.36s || -0m00.17s 0m11.37s | Experiments/SpecEd25519 | 0m11.31s || +0m00.05s 0m07.05s | Specific/GF1305 | 0m07.03s || +0m00.01s 0m03.99s | ModularArithmetic/Pow2BaseProofs | 0m03.99s || +0m00.00s 0m03.74s | ModularArithmetic/Tutorial | 0m03.74s || +0m00.00s 0m03.68s | BaseSystemProofs | 0m03.69s || -0m00.00s 0m03.19s | ModularArithmetic/ModularBaseSystemOpt | 0m03.20s || -0m00.01s 0m02.48s | Util/ZUtil | 0m02.39s || +0m00.08s 0m01.56s | ModularArithmetic/PrimeFieldTheorems | 0m01.60s || -0m00.04s 0m01.56s | Encoding/PointEncodingPre | 0m01.56s || +0m00.00s 0m01.56s | ModularArithmetic/ModularArithmeticTheorems | 0m01.59s || -0m00.03s 0m01.15s | BaseSystem | 0m01.17s || -0m00.02s 0m01.12s | ModularArithmetic/ExtendedBaseVector | 0m01.10s || +0m00.02s 0m00.98s | Experiments/DerivationsOptionRectLetInEncoding | 0m01.00s || -0m00.02s 0m00.93s | ModularArithmetic/BarrettReduction/Z | 0m00.93s || +0m00.00s 0m00.89s | ModularArithmetic/ModularBaseSystemField | 0m00.89s || +0m00.00s 0m00.88s | Util/NumTheoryUtil | 0m00.87s || +0m00.01s 0m00.80s | ModularArithmetic/ModularBaseSystemListProofs | 0m00.82s || -0m00.01s 0m00.74s | Encoding/ModularWordEncodingTheorems | 0m00.68s || +0m00.05s 0m00.67s | Experiments/SpecificCurve25519 | 0m00.69s || -0m00.01s 0m00.63s | ModularArithmetic/ExtPow2BaseMulProofs | 0m00.64s || -0m00.01s 0m00.62s | Testbit | 0m00.64s || -0m00.02s 0m00.60s | Spec/ModularWordEncoding | 0m00.63s || -0m00.03s 0m00.60s | ModularArithmetic/ModularBaseSystemList | 0m00.57s || +0m00.03s 0m00.59s | ModularArithmetic/PseudoMersenneBaseParamProofs | 0m00.59s || +0m00.00s 0m00.59s | Encoding/ModularWordEncodingPre | 0m00.63s || -0m00.04s 0m00.55s | ModularArithmetic/ModularBaseSystem | 0m00.55s || +0m00.00s 0m00.47s | ModularArithmetic/Pre | 0m00.45s || +0m00.01s 0m00.43s | ModularArithmetic/Pow2Base | 0m00.42s || +0m00.01s 0m00.41s | ModularArithmetic/PseudoMersenneBaseParams | 0m00.36s || +0m00.04s 0m00.37s | Spec/ModularArithmetic | 0m00.41s || -0m00.03s --- src/Util/ZUtil.v | 42 ++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 42 insertions(+) (limited to 'src') diff --git a/src/Util/ZUtil.v b/src/Util/ZUtil.v index 8177444a5..7b7daef67 100644 --- a/src/Util/ZUtil.v +++ b/src/Util/ZUtil.v @@ -34,12 +34,16 @@ Create HintDb push_Zpow discriminated. Create HintDb pull_Zpow discriminated. Create HintDb push_Zmul discriminated. Create HintDb pull_Zmul discriminated. +Create HintDb pull_Zmod discriminated. +Create HintDb push_Zmod discriminated. Hint Extern 1 => autorewrite with push_Zopp in * : push_Zopp. Hint Extern 1 => autorewrite with pull_Zopp in * : pull_Zopp. Hint Extern 1 => autorewrite with push_Zpow in * : push_Zpow. Hint Extern 1 => autorewrite with pull_Zpow in * : pull_Zpow. Hint Extern 1 => autorewrite with push_Zmul in * : push_Zmul. Hint Extern 1 => autorewrite with pull_Zmul in * : pull_Zmul. +Hint Extern 1 => autorewrite with pull_Zmod in * : pull_Zmod. +Hint Extern 1 => autorewrite with push_Zmod in * : push_Zmod. Hint Rewrite Z.div_opp_l_nz Z.div_opp_l_z using lia : pull_Zopp. Hint Rewrite Z.mul_opp_l : pull_Zopp. Hint Rewrite <- Z.opp_add_distr : pull_Zopp. @@ -50,6 +54,7 @@ Hint Rewrite Z.pow_sub_r Z.pow_div_l Z.pow_twice_r Z.pow_mul_l Z.pow_add_r using Hint Rewrite <- Z.pow_sub_r Z.pow_div_l Z.pow_mul_l Z.pow_add_r Z.pow_twice_r using lia : pull_Zpow. Hint Rewrite Z.mul_add_distr_l Z.mul_add_distr_r Z.mul_sub_distr_l Z.mul_sub_distr_r : push_Zmul. Hint Rewrite <- Z.mul_add_distr_l Z.mul_add_distr_r Z.mul_sub_distr_l Z.mul_sub_distr_r : pull_Zmul. +Hint Rewrite <- Z.mul_mod Z.add_mod using lia : pull_Zmod. (** For the occasional lemma that can remove the use of [Z.div] *) Create HintDb zstrip_div. @@ -1140,6 +1145,43 @@ Module Z. reflexivity. Qed. + Lemma mul_mod_l a b n : n <> 0 -> (a * b) mod n = ((a mod n) * b) mod n. + Proof. + intros; rewrite (Z.mul_mod a b), (Z.mul_mod (a mod n) b) by lia. + autorewrite with zsimplify; reflexivity. + Qed. + Hint Rewrite <- mul_mod_l using lia : pull_Zmod. + + Lemma mul_mod_r a b n : n <> 0 -> (a * b) mod n = (a * (b mod n)) mod n. + Proof. + intros; rewrite (Z.mul_mod a b), (Z.mul_mod a (b mod n)) by lia. + autorewrite with zsimplify; reflexivity. + Qed. + Hint Rewrite <- mul_mod_r using lia : pull_Zmod. + + Definition NoZMod (x : Z) := True. + Ltac NoZMod := + lazymatch goal with + | [ |- NoZMod (?x mod ?y) ] => fail 0 "Goal has" x "mod" y + | [ |- NoZMod _ ] => constructor + end. + + Lemma mul_mod_push a b n : n <> 0 -> NoZMod a -> NoZMod b -> (a * b) mod n = ((a mod n) * (b mod n)) mod n. + Proof. intros; apply Z.mul_mod; assumption. Qed. + Hint Rewrite mul_mod_push using solve [ NoZMod | lia ] : push_Zmod. + + Lemma add_mod_push a b n : n <> 0 -> NoZMod a -> NoZMod b -> (a + b) mod n = ((a mod n) + (b mod n)) mod n. + Proof. intros; apply Z.add_mod; assumption. Qed. + Hint Rewrite add_mod_push using solve [ NoZMod | lia ] : push_Zmod. + + Lemma mul_mod_l_push a b n : n <> 0 -> NoZMod a -> (a * b) mod n = ((a mod n) * b) mod n. + Proof. intros; apply mul_mod_l; assumption. Qed. + Hint Rewrite mul_mod_l_push using solve [ NoZMod | lia ] : push_Zmod. + + Lemma mul_mod_r_push a b n : n <> 0 -> NoZMod b -> (a * b) mod n = (a * (b mod n)) mod n. + Proof. intros; apply mul_mod_r; assumption. Qed. + Hint Rewrite mul_mod_r_push using solve [ NoZMod | lia ] : push_Zmod. + Section equiv_modulo. Context (N : Z). Definition equiv_modulo x y := x mod N = y mod N. -- cgit v1.2.3