Add some lemmas and hints about Zmod

After    | File Name                                       | Before   || Change
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1m42.02s | Total                                           | 1m42.15s || -0m00.13s
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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

1 files changed, 42 insertions, 0 deletions

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.