Handle Z.pow in {push,pull}_Zmod

6 files changed, 56 insertions, 32 deletions

diff --git a/src/Util/ZUtil/EquivModulo.v b/src/Util/ZUtil/EquivModulo.v
index a67174629..2d8f5aefc 100644
--- a/src/Util/ZUtil/EquivModulo.v
+++ b/src/Util/ZUtil/EquivModulo.v
@@ -5,6 +5,8 @@ Require Import Crypto.Util.Notations.
 Require Import Crypto.Util.ZUtil.Hints.Core.
 Require Import Crypto.Util.ZUtil.Hints.ZArith.
 Require Import Crypto.Util.ZUtil.Modulo.
+Require Import Crypto.Util.ZUtil.Modulo.PullPush.
+
 Local Open Scope Z_scope.
 
 Module Z.
diff --git a/src/Util/ZUtil/Modulo.v b/src/Util/ZUtil/Modulo.v
index be513eb3c..8dea71870 100644
--- a/src/Util/ZUtil/Modulo.v
+++ b/src/Util/ZUtil/Modulo.v
@@ -225,33 +225,6 @@ Module Z.
   Lemma small_mod_eq a b n: a mod n = b mod n -> 0 <= a < n -> a = b mod n.
   Proof. intros; rewrite <-(Z.mod_small a n); auto. Qed.
 
-  Lemma mod_pow_full p q n : (p^q) mod n = ((p mod n)^q) mod n.
-  Proof.
-    destruct (Z_dec' n 0) as [ [H|H] | H]; subst;
-      [
-      | apply Zpower_mod; assumption
-      | rewrite !Zmod_0_r; reflexivity ].
-    { revert H.
-      rewrite <- (Z.opp_involutive (p^q)),
-      <- (Z.opp_involutive ((p mod n)^q)),
-      <- (Z.opp_involutive p),
-      <- (Z.opp_involutive n).
-      generalize (-n); clear n; intros n H.
-      rewrite !Zmod_opp_opp.
-      rewrite !Z.opp_involutive.
-      apply f_equal.
-      destruct (Z.Even_or_Odd q).
-      { rewrite !Z.pow_opp_even by (assumption || omega).
-        destruct (Z.eq_dec (p^q mod n) 0) as [H'|H'], (Z.eq_dec ((-p mod n)^q mod n) 0) as [H''|H''];
-          repeat first [ rewrite Z_mod_zero_opp_full by assumption
-                       | rewrite Z_mod_nz_opp_full by assumption
-                       | reflexivity
-                       | rewrite <- Zpower_mod, Z.pow_opp_even in H'' by (assumption || omega); omega
-                       | rewrite <- Zpower_mod, Z.pow_opp_even in H'' |- * by (assumption || omega); omega ]. }
-      { rewrite Z.pow_opp_odd, !Z.opp_involutive, <- Zpower_mod, Z.pow_opp_odd, ?Z.opp_involutive by (assumption || omega).
-        reflexivity. } }
-  Qed.
-
   Lemma mod_bound_min_max l x u d (H : l <= x <= u)
     : (if l / d =? u / d then Z.min (l mod d) (u mod d) else Z.min 0 (d + 1))
       <= x mod d
diff --git a/src/Util/ZUtil/Modulo/PullPush.v b/src/Util/ZUtil/Modulo/PullPush.v
index dbbe98975..13f8bd24f 100644
--- a/src/Util/ZUtil/Modulo/PullPush.v
+++ b/src/Util/ZUtil/Modulo/PullPush.v
@@ -1,4 +1,5 @@
 Require Import Coq.ZArith.ZArith Coq.micromega.Lia.
+Require Import Coq.ZArith.Znumtheory Coq.ZArith.Zpow_facts.
 Require Import Crypto.Util.ZUtil.Hints.Core.
 Require Import Crypto.Util.ZUtil.ZSimplify.Core.
 Local Open Scope Z_scope.
@@ -89,6 +90,36 @@ Module Z.
   Hint Rewrite lnot_mod_mod : pull_Zmod.
   Hint Resolve lnot_mod_mod : zarith.
 
+  Lemma mod_pow_full p q n : (p^q) mod n = ((p mod n)^q) mod n.
+  Proof.
+    destruct (Z_dec' n 0) as [ [H|H] | H]; subst;
+      [
+      | apply Zpower_mod; assumption
+      | rewrite !Zmod_0_r; reflexivity ].
+    { revert H.
+      rewrite <- (Z.opp_involutive (p^q)),
+      <- (Z.opp_involutive ((p mod n)^q)),
+      <- (Z.opp_involutive p),
+      <- (Z.opp_involutive n).
+      generalize (-n); clear n; intros n H.
+      rewrite !Zmod_opp_opp.
+      rewrite !Z.opp_involutive.
+      apply f_equal.
+      destruct (Z.Even_or_Odd q).
+      { rewrite !Z.pow_opp_even by (assumption || omega).
+        destruct (Z.eq_dec (p^q mod n) 0) as [H'|H'], (Z.eq_dec ((-p mod n)^q mod n) 0) as [H''|H''];
+          repeat first [ rewrite Z_mod_zero_opp_full by assumption
+                       | rewrite Z_mod_nz_opp_full by assumption
+                       | reflexivity
+                       | rewrite <- Zpower_mod, Z.pow_opp_even in H'' by (assumption || omega); omega
+                       | rewrite <- Zpower_mod, Z.pow_opp_even in H'' |- * by (assumption || omega); omega ]. }
+      { rewrite Z.pow_opp_odd, !Z.opp_involutive, <- Zpower_mod, Z.pow_opp_odd, ?Z.opp_involutive by (assumption || omega).
+        reflexivity. } }
+  Qed.
+  Hint Rewrite <- mod_pow_full : pull_Zmod.
+  Hint Resolve mod_pow_full : zarith.
+  Notation pow_mod_full := mod_pow_full.
+
   Definition NoZMod (x : Z) := True.
   Ltac NoZMod :=
     lazymatch goal with
@@ -139,4 +170,8 @@ Module Z.
   Lemma lnot_mod_mod_push v m : NoZMod v -> (Z.lnot v) mod m = (Z.lnot (v mod m) mod m).
   Proof. intros; symmetry; apply lnot_mod_mod. Qed.
   Hint Rewrite lnot_mod_mod_push using solve [ NoZMod ] : push_Zmod.
+
+  Lemma pow_mod_push p q n : NoZMod p -> (p^q) mod n = ((p mod n)^q) mod n.
+  Proof. intros; apply pow_mod_full. Qed.
+  Hint Rewrite pow_mod_push using solve [ NoZMod ] : push_Zmod.
 End Z.
diff --git a/src/Util/ZUtil/Tactics/PullPush/Modulo.v b/src/Util/ZUtil/Tactics/PullPush/Modulo.v
index a2502aeea..4ff81f7ad 100644
--- a/src/Util/ZUtil/Tactics/PullPush/Modulo.v
+++ b/src/Util/ZUtil/Tactics/PullPush/Modulo.v
@@ -78,6 +78,9 @@ Ltac pull_Zmod :=
          | [ |- context[(((-?y) mod ?z)) mod ?z] ]
            => has_no_mod y z;
               rewrite <- (Z.opp_mod_mod y z)
+         | [ |- context[((?x mod ?z)^?y) mod ?z] ]
+           => has_no_mod x z;
+             rewrite <- (Z.pow_mod_full x y z)
          | [ |- context[(?x mod ?z) mod ?z] ]
            => rewrite (Zmod_mod x z)
          | _ => progress autorewrite with pull_Zmod
diff --git a/src/Util/ZUtil/ZSimplify/Core.v b/src/Util/ZUtil/ZSimplify/Core.v
index 6717a5a01..958c44f74 100644
--- a/src/Util/ZUtil/ZSimplify/Core.v
+++ b/src/Util/ZUtil/ZSimplify/Core.v
@@ -1,14 +1,14 @@
 Require Import Coq.ZArith.ZArith.
 Require Export Crypto.Util.ZUtil.Hints.Core.
 
-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 Z_mod_same_full Z.sub_simpl_r Z.sub_simpl_l Z.add_opp_diag_r Z.add_opp_diag_l Zmod_0_l Z.add_simpl_r Z.add_simpl_l Z.opp_0 Zmod_0_r Zmod_mod Z.mul_succ_l Z.mul_succ_r Z.shiftr_0_r Z.shiftr_0_l Z.mod_1_r Zmod_0_l Zmod_0_r Z.shiftl_0_r Z.shiftl_0_l Z.shiftr_0_r Z.shiftr_0_l Z.sub_diag Zdiv_0_r Zdiv_0_l Z.land_0_l Z.land_0_r Z.lor_0_l Z.lor_0_r Z.lnot_0 Z.land_lnot_diag Z.lnot_involutive Z.lxor_lnot_lnot Z.lnot_m1 Z.lxor_m1_l Z.lxor_m1_r Z.add_lnot_diag Z.lor_lnot_diag Z.pow_0_r Z.pow_0_l Z.pow_1_r Z.pow_1_l : zsimplify_fast.
+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 Z_mod_same_full Z.sub_simpl_r Z.sub_simpl_l Z.add_opp_diag_r Z.add_opp_diag_l Zmod_0_l Z.add_simpl_r Z.add_simpl_l Z.opp_0 Zmod_0_r Zmod_mod Z.mul_succ_l Z.mul_succ_r Z.shiftr_0_r Z.shiftr_0_l Z.mod_1_r Zmod_0_l Zmod_0_r Z.shiftl_0_r Z.shiftl_0_l Z.shiftr_0_r Z.shiftr_0_l Z.sub_diag Zdiv_0_r Zdiv_0_l Z.land_0_l Z.land_0_r Z.lor_0_l Z.lor_0_r Z.lnot_0 Z.land_lnot_diag Z.lnot_involutive Z.lxor_lnot_lnot Z.lnot_m1 Z.lxor_m1_l Z.lxor_m1_r Z.add_lnot_diag Z.lor_lnot_diag Z.pow_0_r Z.pow_0_l Z.pow_1_r : zsimplify_fast.
 
-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 Z_mod_same_full Z.sub_simpl_r Z.sub_simpl_l Z.add_opp_diag_r Z.add_opp_diag_l Zmod_0_l Z.add_simpl_r Z.add_simpl_l Z.opp_0 Zmod_0_r Zmod_mod Z.mul_succ_l Z.mul_succ_r Z.shiftr_0_r Z.shiftr_0_l Z.mod_1_r Zmod_0_l Zmod_0_r Z.shiftl_0_r Z.shiftl_0_l Z.shiftr_0_r Z.shiftr_0_l Zplus_minus Z.add_diag Z.abs_0 Z.sgn_0 Nat2Z.inj_0 Z2Nat.inj_0 Zdiv_0_r Zdiv_0_l Z.land_0_l Z.land_0_r Z.lor_0_l Z.lor_0_r Z.lnot_0 Z.land_lnot_diag Z.lnot_involutive Z.lxor_lnot_lnot Z.lnot_m1 Z.lxor_m1_l Z.lxor_m1_r Z.add_lnot_diag Z.lor_lnot_diag Z.pow_0_r Z.pow_0_l Z.pow_1_r Z.pow_1_l : 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 Z.div_0_l Z.mod_mod Z.mod_small Z_mod_zero_opp_full Z.mod_1_l using zutil_arith : zsimplify.
+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 Z_mod_same_full Z.sub_simpl_r Z.sub_simpl_l Z.add_opp_diag_r Z.add_opp_diag_l Zmod_0_l Z.add_simpl_r Z.add_simpl_l Z.opp_0 Zmod_0_r Zmod_mod Z.mul_succ_l Z.mul_succ_r Z.shiftr_0_r Z.shiftr_0_l Z.mod_1_r Zmod_0_l Zmod_0_r Z.shiftl_0_r Z.shiftl_0_l Z.shiftr_0_r Z.shiftr_0_l Zplus_minus Z.add_diag Z.abs_0 Z.sgn_0 Nat2Z.inj_0 Z2Nat.inj_0 Zdiv_0_r Zdiv_0_l Z.land_0_l Z.land_0_r Z.lor_0_l Z.lor_0_r Z.lnot_0 Z.land_lnot_diag Z.lnot_involutive Z.lxor_lnot_lnot Z.lnot_m1 Z.lxor_m1_l Z.lxor_m1_r Z.add_lnot_diag Z.lor_lnot_diag Z.pow_0_r Z.pow_0_l Z.pow_1_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 Z.mod_add Z.div_0_l Z.mod_mod Z.mod_small Z_mod_zero_opp_full Z.mod_1_l Z.pow_1_l using zutil_arith : zsimplify.
 Hint Rewrite <- Z.opp_eq_mul_m1 Z.one_succ Z.two_succ : zsimplify.
 Hint Rewrite <- Z.div_mod using zutil_arith : zsimplify.
 Hint Rewrite (fun x y => proj2 (Z.eqb_neq x y)) using assumption : zsimplify.
-Hint Rewrite Z.mul_0_l Z.mul_0_r Z.mul_1_l Z.mul_1_r Z.add_0_l Z.add_0_r Z.sub_0_l Z.sub_0_r Zdiv_0_l Zdiv_0_r Zdiv_1_r Zmod_0_l Zmod_0_r Zmod_1_r Z.opp_0 Z.abs_0 Z.sgn_0 Nat2Z.inj_0 Z2Nat.inj_0 Z.land_0_l Z.land_0_r Z.lor_0_l Z.lor_0_r Z.lnot_0 Z.land_lnot_diag Z.lnot_involutive Z.lxor_lnot_lnot Z.lnot_m1 Z.lxor_m1_l Z.lxor_m1_r Z.add_lnot_diag Z.lor_lnot_diag Z.pow_0_r Z.pow_0_l Z.pow_1_r Z.pow_1_l : zsimplify_const.
+Hint Rewrite Z.mul_0_l Z.mul_0_r Z.mul_1_l Z.mul_1_r Z.add_0_l Z.add_0_r Z.sub_0_l Z.sub_0_r Zdiv_0_l Zdiv_0_r Zdiv_1_r Zmod_0_l Zmod_0_r Zmod_1_r Z.opp_0 Z.abs_0 Z.sgn_0 Nat2Z.inj_0 Z2Nat.inj_0 Z.land_0_l Z.land_0_r Z.lor_0_l Z.lor_0_r Z.lnot_0 Z.land_lnot_diag Z.lnot_involutive Z.lxor_lnot_lnot Z.lnot_m1 Z.lxor_m1_l Z.lxor_m1_r Z.add_lnot_diag Z.lor_lnot_diag Z.pow_0_r Z.pow_0_l Z.pow_1_r : zsimplify_const.
 
 Hint Rewrite <- Z.one_succ Z.two_succ : zsimplify_const.
 
diff --git a/src/Util/ZUtil/ZSimplify/Simple.v b/src/Util/ZUtil/ZSimplify/Simple.v
index 9b5e0e9c8..94fa05e1a 100644
--- a/src/Util/ZUtil/ZSimplify/Simple.v
+++ b/src/Util/ZUtil/ZSimplify/Simple.v
@@ -3,6 +3,17 @@ Require Import Crypto.Util.ZUtil.Hints.Core.
 Local Open Scope Z_scope.
 
 Module Z.
+  Lemma pow_1_l_Zpos a : 1^Zpos a = 1.
+  Proof. apply Z.pow_1_l; lia. Qed.
+  Hint Rewrite pow_1_l_Zpos : zsimplify_fast zsimplify_const zsimplify.
+  Lemma pow_1_l_0 : 1^Z0 = 1. Proof. reflexivity. Qed.
+  Hint Rewrite pow_1_l_0 : zsimplify_fast zsimplify_const zsimplify.
+  Lemma pow_r_Zneg a b : a^Zneg b = 0. Proof. reflexivity. Qed.
+  Hint Rewrite pow_r_Zneg : zsimplify_fast zsimplify_const zsimplify.
+  Lemma pow_1_l_Zof_nat a : 1^Z.of_nat a = 1.
+  Proof. apply Z.pow_1_l; lia. Qed.
+  Hint Rewrite pow_1_l_Zof_nat : zsimplify_fast zsimplify_const zsimplify.
+
   Lemma sub_same_minus (x y : Z) : x - (x - y) = y.
   Proof. lia. Qed.
   Hint Rewrite sub_same_minus : zsimplify.
@@ -78,5 +89,5 @@ Module Z.
 
   Lemma minus_minus_one : - -1 = 1.
   Proof. reflexivity. Qed.
-  Hint Rewrite minus_minus_one : zsimplify.
+  Hint Rewrite minus_minus_one : zsimplify zsimplify_fast zsimplify_const.
 End Z.