Split off pull_Zmod, push_Zmod from ZUtil

1 files changed, 131 insertions, 0 deletions

diff --git a/src/Util/ZUtil/Modulo/PullPush.v b/src/Util/ZUtil/Modulo/PullPush.v
new file mode 100644
index 000000000..7e3a9b46a
--- /dev/null
+++ b/src/Util/ZUtil/Modulo/PullPush.v
@@ -0,0 +1,131 @@
+Require Import Coq.ZArith.ZArith Coq.micromega.Lia.
+Require Import Crypto.Util.ZUtil.Hints.Core.
+Require Import Crypto.Util.ZUtil.ZSimplify.Core.
+Local Open Scope Z_scope.
+
+Module Z.
+  Lemma mod_r_distr_if (b : bool) x y z : z mod (if b then x else y) = if b then z mod x else z mod y.
+  Proof. destruct b; reflexivity. Qed.
+  Hint Rewrite mod_r_distr_if : push_Zmod.
+  Hint Rewrite <- mod_r_distr_if : pull_Zmod.
+
+  Lemma mod_l_distr_if (b : bool) x y z : (if b then x else y) mod z = if b then x mod z else y mod z.
+  Proof. destruct b; reflexivity. Qed.
+  Hint Rewrite mod_l_distr_if : push_Zmod.
+  Hint Rewrite <- mod_l_distr_if : pull_Zmod.
+
+  (** Version without the [n <> 0] assumption *)
+  Lemma mul_mod_full a b n : (a * b) mod n = ((a mod n) * (b mod n)) mod n.
+  Proof. auto using Zmult_mod. Qed.
+  Hint Rewrite <- mul_mod_full : pull_Zmod.
+  Hint Resolve mul_mod_full : zarith.
+
+  Lemma mul_mod_l a b n : (a * b) mod n = ((a mod n) * b) mod n.
+  Proof.
+    intros; rewrite (mul_mod_full a b), (mul_mod_full (a mod n) b).
+    autorewrite with zsimplify; reflexivity.
+  Qed.
+  Hint Rewrite <- mul_mod_l : pull_Zmod.
+  Hint Resolve mul_mod_l : zarith.
+
+  Lemma mul_mod_r a b n : (a * b) mod n = (a * (b mod n)) mod n.
+  Proof.
+    intros; rewrite (mul_mod_full a b), (mul_mod_full a (b mod n)).
+    autorewrite with zsimplify; reflexivity.
+  Qed.
+  Hint Rewrite <- mul_mod_r : pull_Zmod.
+  Hint Resolve mul_mod_r : zarith.
+
+  Lemma add_mod_full a b n : (a + b) mod n = ((a mod n) + (b mod n)) mod n.
+  Proof. auto using Zplus_mod. Qed.
+  Hint Rewrite <- add_mod_full : pull_Zmod.
+  Hint Resolve add_mod_full : zarith.
+
+  Lemma add_mod_l a b n : (a + b) mod n = ((a mod n) + b) mod n.
+  Proof.
+    intros; rewrite (add_mod_full a b), (add_mod_full (a mod n) b).
+    autorewrite with zsimplify; reflexivity.
+  Qed.
+  Hint Rewrite <- add_mod_l : pull_Zmod.
+  Hint Resolve add_mod_l : zarith.
+
+  Lemma add_mod_r a b n : (a + b) mod n = (a + (b mod n)) mod n.
+  Proof.
+    intros; rewrite (add_mod_full a b), (add_mod_full a (b mod n)).
+    autorewrite with zsimplify; reflexivity.
+  Qed.
+  Hint Rewrite <- add_mod_r : pull_Zmod.
+  Hint Resolve add_mod_r : zarith.
+
+  Lemma opp_mod_mod a n : (-a) mod n = (-(a mod n)) mod n.
+  Proof.
+    intros; destruct (Z_zerop (a mod n)) as [H'|H']; [ rewrite H' | ];
+      [ | rewrite !Z_mod_nz_opp_full ];
+      autorewrite with zsimplify; lia.
+  Qed.
+  Hint Rewrite <- opp_mod_mod : pull_Zmod.
+  Hint Resolve opp_mod_mod : zarith.
+
+  (** Give alternate names for the next three lemmas, for consistency *)
+  Lemma sub_mod_full a b n : (a - b) mod n = ((a mod n) - (b mod n)) mod n.
+  Proof. auto using Zminus_mod. Qed.
+  Hint Rewrite <- sub_mod_full : pull_Zmod.
+  Hint Resolve sub_mod_full : zarith.
+
+  Lemma sub_mod_l a b n : (a - b) mod n = ((a mod n) - b) mod n.
+  Proof. auto using Zminus_mod_idemp_l. Qed.
+  Hint Rewrite <- sub_mod_l : pull_Zmod.
+  Hint Resolve sub_mod_l : zarith.
+
+  Lemma sub_mod_r a b n : (a - b) mod n = (a - (b mod n)) mod n.
+  Proof. auto using Zminus_mod_idemp_r. Qed.
+  Hint Rewrite <- sub_mod_r : pull_Zmod.
+  Hint Resolve sub_mod_r : zarith.
+
+  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 : NoZMod a -> NoZMod b -> (a * b) mod n = ((a mod n) * (b mod n)) mod n.
+  Proof. intros; apply mul_mod_full; assumption. Qed.
+  Hint Rewrite mul_mod_push using solve [ NoZMod ] : push_Zmod.
+
+  Lemma add_mod_push a b n : NoZMod a -> NoZMod b -> (a + b) mod n = ((a mod n) + (b mod n)) mod n.
+  Proof. intros; apply add_mod_full; assumption. Qed.
+  Hint Rewrite add_mod_push using solve [ NoZMod ] : push_Zmod.
+
+  Lemma mul_mod_l_push a b n : 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 ] : push_Zmod.
+
+  Lemma mul_mod_r_push a b n : 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 ] : push_Zmod.
+
+  Lemma add_mod_l_push a b n : NoZMod a -> (a + b) mod n = ((a mod n) + b) mod n.
+  Proof. intros; apply add_mod_l; assumption. Qed.
+  Hint Rewrite add_mod_l_push using solve [ NoZMod ] : push_Zmod.
+
+  Lemma add_mod_r_push a b n : NoZMod b -> (a + b) mod n = (a + (b mod n)) mod n.
+  Proof. intros; apply add_mod_r; assumption. Qed.
+  Hint Rewrite add_mod_r_push using solve [ NoZMod ] : push_Zmod.
+
+  Lemma sub_mod_push a b n : NoZMod a -> NoZMod b -> (a - b) mod n = ((a mod n) - (b mod n)) mod n.
+  Proof. intros; apply Zminus_mod; assumption. Qed.
+  Hint Rewrite sub_mod_push using solve [ NoZMod ] : push_Zmod.
+
+  Lemma sub_mod_l_push a b n : NoZMod a -> (a - b) mod n = ((a mod n) - b) mod n.
+  Proof. intros; symmetry; apply Zminus_mod_idemp_l; assumption. Qed.
+  Hint Rewrite sub_mod_l_push using solve [ NoZMod ] : push_Zmod.
+
+  Lemma sub_mod_r_push a b n : NoZMod b -> (a - b) mod n = (a - (b mod n)) mod n.
+  Proof. intros; symmetry; apply Zminus_mod_idemp_r; assumption. Qed.
+  Hint Rewrite sub_mod_r_push using solve [ NoZMod ] : push_Zmod.
+
+  Lemma opp_mod_mod_push a n : NoZMod a -> (-a) mod n = (-(a mod n)) mod n.
+  Proof. intros; apply opp_mod_mod; assumption. Qed.
+  Hint Rewrite opp_mod_mod using solve [ NoZMod ] : push_Zmod.
+End Z.