Cleanup: mostly moving lemmas to Util files, some moving lemmas to more general contexts.

3 files changed, 7 insertions, 117 deletions

diff --git a/src/Encoding/ModularWordEncodingPre.v b/src/Encoding/ModularWordEncodingPre.v
index cb2f5a045..417344b43 100644
--- a/src/Encoding/ModularWordEncodingPre.v
+++ b/src/Encoding/ModularWordEncodingPre.v
@@ -3,8 +3,7 @@ Require Import Coq.Numbers.Natural.Peano.NPeano.
 Require Import Crypto.ModularArithmetic.PrimeFieldTheorems.
 Require Import Bedrock.Word.
 Require Import Crypto.Tactics.VerdiTactics.
-Require Import Crypto.Util.NatUtil.
-Require Import Crypto.Util.WordUtil.
+Require Import Crypto.Util.ZUtil Crypto.Util.WordUtil.
 Require Import Crypto.Spec.Encoding.
 
 Local Open Scope nat_scope.
@@ -21,45 +20,12 @@ Section ModularWordEncodingPre.
       else None
   .
 
-  (* TODO : move to ZUtil *)
-  Lemma bound_check_N : forall x : F m, (Z.to_N x < Npow2 sz)%N.
-  Proof.
-    intro.
-    pose proof (FieldToZ_range x m_pos) as x_range.
-    rewrite <- Nnat.N2Nat.id.
-    rewrite Npow2_nat.
-    replace (Z.to_N x) with (N.of_nat (Z.to_nat x)) by apply Z_nat_N.
-    apply (Nat2N_inj_lt _ (pow2 sz)).
-    rewrite Zpow_pow2.
-    destruct x_range as [x_low x_high].
-    apply Z2Nat.inj_lt in x_high; try omega.
-    rewrite <- ZUtil.pow_Z2N_Zpow by omega.
-    replace (Z.to_nat 2) with 2%nat by auto.
-    omega.
-  Qed.
-
-  (* TODO: move to WordUtil *)
-  Lemma wordToN_NToWord_idempotent : forall sz n, (n < Npow2 sz)%N ->
-    wordToN (NToWord sz n) = n.
-  Proof.
-    intros.
-    rewrite wordToN_nat, NToWord_nat.
-    rewrite wordToNat_natToWord_idempotent; rewrite Nnat.N2Nat.id; auto.
-  Qed.
-
-  (* TODO: move to WordUtil *)
-  Lemma NToWord_wordToN : forall sz w, NToWord sz (wordToN w) = w.
-  Proof.
-    intros.
-    rewrite wordToN_nat, NToWord_nat, Nnat.Nat2N.id.
-    apply natToWord_wordToNat.
-  Qed.
-
   Lemma Fm_encoding_valid : forall x, Fm_dec (Fm_enc x) = Some x.
   Proof.
     unfold Fm_dec, Fm_enc; intros.
     pose proof (FieldToZ_range x m_pos).
-    rewrite wordToN_NToWord_idempotent by apply bound_check_N.
+    rewrite wordToN_NToWord_idempotent by (apply bound_check_nat_N;
+     assert (Z.to_nat x < Z.to_nat m) by (apply Z2Nat.inj_lt; omega); omega).
     rewrite Z2N.id by omega.
     rewrite ZToField_idempotent.
     break_if; auto; omega.
diff --git a/src/Encoding/ModularWordEncodingTheorems.v b/src/Encoding/ModularWordEncodingTheorems.v
index 83f199d90..a74ccb522 100644
--- a/src/Encoding/ModularWordEncodingTheorems.v
+++ b/src/Encoding/ModularWordEncodingTheorems.v
@@ -1,10 +1,11 @@
 Require Import Coq.ZArith.ZArith Coq.ZArith.Znumtheory.
 Require Import Coq.Numbers.Natural.Peano.NPeano.
 Require Import Crypto.CompleteEdwardsCurve.CompleteEdwardsCurveTheorems.
-Require Import Crypto.ModularArithmetic.PrimeFieldTheorems.
+Require Import Crypto.ModularArithmetic.PrimeFieldTheorems Crypto.ModularArithmetic.ModularArithmeticTheorems.
 Require Import Bedrock.Word.
 Require Import Crypto.Tactics.VerdiTactics.
 Require Import Crypto.Spec.Encoding.
+Require Import Crypto.Util.ZUtil.
 Require Import Crypto.Spec.ModularWordEncoding.
 
 
@@ -47,29 +48,6 @@ Section SignBit.
     rewrite sign_bit_parity; auto.
   Qed.
 
-  (* TODO : move to ZUtil *)
-  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.
-    break_if; rewrite ?Bool.andb_true_r, ?Bool.andb_false_r; auto using Bool.xorb_false_r.
-  Qed.
-
-  (* TODO : move to ModularArithmeticTheorems *)
-  Lemma F_FieldToZ_add_opp : forall x : F m, x <> 0 -> (FieldToZ x + FieldToZ (opp x) = m)%Z.
-  Proof.
-    intros.
-    rewrite FieldToZ_opp.
-    rewrite Z_mod_nz_opp_full, mod_FieldToZ; try omega.
-    rewrite mod_FieldToZ.
-    replace 0%Z with (@FieldToZ m 0) by auto.
-    intro false_eq.
-    rewrite <-F_eq in false_eq.
-    congruence.
-  Qed.
-
   Lemma sign_bit_opp : forall (x : F m), x <> 0 -> negb (sign_bit x) = sign_bit (opp x).
   Proof.
     intros.
diff --git a/src/Encoding/PointEncodingPre.v b/src/Encoding/PointEncodingPre.v
index 9b65d9819..e023da0ea 100644
--- a/src/Encoding/PointEncodingPre.v
+++ b/src/Encoding/PointEncodingPre.v
@@ -118,28 +118,6 @@ Section PointEncoding.
     | None => None
     end.
 
-  (* TODO : move *)
-  Lemma sqrt_mod_q_0 : forall x : F q, sqrt_mod_q x = 0 -> x = 0.
-  Proof.
-    unfold sqrt_mod_q; intros.
-    break_if.
-    - match goal with [ H : ?sqrt_x ^ 2 = x, H' : ?sqrt_x = 0 |- _ ] => rewrite <-H, H' end.
-      ring.
-    - match goal with
-      | [H : sqrt_minus1 * _ = 0 |- _ ]=>
-         apply Fq_mul_zero_why in H; destruct H as [sqrt_minus1_zero | ? ];
-         [ | eapply Fq_root_zero; eauto ]
-      end.
-      unfold sqrt_minus1 in sqrt_minus1_zero.
-      rewrite sqrt_minus1_zero in sqrt_minus1_valid.
-      exfalso.
-      pose proof (@F_opp_spec q 1) as opp_spec_1.
-      rewrite <-sqrt_minus1_valid in opp_spec_1.
-      assert (((1 + 0 ^ 2) : F q) = (1 : F q)) as ring_subst by ring.
-      rewrite ring_subst in *.
-      apply Fq_1_neq_0; assumption.
-  Qed.
-
   Lemma point_coordinates_encoding_canonical : forall w p,
     point_dec_coordinates sign_bit w = Some p -> point_enc_coordinates p = w.
   Proof.
@@ -151,7 +129,7 @@ Section PointEncoding.
       do 2 (break_if; try congruence); inversion coord_dec_Some; subst.
     + destruct (F_eq_dec (sqrt_mod_q (E.solve_for_x2 f1)) 0%F) as [sqrt_0 | ?].
       - rewrite sqrt_0 in *.
-        apply sqrt_mod_q_0 in sqrt_0.
+        apply sqrt_mod_q_root_0 in sqrt_0; try assumption.
         rewrite sqrt_0 in *.
         break_if; [symmetry; auto using Bool.eqb_prop | ].
         rewrite sign_bit_zero in *.
@@ -195,19 +173,6 @@ Proof.
   exact (encoding_valid y).
 Qed.
 
-(*
-Lemma wordToN_enc_neq_opp : forall x, x <> 0 -> (wordToN (enc (opp x)) <> wordToN (enc x))%N.
-Proof.
-  intros x x_nonzero.
-  intro false_eq.
-  apply x_nonzero.
-  apply F_eq_opp_zero; try apply two_lt_q.
-  apply wordToN_inj in false_eq.
-  apply encoding_inj in false_eq.
-  auto.
-Qed.
-*)
-
 Lemma sign_bit_opp_eq_iff : forall x y, y <> 0 ->
   (sign_bit x <> sign_bit y <-> sign_bit x = sign_bit (opp y)).
 Proof.
@@ -242,25 +207,6 @@ Proof.
     reflexivity.
 Qed.
 
-(* TODO : move *)
-Lemma sqrt_mod_q_of_0 : @sqrt_mod_q q 0 = 0.
-Proof.
-  unfold sqrt_mod_q.
-  rewrite !Fq_pow_zero.
-  break_if; ring.
-
-  congruence.
-  intro false_eq.
-  SearchAbout Z.to_N.
-  rewrite <-(N2Z.id 0) in false_eq.
-  rewrite N2Z.inj_0 in false_eq.
-  pose proof (prime_ge_2 q prime_q).
-  apply Z2N.inj in false_eq; zero_bounds.
-  assert (0 < q / 8 + 1)%Z.
-  apply Z.add_nonneg_pos; zero_bounds.
-  omega.
-Qed.
-
 Lemma point_encoding_coordinates_valid : forall p, E.onCurve p ->
    point_dec_coordinates sign_bit (point_enc_coordinates p) = Some p.
 Proof.
@@ -277,7 +223,7 @@ Proof.
     rewrite !Fq_pow_zero, sqrt_mod_q_of_0, Fq_pow_zero by congruence.
     rewrite if_F_eq_dec_if_F_eqb, sign_bit_zero. 
     reflexivity.
-  + assert (sqrt_mod_q (x ^ 2) <> 0) by (intro false_eq; apply sqrt_mod_q_0 in false_eq;
+  + assert (sqrt_mod_q (x ^ 2) <> 0) by (intro false_eq; apply sqrt_mod_q_root_0 in false_eq; try assumption;
       apply Fq_root_zero in false_eq; rewrite false_eq, F_eqb_refl in eqb_x_0; congruence).
     replace (F_eqb (sqrt_mod_q (x ^ 2)) 0) with false by (symmetry;
         apply F_eqb_neq_complete; assumption).