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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 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.
Local Open Scope F_scope.
Section SignBit.
Context {m : Z} {prime_m : prime m} {two_lt_m : (2 < m)%Z} {sz : nat} {bound_check : (Z.to_nat m < 2 ^ sz)%nat}.
Lemma sign_bit_parity : forall x, @sign_bit m sz x = Z.odd x.
Proof.
unfold sign_bit, Fm_enc; intros.
pose proof (shatter_word (NToWord sz (Z.to_N x))) as shatter.
case_eq sz; intros; subst; rewrite shatter.
+ pose proof (prime_ge_2 m prime_m).
simpl in bound_check.
assert (m < 1)%Z by (apply Z2Nat.inj_lt; try omega; assumption).
omega.
+ assert (0 < m)%Z as m_pos by (pose proof prime_ge_2 m prime_m; omega).
pose proof (FieldToZ_range x m_pos).
destruct (FieldToZ x); auto.
- destruct p; auto.
- pose proof (Pos2Z.neg_is_neg p); omega.
Qed.
Lemma sign_bit_zero : @sign_bit m sz 0 = false.
Proof.
rewrite sign_bit_parity; auto.
Qed.
Lemma sign_bit_opp : forall (x : F m), x <> 0 -> negb (@sign_bit m sz x) = @sign_bit m sz (opp x).
Proof.
intros.
pose proof sign_bit_zero as sign_zero.
rewrite !sign_bit_parity in *.
pose proof (F_opp_spec x) as opp_spec_x.
apply F_eq in opp_spec_x.
rewrite FieldToZ_add in opp_spec_x.
rewrite <-opp_spec_x, Z.odd_mod in sign_zero by (pose proof prime_ge_2 m prime_m; omega).
replace (Z.odd m) with true in sign_zero by (destruct (Z.prime_odd_or_2 m prime_m); auto || omega).
rewrite Z.odd_add, F_FieldToZ_add_opp, Z.div_same, Bool.xorb_true_r in sign_zero by assumption || omega.
apply Bool.xorb_eq.
rewrite <-Bool.negb_xorb_l.
assumption.
Qed.
End SignBit.
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