NumTheoryUtil : code cleanup; moved some lemmas to ZUtil.

1 files changed, 126 insertions, 383 deletions

diff --git a/src/Util/NumTheoryUtil.v b/src/Util/NumTheoryUtil.v
index 251f120b0..4f673b545 100644
--- a/src/Util/NumTheoryUtil.v
+++ b/src/Util/NumTheoryUtil.v
@@ -5,270 +5,7 @@ Require Import Coqprime.Zp.
 Require Import VerdiTactics.
 Local Open Scope Z.
 
-(* TODO : integrate Andres's Fermat's little theorem proof *)
-Lemma fermat_little: forall (a p : Z) (prime_p : prime p),
-  a mod p <> 0 -> a ^ (p - 1) mod p = 1.
-Admitted.
-
-Ltac prime_bound := match goal with
-| H : prime ?p |- _ => pose proof (prime_ge_2 p H); try omega
-end.
-
-Lemma prime_minus1_div2_nonneg : forall (x p : Z) (prime_p : prime p),
-  x * 2 + 1 = p -> 0 < x.
-Proof.
-  intros; prime_bound.
-Qed.
-
-Lemma squared_fermat_little: forall (a x p : Z) (prime_p : prime p),
-  x * 2 + 1 = p -> a mod p <> 0 -> (a * a) ^ x mod p = 1.
-Proof.
-  intros.
-  rewrite <- Z.pow_2_r.
-  rewrite <- Z.pow_mul_r by
-    (pose proof (prime_minus1_div2_nonneg x p prime_p H); omega).
-  rewrite Z.mul_comm.
-  replace (x * 2) with (x * 2 + 1 - 1) by omega.
-  rewrite H.
-  apply fermat_little; auto.
-Qed.
-
-Lemma mod_exp_0 : forall a x m, x > 0 -> m > 1 -> a mod m = 0 ->
-  a ^ x mod m = 0.
-Proof.
-  intros.
-  replace x with (Z.of_nat (Z.to_nat x)) in * by (apply Z2Nat.id; omega).
-  induction (Z.to_nat x). {
-    simpl in *; omega.
-  } {
-    rewrite Nat2Z.inj_succ in *.
-    rewrite Z.pow_succ_r by omega.
-    rewrite Z.mul_mod by omega.
-    case_eq n; intros. {
-      subst. simpl.
-      rewrite Zmod_1_l by omega.
-      rewrite H1.
-      apply Zmod_0_l.
-    } {
-      subst.
-      rewrite IHn by (rewrite Nat2Z.inj_succ in *; omega).
-      rewrite H1.
-      auto.
-    }
-  }
-Qed.
-
-Lemma mod_pow : forall (a m b : Z), (0 <= b) -> (m <> 0) ->
-    a ^ b mod m = (a mod m) ^ b mod m.
-Proof.
-  intros; rewrite <- (Z2Nat.id b) by auto.
-  induction (Z.to_nat b); auto.
-  rewrite Nat2Z.inj_succ.
-  do 2 rewrite Z.pow_succ_r by apply Nat2Z.is_nonneg.
-  rewrite Z.mul_mod by auto.
-  rewrite (Z.mul_mod (a mod m) ((a mod m) ^ Z.of_nat n) m) by auto.
-  rewrite <- IHn by auto.
-  rewrite Z.mod_mod by auto.
-  reflexivity.
-Qed.
-
-Lemma euler_criterion_reverse : forall (a x p : Z) (prime_p : prime p),
-  (x * 2 + 1 = p) -> (a mod p <> 0) ->
-  (exists b, b * b mod p = a mod p) -> (a ^ x mod p = 1).
-Proof.
-  intros.
-  destruct H1.
-  assert (x0 mod p <> 0). {
-    intuition.
-    assert (2 > 0) by omega.
-    assert (p > 1) by prime_bound.
-    pose proof (mod_exp_0 x0 2 p H3 H4 H2); clear H3; clear H4.
-    rewrite Z.pow_2_r in H5.
-    rewrite H5 in H1.
-    intuition.
-  }
-  pose proof (squared_fermat_little x0 x p prime_p H H2).
-  assert (0 <= x) by (pose proof (prime_minus1_div2_nonneg x p prime_p H); omega).
-  assert (p <> 0) by prime_bound.
-  rewrite mod_pow by auto.
-  pose proof (mod_pow (x0 * x0) p x H4 H5).
-  rewrite H3 in H6.
-  rewrite H1 in H6.
-  auto.
-Qed.
-(*
-Fixpoint order_mod' y p i r :=
-  match i with
-  | O => r
-  | S i' => if (Zeq_bool (y ^ (Z.of_nat i) mod p) 1)
-            then order_mod' y p i' (Z.of_nat i)
-            else order_mod' y p i' r
-  end.
-
-Definition order_mod y p := order_mod' y p (Z.to_nat (p - 2)) (p - 1).
-
-Lemma order_mod'_le_r : forall y p i r, Z.of_nat i <= r ->
-  order_mod' y p i r <= r.
-Proof.
-  induction i; intros; try (simpl; omega).
-  unfold order_mod'; fold order_mod'.
-  assert (Z.of_nat i <= Z.of_nat (S i)) by (rewrite <- Nat2Z.inj_le; omega).
-  break_if. {
-    specialize (IHi (Z.of_nat (S i)) H0).
-    omega.
-  } {
-    apply IHi.
-    omega.
-  }
-Qed.
-
-Lemma order_mod'_pos : forall y p i r, 1 <= r -> 1 <= order_mod' y p i r.
-Proof.
-  induction i; intros; auto.
-  unfold order_mod'; fold order_mod'.
-  break_if; apply IHi; auto.
-  replace 1 with (Z.of_nat 1) by auto.
-  rewrite <- Nat2Z.inj_le.
-  pose proof (lt_0_Sn i); omega.
-Qed.
-
-Lemma order_mod_bounds : forall y p (prime_p : prime p),
-  1 <= order_mod y p <= p - 1.
-Proof.
-  unfold order_mod; split; intros. {
-    apply order_mod'_pos; prime_bound.
-  } {
-    apply order_mod'_le_r.
-    rewrite Z2Nat.id; prime_bound; omega.
-  }
-Qed.
-
-Lemma order_mod'_id : forall y p i r,
-  y ^ r mod p = 1 ->
-  y ^ (order_mod' y p i r) mod p = 1.
-Proof.
-  induction i; intros; [simpl; auto |].
-  unfold order_mod'; fold order_mod'.
-  break_if; auto.
-  apply IHi.
-  apply Zeq_bool_eq; auto.
-Qed.
-
-Lemma order_mod_id : forall y p (prime_p : prime p), (y mod p <> 0) -> y ^ (order_mod y p) mod p = 1.
-Proof.
-  intros.
-  unfold order_mod; apply order_mod'_id.
-  apply fermat_little; auto.
-Qed.
-
-Lemma order_mod'_upper_bound : forall x y p i r, 1 <= x <= Z.of_nat i ->
-  (Z.of_nat i <= r) -> (y ^ x mod p = 1) -> order_mod' y p i r <= x.
-Proof.
-  induction i; intros; try (simpl in H; omega).
-  unfold order_mod'; fold order_mod'.
-  assert (Z.of_nat i <= Z.of_nat (S i)) by (rewrite <- Nat2Z.inj_le; omega).
-  break_if. {
-    specialize (IHi (Z.of_nat (S i))).
-    destruct (Z.eq_dec x (Z.of_nat (S i))). {
-      rewrite e.
-      apply order_mod'_le_r; auto.
-    } {
-      destruct H.
-      rewrite Nat2Z.inj_succ in H3.
-      apply (Z.le_succ_r x (Z.of_nat i)) in H3.
-      destruct H3; intuition.
-      rewrite Nat2Z.inj_succ.
-      rewrite H3.
-      apply order_mod'_le_r.
-      omega.
-    }
-  } {
-    destruct H.
-    apply Z.le_lteq in H3; destruct H3. {
-      rewrite Nat2Z.inj_succ in H3.
-      apply IHi; omega.
-    } {
-      exfalso.
-      destruct H3.
-      apply Zeq_is_eq_bool in H1.
-      rewrite Heqb in H1.
-      intuition.
-    }
-  }
-Qed.
-
-Lemma order_mod_upper_bound : forall x y p (prime_p : prime p),
-  (1 <= x <= p - 2) ->
-  (y ^ x mod p = 1) -> order_mod y p <= x.
-Proof.
-  unfold order_mod; intros.
-  apply order_mod'_upper_bound; (rewrite Z2Nat.id; omega) || prime_bound.
-Qed.
-
-Lemma order_mod_lowest : forall x y p (prime_p : prime p),
-  1 <= x < order_mod y p -> y ^ x mod p <> 1.
-Proof.
-  intros.
-  intuition.
-  pose proof (order_mod_upper_bound x y p prime_p).
-  assert (1 <= x <= p - 2) as x_bounds. {
-    pose proof (order_mod_bounds y p prime_p).
-    omega.
-  }
-  specialize (H x_bounds H0).
-  omega.
-Qed.
-
-Lemma pow_mod_order : forall x y p (prime_p : prime p), 1 <= x ->
-  y ^ x mod p = 1 -> y ^ (x mod (order_mod y p)) mod p = 1.
-Proof.
-  intros.
-  remember (order_mod y p) as ord.
-  pose proof (order_mod_bounds y p prime_p).
-  assert (0 <= x / ord) by (apply Z.div_pos; omega).
-  assert (y mod p <> 0) by (intuition; apply (mod_exp_0 _ x) in H3; prime_bound).
-  rewrite (Z_div_mod_eq x ord) in H0 by omega.
-  rewrite Z.pow_add_r in H0 by (try apply Z.mul_nonneg_nonneg;
-    try apply Z.mod_pos_bound; omega).
-  rewrite Zmult_mod in H0.
-  rewrite Z.pow_mul_r in H0 by (try apply Z.mod_pos_bound; omega).
-  rewrite mod_pow in H0 by (prime_bound || auto).
-  rewrite Heqord in H0 at 1.
-  rewrite order_mod_id in H0; auto.
-  rewrite Z.pow_1_l in H0 by auto.
-  rewrite Z.mod_1_l in H0 by prime_bound.
-  rewrite Z.mul_1_l in H0.
-  rewrite Z.mod_mod in H0 by prime_bound.
-  auto.
-Qed.
-
-Lemma order_mod_divide : forall x y p (prime_p : prime p), 0 <= x ->
-  y ^ x mod p = 1 -> (order_mod y p | x).
-Proof.
-  intros.
-  pose proof (order_mod_bounds y p prime_p).
-  destruct (Z.eq_dec x 0); [subst; apply Z.divide_0_r|].
-  apply pow_mod_order in H0; (omega || auto).
-  assert (0 < order_mod y p) by omega.
-  apply (Z.mod_pos_bound x _) in H2.
-  pose proof (Z.nonpos_pos_cases (x mod order_mod y p)).
-  destruct H3. {
-    assert (x mod order_mod y p = 0) by omega.
-    apply Z.mod_divide; omega.
-  } {
-    assert (1 <= x mod order_mod y p <= p - 2) by omega.
-    pose proof (order_mod_upper_bound _ y p prime_p H4 H0).
-    omega.
-  }
-Qed.
-
-Lemma e_order_order_mod : forall y p (prime_p : prime p) (lt_1_p : 1 < p),
-  EGroup.e_order Z.eq_dec y (ZPGroup p lt_1_p) = FGroup.g_order (ZPGroup p lt_1_p)
-  -> order_mod y p = p - 1.
-Proof.
-Admitted.
-*)
-
+(* TODO: move somewhere else for lemmas about Coqprime? *)
 Lemma in_ZPGroup_range (n : Z) (n_pos : 1 < n) (p : Z) :
   List.In p (FGroup.s (ZPGroup n n_pos)) -> 1 <= p < n.
 Proof.
@@ -302,118 +39,116 @@ Proof.
   auto.
 Qed.
 
-Lemma exists_primitive_root_power : forall p (prime_p : prime p) (lt_1_p : 1 < p)
-  (p_odd : p <> 2) (* This is provable for p = 2, but we are lazy. *), 
-  (exists y, List.In y (FGroup.s (ZPGroup p lt_1_p))
-  /\ EGroup.e_order Z.eq_dec y (ZPGroup p lt_1_p) = FGroup.g_order (ZPGroup p lt_1_p)
-  /\ (forall x (x_range : 1 <= x <= p - 1), exists j, 0 <= j <= p - 1 /\ y ^ j mod p = x)).
+Section EulerCriterion.
+
+Variable x p : Z.
+Hypothesis prime_p : prime p.
+Hypothesis neq_p_2 : p <> 2. (* Euler's Criterion is also provable with p = 2, but we do not need it and are lazy.*)
+Hypothesis x_id : x * 2 + 1 = p.
+
+
+Lemma lt_1_p : 1 < p. Proof. prime_bound. Qed.
+Lemma x_pos: 0 < x. Proof. prime_bound. Qed.
+Lemma x_nonneg: 0 <= x. Proof. prime_bound. Qed.
+
+Lemma x_id_inv : x = (p - 1) / 2.
 Proof.
-  intros.
-  destruct (Zp_cyclic p lt_1_p prime_p) as [y [y_in_ZPGroup y_order]].
-  exists y; repeat split; auto.
-  intros x x_range.
-  pose proof (in_ZPGroup_range p lt_1_p y y_in_ZPGroup) as y_range1.
-  assert (0 <= y < p) as y_range0 by omega. 
-  pose proof (rel_prime_le_prime y p prime_p y_range1) as rel_prime_y_p.
-  pose proof (in_ZPGroup p lt_1_p y rel_prime_y_p y_range0) as in_ZPGroup_y.
-  destruct (EGroup.support_gpow Z Z.eq_dec (pmult p) y (ZPGroup p lt_1_p) in_ZPGroup_y x) as [k [k_range gpow_y_k]].
-  assert (rel_prime x p) by (apply rel_prime_le_prime; (omega || auto)).
-  assert (List.In x (FGroup.s (ZPGroup p lt_1_p))).
-  apply in_ZPGroup; (omega || auto).
-  apply generator_subgroup_is_group; eauto.
-  exists k; split. {
-    split; try omega.
-    unfold EGroup.e_order in y_order.
-    rewrite y_order in k_range.
-    rewrite <- phi_is_order in k_range.
-    rewrite Euler.prime_phi_n_minus_1 in k_range; (omega || auto).
-  }
-  assert (y mod p = y) as y_small by (apply Z.mod_small; omega).
-  rewrite <- y_small in gpow_y_k.
-  rewrite <- (Zpower_mod_is_gpow y k p lt_1_p) in gpow_y_k by (omega || auto).
-  auto.
+  intros; apply Zeq_plus_swap in x_id.
+  replace (p - 1) with (2 * ((p - 1) / 2)) in x_id by
+    (symmetry; apply Z_div_exact_2; [omega | rewrite <- x_id; apply Z_mod_mult]).
+  ring_simplify in x_id; apply Z.mul_cancel_l in x_id; omega.
 Qed.
 
-Lemma pred_odd : forall x, ~ (2 | x) -> (2 | x - 1).
+Lemma mod_p_order : FGroup.g_order (ZPGroup p lt_1_p) = p - 1.
 Proof.
-  intros.
-  rewrite <- Z.mod_divide in H by omega.
-  rewrite <- Z.mod_divide by omega.
-  pose proof (Zmod_odd x).
-  break_if; intuition.
-  replace x with (Z.succ (x - 1)) in Heqb by omega.
-  rewrite Z.odd_succ in Heqb.
-  pose proof (Zmod_even (x - 1)).
-  rewrite Heqb in H1; simpl in H1; auto.
+  intros; rewrite <- phi_is_order.
+  apply Euler.prime_phi_n_minus_1; auto.
 Qed.
 
-Ltac Zdivide_exists_mul := match goal with
-| [ H : (?a | ?b) |- _ ] => apply Z.mod_divide in H; try apply Zmod_divides in H; destruct H
-| [ |- (?a | ?b) ] => apply Z.mod_divide; try apply Zmod_divides
-end; (omega || auto).
+Lemma p_odd : Z.odd p = true.
+Proof.
+  pose proof (prime_odd_or_2 p prime_p).
+  destruct H; auto.
+Qed.
 
-Lemma prime_pred_divide2 : forall p (prime_p : prime p) (p_odd : p <> 2),
-  (2 | p - 1).
+Lemma prime_pred_even : Z.even (p - 1) = true.
 Proof.
   intros.
-  apply pred_odd.
-  intuition.
-  Zdivide_exists_mul; [ | pose proof (Z.mod_pos_bound p 2); omega].
-  rewrite Z.mul_comm in H.
-  apply Zdivide_intro in H.
-  apply prime_divisors in H; auto.
-  intuition.
-  prime_bound.
+  rewrite <- Z.odd_succ.
+  replace (Z.succ (p - 1)) with p by ring.
+  apply p_odd.
 Qed.
 
-Lemma prime_odd : forall p (prime_p : prime p) (p_odd : p <> 2),
-  Zodd p.
+Lemma fermat_little: forall a (a_nonzero : a mod p <> 0),
+  a ^ (p - 1) mod p = 1.
+Proof.
+Admitted.
+
+Lemma squared_fermat_little: forall a (a_nonzero : a mod p <> 0),
+  (a * a) ^ x mod p = 1.
 Proof.
   intros.
-  pose proof (prime_pred_divide2 p prime_p p_odd).
-  assert (Zeven (p - 1)) by (Zdivide_exists_mul; [rewrite H; apply Zeven_2p
-  | pose proof (Zmod_pos_bound (p - 1) 2); omega]).
-  replace p with (Z.succ (p - 1)) by ring.
-  apply Zodd_Sn; auto.
+  rewrite <- Z.pow_2_r.
+  rewrite <- Z.pow_mul_r by (apply x_nonneg || omega).
+  replace (2 * x) with (x * 2 + 1 - 1) by omega.
+  rewrite x_id.
+  apply fermat_little; auto.
 Qed.
 
-Lemma prime_minus1_div2_exact : forall x p (prime_p : prime p) (p_odd : p <> 2),
-  (x * 2 + 1 = p) -> x = (p - 1) / 2.
+Lemma euler_criterion_reverse : forall a (a_nonzero : a mod p <> 0),
+  (exists b, b * b mod p = a mod p) -> (a ^ x mod p = 1).
 Proof.
-  intros.
-  apply Zeq_plus_swap in H.
-  replace (p - 1) with (2 * ((p - 1) / 2)) in H by
-    (symmetry; apply Z_div_exact_2; try apply Z.mod_divide;
-    try apply prime_pred_divide2; (auto || omega)).
-  ring_simplify in H.
-  apply Z.mul_cancel_l in H; omega.
+  intros ? ? a_square.
+  destruct a_square as [b a_square].
+  assert (b mod p <> 0) as b_nonzero. {
+    intuition.
+    rewrite <- Z.pow_2_r in a_square.
+    rewrite mod_exp_0 in a_square by prime_bound.
+    auto.
+  }
+  pose proof (squared_fermat_little b b_nonzero).
+  rewrite mod_pow in * by prime_bound.
+  rewrite <- a_square; auto.
 Qed.
 
-Lemma divide_mul_div: forall a b c, a <> 0 -> c <> 0 ->
-  (a | b * (a / c)) -> (c | a) -> (c | b).
+Lemma exists_primitive_root_power :
+  (exists y, List.In y (FGroup.s (ZPGroup p lt_1_p))
+  /\ EGroup.e_order Z.eq_dec y (ZPGroup p lt_1_p) = FGroup.g_order (ZPGroup p lt_1_p)
+  /\ (forall a (a_range : 1 <= a <= p - 1), exists j, 0 <= j <= p - 1 /\ y ^ j mod p = a)).
 Proof.
   intros.
-  Zdivide_exists_mul.
-  Zdivide_exists_mul.
-  rewrite H2 in H1.
-  rewrite Z_div_mul' in H1 by auto.
-  replace (b * x) with (x * b) in H1 by ring.
-  replace (c * x * x0) with (x * (x0 * c)) in H1 by ring.
-  rewrite Z.mul_cancel_l in H1 by (intuition; rewrite H3 in H2; ring_simplify in H2; intuition).
-  eapply Zdivide_intro; eauto.
+  destruct (Zp_cyclic p lt_1_p prime_p) as [y [y_in_ZPGroup y_order]].
+  exists y; repeat split; auto.
+  intros.
+  pose proof (in_ZPGroup_range p lt_1_p y y_in_ZPGroup) as y_range1.
+  assert (0 <= y < p) as y_range0 by omega.
+  assert (rel_prime y p) as rel_prime_y_p by (apply rel_prime_le_prime; omega || auto).
+  assert (rel_prime a p) as rel_prime_a_p by (apply rel_prime_le_prime; omega || auto).
+  assert (List.In a (FGroup.s (ZPGroup p lt_1_p))) as a_in_ZPGroup by (apply in_ZPGroup; omega || auto).
+  destruct (EGroup.support_gpow Z Z.eq_dec (pmult p) y (ZPGroup p lt_1_p) y_in_ZPGroup a)
+    as [k [k_range gpow_y_k]]; [apply generator_subgroup_is_group; auto |].
+  exists k; split. {
+    unfold EGroup.e_order in y_order.
+    rewrite y_order in k_range.
+    rewrite mod_p_order in k_range.
+    omega.
+  } {
+    assert (y mod p = y) as y_small by (apply Z.mod_small; omega).
+    rewrite <- y_small in gpow_y_k.
+    rewrite <- (Zpower_mod_is_gpow y k p lt_1_p) in gpow_y_k by (omega || auto).
+    auto.
+  }
 Qed.
 
 Ltac ereplace x := match type of x with ?t =>
   let e := fresh "e" in evar (e:t); replace x with e; subst e end.
 
-Lemma euler_criterion : forall (a x p : Z) (prime_p : prime p)
-  (p_odd : p <> 2) (a_range : 1 <= a <= p - 1) (x_div2_minus1p : x * 2 + 1 = p)
+Lemma euler_criterion : forall a (a_range : 1 <= a <= p - 1)
   (pow_a_x : a ^ x mod p = 1), exists b : Z, b * b mod p = a mod p.
 Proof.
   intros.
-  assert (1 < p) as lt_1_p by prime_bound.
-  destruct (exists_primitive_root_power p prime_p lt_1_p) as [y [in_ZPGroup_y [y_order gpow_y]]]; auto.
-  destruct (gpow_y a a_range) as [j [j_range pow_y_j]].
+  destruct (exists_primitive_root_power) as [y [in_ZPGroup_y [y_order gpow_y]]]; auto.
+  destruct (gpow_y a a_range) as [j [j_range pow_y_j]]; clear gpow_y.
   rewrite mod_pow in pow_a_x by prime_bound.
   replace a with (a mod p) in pow_y_j by (apply Z.mod_small; omega).
   rewrite <- pow_y_j in pow_a_x.
@@ -434,44 +169,47 @@ Proof.
   rewrite <- Z.pow_add_r by (apply Z.div_pos; omega).
   rewrite <- Z_div_plus by omega.
   rewrite Z.mul_comm.
-  rewrite (prime_minus1_div2_exact x p) in divide_mul_j_x; auto.
+  rewrite x_id_inv in divide_mul_j_x; auto.
   apply (divide_mul_div _ j 2) in divide_mul_j_x;
     try (apply prime_pred_divide2 || prime_bound); auto.
   rewrite <- Zdivide_Zdiv_eq by (auto || omega).
   rewrite Zplus_diag_eq_mult_2.
   rewrite Z_div_mult by omega; auto.
+  apply divide2_even_iff.
+  apply prime_pred_even.
 Qed.
 
-Lemma divide2_1mod4 : forall x, x mod 4 = 1 -> 0 <= x -> (2 | x / 2).
+End EulerCriterion.
+
+Lemma divide2_1mod4 : forall x (x_1mod4 : x mod 4 = 1) (x_nonneg : 0 <= x), (2 | x / 2).
 Proof.
   intros.
-  assert (Z.to_nat x mod 4 = 1)%nat. {
-    replace 1%Z with (Z.of_nat 1) in H by auto.
-    replace (x mod 4)%Z with (Z.of_nat (Z.to_nat x mod 4)) in H
+  assert (Z.to_nat x mod 4 = 1)%nat as x_1mod4_nat. {
+    replace 1 with (Z.of_nat 1) in * by auto.
+    replace (x mod 4) with (Z.of_nat (Z.to_nat x mod 4)) in *
       by (rewrite mod_Zmod by omega; rewrite Z2Nat.id; auto).
-    apply Nat2Z.inj in H; auto.
+    apply Nat2Z.inj in x_1mod4; auto.
   }
   remember (Z.to_nat x / 4)%nat as c.
-  pose proof (divide2_1mod4_nat c (Z.to_nat x) Heqc H1).
-  destruct H2.
-  replace 2%nat with (Z.to_nat 2) in H2 by auto.
-  apply inj_eq in H2.
-  rewrite div_Zdiv in H2 by (replace (Z.to_nat 2) with 2%nat by auto; omega).
-  rewrite Nat2Z.inj_mul in H2.
-  do 2 rewrite Z2Nat.id in H2 by (auto || omega).
-  rewrite Z.mul_comm in H2.
-  symmetry in H2.
-  apply Zdivide_intro in H2; auto.
+  destruct (divide2_1mod4_nat c (Z.to_nat x) Heqc x_1mod4_nat) as [k k_id].
+  replace 2%nat with (Z.to_nat 2) in * by auto.
+  apply inj_eq in k_id.
+  rewrite div_Zdiv in k_id by (replace (Z.to_nat 2) with 2%nat by auto; omega).
+  rewrite Nat2Z.inj_mul in k_id.
+  do 2 rewrite Z2Nat.id in k_id by omega.
+  rewrite Z.mul_comm in k_id.
+  symmetry in k_id.
+  apply Zdivide_intro in k_id; auto.
 Qed.
 
 Lemma minus1_even_pow : forall x y, (2 | y) -> (1 < x) -> (0 <= y) -> ((x - 1) ^ y mod x = 1).
 Proof.
-  intros.
-  apply Zdivide_Zdiv_eq in H; try omega.
-  rewrite H.
+  intros ? ? divide_2_y lt_1_x y_nonneg.
+  apply Zdivide_Zdiv_eq in divide_2_y; try omega.
+  rewrite divide_2_y.
   rewrite Z.pow_mul_r by omega.
-  assert ((x - 1)^2 mod x = 1). {
-    replace ((x - 1)^2) with (x ^ 2 - 2 * x + 1)
+  assert ((x - 1) ^ 2 mod x = 1) as square_case. {
+    replace ((x - 1) ^ 2) with (x ^ 2 - 2 * x + 1)
       by (do 2 rewrite Z.pow_2_r; rewrite Z.mul_sub_distr_l; do 2 rewrite Z.mul_sub_distr_r; omega).
     rewrite Zplus_mod.
     rewrite Z.pow_2_r.
@@ -485,24 +223,29 @@ Proof.
   rewrite Z.pow_succ_r by apply Zle_0_nat.
   rewrite Zmult_mod.
   rewrite IHn.
-  rewrite H2.
-  simpl.
-  apply Zmod_1_l; auto.
+  rewrite square_case.
+  simpl; apply Zmod_1_l; auto.
+Qed.
+
+Lemma prime_1mod4_neq2 : forall p (prime_p : prime p), (p mod 4 = 1) -> p <> 2.
+Proof.
+  intros; intuition.
+  assert (4 <> 0)%Z as neq_4_0 by omega.
+  pose proof (Z.div_mod p 4 neq_4_0).
+  omega.
 Qed.
 
 Lemma minus1_square_1mod4 : forall (p : Z) (prime_p : prime p),
   (p mod 4 = 1)%Z -> exists b : Z, (b * b mod p = p - 1)%Z.
 Proof.
   intros.
-  assert (4 <> 0)%Z by omega.
-  pose proof (Z.div_mod p 4 H0).
-  pose proof (prime_ge_2 p (prime_p)).
-  assert (2 | p / 2)%Z by (apply divide2_1mod4; (auto || omega)).
-  replace (p - 1) with ((p - 1) mod p) by (apply Zmod_small; omega).
-  assert (p <> 2) as neq_p_2 by intuition.
-  apply (euler_criterion (p - 1) (p / 2) p prime_p);
-    [ auto | omega | | apply minus1_even_pow; (omega || auto); apply Z_div_pos; omega].
-  pose proof (prime_odd p prime_p neq_p_2).
-  rewrite <- Zdiv2_div.
-  rewrite Zodd_div2; (ring || auto).
+  replace (p - 1) with ((p - 1) mod p) by (apply Zmod_small; split; prime_bound).
+  assert (p <> 2) as neq_p_2 by (apply prime_1mod4_neq2; auto).
+  apply (euler_criterion (p / 2) p prime_p).
+  + auto.
+  + destruct (prime_odd_or_2 p prime_p); intuition.
+    rewrite <- Zdiv2_div.
+    pose proof (Zdiv2_odd_eqn p); break_if; congruence || omega.
+  + prime_bound.
+  + apply minus1_even_pow; [apply divide2_1mod4 | | apply Z_div_pos]; prime_bound.
 Qed.