Theory Gauss
section ‹Gauss' Lemma›
theory Gauss
imports Euler_Criterion
begin
lemma cong_prime_prod_zero_nat:
"[a * b = 0] (mod p) ⟹ prime p ⟹ [a = 0] (mod p) ∨ [b = 0] (mod p)"
for a :: nat
by (auto simp add: cong_altdef_nat prime_dvd_mult_iff)
lemma cong_prime_prod_zero_int:
"[a * b = 0] (mod p) ⟹ prime p ⟹ [a = 0] (mod p) ∨ [b = 0] (mod p)"
for a :: int
by (simp add: cong_0_iff prime_dvd_mult_iff)
locale GAUSS =
fixes p :: "nat"
fixes a :: "int"
assumes p_prime: "prime p"
assumes p_ge_2: "2 < p"
assumes p_a_relprime: "[a ≠ 0](mod p)"
assumes a_nonzero: "0 < a"
begin
definition "A = {0::int <.. ((int p - 1) div 2)}"
definition "B = (λx. x * a) ` A"
definition "C = (λx. x mod p) ` B"
definition "D = C ∩ {.. (int p - 1) div 2}"
definition "E = C ∩ {(int p - 1) div 2 <..}"
definition "F = (λx. (int p - x)) ` E"
subsection ‹Basic properties of p›
lemma odd_p: "odd p"
by (metis p_prime p_ge_2 prime_odd_nat)
lemma p_minus_one_l: "(int p - 1) div 2 < p"
proof -
have "(p - 1) div 2 ≤ (p - 1) div 1"
by (metis div_by_1 div_le_dividend)
also have "… = p - 1" by simp
finally show ?thesis
using p_ge_2 by arith
qed
lemma p_eq2: "int p = (2 * ((int p - 1) div 2)) + 1"
using odd_p p_ge_2 nonzero_mult_div_cancel_left [of 2 "p - 1"] by simp
lemma p_odd_int: obtains z :: int where "int p = 2 * z + 1" "0 < z"
proof
let ?z = "(int p - 1) div 2"
show "int p = 2 * ?z + 1" by (rule p_eq2)
show "0 < ?z"
using p_ge_2 by linarith
qed
subsection ‹Basic Properties of the Gauss Sets›
lemma finite_A: "finite A"
by (auto simp add: A_def)
lemma finite_B: "finite B"
by (auto simp add: B_def finite_A)
lemma finite_C: "finite C"
by (auto simp add: C_def finite_B)
lemma finite_D: "finite D"
by (auto simp add: D_def finite_C)
lemma finite_E: "finite E"
by (auto simp add: E_def finite_C)
lemma finite_F: "finite F"
by (auto simp add: F_def finite_E)
lemma C_eq: "C = D ∪ E"
by (auto simp add: C_def D_def E_def)
lemma A_card_eq: "card A = nat ((int p - 1) div 2)"
by (auto simp add: A_def)
lemma inj_on_xa_A: "inj_on (λx. x * a) A"
using a_nonzero by (simp add: A_def inj_on_def)
definition ResSet :: "int ⇒ int set ⇒ bool"
where "ResSet m X ⟷ (∀y1 y2. y1 ∈ X ∧ y2 ∈ X ∧ [y1 = y2] (mod m) ⟶ y1 = y2)"
lemma ResSet_image:
"0 < m ⟹ ResSet m A ⟹ ∀x ∈ A. ∀y ∈ A. ([f x = f y](mod m) ⟶ x = y) ⟹ ResSet m (f ` A)"
by (auto simp add: ResSet_def)
lemma A_res: "ResSet p A"
using p_ge_2 by (auto simp add: A_def ResSet_def intro!: cong_less_imp_eq_int)
lemma B_res: "ResSet p B"
proof -
have *: "x = y"
if a: "[x * a = y * a] (mod p)"
and b: "0 < x"
and c: "x ≤ (int p - 1) div 2"
and d: "0 < y"
and e: "y ≤ (int p - 1) div 2"
for x y
proof -
from p_a_relprime have "¬ p dvd a"
by (simp add: cong_0_iff)
with p_prime prime_imp_coprime [of _ "nat ¦a¦"]
have "coprime a (int p)"
by (simp_all add: ac_simps)
with a cong_mult_rcancel [of a "int p" x y] have "[x = y] (mod p)"
by simp
with cong_less_imp_eq_int [of x y p] p_minus_one_l
order_le_less_trans [of x "(int p - 1) div 2" p]
order_le_less_trans [of y "(int p - 1) div 2" p]
show ?thesis
by (metis b c cong_less_imp_eq_int d e zero_less_imp_eq_int of_nat_0_le_iff)
qed
show ?thesis
using p_ge_2 p_a_relprime p_minus_one_l
by (metis "*" A_def A_res B_def GAUSS.ResSet_image GAUSS_axioms greaterThanAtMost_iff odd_p odd_pos of_nat_0_less_iff)
qed
lemma SR_B_inj: "inj_on (λx. x mod p) B"
proof -
have False
if a: "x * a mod p = y * a mod p"
and b: "0 < x"
and c: "x ≤ (int p - 1) div 2"
and d: "0 < y"
and e: "y ≤ (int p - 1) div 2"
and f: "x ≠ y"
for x y
proof -
from a have a': "[x * a = y * a](mod p)"
using cong_def by blast
from p_a_relprime have "¬p dvd a"
by (simp add: cong_0_iff)
with p_prime prime_imp_coprime [of _ "nat ¦a¦"]
have "coprime a (int p)"
by (simp_all add: ac_simps)
with a' cong_mult_rcancel [of a "int p" x y]
have "[x = y] (mod p)" by simp
with cong_less_imp_eq_int [of x y p] p_minus_one_l
order_le_less_trans [of x "(int p - 1) div 2" p]
order_le_less_trans [of y "(int p - 1) div 2" p]
have "x = y"
by (metis b c cong_less_imp_eq_int d e zero_less_imp_eq_int of_nat_0_le_iff)
then show ?thesis
by (simp add: f)
qed
then show ?thesis
by (auto simp add: B_def inj_on_def A_def) metis
qed
lemma nonzero_mod_p: "0 < x ⟹ x < int p ⟹ [x ≠ 0](mod p)"
for x :: int
by (simp add: cong_def)
lemma A_ncong_p: "x ∈ A ⟹ [x ≠ 0](mod p)"
by (rule nonzero_mod_p) (auto simp add: A_def)
lemma A_greater_zero: "x ∈ A ⟹ 0 < x"
by (auto simp add: A_def)
lemma B_ncong_p: "x ∈ B ⟹ [x ≠ 0](mod p)"
by (auto simp: B_def p_prime p_a_relprime A_ncong_p dest: cong_prime_prod_zero_int)
lemma B_greater_zero: "x ∈ B ⟹ 0 < x"
using a_nonzero by (auto simp add: B_def A_greater_zero)
lemma B_mod_greater_zero:
"0 < x mod int p" if "x ∈ B"
proof -
from that have "x mod int p ≠ 0"
using B_ncong_p cong_def cong_mult_self_left by blast
moreover from that have "0 < x"
by (rule B_greater_zero)
then have "0 ≤ x mod int p"
by (auto simp add: mod_int_pos_iff intro: neq_le_trans)
ultimately show ?thesis
by simp
qed
lemma C_greater_zero: "y ∈ C ⟹ 0 < y"
by (auto simp add: C_def B_mod_greater_zero)
lemma F_subset: "F ⊆ {x. 0 < x ∧ x ≤ ((int p - 1) div 2)}"
using p_ge_2 by (auto simp add: F_def E_def C_def intro: p_odd_int)
lemma D_subset: "D ⊆ {x. 0 < x ∧ x ≤ ((p - 1) div 2)}"
by (auto simp add: D_def C_greater_zero)
lemma F_eq: "F = {x. ∃y ∈ A. (x = p - ((y * a) mod p) ∧ (int p - 1) div 2 < (y * a) mod p)}"
by (auto simp add: F_def E_def D_def C_def B_def A_def)
lemma D_eq: "D = {x. ∃y ∈ A. (x = (y * a) mod p ∧ (y * a) mod p ≤ (int p - 1) div 2)}"
by (auto simp add: D_def C_def B_def A_def)
lemma all_A_relprime:
"coprime x p" if "x ∈ A"
proof -
from A_ncong_p [OF that] have "¬ int p dvd x"
by (simp add: cong_0_iff)
with p_prime show ?thesis
by (metis (no_types) coprime_commute prime_imp_coprime prime_nat_int_transfer)
qed
lemma A_prod_relprime: "coprime (prod id A) p"
by (auto intro: prod_coprime_left all_A_relprime)
subsection ‹Relationships Between Gauss Sets›
lemma StandardRes_inj_on_ResSet: "ResSet m X ⟹ inj_on (λb. b mod m) X"
by (auto simp add: ResSet_def inj_on_def cong_def)
lemma B_card_eq_A: "card B = card A"
using finite_A by (simp add: finite_A B_def inj_on_xa_A card_image)
lemma B_card_eq: "card B = nat ((int p - 1) div 2)"
by (simp add: B_card_eq_A A_card_eq)
lemma F_card_eq_E: "card F = card E"
using finite_E by (simp add: F_def card_image)
lemma C_card_eq_B: "card C = card B"
proof -
have "inj_on (λx. x mod p) B"
by (metis SR_B_inj)
then show ?thesis
by (metis C_def card_image)
qed
lemma D_E_disj: "D ∩ E = {}"
by (auto simp add: D_def E_def)
lemma C_card_eq_D_plus_E: "card C = card D + card E"
by (auto simp add: C_eq card_Un_disjoint D_E_disj finite_D finite_E)
lemma C_prod_eq_D_times_E: "prod id E * prod id D = prod id C"
by (metis C_eq D_E_disj finite_D finite_E inf_commute prod.union_disjoint sup_commute)
lemma C_B_zcong_prod: "[prod id C = prod id B] (mod p)"
apply (auto simp add: C_def)
apply (insert finite_B SR_B_inj)
apply (drule prod.reindex [of "λx. x mod int p" B id])
apply auto
apply (rule cong_prod)
apply (auto simp add: cong_def)
done
lemma F_Un_D_subset: "(F ∪ D) ⊆ A"
by (intro Un_least subset_trans [OF F_subset] subset_trans [OF D_subset]) (auto simp: A_def)
lemma F_D_disj: "(F ∩ D) = {}"
proof (auto simp add: F_eq D_eq)
fix y z :: int
assume "p - (y * a) mod p = (z * a) mod p"
then have "[(y * a) mod p + (z * a) mod p = 0] (mod p)"
by (metis add.commute diff_eq_eq dvd_refl cong_def dvd_eq_mod_eq_0 mod_0)
moreover have "[y * a = (y * a) mod p] (mod p)"
by (metis cong_def mod_mod_trivial)
ultimately have "[a * (y + z) = 0] (mod p)"
by (metis cong_def mod_add_left_eq mod_add_right_eq mult.commute ring_class.ring_distribs(1))
with p_prime a_nonzero p_a_relprime have a: "[y + z = 0] (mod p)"
by (auto dest!: cong_prime_prod_zero_int)
assume b: "y ∈ A" and c: "z ∈ A"
then have "0 < y + z"
by (auto simp: A_def)
moreover from b c p_eq2 have "y + z < p"
by (auto simp: A_def)
ultimately show False
by (metis a nonzero_mod_p)
qed
lemma F_Un_D_card: "card (F ∪ D) = nat ((p - 1) div 2)"
proof -
have "card (F ∪ D) = card E + card D"
by (auto simp add: finite_F finite_D F_D_disj card_Un_disjoint F_card_eq_E)
then have "card (F ∪ D) = card C"
by (simp add: C_card_eq_D_plus_E)
then show "card (F ∪ D) = nat ((p - 1) div 2)"
by (simp add: C_card_eq_B B_card_eq)
qed
lemma F_Un_D_eq_A: "F ∪ D = A"
using finite_A F_Un_D_subset A_card_eq F_Un_D_card by (auto simp add: card_seteq)
lemma prod_D_F_eq_prod_A: "prod id D * prod id F = prod id A"
by (metis F_D_disj F_Un_D_eq_A Int_commute Un_commute finite_D finite_F prod.union_disjoint)
lemma prod_F_zcong: "[prod id F = ((-1) ^ (card E)) * prod id E] (mod p)"
proof -
have FE: "prod id F = prod ((-) p) E"
using finite_E prod.reindex[OF inj_on_diff_left] by (auto simp add: F_def)
then have "∀x ∈ E. [(p-x) mod p = - x](mod p)"
by (metis cong_def minus_mod_self1 mod_mod_trivial)
then have "[prod ((λx. x mod p) ∘ ((-) p)) E = prod (uminus) E](mod p)"
using finite_E p_ge_2 cong_prod [of E "(λx. x mod p) ∘ ((-) p)" uminus p]
by auto
then have two: "[prod id F = prod (uminus) E](mod p)"
by (metis FE cong_cong_mod_int cong_refl cong_prod minus_mod_self1)
have "prod uminus E = (-1) ^ card E * prod id E"
using finite_E by (induct set: finite) auto
with two show ?thesis
by simp
qed
subsection ‹Gauss' Lemma›
lemma aux: "prod id A * (- 1) ^ card E * a ^ card A * (- 1) ^ card E = prod id A * a ^ card A"
by auto
theorem pre_gauss_lemma: "[a ^ nat((int p - 1) div 2) = (-1) ^ (card E)] (mod p)"
proof -
have "[prod id A = prod id F * prod id D](mod p)"
by (auto simp: prod_D_F_eq_prod_A mult.commute cong del: prod.cong_simp)
then have "[prod id A = ((-1)^(card E) * prod id E) * prod id D] (mod p)"
by (rule cong_trans) (metis cong_scalar_right prod_F_zcong)
then have "[prod id A = ((-1)^(card E) * prod id C)] (mod p)"
using finite_D finite_E by (auto simp add: ac_simps C_prod_eq_D_times_E C_eq D_E_disj prod.union_disjoint)
then have "[prod id A = ((-1)^(card E) * prod id B)] (mod p)"
by (rule cong_trans) (metis C_B_zcong_prod cong_scalar_left)
then have "[prod id A = ((-1)^(card E) * prod id ((λx. x * a) ` A))] (mod p)"
by (simp add: B_def)
then have "[prod id A = ((-1)^(card E) * prod (λx. x * a) A)] (mod p)"
by (simp add: inj_on_xa_A prod.reindex)
moreover have "prod (λx. x * a) A = prod (λx. a) A * prod id A"
using finite_A by (induct set: finite) auto
ultimately have "[prod id A = ((-1)^(card E) * (prod (λx. a) A * prod id A))] (mod p)"
by simp
then have "[prod id A = ((-1)^(card E) * a^(card A) * prod id A)](mod p)"
by (rule cong_trans)
(simp add: cong_scalar_left cong_scalar_right finite_A ac_simps)
then have a: "[prod id A * (-1)^(card E) =
((-1)^(card E) * a^(card A) * prod id A * (-1)^(card E))](mod p)"
by (rule cong_scalar_right)
then have "[prod id A * (-1)^(card E) = prod id A *
(-1)^(card E) * a^(card A) * (-1)^(card E)](mod p)"
by (rule cong_trans) (simp add: a ac_simps)
then have "[prod id A * (-1)^(card E) = prod id A * a^(card A)](mod p)"
by (rule cong_trans) (simp add: aux cong del: prod.cong_simp)
with A_prod_relprime have "[(- 1) ^ card E = a ^ card A](mod p)"
by (metis cong_mult_lcancel)
then show ?thesis
by (simp add: A_card_eq cong_sym)
qed
theorem gauss_lemma: "Legendre a p = (-1) ^ (card E)"
proof -
from euler_criterion p_prime p_ge_2 have "[Legendre a p = a^(nat (((p) - 1) div 2))] (mod p)"
by auto
moreover have "int ((p - 1) div 2) = (int p - 1) div 2"
using p_eq2 by linarith
then have "[a ^ nat (int ((p - 1) div 2)) = a ^ nat ((int p - 1) div 2)] (mod int p)"
by force
ultimately have "[Legendre a p = (-1) ^ (card E)] (mod p)"
using pre_gauss_lemma cong_trans by blast
moreover from p_a_relprime have "Legendre a p = 1 ∨ Legendre a p = -1"
by (auto simp add: Legendre_def)
moreover have "(-1::int) ^ (card E) = 1 ∨ (-1::int) ^ (card E) = -1"
using neg_one_even_power neg_one_odd_power by blast
moreover have "[1 ≠ - 1] (mod int p)"
using cong_iff_dvd_diff [where 'a=int] nonzero_mod_p[of 2] p_odd_int
by fastforce
ultimately show ?thesis
by (auto simp add: cong_sym)
qed
end
end