Theory Quadratic_Reciprocity

theory Quadratic_Reciprocity
imports Gauss
(*  Title:      HOL/Old_Number_Theory/Quadratic_Reciprocity.thy
Authors: Jeremy Avigad, David Gray, and Adam Kramer
*)


header {* The law of Quadratic reciprocity *}

theory Quadratic_Reciprocity
imports Gauss
begin

text {*
Lemmas leading up to the proof of theorem 3.3 in Niven and
Zuckerman's presentation.
*}


context GAUSS
begin

lemma QRLemma1: "a * setsum id A =
p * setsum (%x. ((x * a) div p)) A + setsum id D + setsum id E"

proof -
from finite_A have "a * setsum id A = setsum (%x. a * x) A"
by (auto simp add: setsum_const_mult id_def)
also have "setsum (%x. a * x) = setsum (%x. x * a)"
by (auto simp add: mult_commute)
also have "setsum (%x. x * a) A = setsum id B"
by (simp add: B_def setsum_reindex_id[OF inj_on_xa_A])
also have "... = setsum (%x. p * (x div p) + StandardRes p x) B"
by (auto simp add: StandardRes_def zmod_zdiv_equality)
also have "... = setsum (%x. p * (x div p)) B + setsum (StandardRes p) B"
by (rule setsum_addf)
also have "setsum (StandardRes p) B = setsum id C"
by (auto simp add: C_def setsum_reindex_id[OF SR_B_inj])
also from C_eq have "... = setsum id (D ∪ E)"
by auto
also from finite_D finite_E have "... = setsum id D + setsum id E"
by (rule setsum_Un_disjoint) (auto simp add: D_def E_def)
also have "setsum (%x. p * (x div p)) B =
setsum ((%x. p * (x div p)) o (%x. (x * a))) A"

by (auto simp add: B_def setsum_reindex inj_on_xa_A)
also have "... = setsum (%x. p * ((x * a) div p)) A"
by (auto simp add: o_def)
also from finite_A have "setsum (%x. p * ((x * a) div p)) A =
p * setsum (%x. ((x * a) div p)) A"

by (auto simp add: setsum_const_mult)
finally show ?thesis by arith
qed

lemma QRLemma2: "setsum id A = p * int (card E) - setsum id E +
setsum id D"

proof -
from F_Un_D_eq_A have "setsum id A = setsum id (D ∪ F)"
by (simp add: Un_commute)
also from F_D_disj finite_D finite_F
have "... = setsum id D + setsum id F"
by (auto simp add: Int_commute intro: setsum_Un_disjoint)
also from F_def have "F = (%x. (p - x)) ` E"
by auto
also from finite_E inj_on_pminusx_E have "setsum id ((%x. (p - x)) ` E) =
setsum (%x. (p - x)) E"

by (auto simp add: setsum_reindex)
also from finite_E have "setsum (op - p) E = setsum (%x. p) E - setsum id E"
by (auto simp add: setsum_subtractf id_def)
also from finite_E have "setsum (%x. p) E = p * int(card E)"
by (intro setsum_const)
finally show ?thesis
by arith
qed

lemma QRLemma3: "(a - 1) * setsum id A =
p * (setsum (%x. ((x * a) div p)) A - int(card E)) + 2 * setsum id E"

proof -
have "(a - 1) * setsum id A = a * setsum id A - setsum id A"
by (auto simp add: left_diff_distrib)
also note QRLemma1
also from QRLemma2 have "p * (∑x ∈ A. x * a div p) + setsum id D +
setsum id E - setsum id A =
p * (∑x ∈ A. x * a div p) + setsum id D +
setsum id E - (p * int (card E) - setsum id E + setsum id D)"

by auto
also have "... = p * (∑x ∈ A. x * a div p) -
p * int (card E) + 2 * setsum id E"

by arith
finally show ?thesis
by (auto simp only: right_diff_distrib)
qed

lemma QRLemma4: "a ∈ zOdd ==>
(setsum (%x. ((x * a) div p)) A ∈ zEven) = (int(card E): zEven)"

proof -
assume a_odd: "a ∈ zOdd"
from QRLemma3 have a: "p * (setsum (%x. ((x * a) div p)) A - int(card E)) =
(a - 1) * setsum id A - 2 * setsum id E"

by arith
from a_odd have "a - 1 ∈ zEven"
by (rule odd_minus_one_even)
hence "(a - 1) * setsum id A ∈ zEven"
by (rule even_times_either)
moreover have "2 * setsum id E ∈ zEven"
by (auto simp add: zEven_def)
ultimately have "(a - 1) * setsum id A - 2 * setsum id E ∈ zEven"
by (rule even_minus_even)
with a have "p * (setsum (%x. ((x * a) div p)) A - int(card E)): zEven"
by simp
hence "p ∈ zEven | (setsum (%x. ((x * a) div p)) A - int(card E)): zEven"
by (rule EvenOdd.even_product)
with p_odd have "(setsum (%x. ((x * a) div p)) A - int(card E)): zEven"
by (auto simp add: odd_iff_not_even)
thus ?thesis
by (auto simp only: even_diff [symmetric])
qed

lemma QRLemma5: "a ∈ zOdd ==>
(-1::int)^(card E) = (-1::int)^(nat(setsum (%x. ((x * a) div p)) A))"

proof -
assume "a ∈ zOdd"
from QRLemma4 [OF this] have
"(int(card E): zEven) = (setsum (%x. ((x * a) div p)) A ∈ zEven)" ..
moreover have "0 ≤ int(card E)"
by auto
moreover have "0 ≤ setsum (%x. ((x * a) div p)) A"
proof (intro setsum_nonneg)
show "∀x ∈ A. 0 ≤ x * a div p"
proof
fix x
assume "x ∈ A"
then have "0 ≤ x"
by (auto simp add: A_def)
with a_nonzero have "0 ≤ x * a"
by (auto simp add: zero_le_mult_iff)
with p_g_2 show "0 ≤ x * a div p"
by (auto simp add: pos_imp_zdiv_nonneg_iff)
qed
qed
ultimately have "(-1::int)^nat((int (card E))) =
(-1)^nat(((∑x ∈ A. x * a div p)))"

by (intro neg_one_power_parity, auto)
also have "nat (int(card E)) = card E"
by auto
finally show ?thesis .
qed

end

lemma MainQRLemma: "[| a ∈ zOdd; 0 < a; ~([a = 0] (mod p)); zprime p; 2 < p;
A = {x. 0 < x & x ≤ (p - 1) div 2} |] ==>
(Legendre a p) = (-1::int)^(nat(setsum (%x. ((x * a) div p)) A))"

apply (subst GAUSS.gauss_lemma)
apply (auto simp add: GAUSS_def)
apply (subst GAUSS.QRLemma5)
apply (auto simp add: GAUSS_def)
apply (simp add: GAUSS.A_def [OF GAUSS.intro] GAUSS_def)
done


subsection {* Stuff about S, S1 and S2 *}

locale QRTEMP =
fixes p :: "int"
fixes q :: "int"

assumes p_prime: "zprime p"
assumes p_g_2: "2 < p"
assumes q_prime: "zprime q"
assumes q_g_2: "2 < q"
assumes p_neq_q: "p ≠ q"
begin

definition P_set :: "int set"
where "P_set = {x. 0 < x & x ≤ ((p - 1) div 2) }"

definition Q_set :: "int set"
where "Q_set = {x. 0 < x & x ≤ ((q - 1) div 2) }"

definition S :: "(int * int) set"
where "S = P_set <*> Q_set"

definition S1 :: "(int * int) set"
where "S1 = { (x, y). (x, y):S & ((p * y) < (q * x)) }"

definition S2 :: "(int * int) set"
where "S2 = { (x, y). (x, y):S & ((q * x) < (p * y)) }"

definition f1 :: "int => (int * int) set"
where "f1 j = { (j1, y). (j1, y):S & j1 = j & (y ≤ (q * j) div p) }"

definition f2 :: "int => (int * int) set"
where "f2 j = { (x, j1). (x, j1):S & j1 = j & (x ≤ (p * j) div q) }"

lemma p_fact: "0 < (p - 1) div 2"
proof -
from p_g_2 have "2 ≤ p - 1" by arith
then have "2 div 2 ≤ (p - 1) div 2" by (rule zdiv_mono1, auto)
then show ?thesis by auto
qed

lemma q_fact: "0 < (q - 1) div 2"
proof -
from q_g_2 have "2 ≤ q - 1" by arith
then have "2 div 2 ≤ (q - 1) div 2" by (rule zdiv_mono1, auto)
then show ?thesis by auto
qed

lemma pb_neq_qa:
assumes "1 ≤ b" and "b ≤ (q - 1) div 2"
shows "p * b ≠ q * a"
proof
assume "p * b = q * a"
then have "q dvd (p * b)" by (auto simp add: dvd_def)
with q_prime p_g_2 have "q dvd p | q dvd b"
by (auto simp add: zprime_zdvd_zmult)
moreover have "~ (q dvd p)"
proof
assume "q dvd p"
with p_prime have "q = 1 | q = p"
apply (auto simp add: zprime_def QRTEMP_def)
apply (drule_tac x = q and R = False in allE)
apply (simp add: QRTEMP_def)
apply (subgoal_tac "0 ≤ q", simp add: QRTEMP_def)
apply (insert assms)
apply (auto simp add: QRTEMP_def)
done
with q_g_2 p_neq_q show False by auto
qed
ultimately have "q dvd b" by auto
then have "q ≤ b"
proof -
assume "q dvd b"
moreover from assms have "0 < b" by auto
ultimately show ?thesis using zdvd_bounds [of q b] by auto
qed
with assms have "q ≤ (q - 1) div 2" by auto
then have "2 * q ≤ 2 * ((q - 1) div 2)" by arith
then have "2 * q ≤ q - 1"
proof -
assume a: "2 * q ≤ 2 * ((q - 1) div 2)"
with assms have "q ∈ zOdd" by (auto simp add: QRTEMP_def zprime_zOdd_eq_grt_2)
with odd_minus_one_even have "(q - 1):zEven" by auto
with even_div_2_prop2 have "(q - 1) = 2 * ((q - 1) div 2)" by auto
with a show ?thesis by auto
qed
then have p1: "q ≤ -1" by arith
with q_g_2 show False by auto
qed

lemma P_set_finite: "finite (P_set)"
using p_fact by (auto simp add: P_set_def bdd_int_set_l_le_finite)

lemma Q_set_finite: "finite (Q_set)"
using q_fact by (auto simp add: Q_set_def bdd_int_set_l_le_finite)

lemma S_finite: "finite S"
by (auto simp add: S_def P_set_finite Q_set_finite finite_cartesian_product)

lemma S1_finite: "finite S1"
proof -
have "finite S" by (auto simp add: S_finite)
moreover have "S1 ⊆ S" by (auto simp add: S1_def S_def)
ultimately show ?thesis by (auto simp add: finite_subset)
qed

lemma S2_finite: "finite S2"
proof -
have "finite S" by (auto simp add: S_finite)
moreover have "S2 ⊆ S" by (auto simp add: S2_def S_def)
ultimately show ?thesis by (auto simp add: finite_subset)
qed

lemma P_set_card: "(p - 1) div 2 = int (card (P_set))"
using p_fact by (auto simp add: P_set_def card_bdd_int_set_l_le)

lemma Q_set_card: "(q - 1) div 2 = int (card (Q_set))"
using q_fact by (auto simp add: Q_set_def card_bdd_int_set_l_le)

lemma S_card: "((p - 1) div 2) * ((q - 1) div 2) = int (card(S))"
using P_set_card Q_set_card P_set_finite Q_set_finite
by (auto simp add: S_def zmult_int)

lemma S1_Int_S2_prop: "S1 ∩ S2 = {}"
by (auto simp add: S1_def S2_def)

lemma S1_Union_S2_prop: "S = S1 ∪ S2"
apply (auto simp add: S_def P_set_def Q_set_def S1_def S2_def)
proof -
fix a and b
assume "~ q * a < p * b" and b1: "0 < b" and b2: "b ≤ (q - 1) div 2"
with less_linear have "(p * b < q * a) | (p * b = q * a)" by auto
moreover from pb_neq_qa b1 b2 have "(p * b ≠ q * a)" by auto
ultimately show "p * b < q * a" by auto
qed

lemma card_sum_S1_S2: "((p - 1) div 2) * ((q - 1) div 2) =
int(card(S1)) + int(card(S2))"

proof -
have "((p - 1) div 2) * ((q - 1) div 2) = int (card(S))"
by (auto simp add: S_card)
also have "... = int( card(S1) + card(S2))"
apply (insert S1_finite S2_finite S1_Int_S2_prop S1_Union_S2_prop)
apply (drule card_Un_disjoint, auto)
done
also have "... = int(card(S1)) + int(card(S2))" by auto
finally show ?thesis .
qed

lemma aux1a:
assumes "0 < a" and "a ≤ (p - 1) div 2"
and "0 < b" and "b ≤ (q - 1) div 2"
shows "(p * b < q * a) = (b ≤ q * a div p)"
proof -
have "p * b < q * a ==> b ≤ q * a div p"
proof -
assume "p * b < q * a"
then have "p * b ≤ q * a" by auto
then have "(p * b) div p ≤ (q * a) div p"
by (rule zdiv_mono1) (insert p_g_2, auto)
then show "b ≤ (q * a) div p"
apply (subgoal_tac "p ≠ 0")
apply (frule div_mult_self1_is_id, force)
apply (insert p_g_2, auto)
done
qed
moreover have "b ≤ q * a div p ==> p * b < q * a"
proof -
assume "b ≤ q * a div p"
then have "p * b ≤ p * ((q * a) div p)"
using p_g_2 by (auto simp add: mult_le_cancel_left)
also have "... ≤ q * a"
by (rule zdiv_leq_prop) (insert p_g_2, auto)
finally have "p * b ≤ q * a" .
then have "p * b < q * a | p * b = q * a"
by (simp only: order_le_imp_less_or_eq)
moreover have "p * b ≠ q * a"
by (rule pb_neq_qa) (insert assms, auto)
ultimately show ?thesis by auto
qed
ultimately show ?thesis ..
qed

lemma aux1b:
assumes "0 < a" and "a ≤ (p - 1) div 2"
and "0 < b" and "b ≤ (q - 1) div 2"
shows "(q * a < p * b) = (a ≤ p * b div q)"
proof -
have "q * a < p * b ==> a ≤ p * b div q"
proof -
assume "q * a < p * b"
then have "q * a ≤ p * b" by auto
then have "(q * a) div q ≤ (p * b) div q"
by (rule zdiv_mono1) (insert q_g_2, auto)
then show "a ≤ (p * b) div q"
apply (subgoal_tac "q ≠ 0")
apply (frule div_mult_self1_is_id, force)
apply (insert q_g_2, auto)
done
qed
moreover have "a ≤ p * b div q ==> q * a < p * b"
proof -
assume "a ≤ p * b div q"
then have "q * a ≤ q * ((p * b) div q)"
using q_g_2 by (auto simp add: mult_le_cancel_left)
also have "... ≤ p * b"
by (rule zdiv_leq_prop) (insert q_g_2, auto)
finally have "q * a ≤ p * b" .
then have "q * a < p * b | q * a = p * b"
by (simp only: order_le_imp_less_or_eq)
moreover have "p * b ≠ q * a"
by (rule pb_neq_qa) (insert assms, auto)
ultimately show ?thesis by auto
qed
ultimately show ?thesis ..
qed

lemma (in -) aux2:
assumes "zprime p" and "zprime q" and "2 < p" and "2 < q"
shows "(q * ((p - 1) div 2)) div p ≤ (q - 1) div 2"
proof-
(* Set up what's even and odd *)
from assms have "p ∈ zOdd & q ∈ zOdd"
by (auto simp add: zprime_zOdd_eq_grt_2)
then have even1: "(p - 1):zEven & (q - 1):zEven"
by (auto simp add: odd_minus_one_even)
then have even2: "(2 * p):zEven & ((q - 1) * p):zEven"
by (auto simp add: zEven_def)
then have even3: "(((q - 1) * p) + (2 * p)):zEven"
by (auto simp: EvenOdd.even_plus_even)
(* using these prove it *)
from assms have "q * (p - 1) < ((q - 1) * p) + (2 * p)"
by (auto simp add: int_distrib)
then have "((p - 1) * q) div 2 < (((q - 1) * p) + (2 * p)) div 2"
apply (rule_tac x = "((p - 1) * q)" in even_div_2_l)
by (auto simp add: even3, auto simp add: mult_ac)
also have "((p - 1) * q) div 2 = q * ((p - 1) div 2)"
by (auto simp add: even1 even_prod_div_2)
also have "(((q - 1) * p) + (2 * p)) div 2 = (((q - 1) div 2) * p) + p"
by (auto simp add: even1 even2 even_prod_div_2 even_sum_div_2)
finally show ?thesis
apply (rule_tac x = " q * ((p - 1) div 2)" and
y = "(q - 1) div 2" in div_prop2)
using assms by auto
qed

lemma aux3a: "∀j ∈ P_set. int (card (f1 j)) = (q * j) div p"
proof
fix j
assume j_fact: "j ∈ P_set"
have "int (card (f1 j)) = int (card {y. y ∈ Q_set & y ≤ (q * j) div p})"
proof -
have "finite (f1 j)"
proof -
have "(f1 j) ⊆ S" by (auto simp add: f1_def)
with S_finite show ?thesis by (auto simp add: finite_subset)
qed
moreover have "inj_on (%(x,y). y) (f1 j)"
by (auto simp add: f1_def inj_on_def)
ultimately have "card ((%(x,y). y) ` (f1 j)) = card (f1 j)"
by (auto simp add: f1_def card_image)
moreover have "((%(x,y). y) ` (f1 j)) = {y. y ∈ Q_set & y ≤ (q * j) div p}"
using j_fact by (auto simp add: f1_def S_def Q_set_def P_set_def image_def)
ultimately show ?thesis by (auto simp add: f1_def)
qed
also have "... = int (card {y. 0 < y & y ≤ (q * j) div p})"
proof -
have "{y. y ∈ Q_set & y ≤ (q * j) div p} =
{y. 0 < y & y ≤ (q * j) div p}"

apply (auto simp add: Q_set_def)
proof -
fix x
assume x: "0 < x" "x ≤ q * j div p"
with j_fact P_set_def have "j ≤ (p - 1) div 2" by auto
with q_g_2 have "q * j ≤ q * ((p - 1) div 2)"
by (auto simp add: mult_le_cancel_left)
with p_g_2 have "q * j div p ≤ q * ((p - 1) div 2) div p"
by (auto simp add: zdiv_mono1)
also from QRTEMP_axioms j_fact P_set_def have "... ≤ (q - 1) div 2"
apply simp
apply (insert aux2)
apply (simp add: QRTEMP_def)
done
finally show "x ≤ (q - 1) div 2" using x by auto
qed
then show ?thesis by auto
qed
also have "... = (q * j) div p"
proof -
from j_fact P_set_def have "0 ≤ j" by auto
with q_g_2 have "q * 0 ≤ q * j" by (auto simp only: mult_left_mono)
then have "0 ≤ q * j" by auto
then have "0 div p ≤ (q * j) div p"
apply (rule_tac a = 0 in zdiv_mono1)
apply (insert p_g_2, auto)
done
also have "0 div p = 0" by auto
finally show ?thesis by (auto simp add: card_bdd_int_set_l_le)
qed
finally show "int (card (f1 j)) = q * j div p" .
qed

lemma aux3b: "∀j ∈ Q_set. int (card (f2 j)) = (p * j) div q"
proof
fix j
assume j_fact: "j ∈ Q_set"
have "int (card (f2 j)) = int (card {y. y ∈ P_set & y ≤ (p * j) div q})"
proof -
have "finite (f2 j)"
proof -
have "(f2 j) ⊆ S" by (auto simp add: f2_def)
with S_finite show ?thesis by (auto simp add: finite_subset)
qed
moreover have "inj_on (%(x,y). x) (f2 j)"
by (auto simp add: f2_def inj_on_def)
ultimately have "card ((%(x,y). x) ` (f2 j)) = card (f2 j)"
by (auto simp add: f2_def card_image)
moreover have "((%(x,y). x) ` (f2 j)) = {y. y ∈ P_set & y ≤ (p * j) div q}"
using j_fact by (auto simp add: f2_def S_def Q_set_def P_set_def image_def)
ultimately show ?thesis by (auto simp add: f2_def)
qed
also have "... = int (card {y. 0 < y & y ≤ (p * j) div q})"
proof -
have "{y. y ∈ P_set & y ≤ (p * j) div q} =
{y. 0 < y & y ≤ (p * j) div q}"

apply (auto simp add: P_set_def)
proof -
fix x
assume x: "0 < x" "x ≤ p * j div q"
with j_fact Q_set_def have "j ≤ (q - 1) div 2" by auto
with p_g_2 have "p * j ≤ p * ((q - 1) div 2)"
by (auto simp add: mult_le_cancel_left)
with q_g_2 have "p * j div q ≤ p * ((q - 1) div 2) div q"
by (auto simp add: zdiv_mono1)
also from QRTEMP_axioms j_fact have "... ≤ (p - 1) div 2"
by (auto simp add: aux2 QRTEMP_def)
finally show "x ≤ (p - 1) div 2" using x by auto
qed
then show ?thesis by auto
qed
also have "... = (p * j) div q"
proof -
from j_fact Q_set_def have "0 ≤ j" by auto
with p_g_2 have "p * 0 ≤ p * j" by (auto simp only: mult_left_mono)
then have "0 ≤ p * j" by auto
then have "0 div q ≤ (p * j) div q"
apply (rule_tac a = 0 in zdiv_mono1)
apply (insert q_g_2, auto)
done
also have "0 div q = 0" by auto
finally show ?thesis by (auto simp add: card_bdd_int_set_l_le)
qed
finally show "int (card (f2 j)) = p * j div q" .
qed

lemma S1_card: "int (card(S1)) = setsum (%j. (q * j) div p) P_set"
proof -
have "∀x ∈ P_set. finite (f1 x)"
proof
fix x
have "f1 x ⊆ S" by (auto simp add: f1_def)
with S_finite show "finite (f1 x)" by (auto simp add: finite_subset)
qed
moreover have "(∀x ∈ P_set. ∀y ∈ P_set. x ≠ y --> (f1 x) ∩ (f1 y) = {})"
by (auto simp add: f1_def)
moreover note P_set_finite
ultimately have "int(card (UNION P_set f1)) =
setsum (%x. int(card (f1 x))) P_set"

by(simp add:card_UN_disjoint int_setsum o_def)
moreover have "S1 = UNION P_set f1"
by (auto simp add: f1_def S_def S1_def S2_def P_set_def Q_set_def aux1a)
ultimately have "int(card (S1)) = setsum (%j. int(card (f1 j))) P_set"
by auto
also have "... = setsum (%j. q * j div p) P_set"
using aux3a by(fastforce intro: setsum_cong)
finally show ?thesis .
qed

lemma S2_card: "int (card(S2)) = setsum (%j. (p * j) div q) Q_set"
proof -
have "∀x ∈ Q_set. finite (f2 x)"
proof
fix x
have "f2 x ⊆ S" by (auto simp add: f2_def)
with S_finite show "finite (f2 x)" by (auto simp add: finite_subset)
qed
moreover have "(∀x ∈ Q_set. ∀y ∈ Q_set. x ≠ y -->
(f2 x) ∩ (f2 y) = {})"

by (auto simp add: f2_def)
moreover note Q_set_finite
ultimately have "int(card (UNION Q_set f2)) =
setsum (%x. int(card (f2 x))) Q_set"

by(simp add:card_UN_disjoint int_setsum o_def)
moreover have "S2 = UNION Q_set f2"
by (auto simp add: f2_def S_def S1_def S2_def P_set_def Q_set_def aux1b)
ultimately have "int(card (S2)) = setsum (%j. int(card (f2 j))) Q_set"
by auto
also have "... = setsum (%j. p * j div q) Q_set"
using aux3b by(fastforce intro: setsum_cong)
finally show ?thesis .
qed

lemma S1_carda: "int (card(S1)) =
setsum (%j. (j * q) div p) P_set"

by (auto simp add: S1_card mult_ac)

lemma S2_carda: "int (card(S2)) =
setsum (%j. (j * p) div q) Q_set"

by (auto simp add: S2_card mult_ac)

lemma pq_sum_prop: "(setsum (%j. (j * p) div q) Q_set) +
(setsum (%j. (j * q) div p) P_set) = ((p - 1) div 2) * ((q - 1) div 2)"

proof -
have "(setsum (%j. (j * p) div q) Q_set) +
(setsum (%j. (j * q) div p) P_set) = int (card S2) + int (card S1)"

by (auto simp add: S1_carda S2_carda)
also have "... = int (card S1) + int (card S2)"
by auto
also have "... = ((p - 1) div 2) * ((q - 1) div 2)"
by (auto simp add: card_sum_S1_S2)
finally show ?thesis .
qed


lemma (in -) pq_prime_neq: "[| zprime p; zprime q; p ≠ q |] ==> (~[p = 0] (mod q))"
apply (auto simp add: zcong_eq_zdvd_prop zprime_def)
apply (drule_tac x = q in allE)
apply (drule_tac x = p in allE)
apply auto
done


lemma QR_short: "(Legendre p q) * (Legendre q p) =
(-1::int)^nat(((p - 1) div 2)*((q - 1) div 2))"

proof -
from QRTEMP_axioms have "~([p = 0] (mod q))"
by (auto simp add: pq_prime_neq QRTEMP_def)
with QRTEMP_axioms Q_set_def have a1: "(Legendre p q) = (-1::int) ^
nat(setsum (%x. ((x * p) div q)) Q_set)"

apply (rule_tac p = q in MainQRLemma)
apply (auto simp add: zprime_zOdd_eq_grt_2 QRTEMP_def)
done
from QRTEMP_axioms have "~([q = 0] (mod p))"
apply (rule_tac p = q and q = p in pq_prime_neq)
apply (simp add: QRTEMP_def)+
done
with QRTEMP_axioms P_set_def have a2: "(Legendre q p) =
(-1::int) ^ nat(setsum (%x. ((x * q) div p)) P_set)"

apply (rule_tac p = p in MainQRLemma)
apply (auto simp add: zprime_zOdd_eq_grt_2 QRTEMP_def)
done
from a1 a2 have "(Legendre p q) * (Legendre q p) =
(-1::int) ^ nat(setsum (%x. ((x * p) div q)) Q_set) *
(-1::int) ^ nat(setsum (%x. ((x * q) div p)) P_set)"

by auto
also have "... = (-1::int) ^ (nat(setsum (%x. ((x * p) div q)) Q_set) +
nat(setsum (%x. ((x * q) div p)) P_set))"

by (auto simp add: power_add)
also have "nat(setsum (%x. ((x * p) div q)) Q_set) +
nat(setsum (%x. ((x * q) div p)) P_set) =
nat((setsum (%x. ((x * p) div q)) Q_set) +
(setsum (%x. ((x * q) div p)) P_set))"

apply (rule_tac z = "setsum (%x. ((x * p) div q)) Q_set" in
nat_add_distrib [symmetric])
apply (auto simp add: S1_carda [symmetric] S2_carda [symmetric])
done
also have "... = nat(((p - 1) div 2) * ((q - 1) div 2))"
by (auto simp add: pq_sum_prop)
finally show ?thesis .
qed

end

theorem Quadratic_Reciprocity:
"[| p ∈ zOdd; zprime p; q ∈ zOdd; zprime q;
p ≠ q |]
==> (Legendre p q) * (Legendre q p) =
(-1::int)^nat(((p - 1) div 2)*((q - 1) div 2))"

by (auto simp add: QRTEMP.QR_short zprime_zOdd_eq_grt_2 [symmetric]
QRTEMP_def)

end