(* 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