(* Title: HOL/Old_Number_Theory/Quadratic_Reciprocity.thy Authors: Jeremy Avigad, David Gray, and Adam Kramer *) section ‹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 [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.distrib) also have "setsum (StandardRes p) B = setsum id C" by (auto simp add: C_def setsum.reindex [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.union_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.union_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 (simp add: S_def) 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: ac_simps) 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 ac_simps) lemma S2_carda: "int (card(S2)) = setsum (%j. (j * p) div q) Q_set" by (auto simp add: S2_card ac_simps) 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