--- a/src/HOL/NumberTheory/Quadratic_Reciprocity.thy Fri Sep 11 15:56:51 2009 +1000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,644 +0,0 @@
-(* Title: HOL/NumberTheory/Quadratic_Reciprocity.thy
- ID: $Id$
- 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: zmult_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 \<union> 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 \<union> 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: zdiff_zmult_distrib)
- also note QRLemma1
- also from QRLemma2 have "p * (\<Sum>x \<in> A. x * a div p) + setsum id D +
- setsum id E - setsum id A =
- p * (\<Sum>x \<in> 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 * (\<Sum>x \<in> A. x * a div p) -
- p * int (card E) + 2 * setsum id E"
- by arith
- finally show ?thesis
- by (auto simp only: zdiff_zmult_distrib2)
-qed
-
-lemma QRLemma4: "a \<in> zOdd ==>
- (setsum (%x. ((x * a) div p)) A \<in> zEven) = (int(card E): zEven)"
-proof -
- assume a_odd: "a \<in> 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 \<in> zEven"
- by (rule odd_minus_one_even)
- hence "(a - 1) * setsum id A \<in> zEven"
- by (rule even_times_either)
- moreover have "2 * setsum id E \<in> zEven"
- by (auto simp add: zEven_def)
- ultimately have "(a - 1) * setsum id A - 2 * setsum id E \<in> 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 \<in> 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 \<in> zOdd ==>
- (-1::int)^(card E) = (-1::int)^(nat(setsum (%x. ((x * a) div p)) A))"
-proof -
- assume "a \<in> zOdd"
- from QRLemma4 [OF this] have
- "(int(card E): zEven) = (setsum (%x. ((x * a) div p)) A \<in> zEven)" ..
- moreover have "0 \<le> int(card E)"
- by auto
- moreover have "0 \<le> setsum (%x. ((x * a) div p)) A"
- proof (intro setsum_nonneg)
- show "\<forall>x \<in> A. 0 \<le> x * a div p"
- proof
- fix x
- assume "x \<in> A"
- then have "0 \<le> x"
- by (auto simp add: A_def)
- with a_nonzero have "0 \<le> x * a"
- by (auto simp add: zero_le_mult_iff)
- with p_g_2 show "0 \<le> 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(((\<Sum>x \<in> 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 \<in> zOdd; 0 < a; ~([a = 0] (mod p)); zprime p; 2 < p;
- A = {x. 0 < x & x \<le> (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 \<noteq> q"
-begin
-
-definition
- P_set :: "int set" where
- "P_set = {x. 0 < x & x \<le> ((p - 1) div 2) }"
-
-definition
- Q_set :: "int set" where
- "Q_set = {x. 0 < x & x \<le> ((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 \<le> (q * j) div p) }"
-
-definition
- f2 :: "int => (int * int) set" where
- "f2 j = { (x, j1). (x, j1):S & j1 = j & (x \<le> (p * j) div q) }"
-
-lemma p_fact: "0 < (p - 1) div 2"
-proof -
- from p_g_2 have "2 \<le> p - 1" by arith
- then have "2 div 2 \<le> (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 \<le> q - 1" by arith
- then have "2 div 2 \<le> (q - 1) div 2" by (rule zdiv_mono1, auto)
- then show ?thesis by auto
-qed
-
-lemma pb_neq_qa: "[|1 \<le> b; b \<le> (q - 1) div 2 |] ==>
- (p * b \<noteq> q * a)"
-proof
- assume "p * b = q * a" and "1 \<le> b" and "b \<le> (q - 1) div 2"
- 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 \<le> q", simp add: QRTEMP_def)
- apply (insert prems)
- 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 \<le> b"
- proof -
- assume "q dvd b"
- moreover from prems have "0 < b" by auto
- ultimately show ?thesis using zdvd_bounds [of q b] by auto
- qed
- with prems have "q \<le> (q - 1) div 2" by auto
- then have "2 * q \<le> 2 * ((q - 1) div 2)" by arith
- then have "2 * q \<le> q - 1"
- proof -
- assume "2 * q \<le> 2 * ((q - 1) div 2)"
- with prems have "q \<in> 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 prems show ?thesis by auto
- qed
- then have p1: "q \<le> -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 \<subseteq> 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 \<subseteq> 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 setsum_constant)
-
-lemma S1_Int_S2_prop: "S1 \<inter> S2 = {}"
- by (auto simp add: S1_def S2_def)
-
-lemma S1_Union_S2_prop: "S = S1 \<union> 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 \<le> (q - 1) div 2"
- with zless_linear have "(p * b < q * a) | (p * b = q * a)" by auto
- moreover from pb_neq_qa b1 b2 have "(p * b \<noteq> 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: "[| 0 < a; a \<le> (p - 1) div 2;
- 0 < b; b \<le> (q - 1) div 2 |] ==>
- (p * b < q * a) = (b \<le> q * a div p)"
-proof -
- assume "0 < a" and "a \<le> (p - 1) div 2" and "0 < b" and "b \<le> (q - 1) div 2"
- have "p * b < q * a ==> b \<le> q * a div p"
- proof -
- assume "p * b < q * a"
- then have "p * b \<le> q * a" by auto
- then have "(p * b) div p \<le> (q * a) div p"
- by (rule zdiv_mono1) (insert p_g_2, auto)
- then show "b \<le> (q * a) div p"
- apply (subgoal_tac "p \<noteq> 0")
- apply (frule div_mult_self1_is_id, force)
- apply (insert p_g_2, auto)
- done
- qed
- moreover have "b \<le> q * a div p ==> p * b < q * a"
- proof -
- assume "b \<le> q * a div p"
- then have "p * b \<le> p * ((q * a) div p)"
- using p_g_2 by (auto simp add: mult_le_cancel_left)
- also have "... \<le> q * a"
- by (rule zdiv_leq_prop) (insert p_g_2, auto)
- finally have "p * b \<le> 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 \<noteq> q * a"
- by (rule pb_neq_qa) (insert prems, auto)
- ultimately show ?thesis by auto
- qed
- ultimately show ?thesis ..
-qed
-
-lemma aux1b: "[| 0 < a; a \<le> (p - 1) div 2;
- 0 < b; b \<le> (q - 1) div 2 |] ==>
- (q * a < p * b) = (a \<le> p * b div q)"
-proof -
- assume "0 < a" and "a \<le> (p - 1) div 2" and "0 < b" and "b \<le> (q - 1) div 2"
- have "q * a < p * b ==> a \<le> p * b div q"
- proof -
- assume "q * a < p * b"
- then have "q * a \<le> p * b" by auto
- then have "(q * a) div q \<le> (p * b) div q"
- by (rule zdiv_mono1) (insert q_g_2, auto)
- then show "a \<le> (p * b) div q"
- apply (subgoal_tac "q \<noteq> 0")
- apply (frule div_mult_self1_is_id, force)
- apply (insert q_g_2, auto)
- done
- qed
- moreover have "a \<le> p * b div q ==> q * a < p * b"
- proof -
- assume "a \<le> p * b div q"
- then have "q * a \<le> q * ((p * b) div q)"
- using q_g_2 by (auto simp add: mult_le_cancel_left)
- also have "... \<le> p * b"
- by (rule zdiv_leq_prop) (insert q_g_2, auto)
- finally have "q * a \<le> 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 \<noteq> q * a"
- by (rule pb_neq_qa) (insert prems, auto)
- ultimately show ?thesis by auto
- qed
- ultimately show ?thesis ..
-qed
-
-lemma (in -) aux2: "[| zprime p; zprime q; 2 < p; 2 < q |] ==>
- (q * ((p - 1) div 2)) div p \<le> (q - 1) div 2"
-proof-
- assume "zprime p" and "zprime q" and "2 < p" and "2 < q"
- (* Set up what's even and odd *)
- then have "p \<in> zOdd & q \<in> 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 prems 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: zmult_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 prems by auto
-qed
-
-lemma aux3a: "\<forall>j \<in> P_set. int (card (f1 j)) = (q * j) div p"
-proof
- fix j
- assume j_fact: "j \<in> P_set"
- have "int (card (f1 j)) = int (card {y. y \<in> Q_set & y \<le> (q * j) div p})"
- proof -
- have "finite (f1 j)"
- proof -
- have "(f1 j) \<subseteq> 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 \<in> Q_set & y \<le> (q * j) div p}"
- using prems 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 \<le> (q * j) div p})"
- proof -
- have "{y. y \<in> Q_set & y \<le> (q * j) div p} =
- {y. 0 < y & y \<le> (q * j) div p}"
- apply (auto simp add: Q_set_def)
- proof -
- fix x
- assume "0 < x" and "x \<le> q * j div p"
- with j_fact P_set_def have "j \<le> (p - 1) div 2" by auto
- with q_g_2 have "q * j \<le> q * ((p - 1) div 2)"
- by (auto simp add: mult_le_cancel_left)
- with p_g_2 have "q * j div p \<le> q * ((p - 1) div 2) div p"
- by (auto simp add: zdiv_mono1)
- also from prems P_set_def have "... \<le> (q - 1) div 2"
- apply simp
- apply (insert aux2)
- apply (simp add: QRTEMP_def)
- done
- finally show "x \<le> (q - 1) div 2" using prems by auto
- qed
- then show ?thesis by auto
- qed
- also have "... = (q * j) div p"
- proof -
- from j_fact P_set_def have "0 \<le> j" by auto
- with q_g_2 have "q * 0 \<le> q * j" by (auto simp only: mult_left_mono)
- then have "0 \<le> q * j" by auto
- then have "0 div p \<le> (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: "\<forall>j \<in> Q_set. int (card (f2 j)) = (p * j) div q"
-proof
- fix j
- assume j_fact: "j \<in> Q_set"
- have "int (card (f2 j)) = int (card {y. y \<in> P_set & y \<le> (p * j) div q})"
- proof -
- have "finite (f2 j)"
- proof -
- have "(f2 j) \<subseteq> 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 \<in> P_set & y \<le> (p * j) div q}"
- using prems 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 \<le> (p * j) div q})"
- proof -
- have "{y. y \<in> P_set & y \<le> (p * j) div q} =
- {y. 0 < y & y \<le> (p * j) div q}"
- apply (auto simp add: P_set_def)
- proof -
- fix x
- assume "0 < x" and "x \<le> p * j div q"
- with j_fact Q_set_def have "j \<le> (q - 1) div 2" by auto
- with p_g_2 have "p * j \<le> p * ((q - 1) div 2)"
- by (auto simp add: mult_le_cancel_left)
- with q_g_2 have "p * j div q \<le> p * ((q - 1) div 2) div q"
- by (auto simp add: zdiv_mono1)
- also from prems have "... \<le> (p - 1) div 2"
- by (auto simp add: aux2 QRTEMP_def)
- finally show "x \<le> (p - 1) div 2" using prems by auto
- qed
- then show ?thesis by auto
- qed
- also have "... = (p * j) div q"
- proof -
- from j_fact Q_set_def have "0 \<le> j" by auto
- with p_g_2 have "p * 0 \<le> p * j" by (auto simp only: mult_left_mono)
- then have "0 \<le> p * j" by auto
- then have "0 div q \<le> (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 "\<forall>x \<in> P_set. finite (f1 x)"
- proof
- fix x
- have "f1 x \<subseteq> S" by (auto simp add: f1_def)
- with S_finite show "finite (f1 x)" by (auto simp add: finite_subset)
- qed
- moreover have "(\<forall>x \<in> P_set. \<forall>y \<in> P_set. x \<noteq> y --> (f1 x) \<inter> (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(fastsimp intro: setsum_cong)
- finally show ?thesis .
-qed
-
-lemma S2_card: "int (card(S2)) = setsum (%j. (p * j) div q) Q_set"
-proof -
- have "\<forall>x \<in> Q_set. finite (f2 x)"
- proof
- fix x
- have "f2 x \<subseteq> S" by (auto simp add: f2_def)
- with S_finite show "finite (f2 x)" by (auto simp add: finite_subset)
- qed
- moreover have "(\<forall>x \<in> Q_set. \<forall>y \<in> Q_set. x \<noteq> y -->
- (f2 x) \<inter> (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(fastsimp 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 zmult_ac)
-
-lemma S2_carda: "int (card(S2)) =
- setsum (%j. (j * p) div q) Q_set"
- by (auto simp add: S2_card zmult_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 \<noteq> 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 prems have "~([p = 0] (mod q))"
- by (auto simp add: pq_prime_neq QRTEMP_def)
- with prems 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 prems have "~([q = 0] (mod p))"
- apply (rule_tac p = q and q = p in pq_prime_neq)
- apply (simp add: QRTEMP_def)+
- done
- with prems 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: zpower_zadd_distrib)
- 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 \<in> zOdd; zprime p; q \<in> zOdd; zprime q;
- p \<noteq> 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