paulson@13871: (*  Title:      HOL/Quadratic_Reciprocity/Quadratic_Reciprocity.thy
paulson@13871:     Authors:    Jeremy Avigad, David Gray, and Adam Kramer
paulson@13871:     License:    GPL (GNU GENERAL PUBLIC LICENSE)
paulson@13871: *)
paulson@13871: 
paulson@13871: header {* The law of Quadratic reciprocity *}
paulson@13871: 
paulson@13871: theory Quadratic_Reciprocity = Gauss:;
paulson@13871: 
paulson@13871: (***************************************************************)
paulson@13871: (*                                                             *)
paulson@13871: (*  Lemmas leading up to the proof of theorem 3.3 in           *)
paulson@13871: (*  Niven and Zuckerman's presentation                         *)
paulson@13871: (*                                                             *)
paulson@13871: (***************************************************************)
paulson@13871: 
paulson@13871: lemma (in GAUSS) QRLemma1: "a * setsum id A = 
paulson@13871:   p * setsum (%x. ((x * a) div p)) A + setsum id D + setsum id E";
paulson@13871: proof -;
paulson@13871:   from finite_A have "a * setsum id A = setsum (%x. a * x) A"; 
paulson@13871:     by (auto simp add: setsum_const_mult id_def)
paulson@13871:   also have "setsum (%x. a * x) = setsum (%x. x * a)"; 
paulson@13871:     by (auto simp add: zmult_commute)
paulson@13871:   also; have "setsum (%x. x * a) A = setsum id B";
paulson@13871:     by (auto simp add: B_def sum_prop_id finite_A inj_on_xa_A)
paulson@13871:   also have "... = setsum (%x. p * (x div p) + StandardRes p x) B";
paulson@13871:     apply (rule setsum_same_function)
paulson@13871:     by (auto simp add: finite_B StandardRes_def zmod_zdiv_equality)
paulson@13871:   also have "... = setsum (%x. p * (x div p)) B + setsum (StandardRes p) B";
paulson@13871:     by (rule setsum_addf)
paulson@13871:   also; have "setsum (StandardRes p) B = setsum id C";
paulson@13871:     by (auto simp add: C_def sum_prop_id [THEN sym] finite_B 
paulson@13871:       SR_B_inj)
paulson@13871:   also; from C_eq have "... = setsum id (D \<union> E)";
paulson@13871:     by auto
paulson@13871:   also; from finite_D finite_E have "... = setsum id D + setsum id E";
paulson@13871:     apply (rule setsum_Un_disjoint)
paulson@13871:     by (auto simp add: D_def E_def)
paulson@13871:   also have "setsum (%x. p * (x div p)) B = 
paulson@13871:       setsum ((%x. p * (x div p)) o (%x. (x * a))) A";
paulson@13871:     by (auto simp add: B_def sum_prop finite_A inj_on_xa_A)
paulson@13871:   also have "... = setsum (%x. p * ((x * a) div p)) A";
paulson@13871:     by (auto simp add: o_def)
paulson@13871:   also from finite_A have "setsum (%x. p * ((x * a) div p)) A = 
paulson@13871:     p * setsum (%x. ((x * a) div p)) A";
paulson@13871:     by (auto simp add: setsum_const_mult)
paulson@13871:   finally show ?thesis by arith
paulson@13871: qed;
paulson@13871: 
paulson@13871: lemma (in GAUSS) QRLemma2: "setsum id A = p * int (card E) - setsum id E + 
paulson@13871:   setsum id D"; 
paulson@13871: proof -;
paulson@13871:   from F_Un_D_eq_A have "setsum id A = setsum id (D \<union> F)";
paulson@13871:     by (simp add: Un_commute)
paulson@13871:   also from F_D_disj finite_D finite_F have 
paulson@13871:       "... = setsum id D + setsum id F";
paulson@13871:     apply (simp add: Int_commute)
paulson@13871:     by (intro setsum_Un_disjoint) 
paulson@13871:   also from F_def have "F = (%x. (p - x)) ` E";
paulson@13871:     by auto
paulson@13871:   also from finite_E inj_on_pminusx_E have "setsum id ((%x. (p - x)) ` E) =
paulson@13871:       setsum (%x. (p - x)) E";
paulson@13871:     by (auto simp add: sum_prop)
paulson@13871:   also from finite_E have "setsum (op - p) E = setsum (%x. p) E - setsum id E";
paulson@13871:     by (auto simp add: setsum_minus id_def)
paulson@13871:   also from finite_E have "setsum (%x. p) E = p * int(card E)";
paulson@13871:     by (intro setsum_const)
paulson@13871:   finally show ?thesis;
paulson@13871:     by arith
paulson@13871: qed;
paulson@13871: 
paulson@13871: lemma (in GAUSS) QRLemma3: "(a - 1) * setsum id A = 
paulson@13871:     p * (setsum (%x. ((x * a) div p)) A - int(card E)) + 2 * setsum id E";
paulson@13871: proof -;
paulson@13871:   have "(a - 1) * setsum id A = a * setsum id A - setsum id A";
paulson@13871:     by (auto simp add: zdiff_zmult_distrib)  
paulson@13871:   also note QRLemma1;
paulson@13871:   also; from QRLemma2 have "p * (\<Sum>x \<in> A. x * a div p) + setsum id D + 
paulson@13871:      setsum id E - setsum id A = 
paulson@13871:       p * (\<Sum>x \<in> A. x * a div p) + setsum id D + 
paulson@13871:       setsum id E - (p * int (card E) - setsum id E + setsum id D)";
paulson@13871:     by auto
paulson@13871:   also; have "... = p * (\<Sum>x \<in> A. x * a div p) - 
paulson@13871:       p * int (card E) + 2 * setsum id E"; 
paulson@13871:     by arith
paulson@13871:   finally show ?thesis;
paulson@13871:     by (auto simp only: zdiff_zmult_distrib2)
paulson@13871: qed;
paulson@13871: 
paulson@13871: lemma (in GAUSS) QRLemma4: "a \<in> zOdd ==> 
paulson@13871:     (setsum (%x. ((x * a) div p)) A \<in> zEven) = (int(card E): zEven)";
paulson@13871: proof -;
paulson@13871:   assume a_odd: "a \<in> zOdd";
paulson@13871:   from QRLemma3 have a: "p * (setsum (%x. ((x * a) div p)) A - int(card E)) =
paulson@13871:       (a - 1) * setsum id A - 2 * setsum id E"; 
paulson@13871:     by arith
paulson@13871:   from a_odd have "a - 1 \<in> zEven"
paulson@13871:     by (rule odd_minus_one_even)
paulson@13871:   hence "(a - 1) * setsum id A \<in> zEven";
paulson@13871:     by (rule even_times_either)
paulson@13871:   moreover have "2 * setsum id E \<in> zEven";
paulson@13871:     by (auto simp add: zEven_def)
paulson@13871:   ultimately have "(a - 1) * setsum id A - 2 * setsum id E \<in> zEven"
paulson@13871:     by (rule even_minus_even)
paulson@13871:   with a have "p * (setsum (%x. ((x * a) div p)) A - int(card E)): zEven";
paulson@13871:     by simp
paulson@13871:   hence "p \<in> zEven | (setsum (%x. ((x * a) div p)) A - int(card E)): zEven";
paulson@13871:     by (rule even_product)
paulson@13871:   with p_odd have "(setsum (%x. ((x * a) div p)) A - int(card E)): zEven";
paulson@13871:     by (auto simp add: odd_iff_not_even)
paulson@13871:   thus ?thesis;
paulson@13871:     by (auto simp only: even_diff [THEN sym])
paulson@13871: qed;
paulson@13871: 
paulson@13871: lemma (in GAUSS) QRLemma5: "a \<in> zOdd ==> 
paulson@13871:    (-1::int)^(card E) = (-1::int)^(nat(setsum (%x. ((x * a) div p)) A))";
paulson@13871: proof -;
paulson@13871:   assume "a \<in> zOdd";
paulson@13871:   from QRLemma4 have
paulson@13871:     "(int(card E): zEven) = (setsum (%x. ((x * a) div p)) A \<in> zEven)";..;
paulson@13871:   moreover have "0 \<le> int(card E)";
paulson@13871:     by auto
paulson@13871:   moreover have "0 \<le> setsum (%x. ((x * a) div p)) A";
paulson@13871:     proof (intro setsum_non_neg);
paulson@13871:       from finite_A show "finite A";.;
paulson@13871:       next show "\<forall>x \<in> A. 0 \<le> x * a div p";
paulson@13871:       proof;
paulson@13871:         fix x;
paulson@13871:         assume "x \<in> A";
paulson@13871:         then have "0 \<le> x";
paulson@13871:           by (auto simp add: A_def)
paulson@13871:         with a_nonzero have "0 \<le> x * a";
paulson@14353:           by (auto simp add: zero_le_mult_iff)
paulson@13871:         with p_g_2 show "0 \<le> x * a div p"; 
paulson@13871:           by (auto simp add: pos_imp_zdiv_nonneg_iff)
paulson@13871:       qed;
paulson@13871:     qed;
paulson@13871:   ultimately have "(-1::int)^nat((int (card E))) =
paulson@13871:       (-1)^nat(((\<Sum>x \<in> A. x * a div p)))";
paulson@13871:     by (intro neg_one_power_parity, auto)
paulson@13871:   also have "nat (int(card E)) = card E";
paulson@13871:     by auto
paulson@13871:   finally show ?thesis;.;
paulson@13871: qed;
paulson@13871: 
paulson@13871: lemma MainQRLemma: "[| a \<in> zOdd; 0 < a; ~([a = 0] (mod p));p \<in> zprime; 2 < p;
paulson@13871:   A = {x. 0 < x & x \<le> (p - 1) div 2} |] ==> 
paulson@13871:   (Legendre a p) = (-1::int)^(nat(setsum (%x. ((x * a) div p)) A))";
paulson@13871:   apply (subst GAUSS.gauss_lemma)
paulson@13871:   apply (auto simp add: GAUSS_def)
paulson@13871:   apply (subst GAUSS.QRLemma5)
paulson@13871: by (auto simp add: GAUSS_def)
paulson@13871: 
paulson@13871: (******************************************************************)
paulson@13871: (*                                                                *)
paulson@13871: (* Stuff about S, S1 and S2...                                    *)
paulson@13871: (*                                                                *)
paulson@13871: (******************************************************************)
paulson@13871: 
paulson@13871: locale QRTEMP =
paulson@13871:   fixes p     :: "int"
paulson@13871:   fixes q     :: "int"
paulson@13871:   fixes P_set :: "int set"
paulson@13871:   fixes Q_set :: "int set"
paulson@13871:   fixes S     :: "(int * int) set"
paulson@13871:   fixes S1    :: "(int * int) set"
paulson@13871:   fixes S2    :: "(int * int) set"
paulson@13871:   fixes f1    :: "int => (int * int) set"
paulson@13871:   fixes f2    :: "int => (int * int) set"
paulson@13871: 
paulson@13871:   assumes p_prime: "p \<in> zprime"
paulson@13871:   assumes p_g_2: "2 < p"
paulson@13871:   assumes q_prime: "q \<in> zprime"
paulson@13871:   assumes q_g_2: "2 < q"
paulson@13871:   assumes p_neq_q:      "p \<noteq> q"
paulson@13871: 
paulson@13871:   defines P_set_def: "P_set == {x. 0 < x & x \<le> ((p - 1) div 2) }"
paulson@13871:   defines Q_set_def: "Q_set == {x. 0 < x & x \<le> ((q - 1) div 2) }"
paulson@13871:   defines S_def:     "S     == P_set <*> Q_set"
paulson@13871:   defines S1_def:    "S1    == { (x, y). (x, y):S & ((p * y) < (q * x)) }"
paulson@13871:   defines S2_def:    "S2    == { (x, y). (x, y):S & ((q * x) < (p * y)) }"
paulson@13871:   defines f1_def:    "f1 j  == { (j1, y). (j1, y):S & j1 = j & 
paulson@13871:                                  (y \<le> (q * j) div p) }"
paulson@13871:   defines f2_def:    "f2 j  == { (x, j1). (x, j1):S & j1 = j & 
paulson@13871:                                  (x \<le> (p * j) div q) }";
paulson@13871: 
paulson@13871: lemma (in QRTEMP) p_fact: "0 < (p - 1) div 2";
paulson@13871: proof -;
paulson@13871:   from prems have "2 < p" by (simp add: QRTEMP_def)
paulson@13871:   then have "2 \<le> p - 1" by arith
paulson@13871:   then have "2 div 2 \<le> (p - 1) div 2" by (rule zdiv_mono1, auto)
paulson@13871:   then show ?thesis by auto
paulson@13871: qed;
paulson@13871: 
paulson@13871: lemma (in QRTEMP) q_fact: "0 < (q - 1) div 2";
paulson@13871: proof -;
paulson@13871:   from prems have "2 < q" by (simp add: QRTEMP_def)
paulson@13871:   then have "2 \<le> q - 1" by arith
paulson@13871:   then have "2 div 2 \<le> (q - 1) div 2" by (rule zdiv_mono1, auto)
paulson@13871:   then show ?thesis by auto
paulson@13871: qed;
paulson@13871: 
paulson@13871: lemma (in QRTEMP) pb_neq_qa: "[|1 \<le> b; b \<le> (q - 1) div 2 |] ==> 
paulson@13871:     (p * b \<noteq> q * a)";
paulson@13871: proof;
paulson@13871:   assume "p * b = q * a" and "1 \<le> b" and "b \<le> (q - 1) div 2";
paulson@13871:   then have "q dvd (p * b)" by (auto simp add: dvd_def)
paulson@13871:   with q_prime p_g_2 have "q dvd p | q dvd b";
paulson@13871:     by (auto simp add: zprime_zdvd_zmult)
paulson@13871:   moreover have "~ (q dvd p)";
paulson@13871:   proof;
paulson@13871:     assume "q dvd p";
paulson@13871:     with p_prime have "q = 1 | q = p"
paulson@13871:       apply (auto simp add: zprime_def QRTEMP_def)
paulson@13871:       apply (drule_tac x = q and R = False in allE)
paulson@13871:       apply (simp add: QRTEMP_def)    
paulson@13871:       apply (subgoal_tac "0 \<le> q", simp add: QRTEMP_def)
paulson@13871:       apply (insert prems)
paulson@13871:     by (auto simp add: QRTEMP_def)
paulson@13871:     with q_g_2 p_neq_q show False by auto
paulson@13871:   qed;
paulson@13871:   ultimately have "q dvd b" by auto
paulson@13871:   then have "q \<le> b";
paulson@13871:   proof -;
paulson@13871:     assume "q dvd b";
paulson@13871:     moreover from prems have "0 < b" by auto
paulson@13871:     ultimately show ?thesis by (insert zdvd_bounds [of q b], auto)
paulson@13871:   qed;
paulson@13871:   with prems have "q \<le> (q - 1) div 2" by auto
paulson@13871:   then have "2 * q \<le> 2 * ((q - 1) div 2)" by arith
paulson@13871:   then have "2 * q \<le> q - 1";
paulson@13871:   proof -;
paulson@13871:     assume "2 * q \<le> 2 * ((q - 1) div 2)";
paulson@13871:     with prems have "q \<in> zOdd" by (auto simp add: QRTEMP_def zprime_zOdd_eq_grt_2)
paulson@13871:     with odd_minus_one_even have "(q - 1):zEven" by auto
paulson@13871:     with even_div_2_prop2 have "(q - 1) = 2 * ((q - 1) div 2)" by auto
paulson@13871:     with prems show ?thesis by auto
paulson@13871:   qed;
paulson@13871:   then have p1: "q \<le> -1" by arith
paulson@13871:   with q_g_2 show False by auto
paulson@13871: qed;
paulson@13871: 
paulson@13871: lemma (in QRTEMP) P_set_finite: "finite (P_set)";
paulson@13871:   by (insert p_fact, auto simp add: P_set_def bdd_int_set_l_le_finite)
paulson@13871: 
paulson@13871: lemma (in QRTEMP) Q_set_finite: "finite (Q_set)";
paulson@13871:   by (insert q_fact, auto simp add: Q_set_def bdd_int_set_l_le_finite)
paulson@13871: 
paulson@13871: lemma (in QRTEMP) S_finite: "finite S";
paulson@13871:   by (auto simp add: S_def  P_set_finite Q_set_finite cartesian_product_finite)
paulson@13871: 
paulson@13871: lemma (in QRTEMP) S1_finite: "finite S1";
paulson@13871: proof -;
paulson@13871:   have "finite S" by (auto simp add: S_finite)
paulson@13871:   moreover have "S1 \<subseteq> S" by (auto simp add: S1_def S_def)
paulson@13871:   ultimately show ?thesis by (auto simp add: finite_subset)
paulson@13871: qed;
paulson@13871: 
paulson@13871: lemma (in QRTEMP) S2_finite: "finite S2";
paulson@13871: proof -;
paulson@13871:   have "finite S" by (auto simp add: S_finite)
paulson@13871:   moreover have "S2 \<subseteq> S" by (auto simp add: S2_def S_def)
paulson@13871:   ultimately show ?thesis by (auto simp add: finite_subset)
paulson@13871: qed;
paulson@13871: 
paulson@13871: lemma (in QRTEMP) P_set_card: "(p - 1) div 2 = int (card (P_set))";
paulson@13871:   by (insert p_fact, auto simp add: P_set_def card_bdd_int_set_l_le)
paulson@13871: 
paulson@13871: lemma (in QRTEMP) Q_set_card: "(q - 1) div 2 = int (card (Q_set))";
paulson@13871:   by (insert q_fact, auto simp add: Q_set_def card_bdd_int_set_l_le)
paulson@13871: 
paulson@13871: lemma (in QRTEMP) S_card: "((p - 1) div 2) * ((q - 1) div 2) = int (card(S))";
paulson@13871:   apply (insert P_set_card Q_set_card P_set_finite Q_set_finite)
paulson@13871:   apply (auto simp add: S_def zmult_int)
paulson@13871: done
paulson@13871: 
paulson@13871: lemma (in QRTEMP) S1_Int_S2_prop: "S1 \<inter> S2 = {}";
paulson@13871:   by (auto simp add: S1_def S2_def)
paulson@13871: 
paulson@13871: lemma (in QRTEMP) S1_Union_S2_prop: "S = S1 \<union> S2";
paulson@13871:   apply (auto simp add: S_def P_set_def Q_set_def S1_def S2_def)
paulson@13871:   proof -;
paulson@13871:     fix a and b;
paulson@13871:     assume "~ q * a < p * b" and b1: "0 < b" and b2: "b \<le> (q - 1) div 2";
paulson@13871:     with zless_linear have "(p * b < q * a) | (p * b = q * a)" by auto
paulson@13871:     moreover from pb_neq_qa b1 b2 have "(p * b \<noteq> q * a)" by auto
paulson@13871:     ultimately show "p * b < q * a" by auto
paulson@13871:   qed;
paulson@13871: 
paulson@13871: lemma (in QRTEMP) card_sum_S1_S2: "((p - 1) div 2) * ((q - 1) div 2) = 
paulson@13871:     int(card(S1)) + int(card(S2))";
paulson@13871: proof-;
paulson@13871:   have "((p - 1) div 2) * ((q - 1) div 2) = int (card(S))";
paulson@13871:     by (auto simp add: S_card)
paulson@13871:   also have "... = int( card(S1) + card(S2))";
paulson@13871:     apply (insert S1_finite S2_finite S1_Int_S2_prop S1_Union_S2_prop)
paulson@13871:     apply (drule card_Un_disjoint, auto)
paulson@13871:   done
paulson@13871:   also have "... = int(card(S1)) + int(card(S2))" by auto
paulson@13871:   finally show ?thesis .;
paulson@13871: qed;
paulson@13871: 
paulson@13871: lemma (in QRTEMP) aux1a: "[| 0 < a; a \<le> (p - 1) div 2; 
paulson@13871:                              0 < b; b \<le> (q - 1) div 2 |] ==>
paulson@13871:                           (p * b < q * a) = (b \<le> q * a div p)";
paulson@13871: proof -;
paulson@13871:   assume "0 < a" and "a \<le> (p - 1) div 2" and "0 < b" and "b \<le> (q - 1) div 2";
paulson@13871:   have "p * b < q * a ==> b \<le> q * a div p";
paulson@13871:   proof -;
paulson@13871:     assume "p * b < q * a";
paulson@13871:     then have "p * b \<le> q * a" by auto
paulson@13871:     then have "(p * b) div p \<le> (q * a) div p";
paulson@13871:       by (rule zdiv_mono1, insert p_g_2, auto)
paulson@13871:     then show "b \<le> (q * a) div p";
paulson@13871:       apply (subgoal_tac "p \<noteq> 0")
paulson@13871:       apply (frule zdiv_zmult_self2, force)
paulson@13871:       by (insert p_g_2, auto)
paulson@13871:   qed;
paulson@13871:   moreover have "b \<le> q * a div p ==> p * b < q * a";
paulson@13871:   proof -;
paulson@13871:     assume "b \<le> q * a div p";
paulson@13871:     then have "p * b \<le> p * ((q * a) div p)";
paulson@13871:       by (insert p_g_2, auto simp add: zmult_zle_cancel1)
paulson@13871:     also have "... \<le> q * a";
paulson@13871:       by (rule zdiv_leq_prop, insert p_g_2, auto)
paulson@13871:     finally have "p * b \<le> q * a" .;
paulson@13871:     then have "p * b < q * a | p * b = q * a";
paulson@13871:       by (simp only: order_le_imp_less_or_eq)
paulson@13871:     moreover have "p * b \<noteq> q * a";
paulson@13871:       by (rule  pb_neq_qa, insert prems, auto)
paulson@13871:     ultimately show ?thesis by auto
paulson@13871:   qed;
paulson@13871:   ultimately show ?thesis ..;
paulson@13871: qed;
paulson@13871: 
paulson@13871: lemma (in QRTEMP) aux1b: "[| 0 < a; a \<le> (p - 1) div 2; 
paulson@13871:                              0 < b; b \<le> (q - 1) div 2 |] ==>
paulson@13871:                           (q * a < p * b) = (a \<le> p * b div q)";
paulson@13871: proof -;
paulson@13871:   assume "0 < a" and "a \<le> (p - 1) div 2" and "0 < b" and "b \<le> (q - 1) div 2";
paulson@13871:   have "q * a < p * b ==> a \<le> p * b div q";
paulson@13871:   proof -;
paulson@13871:     assume "q * a < p * b";
paulson@13871:     then have "q * a \<le> p * b" by auto
paulson@13871:     then have "(q * a) div q \<le> (p * b) div q";
paulson@13871:       by (rule zdiv_mono1, insert q_g_2, auto)
paulson@13871:     then show "a \<le> (p * b) div q";
paulson@13871:       apply (subgoal_tac "q \<noteq> 0")
paulson@13871:       apply (frule zdiv_zmult_self2, force)
paulson@13871:       by (insert q_g_2, auto)
paulson@13871:   qed;
paulson@13871:   moreover have "a \<le> p * b div q ==> q * a < p * b";
paulson@13871:   proof -;
paulson@13871:     assume "a \<le> p * b div q";
paulson@13871:     then have "q * a \<le> q * ((p * b) div q)";
paulson@13871:       by (insert q_g_2, auto simp add: zmult_zle_cancel1)
paulson@13871:     also have "... \<le> p * b";
paulson@13871:       by (rule zdiv_leq_prop, insert q_g_2, auto)
paulson@13871:     finally have "q * a \<le> p * b" .;
paulson@13871:     then have "q * a < p * b | q * a = p * b";
paulson@13871:       by (simp only: order_le_imp_less_or_eq)
paulson@13871:     moreover have "p * b \<noteq> q * a";
paulson@13871:       by (rule  pb_neq_qa, insert prems, auto)
paulson@13871:     ultimately show ?thesis by auto
paulson@13871:   qed;
paulson@13871:   ultimately show ?thesis ..;
paulson@13871: qed;
paulson@13871: 
paulson@13871: lemma aux2: "[| p \<in> zprime; q \<in> zprime; 2 < p; 2 < q |] ==> 
paulson@13871:              (q * ((p - 1) div 2)) div p \<le> (q - 1) div 2";
paulson@13871: proof-;
paulson@13871:   assume "p \<in> zprime" and "q \<in> zprime" and "2 < p" and "2 < q";
paulson@13871:   (* Set up what's even and odd *)
paulson@13871:   then have "p \<in> zOdd & q \<in> zOdd";
paulson@13871:     by (auto simp add:  zprime_zOdd_eq_grt_2)
paulson@13871:   then have even1: "(p - 1):zEven & (q - 1):zEven";
paulson@13871:     by (auto simp add: odd_minus_one_even)
paulson@13871:   then have even2: "(2 * p):zEven & ((q - 1) * p):zEven";
paulson@13871:     by (auto simp add: zEven_def)
paulson@13871:   then have even3: "(((q - 1) * p) + (2 * p)):zEven";
paulson@13871:     by (auto simp: even_plus_even)
paulson@13871:   (* using these prove it *)
paulson@13871:   from prems have "q * (p - 1) < ((q - 1) * p) + (2 * p)";
paulson@13871:     by (auto simp add: int_distrib)
paulson@13871:   then have "((p - 1) * q) div 2 < (((q - 1) * p) + (2 * p)) div 2";
paulson@13871:     apply (rule_tac x = "((p - 1) * q)" in even_div_2_l);
paulson@13871:     by (auto simp add: even3, auto simp add: zmult_ac)
paulson@13871:   also have "((p - 1) * q) div 2 = q * ((p - 1) div 2)";
paulson@13871:     by (auto simp add: even1 even_prod_div_2)
paulson@13871:   also have "(((q - 1) * p) + (2 * p)) div 2 = (((q - 1) div 2) * p) + p";
paulson@13871:     by (auto simp add: even1 even2 even_prod_div_2 even_sum_div_2)
paulson@13871:   finally show ?thesis 
paulson@13871:     apply (rule_tac x = " q * ((p - 1) div 2)" and 
paulson@13871:                     y = "(q - 1) div 2" in div_prop2);
paulson@13871:     by (insert prems, auto)
paulson@13871: qed;
paulson@13871: 
paulson@13871: lemma (in QRTEMP) aux3a: "\<forall>j \<in> P_set. int (card (f1 j)) = (q * j) div p";
paulson@13871: proof;
paulson@13871:   fix j;
paulson@13871:   assume j_fact: "j \<in> P_set";
paulson@13871:   have "int (card (f1 j)) = int (card {y. y \<in> Q_set & y \<le> (q * j) div p})";
paulson@13871:   proof -;
paulson@13871:     have "finite (f1 j)";
paulson@13871:     proof -;
paulson@13871:       have "(f1 j) \<subseteq> S" by (auto simp add: f1_def)
paulson@13871:       with S_finite show ?thesis by (auto simp add: finite_subset)
paulson@13871:     qed;
paulson@13871:     moreover have "inj_on (%(x,y). y) (f1 j)";
paulson@13871:       by (auto simp add: f1_def inj_on_def)
paulson@13871:     ultimately have "card ((%(x,y). y) ` (f1 j)) = card  (f1 j)";
paulson@13871:       by (auto simp add: f1_def card_image)
paulson@13871:     moreover have "((%(x,y). y) ` (f1 j)) = {y. y \<in> Q_set & y \<le> (q * j) div p}";
paulson@13871:       by (insert prems, auto simp add: f1_def S_def Q_set_def P_set_def 
paulson@13871:         image_def)
paulson@13871:     ultimately show ?thesis by (auto simp add: f1_def)
paulson@13871:   qed;
paulson@13871:   also have "... = int (card {y. 0 < y & y \<le> (q * j) div p})";
paulson@13871:   proof -;
paulson@13871:     have "{y. y \<in> Q_set & y \<le> (q * j) div p} = 
paulson@13871:         {y. 0 < y & y \<le> (q * j) div p}";
paulson@13871:       apply (auto simp add: Q_set_def)
paulson@13871:       proof -;
paulson@13871:         fix x;
paulson@13871:         assume "0 < x" and "x \<le> q * j div p";
paulson@13871:         with j_fact P_set_def  have "j \<le> (p - 1) div 2"; by auto
paulson@13871:         with q_g_2; have "q * j \<le> q * ((p - 1) div 2)";
paulson@13871:           by (auto simp add: zmult_zle_cancel1)
paulson@13871:         with p_g_2 have "q * j div p \<le> q * ((p - 1) div 2) div p";
paulson@13871:           by (auto simp add: zdiv_mono1)
paulson@13871:         also from prems have "... \<le> (q - 1) div 2";
paulson@13871:           apply simp apply (insert aux2) by (simp add: QRTEMP_def)
paulson@13871:         finally show "x \<le> (q - 1) div 2" by (insert prems, auto)
paulson@13871:       qed;
paulson@13871:     then show ?thesis by auto
paulson@13871:   qed;
paulson@13871:   also have "... = (q * j) div p";
paulson@13871:   proof -;
paulson@13871:     from j_fact P_set_def have "0 \<le> j" by auto
paulson@13871:     with q_g_2 have "q * 0 \<le> q * j" by (auto simp only: zmult_zle_mono2)
paulson@13871:     then have "0 \<le> q * j" by auto
paulson@13871:     then have "0 div p \<le> (q * j) div p";
paulson@13871:       apply (rule_tac a = 0 in zdiv_mono1)
paulson@13871:       by (insert p_g_2, auto)
paulson@13871:     also have "0 div p = 0" by auto
paulson@13871:     finally show ?thesis by (auto simp add: card_bdd_int_set_l_le)
paulson@13871:   qed;
paulson@13871:   finally show "int (card (f1 j)) = q * j div p" .;
paulson@13871: qed;
paulson@13871: 
paulson@13871: lemma (in QRTEMP) aux3b: "\<forall>j \<in> Q_set. int (card (f2 j)) = (p * j) div q";
paulson@13871: proof;
paulson@13871:   fix j;
paulson@13871:   assume j_fact: "j \<in> Q_set";
paulson@13871:   have "int (card (f2 j)) = int (card {y. y \<in> P_set & y \<le> (p * j) div q})";
paulson@13871:   proof -;
paulson@13871:     have "finite (f2 j)";
paulson@13871:     proof -;
paulson@13871:       have "(f2 j) \<subseteq> S" by (auto simp add: f2_def)
paulson@13871:       with S_finite show ?thesis by (auto simp add: finite_subset)
paulson@13871:     qed;
paulson@13871:     moreover have "inj_on (%(x,y). x) (f2 j)";
paulson@13871:       by (auto simp add: f2_def inj_on_def)
paulson@13871:     ultimately have "card ((%(x,y). x) ` (f2 j)) = card  (f2 j)";
paulson@13871:       by (auto simp add: f2_def card_image)
paulson@13871:     moreover have "((%(x,y). x) ` (f2 j)) = {y. y \<in> P_set & y \<le> (p * j) div q}";
paulson@13871:       by (insert prems, auto simp add: f2_def S_def Q_set_def 
paulson@13871:         P_set_def image_def)
paulson@13871:     ultimately show ?thesis by (auto simp add: f2_def)
paulson@13871:   qed;
paulson@13871:   also have "... = int (card {y. 0 < y & y \<le> (p * j) div q})";
paulson@13871:   proof -;
paulson@13871:     have "{y. y \<in> P_set & y \<le> (p * j) div q} = 
paulson@13871:         {y. 0 < y & y \<le> (p * j) div q}";
paulson@13871:       apply (auto simp add: P_set_def)
paulson@13871:       proof -;
paulson@13871:         fix x;
paulson@13871:         assume "0 < x" and "x \<le> p * j div q";
paulson@13871:         with j_fact Q_set_def  have "j \<le> (q - 1) div 2"; by auto
paulson@13871:         with p_g_2; have "p * j \<le> p * ((q - 1) div 2)";
paulson@13871:           by (auto simp add: zmult_zle_cancel1)
paulson@13871:         with q_g_2 have "p * j div q \<le> p * ((q - 1) div 2) div q";
paulson@13871:           by (auto simp add: zdiv_mono1)
paulson@13871:         also from prems have "... \<le> (p - 1) div 2";
paulson@13871:           by (auto simp add: aux2 QRTEMP_def)
paulson@13871:         finally show "x \<le> (p - 1) div 2" by (insert prems, auto)
paulson@13871:       qed;
paulson@13871:     then show ?thesis by auto
paulson@13871:   qed;
paulson@13871:   also have "... = (p * j) div q";
paulson@13871:   proof -;
paulson@13871:     from j_fact Q_set_def have "0 \<le> j" by auto
paulson@13871:     with p_g_2 have "p * 0 \<le> p * j" by (auto simp only: zmult_zle_mono2)
paulson@13871:     then have "0 \<le> p * j" by auto
paulson@13871:     then have "0 div q \<le> (p * j) div q";
paulson@13871:       apply (rule_tac a = 0 in zdiv_mono1)
paulson@13871:       by (insert q_g_2, auto)
paulson@13871:     also have "0 div q = 0" by auto
paulson@13871:     finally show ?thesis by (auto simp add: card_bdd_int_set_l_le)
paulson@13871:   qed;
paulson@13871:   finally show "int (card (f2 j)) = p * j div q" .;
paulson@13871: qed;
paulson@13871: 
paulson@13871: lemma (in QRTEMP) S1_card: "int (card(S1)) = setsum (%j. (q * j) div p) P_set";
paulson@13871: proof -;
paulson@13871:   have "\<forall>x \<in> P_set. finite (f1 x)";
paulson@13871:   proof;
paulson@13871:     fix x;
paulson@13871:     have "f1 x \<subseteq> S" by (auto simp add: f1_def)
paulson@13871:     with S_finite show "finite (f1 x)" by (auto simp add: finite_subset)
paulson@13871:   qed;
paulson@13871:   moreover have "(\<forall>x \<in> P_set. \<forall>y \<in> P_set. x \<noteq> y --> (f1 x) \<inter> (f1 y) = {})";
paulson@13871:     by (auto simp add: f1_def)
paulson@13871:   moreover note P_set_finite;
paulson@13871:   ultimately have "int(card (UNION P_set f1)) = 
paulson@13871:       setsum (%x. int(card (f1 x))) P_set";
paulson@13871:     by (rule_tac A = P_set in int_card_indexed_union_disjoint_sets, auto)
paulson@13871:   moreover have "S1 = UNION P_set f1";
paulson@13871:     by (auto simp add: f1_def S_def S1_def S2_def P_set_def Q_set_def aux1a)
paulson@13871:   ultimately have "int(card (S1)) = setsum (%j. int(card (f1 j))) P_set" 
paulson@13871:     by auto
paulson@13871:   also have "... = setsum (%j. q * j div p) P_set";
paulson@13871:   proof -;
paulson@13871:     note aux3a
paulson@13871:     with  P_set_finite show ?thesis by (rule setsum_same_function)
paulson@13871:   qed;
paulson@13871:   finally show ?thesis .;
paulson@13871: qed;
paulson@13871: 
paulson@13871: lemma (in QRTEMP) S2_card: "int (card(S2)) = setsum (%j. (p * j) div q) Q_set";
paulson@13871: proof -;
paulson@13871:   have "\<forall>x \<in> Q_set. finite (f2 x)";
paulson@13871:   proof;
paulson@13871:     fix x;
paulson@13871:     have "f2 x \<subseteq> S" by (auto simp add: f2_def)
paulson@13871:     with S_finite show "finite (f2 x)" by (auto simp add: finite_subset)
paulson@13871:   qed;
paulson@13871:   moreover have "(\<forall>x \<in> Q_set. \<forall>y \<in> Q_set. x \<noteq> y --> 
paulson@13871:       (f2 x) \<inter> (f2 y) = {})";
paulson@13871:     by (auto simp add: f2_def)
paulson@13871:   moreover note Q_set_finite;
paulson@13871:   ultimately have "int(card (UNION Q_set f2)) = 
paulson@13871:       setsum (%x. int(card (f2 x))) Q_set";
paulson@13871:     by (rule_tac A = Q_set in int_card_indexed_union_disjoint_sets, auto)
paulson@13871:   moreover have "S2 = UNION Q_set f2";
paulson@13871:     by (auto simp add: f2_def S_def S1_def S2_def P_set_def Q_set_def aux1b)
paulson@13871:   ultimately have "int(card (S2)) = setsum (%j. int(card (f2 j))) Q_set" 
paulson@13871:     by auto
paulson@13871:   also have "... = setsum (%j. p * j div q) Q_set";
paulson@13871:   proof -;
paulson@13871:     note aux3b;
paulson@13871:     with Q_set_finite show ?thesis by (rule setsum_same_function)
paulson@13871:   qed;
paulson@13871:   finally show ?thesis .;
paulson@13871: qed;
paulson@13871: 
paulson@13871: lemma (in QRTEMP) S1_carda: "int (card(S1)) = 
paulson@13871:     setsum (%j. (j * q) div p) P_set";
paulson@13871:   by (auto simp add: S1_card zmult_ac)
paulson@13871: 
paulson@13871: lemma (in QRTEMP) S2_carda: "int (card(S2)) = 
paulson@13871:     setsum (%j. (j * p) div q) Q_set";
paulson@13871:   by (auto simp add: S2_card zmult_ac)
paulson@13871: 
paulson@13871: lemma (in QRTEMP) pq_sum_prop: "(setsum (%j. (j * p) div q) Q_set) + 
paulson@13871:     (setsum (%j. (j * q) div p) P_set) = ((p - 1) div 2) * ((q - 1) div 2)";
paulson@13871: proof -;
paulson@13871:   have "(setsum (%j. (j * p) div q) Q_set) + 
paulson@13871:       (setsum (%j. (j * q) div p) P_set) = int (card S2) + int (card S1)";
paulson@13871:     by (auto simp add: S1_carda S2_carda)
paulson@13871:   also have "... = int (card S1) + int (card S2)";
paulson@13871:     by auto
paulson@13871:   also have "... = ((p - 1) div 2) * ((q - 1) div 2)";
paulson@13871:     by (auto simp add: card_sum_S1_S2)
paulson@13871:   finally show ?thesis .;
paulson@13871: qed;
paulson@13871: 
paulson@13871: lemma pq_prime_neq: "[| p \<in> zprime; q \<in> zprime; p \<noteq> q |] ==> (~[p = 0] (mod q))";
paulson@13871:   apply (auto simp add: zcong_eq_zdvd_prop zprime_def)
paulson@13871:   apply (drule_tac x = q in allE)
paulson@13871:   apply (drule_tac x = p in allE)
paulson@13871: by auto
paulson@13871: 
paulson@13871: lemma (in QRTEMP) QR_short: "(Legendre p q) * (Legendre q p) = 
paulson@13871:     (-1::int)^nat(((p - 1) div 2)*((q - 1) div 2))";
paulson@13871: proof -;
paulson@13871:   from prems have "~([p = 0] (mod q))";
paulson@13871:     by (auto simp add: pq_prime_neq QRTEMP_def)
paulson@13871:   with prems have a1: "(Legendre p q) = (-1::int) ^ 
paulson@13871:       nat(setsum (%x. ((x * p) div q)) Q_set)";
paulson@13871:     apply (rule_tac p = q in  MainQRLemma)
paulson@13871:     by (auto simp add: zprime_zOdd_eq_grt_2 QRTEMP_def)
paulson@13871:   from prems have "~([q = 0] (mod p))";
paulson@13871:     apply (rule_tac p = q and q = p in pq_prime_neq)
paulson@13871:     apply (simp add: QRTEMP_def)+;
paulson@13871:     by arith
paulson@13871:   with prems have a2: "(Legendre q p) = 
paulson@13871:       (-1::int) ^ nat(setsum (%x. ((x * q) div p)) P_set)";
paulson@13871:     apply (rule_tac p = p in  MainQRLemma)
paulson@13871:     by (auto simp add: zprime_zOdd_eq_grt_2 QRTEMP_def)
paulson@13871:   from a1 a2 have "(Legendre p q) * (Legendre q p) = 
paulson@13871:       (-1::int) ^ nat(setsum (%x. ((x * p) div q)) Q_set) *
paulson@13871:         (-1::int) ^ nat(setsum (%x. ((x * q) div p)) P_set)";
paulson@13871:     by auto
paulson@13871:   also have "... = (-1::int) ^ (nat(setsum (%x. ((x * p) div q)) Q_set) + 
paulson@13871:                    nat(setsum (%x. ((x * q) div p)) P_set))";
paulson@13871:     by (auto simp add: zpower_zadd_distrib)
paulson@13871:   also have "nat(setsum (%x. ((x * p) div q)) Q_set) + 
paulson@13871:       nat(setsum (%x. ((x * q) div p)) P_set) =
paulson@13871:         nat((setsum (%x. ((x * p) div q)) Q_set) + 
paulson@13871:           (setsum (%x. ((x * q) div p)) P_set))";
paulson@13871:     apply (rule_tac z1 = "setsum (%x. ((x * p) div q)) Q_set" in 
paulson@13871:       nat_add_distrib [THEN sym]);
paulson@13871:     by (auto simp add: S1_carda [THEN sym] S2_carda [THEN sym])
paulson@13871:   also have "... = nat(((p - 1) div 2) * ((q - 1) div 2))";
paulson@13871:     by (auto simp add: pq_sum_prop)
paulson@13871:   finally show ?thesis .;
paulson@13871: qed;
paulson@13871: 
paulson@13871: theorem Quadratic_Reciprocity:
paulson@13871:      "[| p \<in> zOdd; p \<in> zprime; q \<in> zOdd; q \<in> zprime; 
paulson@13871:          p \<noteq> q |] 
paulson@13871:       ==> (Legendre p q) * (Legendre q p) = 
paulson@13871:           (-1::int)^nat(((p - 1) div 2)*((q - 1) div 2))";
paulson@13871:   by (auto simp add: QRTEMP.QR_short zprime_zOdd_eq_grt_2 [THEN sym] 
paulson@13871:                      QRTEMP_def)
paulson@13871: 
paulson@13871: end