changeset 13871 26e5f5e624f6 child 14353 79f9fbef9106
```     1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/src/HOL/NumberTheory/Quadratic_Reciprocity.thy	Thu Mar 20 15:58:25 2003 +0100
1.3 @@ -0,0 +1,628 @@
1.7 +*)
1.8 +
1.10 +
1.12 +
1.13 +(***************************************************************)
1.14 +(*                                                             *)
1.15 +(*  Lemmas leading up to the proof of theorem 3.3 in           *)
1.16 +(*  Niven and Zuckerman's presentation                         *)
1.17 +(*                                                             *)
1.18 +(***************************************************************)
1.19 +
1.20 +lemma (in GAUSS) QRLemma1: "a * setsum id A =
1.21 +  p * setsum (%x. ((x * a) div p)) A + setsum id D + setsum id E";
1.22 +proof -;
1.23 +  from finite_A have "a * setsum id A = setsum (%x. a * x) A";
1.24 +    by (auto simp add: setsum_const_mult id_def)
1.25 +  also have "setsum (%x. a * x) = setsum (%x. x * a)";
1.26 +    by (auto simp add: zmult_commute)
1.27 +  also; have "setsum (%x. x * a) A = setsum id B";
1.28 +    by (auto simp add: B_def sum_prop_id finite_A inj_on_xa_A)
1.29 +  also have "... = setsum (%x. p * (x div p) + StandardRes p x) B";
1.30 +    apply (rule setsum_same_function)
1.31 +    by (auto simp add: finite_B StandardRes_def zmod_zdiv_equality)
1.32 +  also have "... = setsum (%x. p * (x div p)) B + setsum (StandardRes p) B";
1.34 +  also; have "setsum (StandardRes p) B = setsum id C";
1.35 +    by (auto simp add: C_def sum_prop_id [THEN sym] finite_B
1.36 +      SR_B_inj)
1.37 +  also; from C_eq have "... = setsum id (D \<union> E)";
1.38 +    by auto
1.39 +  also; from finite_D finite_E have "... = setsum id D + setsum id E";
1.40 +    apply (rule setsum_Un_disjoint)
1.41 +    by (auto simp add: D_def E_def)
1.42 +  also have "setsum (%x. p * (x div p)) B =
1.43 +      setsum ((%x. p * (x div p)) o (%x. (x * a))) A";
1.44 +    by (auto simp add: B_def sum_prop finite_A inj_on_xa_A)
1.45 +  also have "... = setsum (%x. p * ((x * a) div p)) A";
1.46 +    by (auto simp add: o_def)
1.47 +  also from finite_A have "setsum (%x. p * ((x * a) div p)) A =
1.48 +    p * setsum (%x. ((x * a) div p)) A";
1.49 +    by (auto simp add: setsum_const_mult)
1.50 +  finally show ?thesis by arith
1.51 +qed;
1.52 +
1.53 +lemma (in GAUSS) QRLemma2: "setsum id A = p * int (card E) - setsum id E +
1.54 +  setsum id D";
1.55 +proof -;
1.56 +  from F_Un_D_eq_A have "setsum id A = setsum id (D \<union> F)";
1.57 +    by (simp add: Un_commute)
1.58 +  also from F_D_disj finite_D finite_F have
1.59 +      "... = setsum id D + setsum id F";
1.60 +    apply (simp add: Int_commute)
1.61 +    by (intro setsum_Un_disjoint)
1.62 +  also from F_def have "F = (%x. (p - x)) ` E";
1.63 +    by auto
1.64 +  also from finite_E inj_on_pminusx_E have "setsum id ((%x. (p - x)) ` E) =
1.65 +      setsum (%x. (p - x)) E";
1.66 +    by (auto simp add: sum_prop)
1.67 +  also from finite_E have "setsum (op - p) E = setsum (%x. p) E - setsum id E";
1.68 +    by (auto simp add: setsum_minus id_def)
1.69 +  also from finite_E have "setsum (%x. p) E = p * int(card E)";
1.70 +    by (intro setsum_const)
1.71 +  finally show ?thesis;
1.72 +    by arith
1.73 +qed;
1.74 +
1.75 +lemma (in GAUSS) QRLemma3: "(a - 1) * setsum id A =
1.76 +    p * (setsum (%x. ((x * a) div p)) A - int(card E)) + 2 * setsum id E";
1.77 +proof -;
1.78 +  have "(a - 1) * setsum id A = a * setsum id A - setsum id A";
1.79 +    by (auto simp add: zdiff_zmult_distrib)
1.80 +  also note QRLemma1;
1.81 +  also; from QRLemma2 have "p * (\<Sum>x \<in> A. x * a div p) + setsum id D +
1.82 +     setsum id E - setsum id A =
1.83 +      p * (\<Sum>x \<in> A. x * a div p) + setsum id D +
1.84 +      setsum id E - (p * int (card E) - setsum id E + setsum id D)";
1.85 +    by auto
1.86 +  also; have "... = p * (\<Sum>x \<in> A. x * a div p) -
1.87 +      p * int (card E) + 2 * setsum id E";
1.88 +    by arith
1.89 +  finally show ?thesis;
1.90 +    by (auto simp only: zdiff_zmult_distrib2)
1.91 +qed;
1.92 +
1.93 +lemma (in GAUSS) QRLemma4: "a \<in> zOdd ==>
1.94 +    (setsum (%x. ((x * a) div p)) A \<in> zEven) = (int(card E): zEven)";
1.95 +proof -;
1.96 +  assume a_odd: "a \<in> zOdd";
1.97 +  from QRLemma3 have a: "p * (setsum (%x. ((x * a) div p)) A - int(card E)) =
1.98 +      (a - 1) * setsum id A - 2 * setsum id E";
1.99 +    by arith
1.100 +  from a_odd have "a - 1 \<in> zEven"
1.101 +    by (rule odd_minus_one_even)
1.102 +  hence "(a - 1) * setsum id A \<in> zEven";
1.103 +    by (rule even_times_either)
1.104 +  moreover have "2 * setsum id E \<in> zEven";
1.105 +    by (auto simp add: zEven_def)
1.106 +  ultimately have "(a - 1) * setsum id A - 2 * setsum id E \<in> zEven"
1.107 +    by (rule even_minus_even)
1.108 +  with a have "p * (setsum (%x. ((x * a) div p)) A - int(card E)): zEven";
1.109 +    by simp
1.110 +  hence "p \<in> zEven | (setsum (%x. ((x * a) div p)) A - int(card E)): zEven";
1.111 +    by (rule even_product)
1.112 +  with p_odd have "(setsum (%x. ((x * a) div p)) A - int(card E)): zEven";
1.113 +    by (auto simp add: odd_iff_not_even)
1.114 +  thus ?thesis;
1.115 +    by (auto simp only: even_diff [THEN sym])
1.116 +qed;
1.117 +
1.118 +lemma (in GAUSS) QRLemma5: "a \<in> zOdd ==>
1.119 +   (-1::int)^(card E) = (-1::int)^(nat(setsum (%x. ((x * a) div p)) A))";
1.120 +proof -;
1.121 +  assume "a \<in> zOdd";
1.122 +  from QRLemma4 have
1.123 +    "(int(card E): zEven) = (setsum (%x. ((x * a) div p)) A \<in> zEven)";..;
1.124 +  moreover have "0 \<le> int(card E)";
1.125 +    by auto
1.126 +  moreover have "0 \<le> setsum (%x. ((x * a) div p)) A";
1.127 +    proof (intro setsum_non_neg);
1.128 +      from finite_A show "finite A";.;
1.129 +      next show "\<forall>x \<in> A. 0 \<le> x * a div p";
1.130 +      proof;
1.131 +        fix x;
1.132 +        assume "x \<in> A";
1.133 +        then have "0 \<le> x";
1.134 +          by (auto simp add: A_def)
1.135 +        with a_nonzero have "0 \<le> x * a";
1.136 +          by (auto simp add: int_0_le_mult_iff)
1.137 +        with p_g_2 show "0 \<le> x * a div p";
1.138 +          by (auto simp add: pos_imp_zdiv_nonneg_iff)
1.139 +      qed;
1.140 +    qed;
1.141 +  ultimately have "(-1::int)^nat((int (card E))) =
1.142 +      (-1)^nat(((\<Sum>x \<in> A. x * a div p)))";
1.143 +    by (intro neg_one_power_parity, auto)
1.144 +  also have "nat (int(card E)) = card E";
1.145 +    by auto
1.146 +  finally show ?thesis;.;
1.147 +qed;
1.148 +
1.149 +lemma MainQRLemma: "[| a \<in> zOdd; 0 < a; ~([a = 0] (mod p));p \<in> zprime; 2 < p;
1.150 +  A = {x. 0 < x & x \<le> (p - 1) div 2} |] ==>
1.151 +  (Legendre a p) = (-1::int)^(nat(setsum (%x. ((x * a) div p)) A))";
1.152 +  apply (subst GAUSS.gauss_lemma)
1.153 +  apply (auto simp add: GAUSS_def)
1.154 +  apply (subst GAUSS.QRLemma5)
1.155 +by (auto simp add: GAUSS_def)
1.156 +
1.157 +(******************************************************************)
1.158 +(*                                                                *)
1.159 +(* Stuff about S, S1 and S2...                                    *)
1.160 +(*                                                                *)
1.161 +(******************************************************************)
1.162 +
1.163 +locale QRTEMP =
1.164 +  fixes p     :: "int"
1.165 +  fixes q     :: "int"
1.166 +  fixes P_set :: "int set"
1.167 +  fixes Q_set :: "int set"
1.168 +  fixes S     :: "(int * int) set"
1.169 +  fixes S1    :: "(int * int) set"
1.170 +  fixes S2    :: "(int * int) set"
1.171 +  fixes f1    :: "int => (int * int) set"
1.172 +  fixes f2    :: "int => (int * int) set"
1.173 +
1.174 +  assumes p_prime: "p \<in> zprime"
1.175 +  assumes p_g_2: "2 < p"
1.176 +  assumes q_prime: "q \<in> zprime"
1.177 +  assumes q_g_2: "2 < q"
1.178 +  assumes p_neq_q:      "p \<noteq> q"
1.179 +
1.180 +  defines P_set_def: "P_set == {x. 0 < x & x \<le> ((p - 1) div 2) }"
1.181 +  defines Q_set_def: "Q_set == {x. 0 < x & x \<le> ((q - 1) div 2) }"
1.182 +  defines S_def:     "S     == P_set <*> Q_set"
1.183 +  defines S1_def:    "S1    == { (x, y). (x, y):S & ((p * y) < (q * x)) }"
1.184 +  defines S2_def:    "S2    == { (x, y). (x, y):S & ((q * x) < (p * y)) }"
1.185 +  defines f1_def:    "f1 j  == { (j1, y). (j1, y):S & j1 = j &
1.186 +                                 (y \<le> (q * j) div p) }"
1.187 +  defines f2_def:    "f2 j  == { (x, j1). (x, j1):S & j1 = j &
1.188 +                                 (x \<le> (p * j) div q) }";
1.189 +
1.190 +lemma (in QRTEMP) p_fact: "0 < (p - 1) div 2";
1.191 +proof -;
1.192 +  from prems have "2 < p" by (simp add: QRTEMP_def)
1.193 +  then have "2 \<le> p - 1" by arith
1.194 +  then have "2 div 2 \<le> (p - 1) div 2" by (rule zdiv_mono1, auto)
1.195 +  then show ?thesis by auto
1.196 +qed;
1.197 +
1.198 +lemma (in QRTEMP) q_fact: "0 < (q - 1) div 2";
1.199 +proof -;
1.200 +  from prems have "2 < q" by (simp add: QRTEMP_def)
1.201 +  then have "2 \<le> q - 1" by arith
1.202 +  then have "2 div 2 \<le> (q - 1) div 2" by (rule zdiv_mono1, auto)
1.203 +  then show ?thesis by auto
1.204 +qed;
1.205 +
1.206 +lemma (in QRTEMP) pb_neq_qa: "[|1 \<le> b; b \<le> (q - 1) div 2 |] ==>
1.207 +    (p * b \<noteq> q * a)";
1.208 +proof;
1.209 +  assume "p * b = q * a" and "1 \<le> b" and "b \<le> (q - 1) div 2";
1.210 +  then have "q dvd (p * b)" by (auto simp add: dvd_def)
1.211 +  with q_prime p_g_2 have "q dvd p | q dvd b";
1.212 +    by (auto simp add: zprime_zdvd_zmult)
1.213 +  moreover have "~ (q dvd p)";
1.214 +  proof;
1.215 +    assume "q dvd p";
1.216 +    with p_prime have "q = 1 | q = p"
1.217 +      apply (auto simp add: zprime_def QRTEMP_def)
1.218 +      apply (drule_tac x = q and R = False in allE)
1.219 +      apply (simp add: QRTEMP_def)
1.220 +      apply (subgoal_tac "0 \<le> q", simp add: QRTEMP_def)
1.221 +      apply (insert prems)
1.222 +    by (auto simp add: QRTEMP_def)
1.223 +    with q_g_2 p_neq_q show False by auto
1.224 +  qed;
1.225 +  ultimately have "q dvd b" by auto
1.226 +  then have "q \<le> b";
1.227 +  proof -;
1.228 +    assume "q dvd b";
1.229 +    moreover from prems have "0 < b" by auto
1.230 +    ultimately show ?thesis by (insert zdvd_bounds [of q b], auto)
1.231 +  qed;
1.232 +  with prems have "q \<le> (q - 1) div 2" by auto
1.233 +  then have "2 * q \<le> 2 * ((q - 1) div 2)" by arith
1.234 +  then have "2 * q \<le> q - 1";
1.235 +  proof -;
1.236 +    assume "2 * q \<le> 2 * ((q - 1) div 2)";
1.237 +    with prems have "q \<in> zOdd" by (auto simp add: QRTEMP_def zprime_zOdd_eq_grt_2)
1.238 +    with odd_minus_one_even have "(q - 1):zEven" by auto
1.239 +    with even_div_2_prop2 have "(q - 1) = 2 * ((q - 1) div 2)" by auto
1.240 +    with prems show ?thesis by auto
1.241 +  qed;
1.242 +  then have p1: "q \<le> -1" by arith
1.243 +  with q_g_2 show False by auto
1.244 +qed;
1.245 +
1.246 +lemma (in QRTEMP) P_set_finite: "finite (P_set)";
1.247 +  by (insert p_fact, auto simp add: P_set_def bdd_int_set_l_le_finite)
1.248 +
1.249 +lemma (in QRTEMP) Q_set_finite: "finite (Q_set)";
1.250 +  by (insert q_fact, auto simp add: Q_set_def bdd_int_set_l_le_finite)
1.251 +
1.252 +lemma (in QRTEMP) S_finite: "finite S";
1.253 +  by (auto simp add: S_def  P_set_finite Q_set_finite cartesian_product_finite)
1.254 +
1.255 +lemma (in QRTEMP) S1_finite: "finite S1";
1.256 +proof -;
1.257 +  have "finite S" by (auto simp add: S_finite)
1.258 +  moreover have "S1 \<subseteq> S" by (auto simp add: S1_def S_def)
1.259 +  ultimately show ?thesis by (auto simp add: finite_subset)
1.260 +qed;
1.261 +
1.262 +lemma (in QRTEMP) S2_finite: "finite S2";
1.263 +proof -;
1.264 +  have "finite S" by (auto simp add: S_finite)
1.265 +  moreover have "S2 \<subseteq> S" by (auto simp add: S2_def S_def)
1.266 +  ultimately show ?thesis by (auto simp add: finite_subset)
1.267 +qed;
1.268 +
1.269 +lemma (in QRTEMP) P_set_card: "(p - 1) div 2 = int (card (P_set))";
1.270 +  by (insert p_fact, auto simp add: P_set_def card_bdd_int_set_l_le)
1.271 +
1.272 +lemma (in QRTEMP) Q_set_card: "(q - 1) div 2 = int (card (Q_set))";
1.273 +  by (insert q_fact, auto simp add: Q_set_def card_bdd_int_set_l_le)
1.274 +
1.275 +lemma (in QRTEMP) S_card: "((p - 1) div 2) * ((q - 1) div 2) = int (card(S))";
1.276 +  apply (insert P_set_card Q_set_card P_set_finite Q_set_finite)
1.277 +  apply (auto simp add: S_def zmult_int)
1.278 +done
1.279 +
1.280 +lemma (in QRTEMP) S1_Int_S2_prop: "S1 \<inter> S2 = {}";
1.281 +  by (auto simp add: S1_def S2_def)
1.282 +
1.283 +lemma (in QRTEMP) S1_Union_S2_prop: "S = S1 \<union> S2";
1.284 +  apply (auto simp add: S_def P_set_def Q_set_def S1_def S2_def)
1.285 +  proof -;
1.286 +    fix a and b;
1.287 +    assume "~ q * a < p * b" and b1: "0 < b" and b2: "b \<le> (q - 1) div 2";
1.288 +    with zless_linear have "(p * b < q * a) | (p * b = q * a)" by auto
1.289 +    moreover from pb_neq_qa b1 b2 have "(p * b \<noteq> q * a)" by auto
1.290 +    ultimately show "p * b < q * a" by auto
1.291 +  qed;
1.292 +
1.293 +lemma (in QRTEMP) card_sum_S1_S2: "((p - 1) div 2) * ((q - 1) div 2) =
1.294 +    int(card(S1)) + int(card(S2))";
1.295 +proof-;
1.296 +  have "((p - 1) div 2) * ((q - 1) div 2) = int (card(S))";
1.297 +    by (auto simp add: S_card)
1.298 +  also have "... = int( card(S1) + card(S2))";
1.299 +    apply (insert S1_finite S2_finite S1_Int_S2_prop S1_Union_S2_prop)
1.300 +    apply (drule card_Un_disjoint, auto)
1.301 +  done
1.302 +  also have "... = int(card(S1)) + int(card(S2))" by auto
1.303 +  finally show ?thesis .;
1.304 +qed;
1.305 +
1.306 +lemma (in QRTEMP) aux1a: "[| 0 < a; a \<le> (p - 1) div 2;
1.307 +                             0 < b; b \<le> (q - 1) div 2 |] ==>
1.308 +                          (p * b < q * a) = (b \<le> q * a div p)";
1.309 +proof -;
1.310 +  assume "0 < a" and "a \<le> (p - 1) div 2" and "0 < b" and "b \<le> (q - 1) div 2";
1.311 +  have "p * b < q * a ==> b \<le> q * a div p";
1.312 +  proof -;
1.313 +    assume "p * b < q * a";
1.314 +    then have "p * b \<le> q * a" by auto
1.315 +    then have "(p * b) div p \<le> (q * a) div p";
1.316 +      by (rule zdiv_mono1, insert p_g_2, auto)
1.317 +    then show "b \<le> (q * a) div p";
1.318 +      apply (subgoal_tac "p \<noteq> 0")
1.319 +      apply (frule zdiv_zmult_self2, force)
1.320 +      by (insert p_g_2, auto)
1.321 +  qed;
1.322 +  moreover have "b \<le> q * a div p ==> p * b < q * a";
1.323 +  proof -;
1.324 +    assume "b \<le> q * a div p";
1.325 +    then have "p * b \<le> p * ((q * a) div p)";
1.326 +      by (insert p_g_2, auto simp add: zmult_zle_cancel1)
1.327 +    also have "... \<le> q * a";
1.328 +      by (rule zdiv_leq_prop, insert p_g_2, auto)
1.329 +    finally have "p * b \<le> q * a" .;
1.330 +    then have "p * b < q * a | p * b = q * a";
1.331 +      by (simp only: order_le_imp_less_or_eq)
1.332 +    moreover have "p * b \<noteq> q * a";
1.333 +      by (rule  pb_neq_qa, insert prems, auto)
1.334 +    ultimately show ?thesis by auto
1.335 +  qed;
1.336 +  ultimately show ?thesis ..;
1.337 +qed;
1.338 +
1.339 +lemma (in QRTEMP) aux1b: "[| 0 < a; a \<le> (p - 1) div 2;
1.340 +                             0 < b; b \<le> (q - 1) div 2 |] ==>
1.341 +                          (q * a < p * b) = (a \<le> p * b div q)";
1.342 +proof -;
1.343 +  assume "0 < a" and "a \<le> (p - 1) div 2" and "0 < b" and "b \<le> (q - 1) div 2";
1.344 +  have "q * a < p * b ==> a \<le> p * b div q";
1.345 +  proof -;
1.346 +    assume "q * a < p * b";
1.347 +    then have "q * a \<le> p * b" by auto
1.348 +    then have "(q * a) div q \<le> (p * b) div q";
1.349 +      by (rule zdiv_mono1, insert q_g_2, auto)
1.350 +    then show "a \<le> (p * b) div q";
1.351 +      apply (subgoal_tac "q \<noteq> 0")
1.352 +      apply (frule zdiv_zmult_self2, force)
1.353 +      by (insert q_g_2, auto)
1.354 +  qed;
1.355 +  moreover have "a \<le> p * b div q ==> q * a < p * b";
1.356 +  proof -;
1.357 +    assume "a \<le> p * b div q";
1.358 +    then have "q * a \<le> q * ((p * b) div q)";
1.359 +      by (insert q_g_2, auto simp add: zmult_zle_cancel1)
1.360 +    also have "... \<le> p * b";
1.361 +      by (rule zdiv_leq_prop, insert q_g_2, auto)
1.362 +    finally have "q * a \<le> p * b" .;
1.363 +    then have "q * a < p * b | q * a = p * b";
1.364 +      by (simp only: order_le_imp_less_or_eq)
1.365 +    moreover have "p * b \<noteq> q * a";
1.366 +      by (rule  pb_neq_qa, insert prems, auto)
1.367 +    ultimately show ?thesis by auto
1.368 +  qed;
1.369 +  ultimately show ?thesis ..;
1.370 +qed;
1.371 +
1.372 +lemma aux2: "[| p \<in> zprime; q \<in> zprime; 2 < p; 2 < q |] ==>
1.373 +             (q * ((p - 1) div 2)) div p \<le> (q - 1) div 2";
1.374 +proof-;
1.375 +  assume "p \<in> zprime" and "q \<in> zprime" and "2 < p" and "2 < q";
1.376 +  (* Set up what's even and odd *)
1.377 +  then have "p \<in> zOdd & q \<in> zOdd";
1.378 +    by (auto simp add:  zprime_zOdd_eq_grt_2)
1.379 +  then have even1: "(p - 1):zEven & (q - 1):zEven";
1.380 +    by (auto simp add: odd_minus_one_even)
1.381 +  then have even2: "(2 * p):zEven & ((q - 1) * p):zEven";
1.382 +    by (auto simp add: zEven_def)
1.383 +  then have even3: "(((q - 1) * p) + (2 * p)):zEven";
1.384 +    by (auto simp: even_plus_even)
1.385 +  (* using these prove it *)
1.386 +  from prems have "q * (p - 1) < ((q - 1) * p) + (2 * p)";
1.387 +    by (auto simp add: int_distrib)
1.388 +  then have "((p - 1) * q) div 2 < (((q - 1) * p) + (2 * p)) div 2";
1.389 +    apply (rule_tac x = "((p - 1) * q)" in even_div_2_l);
1.391 +  also have "((p - 1) * q) div 2 = q * ((p - 1) div 2)";
1.392 +    by (auto simp add: even1 even_prod_div_2)
1.393 +  also have "(((q - 1) * p) + (2 * p)) div 2 = (((q - 1) div 2) * p) + p";
1.394 +    by (auto simp add: even1 even2 even_prod_div_2 even_sum_div_2)
1.395 +  finally show ?thesis
1.396 +    apply (rule_tac x = " q * ((p - 1) div 2)" and
1.397 +                    y = "(q - 1) div 2" in div_prop2);
1.398 +    by (insert prems, auto)
1.399 +qed;
1.400 +
1.401 +lemma (in QRTEMP) aux3a: "\<forall>j \<in> P_set. int (card (f1 j)) = (q * j) div p";
1.402 +proof;
1.403 +  fix j;
1.404 +  assume j_fact: "j \<in> P_set";
1.405 +  have "int (card (f1 j)) = int (card {y. y \<in> Q_set & y \<le> (q * j) div p})";
1.406 +  proof -;
1.407 +    have "finite (f1 j)";
1.408 +    proof -;
1.409 +      have "(f1 j) \<subseteq> S" by (auto simp add: f1_def)
1.410 +      with S_finite show ?thesis by (auto simp add: finite_subset)
1.411 +    qed;
1.412 +    moreover have "inj_on (%(x,y). y) (f1 j)";
1.413 +      by (auto simp add: f1_def inj_on_def)
1.414 +    ultimately have "card ((%(x,y). y) ` (f1 j)) = card  (f1 j)";
1.415 +      by (auto simp add: f1_def card_image)
1.416 +    moreover have "((%(x,y). y) ` (f1 j)) = {y. y \<in> Q_set & y \<le> (q * j) div p}";
1.417 +      by (insert prems, auto simp add: f1_def S_def Q_set_def P_set_def
1.418 +        image_def)
1.419 +    ultimately show ?thesis by (auto simp add: f1_def)
1.420 +  qed;
1.421 +  also have "... = int (card {y. 0 < y & y \<le> (q * j) div p})";
1.422 +  proof -;
1.423 +    have "{y. y \<in> Q_set & y \<le> (q * j) div p} =
1.424 +        {y. 0 < y & y \<le> (q * j) div p}";
1.425 +      apply (auto simp add: Q_set_def)
1.426 +      proof -;
1.427 +        fix x;
1.428 +        assume "0 < x" and "x \<le> q * j div p";
1.429 +        with j_fact P_set_def  have "j \<le> (p - 1) div 2"; by auto
1.430 +        with q_g_2; have "q * j \<le> q * ((p - 1) div 2)";
1.431 +          by (auto simp add: zmult_zle_cancel1)
1.432 +        with p_g_2 have "q * j div p \<le> q * ((p - 1) div 2) div p";
1.433 +          by (auto simp add: zdiv_mono1)
1.434 +        also from prems have "... \<le> (q - 1) div 2";
1.435 +          apply simp apply (insert aux2) by (simp add: QRTEMP_def)
1.436 +        finally show "x \<le> (q - 1) div 2" by (insert prems, auto)
1.437 +      qed;
1.438 +    then show ?thesis by auto
1.439 +  qed;
1.440 +  also have "... = (q * j) div p";
1.441 +  proof -;
1.442 +    from j_fact P_set_def have "0 \<le> j" by auto
1.443 +    with q_g_2 have "q * 0 \<le> q * j" by (auto simp only: zmult_zle_mono2)
1.444 +    then have "0 \<le> q * j" by auto
1.445 +    then have "0 div p \<le> (q * j) div p";
1.446 +      apply (rule_tac a = 0 in zdiv_mono1)
1.447 +      by (insert p_g_2, auto)
1.448 +    also have "0 div p = 0" by auto
1.449 +    finally show ?thesis by (auto simp add: card_bdd_int_set_l_le)
1.450 +  qed;
1.451 +  finally show "int (card (f1 j)) = q * j div p" .;
1.452 +qed;
1.453 +
1.454 +lemma (in QRTEMP) aux3b: "\<forall>j \<in> Q_set. int (card (f2 j)) = (p * j) div q";
1.455 +proof;
1.456 +  fix j;
1.457 +  assume j_fact: "j \<in> Q_set";
1.458 +  have "int (card (f2 j)) = int (card {y. y \<in> P_set & y \<le> (p * j) div q})";
1.459 +  proof -;
1.460 +    have "finite (f2 j)";
1.461 +    proof -;
1.462 +      have "(f2 j) \<subseteq> S" by (auto simp add: f2_def)
1.463 +      with S_finite show ?thesis by (auto simp add: finite_subset)
1.464 +    qed;
1.465 +    moreover have "inj_on (%(x,y). x) (f2 j)";
1.466 +      by (auto simp add: f2_def inj_on_def)
1.467 +    ultimately have "card ((%(x,y). x) ` (f2 j)) = card  (f2 j)";
1.468 +      by (auto simp add: f2_def card_image)
1.469 +    moreover have "((%(x,y). x) ` (f2 j)) = {y. y \<in> P_set & y \<le> (p * j) div q}";
1.470 +      by (insert prems, auto simp add: f2_def S_def Q_set_def
1.471 +        P_set_def image_def)
1.472 +    ultimately show ?thesis by (auto simp add: f2_def)
1.473 +  qed;
1.474 +  also have "... = int (card {y. 0 < y & y \<le> (p * j) div q})";
1.475 +  proof -;
1.476 +    have "{y. y \<in> P_set & y \<le> (p * j) div q} =
1.477 +        {y. 0 < y & y \<le> (p * j) div q}";
1.478 +      apply (auto simp add: P_set_def)
1.479 +      proof -;
1.480 +        fix x;
1.481 +        assume "0 < x" and "x \<le> p * j div q";
1.482 +        with j_fact Q_set_def  have "j \<le> (q - 1) div 2"; by auto
1.483 +        with p_g_2; have "p * j \<le> p * ((q - 1) div 2)";
1.484 +          by (auto simp add: zmult_zle_cancel1)
1.485 +        with q_g_2 have "p * j div q \<le> p * ((q - 1) div 2) div q";
1.486 +          by (auto simp add: zdiv_mono1)
1.487 +        also from prems have "... \<le> (p - 1) div 2";
1.488 +          by (auto simp add: aux2 QRTEMP_def)
1.489 +        finally show "x \<le> (p - 1) div 2" by (insert prems, auto)
1.490 +      qed;
1.491 +    then show ?thesis by auto
1.492 +  qed;
1.493 +  also have "... = (p * j) div q";
1.494 +  proof -;
1.495 +    from j_fact Q_set_def have "0 \<le> j" by auto
1.496 +    with p_g_2 have "p * 0 \<le> p * j" by (auto simp only: zmult_zle_mono2)
1.497 +    then have "0 \<le> p * j" by auto
1.498 +    then have "0 div q \<le> (p * j) div q";
1.499 +      apply (rule_tac a = 0 in zdiv_mono1)
1.500 +      by (insert q_g_2, auto)
1.501 +    also have "0 div q = 0" by auto
1.502 +    finally show ?thesis by (auto simp add: card_bdd_int_set_l_le)
1.503 +  qed;
1.504 +  finally show "int (card (f2 j)) = p * j div q" .;
1.505 +qed;
1.506 +
1.507 +lemma (in QRTEMP) S1_card: "int (card(S1)) = setsum (%j. (q * j) div p) P_set";
1.508 +proof -;
1.509 +  have "\<forall>x \<in> P_set. finite (f1 x)";
1.510 +  proof;
1.511 +    fix x;
1.512 +    have "f1 x \<subseteq> S" by (auto simp add: f1_def)
1.513 +    with S_finite show "finite (f1 x)" by (auto simp add: finite_subset)
1.514 +  qed;
1.515 +  moreover have "(\<forall>x \<in> P_set. \<forall>y \<in> P_set. x \<noteq> y --> (f1 x) \<inter> (f1 y) = {})";
1.516 +    by (auto simp add: f1_def)
1.517 +  moreover note P_set_finite;
1.518 +  ultimately have "int(card (UNION P_set f1)) =
1.519 +      setsum (%x. int(card (f1 x))) P_set";
1.520 +    by (rule_tac A = P_set in int_card_indexed_union_disjoint_sets, auto)
1.521 +  moreover have "S1 = UNION P_set f1";
1.522 +    by (auto simp add: f1_def S_def S1_def S2_def P_set_def Q_set_def aux1a)
1.523 +  ultimately have "int(card (S1)) = setsum (%j. int(card (f1 j))) P_set"
1.524 +    by auto
1.525 +  also have "... = setsum (%j. q * j div p) P_set";
1.526 +  proof -;
1.527 +    note aux3a
1.528 +    with  P_set_finite show ?thesis by (rule setsum_same_function)
1.529 +  qed;
1.530 +  finally show ?thesis .;
1.531 +qed;
1.532 +
1.533 +lemma (in QRTEMP) S2_card: "int (card(S2)) = setsum (%j. (p * j) div q) Q_set";
1.534 +proof -;
1.535 +  have "\<forall>x \<in> Q_set. finite (f2 x)";
1.536 +  proof;
1.537 +    fix x;
1.538 +    have "f2 x \<subseteq> S" by (auto simp add: f2_def)
1.539 +    with S_finite show "finite (f2 x)" by (auto simp add: finite_subset)
1.540 +  qed;
1.541 +  moreover have "(\<forall>x \<in> Q_set. \<forall>y \<in> Q_set. x \<noteq> y -->
1.542 +      (f2 x) \<inter> (f2 y) = {})";
1.543 +    by (auto simp add: f2_def)
1.544 +  moreover note Q_set_finite;
1.545 +  ultimately have "int(card (UNION Q_set f2)) =
1.546 +      setsum (%x. int(card (f2 x))) Q_set";
1.547 +    by (rule_tac A = Q_set in int_card_indexed_union_disjoint_sets, auto)
1.548 +  moreover have "S2 = UNION Q_set f2";
1.549 +    by (auto simp add: f2_def S_def S1_def S2_def P_set_def Q_set_def aux1b)
1.550 +  ultimately have "int(card (S2)) = setsum (%j. int(card (f2 j))) Q_set"
1.551 +    by auto
1.552 +  also have "... = setsum (%j. p * j div q) Q_set";
1.553 +  proof -;
1.554 +    note aux3b;
1.555 +    with Q_set_finite show ?thesis by (rule setsum_same_function)
1.556 +  qed;
1.557 +  finally show ?thesis .;
1.558 +qed;
1.559 +
1.560 +lemma (in QRTEMP) S1_carda: "int (card(S1)) =
1.561 +    setsum (%j. (j * q) div p) P_set";
1.562 +  by (auto simp add: S1_card zmult_ac)
1.563 +
1.564 +lemma (in QRTEMP) S2_carda: "int (card(S2)) =
1.565 +    setsum (%j. (j * p) div q) Q_set";
1.566 +  by (auto simp add: S2_card zmult_ac)
1.567 +
1.568 +lemma (in QRTEMP) pq_sum_prop: "(setsum (%j. (j * p) div q) Q_set) +
1.569 +    (setsum (%j. (j * q) div p) P_set) = ((p - 1) div 2) * ((q - 1) div 2)";
1.570 +proof -;
1.571 +  have "(setsum (%j. (j * p) div q) Q_set) +
1.572 +      (setsum (%j. (j * q) div p) P_set) = int (card S2) + int (card S1)";
1.573 +    by (auto simp add: S1_carda S2_carda)
1.574 +  also have "... = int (card S1) + int (card S2)";
1.575 +    by auto
1.576 +  also have "... = ((p - 1) div 2) * ((q - 1) div 2)";
1.577 +    by (auto simp add: card_sum_S1_S2)
1.578 +  finally show ?thesis .;
1.579 +qed;
1.580 +
1.581 +lemma pq_prime_neq: "[| p \<in> zprime; q \<in> zprime; p \<noteq> q |] ==> (~[p = 0] (mod q))";
1.582 +  apply (auto simp add: zcong_eq_zdvd_prop zprime_def)
1.583 +  apply (drule_tac x = q in allE)
1.584 +  apply (drule_tac x = p in allE)
1.585 +by auto
1.586 +
1.587 +lemma (in QRTEMP) QR_short: "(Legendre p q) * (Legendre q p) =
1.588 +    (-1::int)^nat(((p - 1) div 2)*((q - 1) div 2))";
1.589 +proof -;
1.590 +  from prems have "~([p = 0] (mod q))";
1.591 +    by (auto simp add: pq_prime_neq QRTEMP_def)
1.592 +  with prems have a1: "(Legendre p q) = (-1::int) ^
1.593 +      nat(setsum (%x. ((x * p) div q)) Q_set)";
1.594 +    apply (rule_tac p = q in  MainQRLemma)
1.595 +    by (auto simp add: zprime_zOdd_eq_grt_2 QRTEMP_def)
1.596 +  from prems have "~([q = 0] (mod p))";
1.597 +    apply (rule_tac p = q and q = p in pq_prime_neq)
1.598 +    apply (simp add: QRTEMP_def)+;
1.599 +    by arith
1.600 +  with prems have a2: "(Legendre q p) =
1.601 +      (-1::int) ^ nat(setsum (%x. ((x * q) div p)) P_set)";
1.602 +    apply (rule_tac p = p in  MainQRLemma)
1.603 +    by (auto simp add: zprime_zOdd_eq_grt_2 QRTEMP_def)
1.604 +  from a1 a2 have "(Legendre p q) * (Legendre q p) =
1.605 +      (-1::int) ^ nat(setsum (%x. ((x * p) div q)) Q_set) *
1.606 +        (-1::int) ^ nat(setsum (%x. ((x * q) div p)) P_set)";
1.607 +    by auto
1.608 +  also have "... = (-1::int) ^ (nat(setsum (%x. ((x * p) div q)) Q_set) +
1.609 +                   nat(setsum (%x. ((x * q) div p)) P_set))";
1.611 +  also have "nat(setsum (%x. ((x * p) div q)) Q_set) +
1.612 +      nat(setsum (%x. ((x * q) div p)) P_set) =
1.613 +        nat((setsum (%x. ((x * p) div q)) Q_set) +
1.614 +          (setsum (%x. ((x * q) div p)) P_set))";
1.615 +    apply (rule_tac z1 = "setsum (%x. ((x * p) div q)) Q_set" in
1.617 +    by (auto simp add: S1_carda [THEN sym] S2_carda [THEN sym])
1.618 +  also have "... = nat(((p - 1) div 2) * ((q - 1) div 2))";
1.619 +    by (auto simp add: pq_sum_prop)
1.620 +  finally show ?thesis .;
1.621 +qed;
1.622 +