author paulson Mon Jan 12 16:51:45 2004 +0100 (2004-01-12) changeset 14353 79f9fbef9106 parent 13871 26e5f5e624f6 child 14387 e96d5c42c4b0 permissions -rw-r--r--
Added lemmas to Ring_and_Field with slightly modified simplification rules

Deleted some little-used integer theorems, replacing them by the generic ones
in Ring_and_Field

Consolidated integer powers
```     1 (*  Title:      HOL/Quadratic_Reciprocity/Quadratic_Reciprocity.thy
```
```     2     Authors:    Jeremy Avigad, David Gray, and Adam Kramer
```
```     3     License:    GPL (GNU GENERAL PUBLIC LICENSE)
```
```     4 *)
```
```     5
```
```     6 header {* The law of Quadratic reciprocity *}
```
```     7
```
```     8 theory Quadratic_Reciprocity = Gauss:;
```
```     9
```
```    10 (***************************************************************)
```
```    11 (*                                                             *)
```
```    12 (*  Lemmas leading up to the proof of theorem 3.3 in           *)
```
```    13 (*  Niven and Zuckerman's presentation                         *)
```
```    14 (*                                                             *)
```
```    15 (***************************************************************)
```
```    16
```
```    17 lemma (in GAUSS) QRLemma1: "a * setsum id A =
```
```    18   p * setsum (%x. ((x * a) div p)) A + setsum id D + setsum id E";
```
```    19 proof -;
```
```    20   from finite_A have "a * setsum id A = setsum (%x. a * x) A";
```
```    21     by (auto simp add: setsum_const_mult id_def)
```
```    22   also have "setsum (%x. a * x) = setsum (%x. x * a)";
```
```    23     by (auto simp add: zmult_commute)
```
```    24   also; have "setsum (%x. x * a) A = setsum id B";
```
```    25     by (auto simp add: B_def sum_prop_id finite_A inj_on_xa_A)
```
```    26   also have "... = setsum (%x. p * (x div p) + StandardRes p x) B";
```
```    27     apply (rule setsum_same_function)
```
```    28     by (auto simp add: finite_B StandardRes_def zmod_zdiv_equality)
```
```    29   also have "... = setsum (%x. p * (x div p)) B + setsum (StandardRes p) B";
```
```    30     by (rule setsum_addf)
```
```    31   also; have "setsum (StandardRes p) B = setsum id C";
```
```    32     by (auto simp add: C_def sum_prop_id [THEN sym] finite_B
```
```    33       SR_B_inj)
```
```    34   also; from C_eq have "... = setsum id (D \<union> E)";
```
```    35     by auto
```
```    36   also; from finite_D finite_E have "... = setsum id D + setsum id E";
```
```    37     apply (rule setsum_Un_disjoint)
```
```    38     by (auto simp add: D_def E_def)
```
```    39   also have "setsum (%x. p * (x div p)) B =
```
```    40       setsum ((%x. p * (x div p)) o (%x. (x * a))) A";
```
```    41     by (auto simp add: B_def sum_prop finite_A inj_on_xa_A)
```
```    42   also have "... = setsum (%x. p * ((x * a) div p)) A";
```
```    43     by (auto simp add: o_def)
```
```    44   also from finite_A have "setsum (%x. p * ((x * a) div p)) A =
```
```    45     p * setsum (%x. ((x * a) div p)) A";
```
```    46     by (auto simp add: setsum_const_mult)
```
```    47   finally show ?thesis by arith
```
```    48 qed;
```
```    49
```
```    50 lemma (in GAUSS) QRLemma2: "setsum id A = p * int (card E) - setsum id E +
```
```    51   setsum id D";
```
```    52 proof -;
```
```    53   from F_Un_D_eq_A have "setsum id A = setsum id (D \<union> F)";
```
```    54     by (simp add: Un_commute)
```
```    55   also from F_D_disj finite_D finite_F have
```
```    56       "... = setsum id D + setsum id F";
```
```    57     apply (simp add: Int_commute)
```
```    58     by (intro setsum_Un_disjoint)
```
```    59   also from F_def have "F = (%x. (p - x)) ` E";
```
```    60     by auto
```
```    61   also from finite_E inj_on_pminusx_E have "setsum id ((%x. (p - x)) ` E) =
```
```    62       setsum (%x. (p - x)) E";
```
```    63     by (auto simp add: sum_prop)
```
```    64   also from finite_E have "setsum (op - p) E = setsum (%x. p) E - setsum id E";
```
```    65     by (auto simp add: setsum_minus id_def)
```
```    66   also from finite_E have "setsum (%x. p) E = p * int(card E)";
```
```    67     by (intro setsum_const)
```
```    68   finally show ?thesis;
```
```    69     by arith
```
```    70 qed;
```
```    71
```
```    72 lemma (in GAUSS) QRLemma3: "(a - 1) * setsum id A =
```
```    73     p * (setsum (%x. ((x * a) div p)) A - int(card E)) + 2 * setsum id E";
```
```    74 proof -;
```
```    75   have "(a - 1) * setsum id A = a * setsum id A - setsum id A";
```
```    76     by (auto simp add: zdiff_zmult_distrib)
```
```    77   also note QRLemma1;
```
```    78   also; from QRLemma2 have "p * (\<Sum>x \<in> A. x * a div p) + setsum id D +
```
```    79      setsum id E - setsum id A =
```
```    80       p * (\<Sum>x \<in> A. x * a div p) + setsum id D +
```
```    81       setsum id E - (p * int (card E) - setsum id E + setsum id D)";
```
```    82     by auto
```
```    83   also; have "... = p * (\<Sum>x \<in> A. x * a div p) -
```
```    84       p * int (card E) + 2 * setsum id E";
```
```    85     by arith
```
```    86   finally show ?thesis;
```
```    87     by (auto simp only: zdiff_zmult_distrib2)
```
```    88 qed;
```
```    89
```
```    90 lemma (in GAUSS) QRLemma4: "a \<in> zOdd ==>
```
```    91     (setsum (%x. ((x * a) div p)) A \<in> zEven) = (int(card E): zEven)";
```
```    92 proof -;
```
```    93   assume a_odd: "a \<in> zOdd";
```
```    94   from QRLemma3 have a: "p * (setsum (%x. ((x * a) div p)) A - int(card E)) =
```
```    95       (a - 1) * setsum id A - 2 * setsum id E";
```
```    96     by arith
```
```    97   from a_odd have "a - 1 \<in> zEven"
```
```    98     by (rule odd_minus_one_even)
```
```    99   hence "(a - 1) * setsum id A \<in> zEven";
```
```   100     by (rule even_times_either)
```
```   101   moreover have "2 * setsum id E \<in> zEven";
```
```   102     by (auto simp add: zEven_def)
```
```   103   ultimately have "(a - 1) * setsum id A - 2 * setsum id E \<in> zEven"
```
```   104     by (rule even_minus_even)
```
```   105   with a have "p * (setsum (%x. ((x * a) div p)) A - int(card E)): zEven";
```
```   106     by simp
```
```   107   hence "p \<in> zEven | (setsum (%x. ((x * a) div p)) A - int(card E)): zEven";
```
```   108     by (rule even_product)
```
```   109   with p_odd have "(setsum (%x. ((x * a) div p)) A - int(card E)): zEven";
```
```   110     by (auto simp add: odd_iff_not_even)
```
```   111   thus ?thesis;
```
```   112     by (auto simp only: even_diff [THEN sym])
```
```   113 qed;
```
```   114
```
```   115 lemma (in GAUSS) QRLemma5: "a \<in> zOdd ==>
```
```   116    (-1::int)^(card E) = (-1::int)^(nat(setsum (%x. ((x * a) div p)) A))";
```
```   117 proof -;
```
```   118   assume "a \<in> zOdd";
```
```   119   from QRLemma4 have
```
```   120     "(int(card E): zEven) = (setsum (%x. ((x * a) div p)) A \<in> zEven)";..;
```
```   121   moreover have "0 \<le> int(card E)";
```
```   122     by auto
```
```   123   moreover have "0 \<le> setsum (%x. ((x * a) div p)) A";
```
```   124     proof (intro setsum_non_neg);
```
```   125       from finite_A show "finite A";.;
```
```   126       next show "\<forall>x \<in> A. 0 \<le> x * a div p";
```
```   127       proof;
```
```   128         fix x;
```
```   129         assume "x \<in> A";
```
```   130         then have "0 \<le> x";
```
```   131           by (auto simp add: A_def)
```
```   132         with a_nonzero have "0 \<le> x * a";
```
```   133           by (auto simp add: zero_le_mult_iff)
```
```   134         with p_g_2 show "0 \<le> x * a div p";
```
```   135           by (auto simp add: pos_imp_zdiv_nonneg_iff)
```
```   136       qed;
```
```   137     qed;
```
```   138   ultimately have "(-1::int)^nat((int (card E))) =
```
```   139       (-1)^nat(((\<Sum>x \<in> A. x * a div p)))";
```
```   140     by (intro neg_one_power_parity, auto)
```
```   141   also have "nat (int(card E)) = card E";
```
```   142     by auto
```
```   143   finally show ?thesis;.;
```
```   144 qed;
```
```   145
```
```   146 lemma MainQRLemma: "[| a \<in> zOdd; 0 < a; ~([a = 0] (mod p));p \<in> zprime; 2 < p;
```
```   147   A = {x. 0 < x & x \<le> (p - 1) div 2} |] ==>
```
```   148   (Legendre a p) = (-1::int)^(nat(setsum (%x. ((x * a) div p)) A))";
```
```   149   apply (subst GAUSS.gauss_lemma)
```
```   150   apply (auto simp add: GAUSS_def)
```
```   151   apply (subst GAUSS.QRLemma5)
```
```   152 by (auto simp add: GAUSS_def)
```
```   153
```
```   154 (******************************************************************)
```
```   155 (*                                                                *)
```
```   156 (* Stuff about S, S1 and S2...                                    *)
```
```   157 (*                                                                *)
```
```   158 (******************************************************************)
```
```   159
```
```   160 locale QRTEMP =
```
```   161   fixes p     :: "int"
```
```   162   fixes q     :: "int"
```
```   163   fixes P_set :: "int set"
```
```   164   fixes Q_set :: "int set"
```
```   165   fixes S     :: "(int * int) set"
```
```   166   fixes S1    :: "(int * int) set"
```
```   167   fixes S2    :: "(int * int) set"
```
```   168   fixes f1    :: "int => (int * int) set"
```
```   169   fixes f2    :: "int => (int * int) set"
```
```   170
```
```   171   assumes p_prime: "p \<in> zprime"
```
```   172   assumes p_g_2: "2 < p"
```
```   173   assumes q_prime: "q \<in> zprime"
```
```   174   assumes q_g_2: "2 < q"
```
```   175   assumes p_neq_q:      "p \<noteq> q"
```
```   176
```
```   177   defines P_set_def: "P_set == {x. 0 < x & x \<le> ((p - 1) div 2) }"
```
```   178   defines Q_set_def: "Q_set == {x. 0 < x & x \<le> ((q - 1) div 2) }"
```
```   179   defines S_def:     "S     == P_set <*> Q_set"
```
```   180   defines S1_def:    "S1    == { (x, y). (x, y):S & ((p * y) < (q * x)) }"
```
```   181   defines S2_def:    "S2    == { (x, y). (x, y):S & ((q * x) < (p * y)) }"
```
```   182   defines f1_def:    "f1 j  == { (j1, y). (j1, y):S & j1 = j &
```
```   183                                  (y \<le> (q * j) div p) }"
```
```   184   defines f2_def:    "f2 j  == { (x, j1). (x, j1):S & j1 = j &
```
```   185                                  (x \<le> (p * j) div q) }";
```
```   186
```
```   187 lemma (in QRTEMP) p_fact: "0 < (p - 1) div 2";
```
```   188 proof -;
```
```   189   from prems have "2 < p" by (simp add: QRTEMP_def)
```
```   190   then have "2 \<le> p - 1" by arith
```
```   191   then have "2 div 2 \<le> (p - 1) div 2" by (rule zdiv_mono1, auto)
```
```   192   then show ?thesis by auto
```
```   193 qed;
```
```   194
```
```   195 lemma (in QRTEMP) q_fact: "0 < (q - 1) div 2";
```
```   196 proof -;
```
```   197   from prems have "2 < q" by (simp add: QRTEMP_def)
```
```   198   then have "2 \<le> q - 1" by arith
```
```   199   then have "2 div 2 \<le> (q - 1) div 2" by (rule zdiv_mono1, auto)
```
```   200   then show ?thesis by auto
```
```   201 qed;
```
```   202
```
```   203 lemma (in QRTEMP) pb_neq_qa: "[|1 \<le> b; b \<le> (q - 1) div 2 |] ==>
```
```   204     (p * b \<noteq> q * a)";
```
```   205 proof;
```
```   206   assume "p * b = q * a" and "1 \<le> b" and "b \<le> (q - 1) div 2";
```
```   207   then have "q dvd (p * b)" by (auto simp add: dvd_def)
```
```   208   with q_prime p_g_2 have "q dvd p | q dvd b";
```
```   209     by (auto simp add: zprime_zdvd_zmult)
```
```   210   moreover have "~ (q dvd p)";
```
```   211   proof;
```
```   212     assume "q dvd p";
```
```   213     with p_prime have "q = 1 | q = p"
```
```   214       apply (auto simp add: zprime_def QRTEMP_def)
```
```   215       apply (drule_tac x = q and R = False in allE)
```
```   216       apply (simp add: QRTEMP_def)
```
```   217       apply (subgoal_tac "0 \<le> q", simp add: QRTEMP_def)
```
```   218       apply (insert prems)
```
```   219     by (auto simp add: QRTEMP_def)
```
```   220     with q_g_2 p_neq_q show False by auto
```
```   221   qed;
```
```   222   ultimately have "q dvd b" by auto
```
```   223   then have "q \<le> b";
```
```   224   proof -;
```
```   225     assume "q dvd b";
```
```   226     moreover from prems have "0 < b" by auto
```
```   227     ultimately show ?thesis by (insert zdvd_bounds [of q b], auto)
```
```   228   qed;
```
```   229   with prems have "q \<le> (q - 1) div 2" by auto
```
```   230   then have "2 * q \<le> 2 * ((q - 1) div 2)" by arith
```
```   231   then have "2 * q \<le> q - 1";
```
```   232   proof -;
```
```   233     assume "2 * q \<le> 2 * ((q - 1) div 2)";
```
```   234     with prems have "q \<in> zOdd" by (auto simp add: QRTEMP_def zprime_zOdd_eq_grt_2)
```
```   235     with odd_minus_one_even have "(q - 1):zEven" by auto
```
```   236     with even_div_2_prop2 have "(q - 1) = 2 * ((q - 1) div 2)" by auto
```
```   237     with prems show ?thesis by auto
```
```   238   qed;
```
```   239   then have p1: "q \<le> -1" by arith
```
```   240   with q_g_2 show False by auto
```
```   241 qed;
```
```   242
```
```   243 lemma (in QRTEMP) P_set_finite: "finite (P_set)";
```
```   244   by (insert p_fact, auto simp add: P_set_def bdd_int_set_l_le_finite)
```
```   245
```
```   246 lemma (in QRTEMP) Q_set_finite: "finite (Q_set)";
```
```   247   by (insert q_fact, auto simp add: Q_set_def bdd_int_set_l_le_finite)
```
```   248
```
```   249 lemma (in QRTEMP) S_finite: "finite S";
```
```   250   by (auto simp add: S_def  P_set_finite Q_set_finite cartesian_product_finite)
```
```   251
```
```   252 lemma (in QRTEMP) S1_finite: "finite S1";
```
```   253 proof -;
```
```   254   have "finite S" by (auto simp add: S_finite)
```
```   255   moreover have "S1 \<subseteq> S" by (auto simp add: S1_def S_def)
```
```   256   ultimately show ?thesis by (auto simp add: finite_subset)
```
```   257 qed;
```
```   258
```
```   259 lemma (in QRTEMP) S2_finite: "finite S2";
```
```   260 proof -;
```
```   261   have "finite S" by (auto simp add: S_finite)
```
```   262   moreover have "S2 \<subseteq> S" by (auto simp add: S2_def S_def)
```
```   263   ultimately show ?thesis by (auto simp add: finite_subset)
```
```   264 qed;
```
```   265
```
```   266 lemma (in QRTEMP) P_set_card: "(p - 1) div 2 = int (card (P_set))";
```
```   267   by (insert p_fact, auto simp add: P_set_def card_bdd_int_set_l_le)
```
```   268
```
```   269 lemma (in QRTEMP) Q_set_card: "(q - 1) div 2 = int (card (Q_set))";
```
```   270   by (insert q_fact, auto simp add: Q_set_def card_bdd_int_set_l_le)
```
```   271
```
```   272 lemma (in QRTEMP) S_card: "((p - 1) div 2) * ((q - 1) div 2) = int (card(S))";
```
```   273   apply (insert P_set_card Q_set_card P_set_finite Q_set_finite)
```
```   274   apply (auto simp add: S_def zmult_int)
```
```   275 done
```
```   276
```
```   277 lemma (in QRTEMP) S1_Int_S2_prop: "S1 \<inter> S2 = {}";
```
```   278   by (auto simp add: S1_def S2_def)
```
```   279
```
```   280 lemma (in QRTEMP) S1_Union_S2_prop: "S = S1 \<union> S2";
```
```   281   apply (auto simp add: S_def P_set_def Q_set_def S1_def S2_def)
```
```   282   proof -;
```
```   283     fix a and b;
```
```   284     assume "~ q * a < p * b" and b1: "0 < b" and b2: "b \<le> (q - 1) div 2";
```
```   285     with zless_linear have "(p * b < q * a) | (p * b = q * a)" by auto
```
```   286     moreover from pb_neq_qa b1 b2 have "(p * b \<noteq> q * a)" by auto
```
```   287     ultimately show "p * b < q * a" by auto
```
```   288   qed;
```
```   289
```
```   290 lemma (in QRTEMP) card_sum_S1_S2: "((p - 1) div 2) * ((q - 1) div 2) =
```
```   291     int(card(S1)) + int(card(S2))";
```
```   292 proof-;
```
```   293   have "((p - 1) div 2) * ((q - 1) div 2) = int (card(S))";
```
```   294     by (auto simp add: S_card)
```
```   295   also have "... = int( card(S1) + card(S2))";
```
```   296     apply (insert S1_finite S2_finite S1_Int_S2_prop S1_Union_S2_prop)
```
```   297     apply (drule card_Un_disjoint, auto)
```
```   298   done
```
```   299   also have "... = int(card(S1)) + int(card(S2))" by auto
```
```   300   finally show ?thesis .;
```
```   301 qed;
```
```   302
```
```   303 lemma (in QRTEMP) aux1a: "[| 0 < a; a \<le> (p - 1) div 2;
```
```   304                              0 < b; b \<le> (q - 1) div 2 |] ==>
```
```   305                           (p * b < q * a) = (b \<le> q * a div p)";
```
```   306 proof -;
```
```   307   assume "0 < a" and "a \<le> (p - 1) div 2" and "0 < b" and "b \<le> (q - 1) div 2";
```
```   308   have "p * b < q * a ==> b \<le> q * a div p";
```
```   309   proof -;
```
```   310     assume "p * b < q * a";
```
```   311     then have "p * b \<le> q * a" by auto
```
```   312     then have "(p * b) div p \<le> (q * a) div p";
```
```   313       by (rule zdiv_mono1, insert p_g_2, auto)
```
```   314     then show "b \<le> (q * a) div p";
```
```   315       apply (subgoal_tac "p \<noteq> 0")
```
```   316       apply (frule zdiv_zmult_self2, force)
```
```   317       by (insert p_g_2, auto)
```
```   318   qed;
```
```   319   moreover have "b \<le> q * a div p ==> p * b < q * a";
```
```   320   proof -;
```
```   321     assume "b \<le> q * a div p";
```
```   322     then have "p * b \<le> p * ((q * a) div p)";
```
```   323       by (insert p_g_2, auto simp add: zmult_zle_cancel1)
```
```   324     also have "... \<le> q * a";
```
```   325       by (rule zdiv_leq_prop, insert p_g_2, auto)
```
```   326     finally have "p * b \<le> q * a" .;
```
```   327     then have "p * b < q * a | p * b = q * a";
```
```   328       by (simp only: order_le_imp_less_or_eq)
```
```   329     moreover have "p * b \<noteq> q * a";
```
```   330       by (rule  pb_neq_qa, insert prems, auto)
```
```   331     ultimately show ?thesis by auto
```
```   332   qed;
```
```   333   ultimately show ?thesis ..;
```
```   334 qed;
```
```   335
```
```   336 lemma (in QRTEMP) aux1b: "[| 0 < a; a \<le> (p - 1) div 2;
```
```   337                              0 < b; b \<le> (q - 1) div 2 |] ==>
```
```   338                           (q * a < p * b) = (a \<le> p * b div q)";
```
```   339 proof -;
```
```   340   assume "0 < a" and "a \<le> (p - 1) div 2" and "0 < b" and "b \<le> (q - 1) div 2";
```
```   341   have "q * a < p * b ==> a \<le> p * b div q";
```
```   342   proof -;
```
```   343     assume "q * a < p * b";
```
```   344     then have "q * a \<le> p * b" by auto
```
```   345     then have "(q * a) div q \<le> (p * b) div q";
```
```   346       by (rule zdiv_mono1, insert q_g_2, auto)
```
```   347     then show "a \<le> (p * b) div q";
```
```   348       apply (subgoal_tac "q \<noteq> 0")
```
```   349       apply (frule zdiv_zmult_self2, force)
```
```   350       by (insert q_g_2, auto)
```
```   351   qed;
```
```   352   moreover have "a \<le> p * b div q ==> q * a < p * b";
```
```   353   proof -;
```
```   354     assume "a \<le> p * b div q";
```
```   355     then have "q * a \<le> q * ((p * b) div q)";
```
```   356       by (insert q_g_2, auto simp add: zmult_zle_cancel1)
```
```   357     also have "... \<le> p * b";
```
```   358       by (rule zdiv_leq_prop, insert q_g_2, auto)
```
```   359     finally have "q * a \<le> p * b" .;
```
```   360     then have "q * a < p * b | q * a = p * b";
```
```   361       by (simp only: order_le_imp_less_or_eq)
```
```   362     moreover have "p * b \<noteq> q * a";
```
```   363       by (rule  pb_neq_qa, insert prems, auto)
```
```   364     ultimately show ?thesis by auto
```
```   365   qed;
```
```   366   ultimately show ?thesis ..;
```
```   367 qed;
```
```   368
```
```   369 lemma aux2: "[| p \<in> zprime; q \<in> zprime; 2 < p; 2 < q |] ==>
```
```   370              (q * ((p - 1) div 2)) div p \<le> (q - 1) div 2";
```
```   371 proof-;
```
```   372   assume "p \<in> zprime" and "q \<in> zprime" and "2 < p" and "2 < q";
```
```   373   (* Set up what's even and odd *)
```
```   374   then have "p \<in> zOdd & q \<in> zOdd";
```
```   375     by (auto simp add:  zprime_zOdd_eq_grt_2)
```
```   376   then have even1: "(p - 1):zEven & (q - 1):zEven";
```
```   377     by (auto simp add: odd_minus_one_even)
```
```   378   then have even2: "(2 * p):zEven & ((q - 1) * p):zEven";
```
```   379     by (auto simp add: zEven_def)
```
```   380   then have even3: "(((q - 1) * p) + (2 * p)):zEven";
```
```   381     by (auto simp: even_plus_even)
```
```   382   (* using these prove it *)
```
```   383   from prems have "q * (p - 1) < ((q - 1) * p) + (2 * p)";
```
```   384     by (auto simp add: int_distrib)
```
```   385   then have "((p - 1) * q) div 2 < (((q - 1) * p) + (2 * p)) div 2";
```
```   386     apply (rule_tac x = "((p - 1) * q)" in even_div_2_l);
```
```   387     by (auto simp add: even3, auto simp add: zmult_ac)
```
```   388   also have "((p - 1) * q) div 2 = q * ((p - 1) div 2)";
```
```   389     by (auto simp add: even1 even_prod_div_2)
```
```   390   also have "(((q - 1) * p) + (2 * p)) div 2 = (((q - 1) div 2) * p) + p";
```
```   391     by (auto simp add: even1 even2 even_prod_div_2 even_sum_div_2)
```
```   392   finally show ?thesis
```
```   393     apply (rule_tac x = " q * ((p - 1) div 2)" and
```
```   394                     y = "(q - 1) div 2" in div_prop2);
```
```   395     by (insert prems, auto)
```
```   396 qed;
```
```   397
```
```   398 lemma (in QRTEMP) aux3a: "\<forall>j \<in> P_set. int (card (f1 j)) = (q * j) div p";
```
```   399 proof;
```
```   400   fix j;
```
```   401   assume j_fact: "j \<in> P_set";
```
```   402   have "int (card (f1 j)) = int (card {y. y \<in> Q_set & y \<le> (q * j) div p})";
```
```   403   proof -;
```
```   404     have "finite (f1 j)";
```
```   405     proof -;
```
```   406       have "(f1 j) \<subseteq> S" by (auto simp add: f1_def)
```
```   407       with S_finite show ?thesis by (auto simp add: finite_subset)
```
```   408     qed;
```
```   409     moreover have "inj_on (%(x,y). y) (f1 j)";
```
```   410       by (auto simp add: f1_def inj_on_def)
```
```   411     ultimately have "card ((%(x,y). y) ` (f1 j)) = card  (f1 j)";
```
```   412       by (auto simp add: f1_def card_image)
```
```   413     moreover have "((%(x,y). y) ` (f1 j)) = {y. y \<in> Q_set & y \<le> (q * j) div p}";
```
```   414       by (insert prems, auto simp add: f1_def S_def Q_set_def P_set_def
```
```   415         image_def)
```
```   416     ultimately show ?thesis by (auto simp add: f1_def)
```
```   417   qed;
```
```   418   also have "... = int (card {y. 0 < y & y \<le> (q * j) div p})";
```
```   419   proof -;
```
```   420     have "{y. y \<in> Q_set & y \<le> (q * j) div p} =
```
```   421         {y. 0 < y & y \<le> (q * j) div p}";
```
```   422       apply (auto simp add: Q_set_def)
```
```   423       proof -;
```
```   424         fix x;
```
```   425         assume "0 < x" and "x \<le> q * j div p";
```
```   426         with j_fact P_set_def  have "j \<le> (p - 1) div 2"; by auto
```
```   427         with q_g_2; have "q * j \<le> q * ((p - 1) div 2)";
```
```   428           by (auto simp add: zmult_zle_cancel1)
```
```   429         with p_g_2 have "q * j div p \<le> q * ((p - 1) div 2) div p";
```
```   430           by (auto simp add: zdiv_mono1)
```
```   431         also from prems have "... \<le> (q - 1) div 2";
```
```   432           apply simp apply (insert aux2) by (simp add: QRTEMP_def)
```
```   433         finally show "x \<le> (q - 1) div 2" by (insert prems, auto)
```
```   434       qed;
```
```   435     then show ?thesis by auto
```
```   436   qed;
```
```   437   also have "... = (q * j) div p";
```
```   438   proof -;
```
```   439     from j_fact P_set_def have "0 \<le> j" by auto
```
```   440     with q_g_2 have "q * 0 \<le> q * j" by (auto simp only: zmult_zle_mono2)
```
```   441     then have "0 \<le> q * j" by auto
```
```   442     then have "0 div p \<le> (q * j) div p";
```
```   443       apply (rule_tac a = 0 in zdiv_mono1)
```
```   444       by (insert p_g_2, auto)
```
```   445     also have "0 div p = 0" by auto
```
```   446     finally show ?thesis by (auto simp add: card_bdd_int_set_l_le)
```
```   447   qed;
```
```   448   finally show "int (card (f1 j)) = q * j div p" .;
```
```   449 qed;
```
```   450
```
```   451 lemma (in QRTEMP) aux3b: "\<forall>j \<in> Q_set. int (card (f2 j)) = (p * j) div q";
```
```   452 proof;
```
```   453   fix j;
```
```   454   assume j_fact: "j \<in> Q_set";
```
```   455   have "int (card (f2 j)) = int (card {y. y \<in> P_set & y \<le> (p * j) div q})";
```
```   456   proof -;
```
```   457     have "finite (f2 j)";
```
```   458     proof -;
```
```   459       have "(f2 j) \<subseteq> S" by (auto simp add: f2_def)
```
```   460       with S_finite show ?thesis by (auto simp add: finite_subset)
```
```   461     qed;
```
```   462     moreover have "inj_on (%(x,y). x) (f2 j)";
```
```   463       by (auto simp add: f2_def inj_on_def)
```
```   464     ultimately have "card ((%(x,y). x) ` (f2 j)) = card  (f2 j)";
```
```   465       by (auto simp add: f2_def card_image)
```
```   466     moreover have "((%(x,y). x) ` (f2 j)) = {y. y \<in> P_set & y \<le> (p * j) div q}";
```
```   467       by (insert prems, auto simp add: f2_def S_def Q_set_def
```
```   468         P_set_def image_def)
```
```   469     ultimately show ?thesis by (auto simp add: f2_def)
```
```   470   qed;
```
```   471   also have "... = int (card {y. 0 < y & y \<le> (p * j) div q})";
```
```   472   proof -;
```
```   473     have "{y. y \<in> P_set & y \<le> (p * j) div q} =
```
```   474         {y. 0 < y & y \<le> (p * j) div q}";
```
```   475       apply (auto simp add: P_set_def)
```
```   476       proof -;
```
```   477         fix x;
```
```   478         assume "0 < x" and "x \<le> p * j div q";
```
```   479         with j_fact Q_set_def  have "j \<le> (q - 1) div 2"; by auto
```
```   480         with p_g_2; have "p * j \<le> p * ((q - 1) div 2)";
```
```   481           by (auto simp add: zmult_zle_cancel1)
```
```   482         with q_g_2 have "p * j div q \<le> p * ((q - 1) div 2) div q";
```
```   483           by (auto simp add: zdiv_mono1)
```
```   484         also from prems have "... \<le> (p - 1) div 2";
```
```   485           by (auto simp add: aux2 QRTEMP_def)
```
```   486         finally show "x \<le> (p - 1) div 2" by (insert prems, auto)
```
```   487       qed;
```
```   488     then show ?thesis by auto
```
```   489   qed;
```
```   490   also have "... = (p * j) div q";
```
```   491   proof -;
```
```   492     from j_fact Q_set_def have "0 \<le> j" by auto
```
```   493     with p_g_2 have "p * 0 \<le> p * j" by (auto simp only: zmult_zle_mono2)
```
```   494     then have "0 \<le> p * j" by auto
```
```   495     then have "0 div q \<le> (p * j) div q";
```
```   496       apply (rule_tac a = 0 in zdiv_mono1)
```
```   497       by (insert q_g_2, auto)
```
```   498     also have "0 div q = 0" by auto
```
```   499     finally show ?thesis by (auto simp add: card_bdd_int_set_l_le)
```
```   500   qed;
```
```   501   finally show "int (card (f2 j)) = p * j div q" .;
```
```   502 qed;
```
```   503
```
```   504 lemma (in QRTEMP) S1_card: "int (card(S1)) = setsum (%j. (q * j) div p) P_set";
```
```   505 proof -;
```
```   506   have "\<forall>x \<in> P_set. finite (f1 x)";
```
```   507   proof;
```
```   508     fix x;
```
```   509     have "f1 x \<subseteq> S" by (auto simp add: f1_def)
```
```   510     with S_finite show "finite (f1 x)" by (auto simp add: finite_subset)
```
```   511   qed;
```
```   512   moreover have "(\<forall>x \<in> P_set. \<forall>y \<in> P_set. x \<noteq> y --> (f1 x) \<inter> (f1 y) = {})";
```
```   513     by (auto simp add: f1_def)
```
```   514   moreover note P_set_finite;
```
```   515   ultimately have "int(card (UNION P_set f1)) =
```
```   516       setsum (%x. int(card (f1 x))) P_set";
```
```   517     by (rule_tac A = P_set in int_card_indexed_union_disjoint_sets, auto)
```
```   518   moreover have "S1 = UNION P_set f1";
```
```   519     by (auto simp add: f1_def S_def S1_def S2_def P_set_def Q_set_def aux1a)
```
```   520   ultimately have "int(card (S1)) = setsum (%j. int(card (f1 j))) P_set"
```
```   521     by auto
```
```   522   also have "... = setsum (%j. q * j div p) P_set";
```
```   523   proof -;
```
```   524     note aux3a
```
```   525     with  P_set_finite show ?thesis by (rule setsum_same_function)
```
```   526   qed;
```
```   527   finally show ?thesis .;
```
```   528 qed;
```
```   529
```
```   530 lemma (in QRTEMP) S2_card: "int (card(S2)) = setsum (%j. (p * j) div q) Q_set";
```
```   531 proof -;
```
```   532   have "\<forall>x \<in> Q_set. finite (f2 x)";
```
```   533   proof;
```
```   534     fix x;
```
```   535     have "f2 x \<subseteq> S" by (auto simp add: f2_def)
```
```   536     with S_finite show "finite (f2 x)" by (auto simp add: finite_subset)
```
```   537   qed;
```
```   538   moreover have "(\<forall>x \<in> Q_set. \<forall>y \<in> Q_set. x \<noteq> y -->
```
```   539       (f2 x) \<inter> (f2 y) = {})";
```
```   540     by (auto simp add: f2_def)
```
```   541   moreover note Q_set_finite;
```
```   542   ultimately have "int(card (UNION Q_set f2)) =
```
```   543       setsum (%x. int(card (f2 x))) Q_set";
```
```   544     by (rule_tac A = Q_set in int_card_indexed_union_disjoint_sets, auto)
```
```   545   moreover have "S2 = UNION Q_set f2";
```
```   546     by (auto simp add: f2_def S_def S1_def S2_def P_set_def Q_set_def aux1b)
```
```   547   ultimately have "int(card (S2)) = setsum (%j. int(card (f2 j))) Q_set"
```
```   548     by auto
```
```   549   also have "... = setsum (%j. p * j div q) Q_set";
```
```   550   proof -;
```
```   551     note aux3b;
```
```   552     with Q_set_finite show ?thesis by (rule setsum_same_function)
```
```   553   qed;
```
```   554   finally show ?thesis .;
```
```   555 qed;
```
```   556
```
```   557 lemma (in QRTEMP) S1_carda: "int (card(S1)) =
```
```   558     setsum (%j. (j * q) div p) P_set";
```
```   559   by (auto simp add: S1_card zmult_ac)
```
```   560
```
```   561 lemma (in QRTEMP) S2_carda: "int (card(S2)) =
```
```   562     setsum (%j. (j * p) div q) Q_set";
```
```   563   by (auto simp add: S2_card zmult_ac)
```
```   564
```
```   565 lemma (in QRTEMP) pq_sum_prop: "(setsum (%j. (j * p) div q) Q_set) +
```
```   566     (setsum (%j. (j * q) div p) P_set) = ((p - 1) div 2) * ((q - 1) div 2)";
```
```   567 proof -;
```
```   568   have "(setsum (%j. (j * p) div q) Q_set) +
```
```   569       (setsum (%j. (j * q) div p) P_set) = int (card S2) + int (card S1)";
```
```   570     by (auto simp add: S1_carda S2_carda)
```
```   571   also have "... = int (card S1) + int (card S2)";
```
```   572     by auto
```
```   573   also have "... = ((p - 1) div 2) * ((q - 1) div 2)";
```
```   574     by (auto simp add: card_sum_S1_S2)
```
```   575   finally show ?thesis .;
```
```   576 qed;
```
```   577
```
```   578 lemma pq_prime_neq: "[| p \<in> zprime; q \<in> zprime; p \<noteq> q |] ==> (~[p = 0] (mod q))";
```
```   579   apply (auto simp add: zcong_eq_zdvd_prop zprime_def)
```
```   580   apply (drule_tac x = q in allE)
```
```   581   apply (drule_tac x = p in allE)
```
```   582 by auto
```
```   583
```
```   584 lemma (in QRTEMP) QR_short: "(Legendre p q) * (Legendre q p) =
```
```   585     (-1::int)^nat(((p - 1) div 2)*((q - 1) div 2))";
```
```   586 proof -;
```
```   587   from prems have "~([p = 0] (mod q))";
```
```   588     by (auto simp add: pq_prime_neq QRTEMP_def)
```
```   589   with prems have a1: "(Legendre p q) = (-1::int) ^
```
```   590       nat(setsum (%x. ((x * p) div q)) Q_set)";
```
```   591     apply (rule_tac p = q in  MainQRLemma)
```
```   592     by (auto simp add: zprime_zOdd_eq_grt_2 QRTEMP_def)
```
```   593   from prems have "~([q = 0] (mod p))";
```
```   594     apply (rule_tac p = q and q = p in pq_prime_neq)
```
```   595     apply (simp add: QRTEMP_def)+;
```
```   596     by arith
```
```   597   with prems have a2: "(Legendre q p) =
```
```   598       (-1::int) ^ nat(setsum (%x. ((x * q) div p)) P_set)";
```
```   599     apply (rule_tac p = p in  MainQRLemma)
```
```   600     by (auto simp add: zprime_zOdd_eq_grt_2 QRTEMP_def)
```
```   601   from a1 a2 have "(Legendre p q) * (Legendre q p) =
```
```   602       (-1::int) ^ nat(setsum (%x. ((x * p) div q)) Q_set) *
```
```   603         (-1::int) ^ nat(setsum (%x. ((x * q) div p)) P_set)";
```
```   604     by auto
```
```   605   also have "... = (-1::int) ^ (nat(setsum (%x. ((x * p) div q)) Q_set) +
```
```   606                    nat(setsum (%x. ((x * q) div p)) P_set))";
```
```   607     by (auto simp add: zpower_zadd_distrib)
```
```   608   also have "nat(setsum (%x. ((x * p) div q)) Q_set) +
```
```   609       nat(setsum (%x. ((x * q) div p)) P_set) =
```
```   610         nat((setsum (%x. ((x * p) div q)) Q_set) +
```
```   611           (setsum (%x. ((x * q) div p)) P_set))";
```
```   612     apply (rule_tac z1 = "setsum (%x. ((x * p) div q)) Q_set" in
```
```   613       nat_add_distrib [THEN sym]);
```
```   614     by (auto simp add: S1_carda [THEN sym] S2_carda [THEN sym])
```
```   615   also have "... = nat(((p - 1) div 2) * ((q - 1) div 2))";
```
```   616     by (auto simp add: pq_sum_prop)
```
```   617   finally show ?thesis .;
```
```   618 qed;
```
```   619
```
```   620 theorem Quadratic_Reciprocity:
```
```   621      "[| p \<in> zOdd; p \<in> zprime; q \<in> zOdd; q \<in> zprime;
```
```   622          p \<noteq> q |]
```
```   623       ==> (Legendre p q) * (Legendre q p) =
```
```   624           (-1::int)^nat(((p - 1) div 2)*((q - 1) div 2))";
```
```   625   by (auto simp add: QRTEMP.QR_short zprime_zOdd_eq_grt_2 [THEN sym]
```
```   626                      QRTEMP_def)
```
```   627
```
```   628 end
```