author blanchet Thu Sep 11 18:54:36 2014 +0200 (2014-09-11) changeset 58306 117ba6cbe414 parent 57514 bdc2c6b40bf2 child 58889 5b7a9633cfa8 permissions -rw-r--r--
renamed 'rep_datatype' to 'old_rep_datatype' (HOL)
```     1 (*  Title:      HOL/Old_Number_Theory/Quadratic_Reciprocity.thy
```
```     2     Authors:    Jeremy Avigad, David Gray, and Adam Kramer
```
```     3 *)
```
```     4
```
```     5 header {* The law of Quadratic reciprocity *}
```
```     6
```
```     7 theory Quadratic_Reciprocity
```
```     8 imports Gauss
```
```     9 begin
```
```    10
```
```    11 text {*
```
```    12   Lemmas leading up to the proof of theorem 3.3 in Niven and
```
```    13   Zuckerman's presentation.
```
```    14 *}
```
```    15
```
```    16 context GAUSS
```
```    17 begin
```
```    18
```
```    19 lemma QRLemma1: "a * setsum id A =
```
```    20   p * setsum (%x. ((x * a) div p)) A + setsum id D + setsum id E"
```
```    21 proof -
```
```    22   from finite_A have "a * setsum id A = setsum (%x. a * x) A"
```
```    23     by (auto simp add: setsum_const_mult id_def)
```
```    24   also have "setsum (%x. a * x) = setsum (%x. x * a)"
```
```    25     by (auto simp add: mult.commute)
```
```    26   also have "setsum (%x. x * a) A = setsum id B"
```
```    27     by (simp add: B_def setsum.reindex [OF inj_on_xa_A])
```
```    28   also have "... = setsum (%x. p * (x div p) + StandardRes p x) B"
```
```    29     by (auto simp add: StandardRes_def zmod_zdiv_equality)
```
```    30   also have "... = setsum (%x. p * (x div p)) B + setsum (StandardRes p) B"
```
```    31     by (rule setsum.distrib)
```
```    32   also have "setsum (StandardRes p) B = setsum id C"
```
```    33     by (auto simp add: C_def setsum.reindex [OF 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     by (rule setsum.union_disjoint) (auto simp add: D_def E_def)
```
```    38   also have "setsum (%x. p * (x div p)) B =
```
```    39       setsum ((%x. p * (x div p)) o (%x. (x * a))) A"
```
```    40     by (auto simp add: B_def setsum.reindex inj_on_xa_A)
```
```    41   also have "... = setsum (%x. p * ((x * a) div p)) A"
```
```    42     by (auto simp add: o_def)
```
```    43   also from finite_A have "setsum (%x. p * ((x * a) div p)) A =
```
```    44     p * setsum (%x. ((x * a) div p)) A"
```
```    45     by (auto simp add: setsum_const_mult)
```
```    46   finally show ?thesis by arith
```
```    47 qed
```
```    48
```
```    49 lemma QRLemma2: "setsum id A = p * int (card E) - setsum id E +
```
```    50   setsum id D"
```
```    51 proof -
```
```    52   from F_Un_D_eq_A have "setsum id A = setsum id (D \<union> F)"
```
```    53     by (simp add: Un_commute)
```
```    54   also from F_D_disj finite_D finite_F
```
```    55   have "... = setsum id D + setsum id F"
```
```    56     by (auto simp add: Int_commute intro: setsum.union_disjoint)
```
```    57   also from F_def have "F = (%x. (p - x)) ` E"
```
```    58     by auto
```
```    59   also from finite_E inj_on_pminusx_E have "setsum id ((%x. (p - x)) ` E) =
```
```    60       setsum (%x. (p - x)) E"
```
```    61     by (auto simp add: setsum.reindex)
```
```    62   also from finite_E have "setsum (op - p) E = setsum (%x. p) E - setsum id E"
```
```    63     by (auto simp add: setsum_subtractf id_def)
```
```    64   also from finite_E have "setsum (%x. p) E = p * int(card E)"
```
```    65     by (intro setsum_const)
```
```    66   finally show ?thesis
```
```    67     by arith
```
```    68 qed
```
```    69
```
```    70 lemma QRLemma3: "(a - 1) * setsum id A =
```
```    71     p * (setsum (%x. ((x * a) div p)) A - int(card E)) + 2 * setsum id E"
```
```    72 proof -
```
```    73   have "(a - 1) * setsum id A = a * setsum id A - setsum id A"
```
```    74     by (auto simp add: left_diff_distrib)
```
```    75   also note QRLemma1
```
```    76   also from QRLemma2 have "p * (\<Sum>x \<in> A. x * a div p) + setsum id D +
```
```    77      setsum id E - setsum id A =
```
```    78       p * (\<Sum>x \<in> A. x * a div p) + setsum id D +
```
```    79       setsum id E - (p * int (card E) - setsum id E + setsum id D)"
```
```    80     by auto
```
```    81   also have "... = p * (\<Sum>x \<in> A. x * a div p) -
```
```    82       p * int (card E) + 2 * setsum id E"
```
```    83     by arith
```
```    84   finally show ?thesis
```
```    85     by (auto simp only: right_diff_distrib)
```
```    86 qed
```
```    87
```
```    88 lemma QRLemma4: "a \<in> zOdd ==>
```
```    89     (setsum (%x. ((x * a) div p)) A \<in> zEven) = (int(card E): zEven)"
```
```    90 proof -
```
```    91   assume a_odd: "a \<in> zOdd"
```
```    92   from QRLemma3 have a: "p * (setsum (%x. ((x * a) div p)) A - int(card E)) =
```
```    93       (a - 1) * setsum id A - 2 * setsum id E"
```
```    94     by arith
```
```    95   from a_odd have "a - 1 \<in> zEven"
```
```    96     by (rule odd_minus_one_even)
```
```    97   hence "(a - 1) * setsum id A \<in> zEven"
```
```    98     by (rule even_times_either)
```
```    99   moreover have "2 * setsum id E \<in> zEven"
```
```   100     by (auto simp add: zEven_def)
```
```   101   ultimately have "(a - 1) * setsum id A - 2 * setsum id E \<in> zEven"
```
```   102     by (rule even_minus_even)
```
```   103   with a have "p * (setsum (%x. ((x * a) div p)) A - int(card E)): zEven"
```
```   104     by simp
```
```   105   hence "p \<in> zEven | (setsum (%x. ((x * a) div p)) A - int(card E)): zEven"
```
```   106     by (rule EvenOdd.even_product)
```
```   107   with p_odd have "(setsum (%x. ((x * a) div p)) A - int(card E)): zEven"
```
```   108     by (auto simp add: odd_iff_not_even)
```
```   109   thus ?thesis
```
```   110     by (auto simp only: even_diff [symmetric])
```
```   111 qed
```
```   112
```
```   113 lemma QRLemma5: "a \<in> zOdd ==>
```
```   114    (-1::int)^(card E) = (-1::int)^(nat(setsum (%x. ((x * a) div p)) A))"
```
```   115 proof -
```
```   116   assume "a \<in> zOdd"
```
```   117   from QRLemma4 [OF this] have
```
```   118     "(int(card E): zEven) = (setsum (%x. ((x * a) div p)) A \<in> zEven)" ..
```
```   119   moreover have "0 \<le> int(card E)"
```
```   120     by auto
```
```   121   moreover have "0 \<le> setsum (%x. ((x * a) div p)) A"
```
```   122     proof (intro setsum_nonneg)
```
```   123       show "\<forall>x \<in> A. 0 \<le> x * a div p"
```
```   124       proof
```
```   125         fix x
```
```   126         assume "x \<in> A"
```
```   127         then have "0 \<le> x"
```
```   128           by (auto simp add: A_def)
```
```   129         with a_nonzero have "0 \<le> x * a"
```
```   130           by (auto simp add: zero_le_mult_iff)
```
```   131         with p_g_2 show "0 \<le> x * a div p"
```
```   132           by (auto simp add: pos_imp_zdiv_nonneg_iff)
```
```   133       qed
```
```   134     qed
```
```   135   ultimately have "(-1::int)^nat((int (card E))) =
```
```   136       (-1)^nat(((\<Sum>x \<in> A. x * a div p)))"
```
```   137     by (intro neg_one_power_parity, auto)
```
```   138   also have "nat (int(card E)) = card E"
```
```   139     by auto
```
```   140   finally show ?thesis .
```
```   141 qed
```
```   142
```
```   143 end
```
```   144
```
```   145 lemma MainQRLemma: "[| a \<in> zOdd; 0 < a; ~([a = 0] (mod p)); zprime p; 2 < p;
```
```   146   A = {x. 0 < x & x \<le> (p - 1) div 2} |] ==>
```
```   147   (Legendre a p) = (-1::int)^(nat(setsum (%x. ((x * a) div p)) A))"
```
```   148   apply (subst GAUSS.gauss_lemma)
```
```   149   apply (auto simp add: GAUSS_def)
```
```   150   apply (subst GAUSS.QRLemma5)
```
```   151   apply (auto simp add: GAUSS_def)
```
```   152   apply (simp add: GAUSS.A_def [OF GAUSS.intro] GAUSS_def)
```
```   153   done
```
```   154
```
```   155
```
```   156 subsection {* Stuff about S, S1 and S2 *}
```
```   157
```
```   158 locale QRTEMP =
```
```   159   fixes p     :: "int"
```
```   160   fixes q     :: "int"
```
```   161
```
```   162   assumes p_prime: "zprime p"
```
```   163   assumes p_g_2: "2 < p"
```
```   164   assumes q_prime: "zprime q"
```
```   165   assumes q_g_2: "2 < q"
```
```   166   assumes p_neq_q:      "p \<noteq> q"
```
```   167 begin
```
```   168
```
```   169 definition P_set :: "int set"
```
```   170   where "P_set = {x. 0 < x & x \<le> ((p - 1) div 2) }"
```
```   171
```
```   172 definition Q_set :: "int set"
```
```   173   where "Q_set = {x. 0 < x & x \<le> ((q - 1) div 2) }"
```
```   174
```
```   175 definition S :: "(int * int) set"
```
```   176   where "S = P_set <*> Q_set"
```
```   177
```
```   178 definition S1 :: "(int * int) set"
```
```   179   where "S1 = { (x, y). (x, y):S & ((p * y) < (q * x)) }"
```
```   180
```
```   181 definition S2 :: "(int * int) set"
```
```   182   where "S2 = { (x, y). (x, y):S & ((q * x) < (p * y)) }"
```
```   183
```
```   184 definition f1 :: "int => (int * int) set"
```
```   185   where "f1 j = { (j1, y). (j1, y):S & j1 = j & (y \<le> (q * j) div p) }"
```
```   186
```
```   187 definition f2 :: "int => (int * int) set"
```
```   188   where "f2 j = { (x, j1). (x, j1):S & j1 = j & (x \<le> (p * j) div q) }"
```
```   189
```
```   190 lemma p_fact: "0 < (p - 1) div 2"
```
```   191 proof -
```
```   192   from p_g_2 have "2 \<le> p - 1" by arith
```
```   193   then have "2 div 2 \<le> (p - 1) div 2" by (rule zdiv_mono1, auto)
```
```   194   then show ?thesis by auto
```
```   195 qed
```
```   196
```
```   197 lemma q_fact: "0 < (q - 1) div 2"
```
```   198 proof -
```
```   199   from q_g_2 have "2 \<le> q - 1" by arith
```
```   200   then have "2 div 2 \<le> (q - 1) div 2" by (rule zdiv_mono1, auto)
```
```   201   then show ?thesis by auto
```
```   202 qed
```
```   203
```
```   204 lemma pb_neq_qa:
```
```   205   assumes "1 \<le> b" and "b \<le> (q - 1) div 2"
```
```   206   shows "p * b \<noteq> q * a"
```
```   207 proof
```
```   208   assume "p * b = q * a"
```
```   209   then have "q dvd (p * b)" by (auto simp add: dvd_def)
```
```   210   with q_prime p_g_2 have "q dvd p | q dvd b"
```
```   211     by (auto simp add: zprime_zdvd_zmult)
```
```   212   moreover have "~ (q dvd p)"
```
```   213   proof
```
```   214     assume "q dvd p"
```
```   215     with p_prime have "q = 1 | q = p"
```
```   216       apply (auto simp add: zprime_def QRTEMP_def)
```
```   217       apply (drule_tac x = q and R = False in allE)
```
```   218       apply (simp add: QRTEMP_def)
```
```   219       apply (subgoal_tac "0 \<le> q", simp add: QRTEMP_def)
```
```   220       apply (insert assms)
```
```   221       apply (auto simp add: QRTEMP_def)
```
```   222       done
```
```   223     with q_g_2 p_neq_q show False by auto
```
```   224   qed
```
```   225   ultimately have "q dvd b" by auto
```
```   226   then have "q \<le> b"
```
```   227   proof -
```
```   228     assume "q dvd b"
```
```   229     moreover from assms have "0 < b" by auto
```
```   230     ultimately show ?thesis using zdvd_bounds [of q b] by auto
```
```   231   qed
```
```   232   with assms have "q \<le> (q - 1) div 2" by auto
```
```   233   then have "2 * q \<le> 2 * ((q - 1) div 2)" by arith
```
```   234   then have "2 * q \<le> q - 1"
```
```   235   proof -
```
```   236     assume a: "2 * q \<le> 2 * ((q - 1) div 2)"
```
```   237     with assms have "q \<in> zOdd" by (auto simp add: QRTEMP_def zprime_zOdd_eq_grt_2)
```
```   238     with odd_minus_one_even have "(q - 1):zEven" by auto
```
```   239     with even_div_2_prop2 have "(q - 1) = 2 * ((q - 1) div 2)" by auto
```
```   240     with a show ?thesis by auto
```
```   241   qed
```
```   242   then have p1: "q \<le> -1" by arith
```
```   243   with q_g_2 show False by auto
```
```   244 qed
```
```   245
```
```   246 lemma P_set_finite: "finite (P_set)"
```
```   247   using p_fact by (auto simp add: P_set_def bdd_int_set_l_le_finite)
```
```   248
```
```   249 lemma Q_set_finite: "finite (Q_set)"
```
```   250   using q_fact by (auto simp add: Q_set_def bdd_int_set_l_le_finite)
```
```   251
```
```   252 lemma S_finite: "finite S"
```
```   253   by (auto simp add: S_def  P_set_finite Q_set_finite finite_cartesian_product)
```
```   254
```
```   255 lemma S1_finite: "finite S1"
```
```   256 proof -
```
```   257   have "finite S" by (auto simp add: S_finite)
```
```   258   moreover have "S1 \<subseteq> S" by (auto simp add: S1_def S_def)
```
```   259   ultimately show ?thesis by (auto simp add: finite_subset)
```
```   260 qed
```
```   261
```
```   262 lemma S2_finite: "finite S2"
```
```   263 proof -
```
```   264   have "finite S" by (auto simp add: S_finite)
```
```   265   moreover have "S2 \<subseteq> S" by (auto simp add: S2_def S_def)
```
```   266   ultimately show ?thesis by (auto simp add: finite_subset)
```
```   267 qed
```
```   268
```
```   269 lemma P_set_card: "(p - 1) div 2 = int (card (P_set))"
```
```   270   using p_fact by (auto simp add: P_set_def card_bdd_int_set_l_le)
```
```   271
```
```   272 lemma Q_set_card: "(q - 1) div 2 = int (card (Q_set))"
```
```   273   using q_fact by (auto simp add: Q_set_def card_bdd_int_set_l_le)
```
```   274
```
```   275 lemma S_card: "((p - 1) div 2) * ((q - 1) div 2) = int (card(S))"
```
```   276   using P_set_card Q_set_card P_set_finite Q_set_finite
```
```   277   by (auto simp add: S_def zmult_int)
```
```   278
```
```   279 lemma S1_Int_S2_prop: "S1 \<inter> S2 = {}"
```
```   280   by (auto simp add: S1_def S2_def)
```
```   281
```
```   282 lemma S1_Union_S2_prop: "S = S1 \<union> S2"
```
```   283   apply (auto simp add: S_def P_set_def Q_set_def S1_def S2_def)
```
```   284 proof -
```
```   285   fix a and b
```
```   286   assume "~ q * a < p * b" and b1: "0 < b" and b2: "b \<le> (q - 1) div 2"
```
```   287   with less_linear have "(p * b < q * a) | (p * b = q * a)" by auto
```
```   288   moreover from pb_neq_qa b1 b2 have "(p * b \<noteq> q * a)" by auto
```
```   289   ultimately show "p * b < q * a" by auto
```
```   290 qed
```
```   291
```
```   292 lemma card_sum_S1_S2: "((p - 1) div 2) * ((q - 1) div 2) =
```
```   293     int(card(S1)) + int(card(S2))"
```
```   294 proof -
```
```   295   have "((p - 1) div 2) * ((q - 1) div 2) = int (card(S))"
```
```   296     by (auto simp add: S_card)
```
```   297   also have "... = int( card(S1) + card(S2))"
```
```   298     apply (insert S1_finite S2_finite S1_Int_S2_prop S1_Union_S2_prop)
```
```   299     apply (drule card_Un_disjoint, auto)
```
```   300     done
```
```   301   also have "... = int(card(S1)) + int(card(S2))" by auto
```
```   302   finally show ?thesis .
```
```   303 qed
```
```   304
```
```   305 lemma aux1a:
```
```   306   assumes "0 < a" and "a \<le> (p - 1) div 2"
```
```   307     and "0 < b" and "b \<le> (q - 1) div 2"
```
```   308   shows "(p * b < q * a) = (b \<le> q * a div p)"
```
```   309 proof -
```
```   310   have "p * b < q * a ==> b \<le> q * a div p"
```
```   311   proof -
```
```   312     assume "p * b < q * a"
```
```   313     then have "p * b \<le> q * a" by auto
```
```   314     then have "(p * b) div p \<le> (q * a) div p"
```
```   315       by (rule zdiv_mono1) (insert p_g_2, auto)
```
```   316     then show "b \<le> (q * a) div p"
```
```   317       apply (subgoal_tac "p \<noteq> 0")
```
```   318       apply (frule div_mult_self1_is_id, force)
```
```   319       apply (insert p_g_2, auto)
```
```   320       done
```
```   321   qed
```
```   322   moreover have "b \<le> q * a div p ==> p * b < q * a"
```
```   323   proof -
```
```   324     assume "b \<le> q * a div p"
```
```   325     then have "p * b \<le> p * ((q * a) div p)"
```
```   326       using p_g_2 by (auto simp add: mult_le_cancel_left)
```
```   327     also have "... \<le> q * a"
```
```   328       by (rule zdiv_leq_prop) (insert p_g_2, auto)
```
```   329     finally have "p * b \<le> q * a" .
```
```   330     then have "p * b < q * a | p * b = q * a"
```
```   331       by (simp only: order_le_imp_less_or_eq)
```
```   332     moreover have "p * b \<noteq> q * a"
```
```   333       by (rule pb_neq_qa) (insert assms, auto)
```
```   334     ultimately show ?thesis by auto
```
```   335   qed
```
```   336   ultimately show ?thesis ..
```
```   337 qed
```
```   338
```
```   339 lemma aux1b:
```
```   340   assumes "0 < a" and "a \<le> (p - 1) div 2"
```
```   341     and "0 < b" and "b \<le> (q - 1) div 2"
```
```   342   shows "(q * a < p * b) = (a \<le> p * b div q)"
```
```   343 proof -
```
```   344   have "q * a < p * b ==> a \<le> p * b div q"
```
```   345   proof -
```
```   346     assume "q * a < p * b"
```
```   347     then have "q * a \<le> p * b" by auto
```
```   348     then have "(q * a) div q \<le> (p * b) div q"
```
```   349       by (rule zdiv_mono1) (insert q_g_2, auto)
```
```   350     then show "a \<le> (p * b) div q"
```
```   351       apply (subgoal_tac "q \<noteq> 0")
```
```   352       apply (frule div_mult_self1_is_id, force)
```
```   353       apply (insert q_g_2, auto)
```
```   354       done
```
```   355   qed
```
```   356   moreover have "a \<le> p * b div q ==> q * a < p * b"
```
```   357   proof -
```
```   358     assume "a \<le> p * b div q"
```
```   359     then have "q * a \<le> q * ((p * b) div q)"
```
```   360       using q_g_2 by (auto simp add: mult_le_cancel_left)
```
```   361     also have "... \<le> p * b"
```
```   362       by (rule zdiv_leq_prop) (insert q_g_2, auto)
```
```   363     finally have "q * a \<le> p * b" .
```
```   364     then have "q * a < p * b | q * a = p * b"
```
```   365       by (simp only: order_le_imp_less_or_eq)
```
```   366     moreover have "p * b \<noteq> q * a"
```
```   367       by (rule  pb_neq_qa) (insert assms, auto)
```
```   368     ultimately show ?thesis by auto
```
```   369   qed
```
```   370   ultimately show ?thesis ..
```
```   371 qed
```
```   372
```
```   373 lemma (in -) aux2:
```
```   374   assumes "zprime p" and "zprime q" and "2 < p" and "2 < q"
```
```   375   shows "(q * ((p - 1) div 2)) div p \<le> (q - 1) div 2"
```
```   376 proof-
```
```   377   (* Set up what's even and odd *)
```
```   378   from assms have "p \<in> zOdd & q \<in> zOdd"
```
```   379     by (auto simp add:  zprime_zOdd_eq_grt_2)
```
```   380   then have even1: "(p - 1):zEven & (q - 1):zEven"
```
```   381     by (auto simp add: odd_minus_one_even)
```
```   382   then have even2: "(2 * p):zEven & ((q - 1) * p):zEven"
```
```   383     by (auto simp add: zEven_def)
```
```   384   then have even3: "(((q - 1) * p) + (2 * p)):zEven"
```
```   385     by (auto simp: EvenOdd.even_plus_even)
```
```   386   (* using these prove it *)
```
```   387   from assms have "q * (p - 1) < ((q - 1) * p) + (2 * p)"
```
```   388     by (auto simp add: int_distrib)
```
```   389   then have "((p - 1) * q) div 2 < (((q - 1) * p) + (2 * p)) div 2"
```
```   390     apply (rule_tac x = "((p - 1) * q)" in even_div_2_l)
```
```   391     by (auto simp add: even3, auto simp add: ac_simps)
```
```   392   also have "((p - 1) * q) div 2 = q * ((p - 1) div 2)"
```
```   393     by (auto simp add: even1 even_prod_div_2)
```
```   394   also have "(((q - 1) * p) + (2 * p)) div 2 = (((q - 1) div 2) * p) + p"
```
```   395     by (auto simp add: even1 even2 even_prod_div_2 even_sum_div_2)
```
```   396   finally show ?thesis
```
```   397     apply (rule_tac x = " q * ((p - 1) div 2)" and
```
```   398                     y = "(q - 1) div 2" in div_prop2)
```
```   399     using assms by auto
```
```   400 qed
```
```   401
```
```   402 lemma aux3a: "\<forall>j \<in> P_set. int (card (f1 j)) = (q * j) div p"
```
```   403 proof
```
```   404   fix j
```
```   405   assume j_fact: "j \<in> P_set"
```
```   406   have "int (card (f1 j)) = int (card {y. y \<in> Q_set & y \<le> (q * j) div p})"
```
```   407   proof -
```
```   408     have "finite (f1 j)"
```
```   409     proof -
```
```   410       have "(f1 j) \<subseteq> S" by (auto simp add: f1_def)
```
```   411       with S_finite show ?thesis by (auto simp add: finite_subset)
```
```   412     qed
```
```   413     moreover have "inj_on (%(x,y). y) (f1 j)"
```
```   414       by (auto simp add: f1_def inj_on_def)
```
```   415     ultimately have "card ((%(x,y). y) ` (f1 j)) = card  (f1 j)"
```
```   416       by (auto simp add: f1_def card_image)
```
```   417     moreover have "((%(x,y). y) ` (f1 j)) = {y. y \<in> Q_set & y \<le> (q * j) div p}"
```
```   418       using j_fact by (auto simp add: f1_def S_def Q_set_def P_set_def image_def)
```
```   419     ultimately show ?thesis by (auto simp add: f1_def)
```
```   420   qed
```
```   421   also have "... = int (card {y. 0 < y & y \<le> (q * j) div p})"
```
```   422   proof -
```
```   423     have "{y. y \<in> Q_set & y \<le> (q * j) div p} =
```
```   424         {y. 0 < y & y \<le> (q * j) div p}"
```
```   425       apply (auto simp add: Q_set_def)
```
```   426     proof -
```
```   427       fix x
```
```   428       assume x: "0 < x" "x \<le> q * j div p"
```
```   429       with j_fact P_set_def  have "j \<le> (p - 1) div 2" by auto
```
```   430       with q_g_2 have "q * j \<le> q * ((p - 1) div 2)"
```
```   431         by (auto simp add: mult_le_cancel_left)
```
```   432       with p_g_2 have "q * j div p \<le> q * ((p - 1) div 2) div p"
```
```   433         by (auto simp add: zdiv_mono1)
```
```   434       also from QRTEMP_axioms j_fact P_set_def have "... \<le> (q - 1) div 2"
```
```   435         apply simp
```
```   436         apply (insert aux2)
```
```   437         apply (simp add: QRTEMP_def)
```
```   438         done
```
```   439       finally show "x \<le> (q - 1) div 2" using x by auto
```
```   440     qed
```
```   441     then show ?thesis by auto
```
```   442   qed
```
```   443   also have "... = (q * j) div p"
```
```   444   proof -
```
```   445     from j_fact P_set_def have "0 \<le> j" by auto
```
```   446     with q_g_2 have "q * 0 \<le> q * j" by (auto simp only: mult_left_mono)
```
```   447     then have "0 \<le> q * j" by auto
```
```   448     then have "0 div p \<le> (q * j) div p"
```
```   449       apply (rule_tac a = 0 in zdiv_mono1)
```
```   450       apply (insert p_g_2, auto)
```
```   451       done
```
```   452     also have "0 div p = 0" by auto
```
```   453     finally show ?thesis by (auto simp add: card_bdd_int_set_l_le)
```
```   454   qed
```
```   455   finally show "int (card (f1 j)) = q * j div p" .
```
```   456 qed
```
```   457
```
```   458 lemma aux3b: "\<forall>j \<in> Q_set. int (card (f2 j)) = (p * j) div q"
```
```   459 proof
```
```   460   fix j
```
```   461   assume j_fact: "j \<in> Q_set"
```
```   462   have "int (card (f2 j)) = int (card {y. y \<in> P_set & y \<le> (p * j) div q})"
```
```   463   proof -
```
```   464     have "finite (f2 j)"
```
```   465     proof -
```
```   466       have "(f2 j) \<subseteq> S" by (auto simp add: f2_def)
```
```   467       with S_finite show ?thesis by (auto simp add: finite_subset)
```
```   468     qed
```
```   469     moreover have "inj_on (%(x,y). x) (f2 j)"
```
```   470       by (auto simp add: f2_def inj_on_def)
```
```   471     ultimately have "card ((%(x,y). x) ` (f2 j)) = card  (f2 j)"
```
```   472       by (auto simp add: f2_def card_image)
```
```   473     moreover have "((%(x,y). x) ` (f2 j)) = {y. y \<in> P_set & y \<le> (p * j) div q}"
```
```   474       using j_fact by (auto simp add: f2_def S_def Q_set_def P_set_def image_def)
```
```   475     ultimately show ?thesis by (auto simp add: f2_def)
```
```   476   qed
```
```   477   also have "... = int (card {y. 0 < y & y \<le> (p * j) div q})"
```
```   478   proof -
```
```   479     have "{y. y \<in> P_set & y \<le> (p * j) div q} =
```
```   480         {y. 0 < y & y \<le> (p * j) div q}"
```
```   481       apply (auto simp add: P_set_def)
```
```   482     proof -
```
```   483       fix x
```
```   484       assume x: "0 < x" "x \<le> p * j div q"
```
```   485       with j_fact Q_set_def  have "j \<le> (q - 1) div 2" by auto
```
```   486       with p_g_2 have "p * j \<le> p * ((q - 1) div 2)"
```
```   487         by (auto simp add: mult_le_cancel_left)
```
```   488       with q_g_2 have "p * j div q \<le> p * ((q - 1) div 2) div q"
```
```   489         by (auto simp add: zdiv_mono1)
```
```   490       also from QRTEMP_axioms j_fact have "... \<le> (p - 1) div 2"
```
```   491         by (auto simp add: aux2 QRTEMP_def)
```
```   492       finally show "x \<le> (p - 1) div 2" using x by auto
```
```   493       qed
```
```   494     then show ?thesis by auto
```
```   495   qed
```
```   496   also have "... = (p * j) div q"
```
```   497   proof -
```
```   498     from j_fact Q_set_def have "0 \<le> j" by auto
```
```   499     with p_g_2 have "p * 0 \<le> p * j" by (auto simp only: mult_left_mono)
```
```   500     then have "0 \<le> p * j" by auto
```
```   501     then have "0 div q \<le> (p * j) div q"
```
```   502       apply (rule_tac a = 0 in zdiv_mono1)
```
```   503       apply (insert q_g_2, auto)
```
```   504       done
```
```   505     also have "0 div q = 0" by auto
```
```   506     finally show ?thesis by (auto simp add: card_bdd_int_set_l_le)
```
```   507   qed
```
```   508   finally show "int (card (f2 j)) = p * j div q" .
```
```   509 qed
```
```   510
```
```   511 lemma S1_card: "int (card(S1)) = setsum (%j. (q * j) div p) P_set"
```
```   512 proof -
```
```   513   have "\<forall>x \<in> P_set. finite (f1 x)"
```
```   514   proof
```
```   515     fix x
```
```   516     have "f1 x \<subseteq> S" by (auto simp add: f1_def)
```
```   517     with S_finite show "finite (f1 x)" by (auto simp add: finite_subset)
```
```   518   qed
```
```   519   moreover have "(\<forall>x \<in> P_set. \<forall>y \<in> P_set. x \<noteq> y --> (f1 x) \<inter> (f1 y) = {})"
```
```   520     by (auto simp add: f1_def)
```
```   521   moreover note P_set_finite
```
```   522   ultimately have "int(card (UNION P_set f1)) =
```
```   523       setsum (%x. int(card (f1 x))) P_set"
```
```   524     by(simp add:card_UN_disjoint int_setsum o_def)
```
```   525   moreover have "S1 = UNION P_set f1"
```
```   526     by (auto simp add: f1_def S_def S1_def S2_def P_set_def Q_set_def aux1a)
```
```   527   ultimately have "int(card (S1)) = setsum (%j. int(card (f1 j))) P_set"
```
```   528     by auto
```
```   529   also have "... = setsum (%j. q * j div p) P_set"
```
```   530     using aux3a by(fastforce intro: setsum.cong)
```
```   531   finally show ?thesis .
```
```   532 qed
```
```   533
```
```   534 lemma S2_card: "int (card(S2)) = setsum (%j. (p * j) div q) Q_set"
```
```   535 proof -
```
```   536   have "\<forall>x \<in> Q_set. finite (f2 x)"
```
```   537   proof
```
```   538     fix x
```
```   539     have "f2 x \<subseteq> S" by (auto simp add: f2_def)
```
```   540     with S_finite show "finite (f2 x)" by (auto simp add: finite_subset)
```
```   541   qed
```
```   542   moreover have "(\<forall>x \<in> Q_set. \<forall>y \<in> Q_set. x \<noteq> y -->
```
```   543       (f2 x) \<inter> (f2 y) = {})"
```
```   544     by (auto simp add: f2_def)
```
```   545   moreover note Q_set_finite
```
```   546   ultimately have "int(card (UNION Q_set f2)) =
```
```   547       setsum (%x. int(card (f2 x))) Q_set"
```
```   548     by(simp add:card_UN_disjoint int_setsum o_def)
```
```   549   moreover have "S2 = UNION Q_set f2"
```
```   550     by (auto simp add: f2_def S_def S1_def S2_def P_set_def Q_set_def aux1b)
```
```   551   ultimately have "int(card (S2)) = setsum (%j. int(card (f2 j))) Q_set"
```
```   552     by auto
```
```   553   also have "... = setsum (%j. p * j div q) Q_set"
```
```   554     using aux3b by(fastforce intro: setsum.cong)
```
```   555   finally show ?thesis .
```
```   556 qed
```
```   557
```
```   558 lemma S1_carda: "int (card(S1)) =
```
```   559     setsum (%j. (j * q) div p) P_set"
```
```   560   by (auto simp add: S1_card ac_simps)
```
```   561
```
```   562 lemma S2_carda: "int (card(S2)) =
```
```   563     setsum (%j. (j * p) div q) Q_set"
```
```   564   by (auto simp add: S2_card ac_simps)
```
```   565
```
```   566 lemma pq_sum_prop: "(setsum (%j. (j * p) div q) Q_set) +
```
```   567     (setsum (%j. (j * q) div p) P_set) = ((p - 1) div 2) * ((q - 1) div 2)"
```
```   568 proof -
```
```   569   have "(setsum (%j. (j * p) div q) Q_set) +
```
```   570       (setsum (%j. (j * q) div p) P_set) = int (card S2) + int (card S1)"
```
```   571     by (auto simp add: S1_carda S2_carda)
```
```   572   also have "... = int (card S1) + int (card S2)"
```
```   573     by auto
```
```   574   also have "... = ((p - 1) div 2) * ((q - 1) div 2)"
```
```   575     by (auto simp add: card_sum_S1_S2)
```
```   576   finally show ?thesis .
```
```   577 qed
```
```   578
```
```   579
```
```   580 lemma (in -) pq_prime_neq: "[| zprime p; zprime q; p \<noteq> q |] ==> (~[p = 0] (mod q))"
```
```   581   apply (auto simp add: zcong_eq_zdvd_prop zprime_def)
```
```   582   apply (drule_tac x = q in allE)
```
```   583   apply (drule_tac x = p in allE)
```
```   584   apply auto
```
```   585   done
```
```   586
```
```   587
```
```   588 lemma QR_short: "(Legendre p q) * (Legendre q p) =
```
```   589     (-1::int)^nat(((p - 1) div 2)*((q - 1) div 2))"
```
```   590 proof -
```
```   591   from QRTEMP_axioms have "~([p = 0] (mod q))"
```
```   592     by (auto simp add: pq_prime_neq QRTEMP_def)
```
```   593   with QRTEMP_axioms Q_set_def have a1: "(Legendre p q) = (-1::int) ^
```
```   594       nat(setsum (%x. ((x * p) div q)) Q_set)"
```
```   595     apply (rule_tac p = q in  MainQRLemma)
```
```   596     apply (auto simp add: zprime_zOdd_eq_grt_2 QRTEMP_def)
```
```   597     done
```
```   598   from QRTEMP_axioms have "~([q = 0] (mod p))"
```
```   599     apply (rule_tac p = q and q = p in pq_prime_neq)
```
```   600     apply (simp add: QRTEMP_def)+
```
```   601     done
```
```   602   with QRTEMP_axioms P_set_def have a2: "(Legendre q p) =
```
```   603       (-1::int) ^ nat(setsum (%x. ((x * q) div p)) P_set)"
```
```   604     apply (rule_tac p = p in  MainQRLemma)
```
```   605     apply (auto simp add: zprime_zOdd_eq_grt_2 QRTEMP_def)
```
```   606     done
```
```   607   from a1 a2 have "(Legendre p q) * (Legendre q p) =
```
```   608       (-1::int) ^ nat(setsum (%x. ((x * p) div q)) Q_set) *
```
```   609         (-1::int) ^ nat(setsum (%x. ((x * q) div p)) P_set)"
```
```   610     by auto
```
```   611   also have "... = (-1::int) ^ (nat(setsum (%x. ((x * p) div q)) Q_set) +
```
```   612                    nat(setsum (%x. ((x * q) div p)) P_set))"
```
```   613     by (auto simp add: power_add)
```
```   614   also have "nat(setsum (%x. ((x * p) div q)) Q_set) +
```
```   615       nat(setsum (%x. ((x * q) div p)) P_set) =
```
```   616         nat((setsum (%x. ((x * p) div q)) Q_set) +
```
```   617           (setsum (%x. ((x * q) div p)) P_set))"
```
```   618     apply (rule_tac z = "setsum (%x. ((x * p) div q)) Q_set" in
```
```   619       nat_add_distrib [symmetric])
```
```   620     apply (auto simp add: S1_carda [symmetric] S2_carda [symmetric])
```
```   621     done
```
```   622   also have "... = nat(((p - 1) div 2) * ((q - 1) div 2))"
```
```   623     by (auto simp add: pq_sum_prop)
```
```   624   finally show ?thesis .
```
```   625 qed
```
```   626
```
```   627 end
```
```   628
```
```   629 theorem Quadratic_Reciprocity:
```
```   630      "[| p \<in> zOdd; zprime p; q \<in> zOdd; zprime q;
```
```   631          p \<noteq> q |]
```
```   632       ==> (Legendre p q) * (Legendre q p) =
```
```   633           (-1::int)^nat(((p - 1) div 2)*((q - 1) div 2))"
```
```   634   by (auto simp add: QRTEMP.QR_short zprime_zOdd_eq_grt_2 [symmetric]
```
```   635                      QRTEMP_def)
```
```   636
```
```   637 end
```