src/HOL/Old_Number_Theory/Quadratic_Reciprocity.thy
author wenzelm
Sun Dec 27 22:07:17 2015 +0100 (2015-12-27)
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     1 (*  Title:      HOL/Old_Number_Theory/Quadratic_Reciprocity.thy
     2     Authors:    Jeremy Avigad, David Gray, and Adam Kramer
     3 *)
     4 
     5 section \<open>The law of Quadratic reciprocity\<close>
     6 
     7 theory Quadratic_Reciprocity
     8 imports Gauss
     9 begin
    10 
    11 text \<open>
    12   Lemmas leading up to the proof of theorem 3.3 in Niven and
    13   Zuckerman's presentation.
    14 \<close>
    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 \<open>Stuff about S, S1 and S2\<close>
   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 \<times> 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 (simp add: S_def)
   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