src/HOL/NumberTheory/Quadratic_Reciprocity.thy
author paulson
Tue Jun 28 15:27:45 2005 +0200 (2005-06-28)
changeset 16587 b34c8aa657a5
parent 15541 206d779ba96d
child 16663 13e9c402308b
permissions -rw-r--r--
Constant "If" is now local
paulson@13871
     1
(*  Title:      HOL/Quadratic_Reciprocity/Quadratic_Reciprocity.thy
kleing@14981
     2
    ID:         $Id$
paulson@13871
     3
    Authors:    Jeremy Avigad, David Gray, and Adam Kramer
paulson@13871
     4
*)
paulson@13871
     5
paulson@13871
     6
header {* The law of Quadratic reciprocity *}
paulson@13871
     7
nipkow@15392
     8
theory Quadratic_Reciprocity
nipkow@15392
     9
imports Gauss
nipkow@15392
    10
begin
paulson@13871
    11
paulson@13871
    12
(***************************************************************)
paulson@13871
    13
(*                                                             *)
paulson@13871
    14
(*  Lemmas leading up to the proof of theorem 3.3 in           *)
paulson@13871
    15
(*  Niven and Zuckerman's presentation                         *)
paulson@13871
    16
(*                                                             *)
paulson@13871
    17
(***************************************************************)
paulson@13871
    18
paulson@13871
    19
lemma (in GAUSS) QRLemma1: "a * setsum id A = 
nipkow@15392
    20
  p * setsum (%x. ((x * a) div p)) A + setsum id D + setsum id E"
nipkow@15392
    21
proof -
nipkow@15392
    22
  from finite_A have "a * setsum id A = setsum (%x. a * x) A" 
paulson@13871
    23
    by (auto simp add: setsum_const_mult id_def)
nipkow@15392
    24
  also have "setsum (%x. a * x) = setsum (%x. x * a)" 
paulson@13871
    25
    by (auto simp add: zmult_commute)
nipkow@15392
    26
  also have "setsum (%x. x * a) A = setsum id B"
nipkow@15392
    27
    by (auto simp add: B_def setsum_reindex_id finite_A inj_on_xa_A)
nipkow@15392
    28
  also have "... = setsum (%x. p * (x div p) + StandardRes p x) B"
nipkow@15392
    29
    apply (rule setsum_cong)
paulson@13871
    30
    by (auto simp add: finite_B StandardRes_def zmod_zdiv_equality)
nipkow@15392
    31
  also have "... = setsum (%x. p * (x div p)) B + setsum (StandardRes p) B"
paulson@13871
    32
    by (rule setsum_addf)
nipkow@15392
    33
  also have "setsum (StandardRes p) B = setsum id C"
nipkow@15392
    34
    by (auto simp add: C_def setsum_reindex_id [THEN sym] finite_B 
paulson@13871
    35
      SR_B_inj)
nipkow@15392
    36
  also from C_eq have "... = setsum id (D \<union> E)"
paulson@13871
    37
    by auto
nipkow@15392
    38
  also from finite_D finite_E have "... = setsum id D + setsum id E"
paulson@13871
    39
    apply (rule setsum_Un_disjoint)
paulson@13871
    40
    by (auto simp add: D_def E_def)
paulson@13871
    41
  also have "setsum (%x. p * (x div p)) B = 
nipkow@15392
    42
      setsum ((%x. p * (x div p)) o (%x. (x * a))) A"
nipkow@15392
    43
    by (auto simp add: B_def setsum_reindex finite_A inj_on_xa_A)
nipkow@15392
    44
  also have "... = setsum (%x. p * ((x * a) div p)) A"
paulson@13871
    45
    by (auto simp add: o_def)
paulson@13871
    46
  also from finite_A have "setsum (%x. p * ((x * a) div p)) A = 
nipkow@15392
    47
    p * setsum (%x. ((x * a) div p)) A"
paulson@13871
    48
    by (auto simp add: setsum_const_mult)
paulson@13871
    49
  finally show ?thesis by arith
nipkow@15392
    50
qed
paulson@13871
    51
paulson@13871
    52
lemma (in GAUSS) QRLemma2: "setsum id A = p * int (card E) - setsum id E + 
nipkow@15392
    53
  setsum id D" 
nipkow@15392
    54
proof -
nipkow@15392
    55
  from F_Un_D_eq_A have "setsum id A = setsum id (D \<union> F)"
paulson@13871
    56
    by (simp add: Un_commute)
paulson@13871
    57
  also from F_D_disj finite_D finite_F have 
nipkow@15392
    58
      "... = setsum id D + setsum id F"
paulson@13871
    59
    apply (simp add: Int_commute)
paulson@13871
    60
    by (intro setsum_Un_disjoint) 
nipkow@15392
    61
  also from F_def have "F = (%x. (p - x)) ` E"
paulson@13871
    62
    by auto
paulson@13871
    63
  also from finite_E inj_on_pminusx_E have "setsum id ((%x. (p - x)) ` E) =
nipkow@15392
    64
      setsum (%x. (p - x)) E"
nipkow@15392
    65
    by (auto simp add: setsum_reindex)
nipkow@15392
    66
  also from finite_E have "setsum (op - p) E = setsum (%x. p) E - setsum id E"
nipkow@15392
    67
    by (auto simp add: setsum_subtractf id_def)
nipkow@15392
    68
  also from finite_E have "setsum (%x. p) E = p * int(card E)"
paulson@13871
    69
    by (intro setsum_const)
nipkow@15392
    70
  finally show ?thesis
paulson@13871
    71
    by arith
nipkow@15392
    72
qed
paulson@13871
    73
paulson@13871
    74
lemma (in GAUSS) QRLemma3: "(a - 1) * setsum id A = 
nipkow@15392
    75
    p * (setsum (%x. ((x * a) div p)) A - int(card E)) + 2 * setsum id E"
nipkow@15392
    76
proof -
nipkow@15392
    77
  have "(a - 1) * setsum id A = a * setsum id A - setsum id A"
paulson@13871
    78
    by (auto simp add: zdiff_zmult_distrib)  
nipkow@15392
    79
  also note QRLemma1
nipkow@15392
    80
  also from QRLemma2 have "p * (\<Sum>x \<in> A. x * a div p) + setsum id D + 
paulson@13871
    81
     setsum id E - setsum id A = 
paulson@13871
    82
      p * (\<Sum>x \<in> A. x * a div p) + setsum id D + 
nipkow@15392
    83
      setsum id E - (p * int (card E) - setsum id E + setsum id D)"
paulson@13871
    84
    by auto
nipkow@15392
    85
  also have "... = p * (\<Sum>x \<in> A. x * a div p) - 
nipkow@15392
    86
      p * int (card E) + 2 * setsum id E" 
paulson@13871
    87
    by arith
nipkow@15392
    88
  finally show ?thesis
paulson@13871
    89
    by (auto simp only: zdiff_zmult_distrib2)
nipkow@15392
    90
qed
paulson@13871
    91
paulson@13871
    92
lemma (in GAUSS) QRLemma4: "a \<in> zOdd ==> 
nipkow@15392
    93
    (setsum (%x. ((x * a) div p)) A \<in> zEven) = (int(card E): zEven)"
nipkow@15392
    94
proof -
nipkow@15392
    95
  assume a_odd: "a \<in> zOdd"
paulson@13871
    96
  from QRLemma3 have a: "p * (setsum (%x. ((x * a) div p)) A - int(card E)) =
nipkow@15392
    97
      (a - 1) * setsum id A - 2 * setsum id E" 
paulson@13871
    98
    by arith
paulson@13871
    99
  from a_odd have "a - 1 \<in> zEven"
paulson@13871
   100
    by (rule odd_minus_one_even)
nipkow@15392
   101
  hence "(a - 1) * setsum id A \<in> zEven"
paulson@13871
   102
    by (rule even_times_either)
nipkow@15392
   103
  moreover have "2 * setsum id E \<in> zEven"
paulson@13871
   104
    by (auto simp add: zEven_def)
paulson@13871
   105
  ultimately have "(a - 1) * setsum id A - 2 * setsum id E \<in> zEven"
paulson@13871
   106
    by (rule even_minus_even)
nipkow@15392
   107
  with a have "p * (setsum (%x. ((x * a) div p)) A - int(card E)): zEven"
paulson@13871
   108
    by simp
nipkow@15392
   109
  hence "p \<in> zEven | (setsum (%x. ((x * a) div p)) A - int(card E)): zEven"
paulson@14434
   110
    by (rule EvenOdd.even_product)
nipkow@15392
   111
  with p_odd have "(setsum (%x. ((x * a) div p)) A - int(card E)): zEven"
paulson@13871
   112
    by (auto simp add: odd_iff_not_even)
nipkow@15392
   113
  thus ?thesis
paulson@13871
   114
    by (auto simp only: even_diff [THEN sym])
nipkow@15392
   115
qed
paulson@13871
   116
paulson@13871
   117
lemma (in GAUSS) QRLemma5: "a \<in> zOdd ==> 
nipkow@15392
   118
   (-1::int)^(card E) = (-1::int)^(nat(setsum (%x. ((x * a) div p)) A))"
nipkow@15392
   119
proof -
nipkow@15392
   120
  assume "a \<in> zOdd"
paulson@13871
   121
  from QRLemma4 have
nipkow@15392
   122
    "(int(card E): zEven) = (setsum (%x. ((x * a) div p)) A \<in> zEven)"..
nipkow@15392
   123
  moreover have "0 \<le> int(card E)"
paulson@13871
   124
    by auto
nipkow@15392
   125
  moreover have "0 \<le> setsum (%x. ((x * a) div p)) A"
nipkow@15392
   126
    proof (intro setsum_nonneg)
nipkow@15537
   127
      show "\<forall>x \<in> A. 0 \<le> x * a div p"
nipkow@15392
   128
      proof
nipkow@15392
   129
        fix x
nipkow@15392
   130
        assume "x \<in> A"
nipkow@15392
   131
        then have "0 \<le> x"
paulson@13871
   132
          by (auto simp add: A_def)
nipkow@15392
   133
        with a_nonzero have "0 \<le> x * a"
paulson@14353
   134
          by (auto simp add: zero_le_mult_iff)
nipkow@15392
   135
        with p_g_2 show "0 \<le> x * a div p" 
paulson@13871
   136
          by (auto simp add: pos_imp_zdiv_nonneg_iff)
nipkow@15392
   137
      qed
nipkow@15392
   138
    qed
paulson@13871
   139
  ultimately have "(-1::int)^nat((int (card E))) =
nipkow@15392
   140
      (-1)^nat(((\<Sum>x \<in> A. x * a div p)))"
paulson@13871
   141
    by (intro neg_one_power_parity, auto)
nipkow@15392
   142
  also have "nat (int(card E)) = card E"
paulson@13871
   143
    by auto
nipkow@15392
   144
  finally show ?thesis .
nipkow@15392
   145
qed
paulson@13871
   146
paulson@13871
   147
lemma MainQRLemma: "[| a \<in> zOdd; 0 < a; ~([a = 0] (mod p));p \<in> zprime; 2 < p;
paulson@13871
   148
  A = {x. 0 < x & x \<le> (p - 1) div 2} |] ==> 
nipkow@15392
   149
  (Legendre a p) = (-1::int)^(nat(setsum (%x. ((x * a) div p)) A))"
paulson@13871
   150
  apply (subst GAUSS.gauss_lemma)
paulson@13871
   151
  apply (auto simp add: GAUSS_def)
paulson@13871
   152
  apply (subst GAUSS.QRLemma5)
paulson@13871
   153
by (auto simp add: GAUSS_def)
paulson@13871
   154
paulson@13871
   155
(******************************************************************)
paulson@13871
   156
(*                                                                *)
paulson@13871
   157
(* Stuff about S, S1 and S2...                                    *)
paulson@13871
   158
(*                                                                *)
paulson@13871
   159
(******************************************************************)
paulson@13871
   160
paulson@13871
   161
locale QRTEMP =
paulson@13871
   162
  fixes p     :: "int"
paulson@13871
   163
  fixes q     :: "int"
paulson@13871
   164
  fixes P_set :: "int set"
paulson@13871
   165
  fixes Q_set :: "int set"
paulson@13871
   166
  fixes S     :: "(int * int) set"
paulson@13871
   167
  fixes S1    :: "(int * int) set"
paulson@13871
   168
  fixes S2    :: "(int * int) set"
paulson@13871
   169
  fixes f1    :: "int => (int * int) set"
paulson@13871
   170
  fixes f2    :: "int => (int * int) set"
paulson@13871
   171
paulson@13871
   172
  assumes p_prime: "p \<in> zprime"
paulson@13871
   173
  assumes p_g_2: "2 < p"
paulson@13871
   174
  assumes q_prime: "q \<in> zprime"
paulson@13871
   175
  assumes q_g_2: "2 < q"
paulson@13871
   176
  assumes p_neq_q:      "p \<noteq> q"
paulson@13871
   177
paulson@13871
   178
  defines P_set_def: "P_set == {x. 0 < x & x \<le> ((p - 1) div 2) }"
paulson@13871
   179
  defines Q_set_def: "Q_set == {x. 0 < x & x \<le> ((q - 1) div 2) }"
paulson@13871
   180
  defines S_def:     "S     == P_set <*> Q_set"
paulson@13871
   181
  defines S1_def:    "S1    == { (x, y). (x, y):S & ((p * y) < (q * x)) }"
paulson@13871
   182
  defines S2_def:    "S2    == { (x, y). (x, y):S & ((q * x) < (p * y)) }"
paulson@13871
   183
  defines f1_def:    "f1 j  == { (j1, y). (j1, y):S & j1 = j & 
paulson@13871
   184
                                 (y \<le> (q * j) div p) }"
paulson@13871
   185
  defines f2_def:    "f2 j  == { (x, j1). (x, j1):S & j1 = j & 
nipkow@15392
   186
                                 (x \<le> (p * j) div q) }"
paulson@13871
   187
nipkow@15392
   188
lemma (in QRTEMP) p_fact: "0 < (p - 1) div 2"
nipkow@15392
   189
proof -
paulson@13871
   190
  from prems have "2 < p" by (simp add: QRTEMP_def)
paulson@13871
   191
  then have "2 \<le> p - 1" by arith
paulson@13871
   192
  then have "2 div 2 \<le> (p - 1) div 2" by (rule zdiv_mono1, auto)
paulson@13871
   193
  then show ?thesis by auto
nipkow@15392
   194
qed
paulson@13871
   195
nipkow@15392
   196
lemma (in QRTEMP) q_fact: "0 < (q - 1) div 2"
nipkow@15392
   197
proof -
paulson@13871
   198
  from prems have "2 < q" by (simp add: QRTEMP_def)
paulson@13871
   199
  then have "2 \<le> q - 1" by arith
paulson@13871
   200
  then have "2 div 2 \<le> (q - 1) div 2" by (rule zdiv_mono1, auto)
paulson@13871
   201
  then show ?thesis by auto
nipkow@15392
   202
qed
paulson@13871
   203
paulson@13871
   204
lemma (in QRTEMP) pb_neq_qa: "[|1 \<le> b; b \<le> (q - 1) div 2 |] ==> 
nipkow@15392
   205
    (p * b \<noteq> q * a)"
nipkow@15392
   206
proof
nipkow@15392
   207
  assume "p * b = q * a" and "1 \<le> b" and "b \<le> (q - 1) div 2"
paulson@13871
   208
  then have "q dvd (p * b)" by (auto simp add: dvd_def)
nipkow@15392
   209
  with q_prime p_g_2 have "q dvd p | q dvd b"
paulson@13871
   210
    by (auto simp add: zprime_zdvd_zmult)
nipkow@15392
   211
  moreover have "~ (q dvd p)"
nipkow@15392
   212
  proof
nipkow@15392
   213
    assume "q dvd p"
paulson@13871
   214
    with p_prime have "q = 1 | q = p"
paulson@13871
   215
      apply (auto simp add: zprime_def QRTEMP_def)
paulson@13871
   216
      apply (drule_tac x = q and R = False in allE)
paulson@13871
   217
      apply (simp add: QRTEMP_def)    
paulson@13871
   218
      apply (subgoal_tac "0 \<le> q", simp add: QRTEMP_def)
paulson@13871
   219
      apply (insert prems)
paulson@13871
   220
    by (auto simp add: QRTEMP_def)
paulson@13871
   221
    with q_g_2 p_neq_q show False by auto
nipkow@15392
   222
  qed
paulson@13871
   223
  ultimately have "q dvd b" by auto
nipkow@15392
   224
  then have "q \<le> b"
nipkow@15392
   225
  proof -
nipkow@15392
   226
    assume "q dvd b"
paulson@13871
   227
    moreover from prems have "0 < b" by auto
paulson@13871
   228
    ultimately show ?thesis by (insert zdvd_bounds [of q b], auto)
nipkow@15392
   229
  qed
paulson@13871
   230
  with prems have "q \<le> (q - 1) div 2" by auto
paulson@13871
   231
  then have "2 * q \<le> 2 * ((q - 1) div 2)" by arith
nipkow@15392
   232
  then have "2 * q \<le> q - 1"
nipkow@15392
   233
  proof -
nipkow@15392
   234
    assume "2 * q \<le> 2 * ((q - 1) div 2)"
paulson@13871
   235
    with prems have "q \<in> zOdd" by (auto simp add: QRTEMP_def zprime_zOdd_eq_grt_2)
paulson@13871
   236
    with odd_minus_one_even have "(q - 1):zEven" by auto
paulson@13871
   237
    with even_div_2_prop2 have "(q - 1) = 2 * ((q - 1) div 2)" by auto
paulson@13871
   238
    with prems show ?thesis by auto
nipkow@15392
   239
  qed
paulson@13871
   240
  then have p1: "q \<le> -1" by arith
paulson@13871
   241
  with q_g_2 show False by auto
nipkow@15392
   242
qed
paulson@13871
   243
nipkow@15392
   244
lemma (in QRTEMP) P_set_finite: "finite (P_set)"
paulson@13871
   245
  by (insert p_fact, auto simp add: P_set_def bdd_int_set_l_le_finite)
paulson@13871
   246
nipkow@15392
   247
lemma (in QRTEMP) Q_set_finite: "finite (Q_set)"
paulson@13871
   248
  by (insert q_fact, auto simp add: Q_set_def bdd_int_set_l_le_finite)
paulson@13871
   249
nipkow@15392
   250
lemma (in QRTEMP) S_finite: "finite S"
nipkow@15402
   251
  by (auto simp add: S_def  P_set_finite Q_set_finite finite_cartesian_product)
paulson@13871
   252
nipkow@15392
   253
lemma (in QRTEMP) S1_finite: "finite S1"
nipkow@15392
   254
proof -
paulson@13871
   255
  have "finite S" by (auto simp add: S_finite)
paulson@13871
   256
  moreover have "S1 \<subseteq> S" by (auto simp add: S1_def S_def)
paulson@13871
   257
  ultimately show ?thesis by (auto simp add: finite_subset)
nipkow@15392
   258
qed
paulson@13871
   259
nipkow@15392
   260
lemma (in QRTEMP) S2_finite: "finite S2"
nipkow@15392
   261
proof -
paulson@13871
   262
  have "finite S" by (auto simp add: S_finite)
paulson@13871
   263
  moreover have "S2 \<subseteq> S" by (auto simp add: S2_def S_def)
paulson@13871
   264
  ultimately show ?thesis by (auto simp add: finite_subset)
nipkow@15392
   265
qed
paulson@13871
   266
nipkow@15392
   267
lemma (in QRTEMP) P_set_card: "(p - 1) div 2 = int (card (P_set))"
paulson@13871
   268
  by (insert p_fact, auto simp add: P_set_def card_bdd_int_set_l_le)
paulson@13871
   269
nipkow@15392
   270
lemma (in QRTEMP) Q_set_card: "(q - 1) div 2 = int (card (Q_set))"
paulson@13871
   271
  by (insert q_fact, auto simp add: Q_set_def card_bdd_int_set_l_le)
paulson@13871
   272
nipkow@15392
   273
lemma (in QRTEMP) S_card: "((p - 1) div 2) * ((q - 1) div 2) = int (card(S))"
paulson@13871
   274
  apply (insert P_set_card Q_set_card P_set_finite Q_set_finite)
nipkow@15541
   275
  apply (auto simp add: S_def zmult_int setsum_constant)
paulson@13871
   276
done
paulson@13871
   277
nipkow@15392
   278
lemma (in QRTEMP) S1_Int_S2_prop: "S1 \<inter> S2 = {}"
paulson@13871
   279
  by (auto simp add: S1_def S2_def)
paulson@13871
   280
nipkow@15392
   281
lemma (in QRTEMP) S1_Union_S2_prop: "S = S1 \<union> S2"
paulson@13871
   282
  apply (auto simp add: S_def P_set_def Q_set_def S1_def S2_def)
nipkow@15392
   283
  proof -
nipkow@15392
   284
    fix a and b
nipkow@15392
   285
    assume "~ q * a < p * b" and b1: "0 < b" and b2: "b \<le> (q - 1) div 2"
paulson@13871
   286
    with zless_linear have "(p * b < q * a) | (p * b = q * a)" by auto
paulson@13871
   287
    moreover from pb_neq_qa b1 b2 have "(p * b \<noteq> q * a)" by auto
paulson@13871
   288
    ultimately show "p * b < q * a" by auto
nipkow@15392
   289
  qed
paulson@13871
   290
paulson@13871
   291
lemma (in QRTEMP) card_sum_S1_S2: "((p - 1) div 2) * ((q - 1) div 2) = 
nipkow@15392
   292
    int(card(S1)) + int(card(S2))"
nipkow@15392
   293
proof-
nipkow@15392
   294
  have "((p - 1) div 2) * ((q - 1) div 2) = int (card(S))"
paulson@13871
   295
    by (auto simp add: S_card)
nipkow@15392
   296
  also have "... = int( card(S1) + card(S2))"
paulson@13871
   297
    apply (insert S1_finite S2_finite S1_Int_S2_prop S1_Union_S2_prop)
paulson@13871
   298
    apply (drule card_Un_disjoint, auto)
paulson@13871
   299
  done
paulson@13871
   300
  also have "... = int(card(S1)) + int(card(S2))" by auto
nipkow@15392
   301
  finally show ?thesis .
nipkow@15392
   302
qed
paulson@13871
   303
paulson@13871
   304
lemma (in QRTEMP) aux1a: "[| 0 < a; a \<le> (p - 1) div 2; 
paulson@13871
   305
                             0 < b; b \<le> (q - 1) div 2 |] ==>
nipkow@15392
   306
                          (p * b < q * a) = (b \<le> q * a div p)"
nipkow@15392
   307
proof -
nipkow@15392
   308
  assume "0 < a" and "a \<le> (p - 1) div 2" and "0 < b" and "b \<le> (q - 1) div 2"
nipkow@15392
   309
  have "p * b < q * a ==> b \<le> q * a div p"
nipkow@15392
   310
  proof -
nipkow@15392
   311
    assume "p * b < q * a"
paulson@13871
   312
    then have "p * b \<le> q * a" by auto
nipkow@15392
   313
    then have "(p * b) div p \<le> (q * a) div p"
paulson@13871
   314
      by (rule zdiv_mono1, insert p_g_2, auto)
nipkow@15392
   315
    then show "b \<le> (q * a) div p"
paulson@13871
   316
      apply (subgoal_tac "p \<noteq> 0")
paulson@13871
   317
      apply (frule zdiv_zmult_self2, force)
paulson@13871
   318
      by (insert p_g_2, auto)
nipkow@15392
   319
  qed
nipkow@15392
   320
  moreover have "b \<le> q * a div p ==> p * b < q * a"
nipkow@15392
   321
  proof -
nipkow@15392
   322
    assume "b \<le> q * a div p"
nipkow@15392
   323
    then have "p * b \<le> p * ((q * a) div p)"
paulson@14387
   324
      by (insert p_g_2, auto simp add: mult_le_cancel_left)
nipkow@15392
   325
    also have "... \<le> q * a"
paulson@13871
   326
      by (rule zdiv_leq_prop, insert p_g_2, auto)
nipkow@15392
   327
    finally have "p * b \<le> q * a" .
nipkow@15392
   328
    then have "p * b < q * a | p * b = q * a"
paulson@13871
   329
      by (simp only: order_le_imp_less_or_eq)
nipkow@15392
   330
    moreover have "p * b \<noteq> q * a"
paulson@13871
   331
      by (rule  pb_neq_qa, insert prems, auto)
paulson@13871
   332
    ultimately show ?thesis by auto
nipkow@15392
   333
  qed
nipkow@15392
   334
  ultimately show ?thesis ..
nipkow@15392
   335
qed
paulson@13871
   336
paulson@13871
   337
lemma (in QRTEMP) aux1b: "[| 0 < a; a \<le> (p - 1) div 2; 
paulson@13871
   338
                             0 < b; b \<le> (q - 1) div 2 |] ==>
nipkow@15392
   339
                          (q * a < p * b) = (a \<le> p * b div q)"
nipkow@15392
   340
proof -
nipkow@15392
   341
  assume "0 < a" and "a \<le> (p - 1) div 2" and "0 < b" and "b \<le> (q - 1) div 2"
nipkow@15392
   342
  have "q * a < p * b ==> a \<le> p * b div q"
nipkow@15392
   343
  proof -
nipkow@15392
   344
    assume "q * a < p * b"
paulson@13871
   345
    then have "q * a \<le> p * b" by auto
nipkow@15392
   346
    then have "(q * a) div q \<le> (p * b) div q"
paulson@13871
   347
      by (rule zdiv_mono1, insert q_g_2, auto)
nipkow@15392
   348
    then show "a \<le> (p * b) div q"
paulson@13871
   349
      apply (subgoal_tac "q \<noteq> 0")
paulson@13871
   350
      apply (frule zdiv_zmult_self2, force)
paulson@13871
   351
      by (insert q_g_2, auto)
nipkow@15392
   352
  qed
nipkow@15392
   353
  moreover have "a \<le> p * b div q ==> q * a < p * b"
nipkow@15392
   354
  proof -
nipkow@15392
   355
    assume "a \<le> p * b div q"
nipkow@15392
   356
    then have "q * a \<le> q * ((p * b) div q)"
paulson@14387
   357
      by (insert q_g_2, auto simp add: mult_le_cancel_left)
nipkow@15392
   358
    also have "... \<le> p * b"
paulson@13871
   359
      by (rule zdiv_leq_prop, insert q_g_2, auto)
nipkow@15392
   360
    finally have "q * a \<le> p * b" .
nipkow@15392
   361
    then have "q * a < p * b | q * a = p * b"
paulson@13871
   362
      by (simp only: order_le_imp_less_or_eq)
nipkow@15392
   363
    moreover have "p * b \<noteq> q * a"
paulson@13871
   364
      by (rule  pb_neq_qa, insert prems, auto)
paulson@13871
   365
    ultimately show ?thesis by auto
nipkow@15392
   366
  qed
nipkow@15392
   367
  ultimately show ?thesis ..
nipkow@15392
   368
qed
paulson@13871
   369
paulson@13871
   370
lemma aux2: "[| p \<in> zprime; q \<in> zprime; 2 < p; 2 < q |] ==> 
nipkow@15392
   371
             (q * ((p - 1) div 2)) div p \<le> (q - 1) div 2"
nipkow@15392
   372
proof-
nipkow@15392
   373
  assume "p \<in> zprime" and "q \<in> zprime" and "2 < p" and "2 < q"
paulson@13871
   374
  (* Set up what's even and odd *)
nipkow@15392
   375
  then have "p \<in> zOdd & q \<in> zOdd"
paulson@13871
   376
    by (auto simp add:  zprime_zOdd_eq_grt_2)
nipkow@15392
   377
  then have even1: "(p - 1):zEven & (q - 1):zEven"
paulson@13871
   378
    by (auto simp add: odd_minus_one_even)
nipkow@15392
   379
  then have even2: "(2 * p):zEven & ((q - 1) * p):zEven"
paulson@13871
   380
    by (auto simp add: zEven_def)
nipkow@15392
   381
  then have even3: "(((q - 1) * p) + (2 * p)):zEven"
paulson@14434
   382
    by (auto simp: EvenOdd.even_plus_even)
paulson@13871
   383
  (* using these prove it *)
nipkow@15392
   384
  from prems have "q * (p - 1) < ((q - 1) * p) + (2 * p)"
paulson@13871
   385
    by (auto simp add: int_distrib)
nipkow@15392
   386
  then have "((p - 1) * q) div 2 < (((q - 1) * p) + (2 * p)) div 2"
nipkow@15392
   387
    apply (rule_tac x = "((p - 1) * q)" in even_div_2_l)
paulson@13871
   388
    by (auto simp add: even3, auto simp add: zmult_ac)
nipkow@15392
   389
  also have "((p - 1) * q) div 2 = q * ((p - 1) div 2)"
paulson@13871
   390
    by (auto simp add: even1 even_prod_div_2)
nipkow@15392
   391
  also have "(((q - 1) * p) + (2 * p)) div 2 = (((q - 1) div 2) * p) + p"
paulson@13871
   392
    by (auto simp add: even1 even2 even_prod_div_2 even_sum_div_2)
paulson@13871
   393
  finally show ?thesis 
paulson@13871
   394
    apply (rule_tac x = " q * ((p - 1) div 2)" and 
nipkow@15392
   395
                    y = "(q - 1) div 2" in div_prop2)
paulson@13871
   396
    by (insert prems, auto)
nipkow@15392
   397
qed
paulson@13871
   398
nipkow@15392
   399
lemma (in QRTEMP) aux3a: "\<forall>j \<in> P_set. int (card (f1 j)) = (q * j) div p"
nipkow@15392
   400
proof
nipkow@15392
   401
  fix j
nipkow@15392
   402
  assume j_fact: "j \<in> P_set"
nipkow@15392
   403
  have "int (card (f1 j)) = int (card {y. y \<in> Q_set & y \<le> (q * j) div p})"
nipkow@15392
   404
  proof -
nipkow@15392
   405
    have "finite (f1 j)"
nipkow@15392
   406
    proof -
paulson@13871
   407
      have "(f1 j) \<subseteq> S" by (auto simp add: f1_def)
paulson@13871
   408
      with S_finite show ?thesis by (auto simp add: finite_subset)
nipkow@15392
   409
    qed
nipkow@15392
   410
    moreover have "inj_on (%(x,y). y) (f1 j)"
paulson@13871
   411
      by (auto simp add: f1_def inj_on_def)
nipkow@15392
   412
    ultimately have "card ((%(x,y). y) ` (f1 j)) = card  (f1 j)"
paulson@13871
   413
      by (auto simp add: f1_def card_image)
nipkow@15392
   414
    moreover have "((%(x,y). y) ` (f1 j)) = {y. y \<in> Q_set & y \<le> (q * j) div p}"
paulson@13871
   415
      by (insert prems, auto simp add: f1_def S_def Q_set_def P_set_def 
paulson@13871
   416
        image_def)
paulson@13871
   417
    ultimately show ?thesis by (auto simp add: f1_def)
nipkow@15392
   418
  qed
nipkow@15392
   419
  also have "... = int (card {y. 0 < y & y \<le> (q * j) div p})"
nipkow@15392
   420
  proof -
paulson@13871
   421
    have "{y. y \<in> Q_set & y \<le> (q * j) div p} = 
nipkow@15392
   422
        {y. 0 < y & y \<le> (q * j) div p}"
paulson@13871
   423
      apply (auto simp add: Q_set_def)
nipkow@15392
   424
      proof -
nipkow@15392
   425
        fix x
nipkow@15392
   426
        assume "0 < x" and "x \<le> q * j div p"
nipkow@15392
   427
        with j_fact P_set_def  have "j \<le> (p - 1) div 2" by auto
nipkow@15392
   428
        with q_g_2 have "q * j \<le> q * ((p - 1) div 2)"
paulson@14387
   429
          by (auto simp add: mult_le_cancel_left)
nipkow@15392
   430
        with p_g_2 have "q * j div p \<le> q * ((p - 1) div 2) div p"
paulson@13871
   431
          by (auto simp add: zdiv_mono1)
nipkow@15392
   432
        also from prems have "... \<le> (q - 1) div 2"
paulson@13871
   433
          apply simp apply (insert aux2) by (simp add: QRTEMP_def)
paulson@13871
   434
        finally show "x \<le> (q - 1) div 2" by (insert prems, auto)
nipkow@15392
   435
      qed
paulson@13871
   436
    then show ?thesis by auto
nipkow@15392
   437
  qed
nipkow@15392
   438
  also have "... = (q * j) div p"
nipkow@15392
   439
  proof -
paulson@13871
   440
    from j_fact P_set_def have "0 \<le> j" by auto
paulson@14387
   441
    with q_g_2 have "q * 0 \<le> q * j" by (auto simp only: mult_left_mono)
paulson@13871
   442
    then have "0 \<le> q * j" by auto
nipkow@15392
   443
    then have "0 div p \<le> (q * j) div p"
paulson@13871
   444
      apply (rule_tac a = 0 in zdiv_mono1)
paulson@13871
   445
      by (insert p_g_2, auto)
paulson@13871
   446
    also have "0 div p = 0" by auto
paulson@13871
   447
    finally show ?thesis by (auto simp add: card_bdd_int_set_l_le)
nipkow@15392
   448
  qed
nipkow@15392
   449
  finally show "int (card (f1 j)) = q * j div p" .
nipkow@15392
   450
qed
paulson@13871
   451
nipkow@15392
   452
lemma (in QRTEMP) aux3b: "\<forall>j \<in> Q_set. int (card (f2 j)) = (p * j) div q"
nipkow@15392
   453
proof
nipkow@15392
   454
  fix j
nipkow@15392
   455
  assume j_fact: "j \<in> Q_set"
nipkow@15392
   456
  have "int (card (f2 j)) = int (card {y. y \<in> P_set & y \<le> (p * j) div q})"
nipkow@15392
   457
  proof -
nipkow@15392
   458
    have "finite (f2 j)"
nipkow@15392
   459
    proof -
paulson@13871
   460
      have "(f2 j) \<subseteq> S" by (auto simp add: f2_def)
paulson@13871
   461
      with S_finite show ?thesis by (auto simp add: finite_subset)
nipkow@15392
   462
    qed
nipkow@15392
   463
    moreover have "inj_on (%(x,y). x) (f2 j)"
paulson@13871
   464
      by (auto simp add: f2_def inj_on_def)
nipkow@15392
   465
    ultimately have "card ((%(x,y). x) ` (f2 j)) = card  (f2 j)"
paulson@13871
   466
      by (auto simp add: f2_def card_image)
nipkow@15392
   467
    moreover have "((%(x,y). x) ` (f2 j)) = {y. y \<in> P_set & y \<le> (p * j) div q}"
paulson@13871
   468
      by (insert prems, auto simp add: f2_def S_def Q_set_def 
paulson@13871
   469
        P_set_def image_def)
paulson@13871
   470
    ultimately show ?thesis by (auto simp add: f2_def)
nipkow@15392
   471
  qed
nipkow@15392
   472
  also have "... = int (card {y. 0 < y & y \<le> (p * j) div q})"
nipkow@15392
   473
  proof -
paulson@13871
   474
    have "{y. y \<in> P_set & y \<le> (p * j) div q} = 
nipkow@15392
   475
        {y. 0 < y & y \<le> (p * j) div q}"
paulson@13871
   476
      apply (auto simp add: P_set_def)
nipkow@15392
   477
      proof -
nipkow@15392
   478
        fix x
nipkow@15392
   479
        assume "0 < x" and "x \<le> p * j div q"
nipkow@15392
   480
        with j_fact Q_set_def  have "j \<le> (q - 1) div 2" by auto
nipkow@15392
   481
        with p_g_2 have "p * j \<le> p * ((q - 1) div 2)"
paulson@14387
   482
          by (auto simp add: mult_le_cancel_left)
nipkow@15392
   483
        with q_g_2 have "p * j div q \<le> p * ((q - 1) div 2) div q"
paulson@13871
   484
          by (auto simp add: zdiv_mono1)
nipkow@15392
   485
        also from prems have "... \<le> (p - 1) div 2"
paulson@13871
   486
          by (auto simp add: aux2 QRTEMP_def)
paulson@13871
   487
        finally show "x \<le> (p - 1) div 2" by (insert prems, auto)
nipkow@15392
   488
      qed
paulson@13871
   489
    then show ?thesis by auto
nipkow@15392
   490
  qed
nipkow@15392
   491
  also have "... = (p * j) div q"
nipkow@15392
   492
  proof -
paulson@13871
   493
    from j_fact Q_set_def have "0 \<le> j" by auto
paulson@14387
   494
    with p_g_2 have "p * 0 \<le> p * j" by (auto simp only: mult_left_mono)
paulson@13871
   495
    then have "0 \<le> p * j" by auto
nipkow@15392
   496
    then have "0 div q \<le> (p * j) div q"
paulson@13871
   497
      apply (rule_tac a = 0 in zdiv_mono1)
paulson@13871
   498
      by (insert q_g_2, auto)
paulson@13871
   499
    also have "0 div q = 0" by auto
paulson@13871
   500
    finally show ?thesis by (auto simp add: card_bdd_int_set_l_le)
nipkow@15392
   501
  qed
nipkow@15392
   502
  finally show "int (card (f2 j)) = p * j div q" .
nipkow@15392
   503
qed
paulson@13871
   504
nipkow@15392
   505
lemma (in QRTEMP) S1_card: "int (card(S1)) = setsum (%j. (q * j) div p) P_set"
nipkow@15392
   506
proof -
nipkow@15392
   507
  have "\<forall>x \<in> P_set. finite (f1 x)"
nipkow@15392
   508
  proof
nipkow@15392
   509
    fix x
paulson@13871
   510
    have "f1 x \<subseteq> S" by (auto simp add: f1_def)
paulson@13871
   511
    with S_finite show "finite (f1 x)" by (auto simp add: finite_subset)
nipkow@15392
   512
  qed
nipkow@15392
   513
  moreover have "(\<forall>x \<in> P_set. \<forall>y \<in> P_set. x \<noteq> y --> (f1 x) \<inter> (f1 y) = {})"
paulson@13871
   514
    by (auto simp add: f1_def)
nipkow@15392
   515
  moreover note P_set_finite
paulson@13871
   516
  ultimately have "int(card (UNION P_set f1)) = 
nipkow@15392
   517
      setsum (%x. int(card (f1 x))) P_set"
nipkow@15402
   518
    by(simp add:card_UN_disjoint int_setsum o_def)
nipkow@15392
   519
  moreover have "S1 = UNION P_set f1"
paulson@13871
   520
    by (auto simp add: f1_def S_def S1_def S2_def P_set_def Q_set_def aux1a)
paulson@13871
   521
  ultimately have "int(card (S1)) = setsum (%j. int(card (f1 j))) P_set" 
paulson@13871
   522
    by auto
nipkow@15392
   523
  also have "... = setsum (%j. q * j div p) P_set"
nipkow@15392
   524
    using aux3a by(fastsimp intro: setsum_cong)
nipkow@15392
   525
  finally show ?thesis .
nipkow@15392
   526
qed
paulson@13871
   527
nipkow@15392
   528
lemma (in QRTEMP) S2_card: "int (card(S2)) = setsum (%j. (p * j) div q) Q_set"
nipkow@15392
   529
proof -
nipkow@15392
   530
  have "\<forall>x \<in> Q_set. finite (f2 x)"
nipkow@15392
   531
  proof
nipkow@15392
   532
    fix x
paulson@13871
   533
    have "f2 x \<subseteq> S" by (auto simp add: f2_def)
paulson@13871
   534
    with S_finite show "finite (f2 x)" by (auto simp add: finite_subset)
nipkow@15392
   535
  qed
paulson@13871
   536
  moreover have "(\<forall>x \<in> Q_set. \<forall>y \<in> Q_set. x \<noteq> y --> 
nipkow@15392
   537
      (f2 x) \<inter> (f2 y) = {})"
paulson@13871
   538
    by (auto simp add: f2_def)
nipkow@15392
   539
  moreover note Q_set_finite
paulson@13871
   540
  ultimately have "int(card (UNION Q_set f2)) = 
nipkow@15392
   541
      setsum (%x. int(card (f2 x))) Q_set"
nipkow@15402
   542
    by(simp add:card_UN_disjoint int_setsum o_def)
nipkow@15392
   543
  moreover have "S2 = UNION Q_set f2"
paulson@13871
   544
    by (auto simp add: f2_def S_def S1_def S2_def P_set_def Q_set_def aux1b)
paulson@13871
   545
  ultimately have "int(card (S2)) = setsum (%j. int(card (f2 j))) Q_set" 
paulson@13871
   546
    by auto
nipkow@15392
   547
  also have "... = setsum (%j. p * j div q) Q_set"
nipkow@15392
   548
    using aux3b by(fastsimp intro: setsum_cong)
nipkow@15392
   549
  finally show ?thesis .
nipkow@15392
   550
qed
paulson@13871
   551
paulson@13871
   552
lemma (in QRTEMP) S1_carda: "int (card(S1)) = 
nipkow@15392
   553
    setsum (%j. (j * q) div p) P_set"
paulson@13871
   554
  by (auto simp add: S1_card zmult_ac)
paulson@13871
   555
paulson@13871
   556
lemma (in QRTEMP) S2_carda: "int (card(S2)) = 
nipkow@15392
   557
    setsum (%j. (j * p) div q) Q_set"
paulson@13871
   558
  by (auto simp add: S2_card zmult_ac)
paulson@13871
   559
paulson@13871
   560
lemma (in QRTEMP) pq_sum_prop: "(setsum (%j. (j * p) div q) Q_set) + 
nipkow@15392
   561
    (setsum (%j. (j * q) div p) P_set) = ((p - 1) div 2) * ((q - 1) div 2)"
nipkow@15392
   562
proof -
paulson@13871
   563
  have "(setsum (%j. (j * p) div q) Q_set) + 
nipkow@15392
   564
      (setsum (%j. (j * q) div p) P_set) = int (card S2) + int (card S1)"
paulson@13871
   565
    by (auto simp add: S1_carda S2_carda)
nipkow@15392
   566
  also have "... = int (card S1) + int (card S2)"
paulson@13871
   567
    by auto
nipkow@15392
   568
  also have "... = ((p - 1) div 2) * ((q - 1) div 2)"
paulson@13871
   569
    by (auto simp add: card_sum_S1_S2)
nipkow@15392
   570
  finally show ?thesis .
nipkow@15392
   571
qed
paulson@13871
   572
nipkow@15392
   573
lemma pq_prime_neq: "[| p \<in> zprime; q \<in> zprime; p \<noteq> q |] ==> (~[p = 0] (mod q))"
paulson@13871
   574
  apply (auto simp add: zcong_eq_zdvd_prop zprime_def)
paulson@13871
   575
  apply (drule_tac x = q in allE)
paulson@13871
   576
  apply (drule_tac x = p in allE)
paulson@13871
   577
by auto
paulson@13871
   578
paulson@13871
   579
lemma (in QRTEMP) QR_short: "(Legendre p q) * (Legendre q p) = 
nipkow@15392
   580
    (-1::int)^nat(((p - 1) div 2)*((q - 1) div 2))"
nipkow@15392
   581
proof -
nipkow@15392
   582
  from prems have "~([p = 0] (mod q))"
paulson@13871
   583
    by (auto simp add: pq_prime_neq QRTEMP_def)
paulson@13871
   584
  with prems have a1: "(Legendre p q) = (-1::int) ^ 
nipkow@15392
   585
      nat(setsum (%x. ((x * p) div q)) Q_set)"
paulson@13871
   586
    apply (rule_tac p = q in  MainQRLemma)
paulson@13871
   587
    by (auto simp add: zprime_zOdd_eq_grt_2 QRTEMP_def)
nipkow@15392
   588
  from prems have "~([q = 0] (mod p))"
paulson@13871
   589
    apply (rule_tac p = q and q = p in pq_prime_neq)
nipkow@15392
   590
    apply (simp add: QRTEMP_def)+
paulson@13871
   591
    by arith
paulson@13871
   592
  with prems have a2: "(Legendre q p) = 
nipkow@15392
   593
      (-1::int) ^ nat(setsum (%x. ((x * q) div p)) P_set)"
paulson@13871
   594
    apply (rule_tac p = p in  MainQRLemma)
paulson@13871
   595
    by (auto simp add: zprime_zOdd_eq_grt_2 QRTEMP_def)
paulson@13871
   596
  from a1 a2 have "(Legendre p q) * (Legendre q p) = 
paulson@13871
   597
      (-1::int) ^ nat(setsum (%x. ((x * p) div q)) Q_set) *
nipkow@15392
   598
        (-1::int) ^ nat(setsum (%x. ((x * q) div p)) P_set)"
paulson@13871
   599
    by auto
paulson@13871
   600
  also have "... = (-1::int) ^ (nat(setsum (%x. ((x * p) div q)) Q_set) + 
nipkow@15392
   601
                   nat(setsum (%x. ((x * q) div p)) P_set))"
paulson@13871
   602
    by (auto simp add: zpower_zadd_distrib)
paulson@13871
   603
  also have "nat(setsum (%x. ((x * p) div q)) Q_set) + 
paulson@13871
   604
      nat(setsum (%x. ((x * q) div p)) P_set) =
paulson@13871
   605
        nat((setsum (%x. ((x * p) div q)) Q_set) + 
nipkow@15392
   606
          (setsum (%x. ((x * q) div p)) P_set))"
paulson@13871
   607
    apply (rule_tac z1 = "setsum (%x. ((x * p) div q)) Q_set" in 
nipkow@15392
   608
      nat_add_distrib [THEN sym])
paulson@13871
   609
    by (auto simp add: S1_carda [THEN sym] S2_carda [THEN sym])
nipkow@15392
   610
  also have "... = nat(((p - 1) div 2) * ((q - 1) div 2))"
paulson@13871
   611
    by (auto simp add: pq_sum_prop)
nipkow@15392
   612
  finally show ?thesis .
nipkow@15392
   613
qed
paulson@13871
   614
paulson@13871
   615
theorem Quadratic_Reciprocity:
paulson@13871
   616
     "[| p \<in> zOdd; p \<in> zprime; q \<in> zOdd; q \<in> zprime; 
paulson@13871
   617
         p \<noteq> q |] 
paulson@13871
   618
      ==> (Legendre p q) * (Legendre q p) = 
nipkow@15392
   619
          (-1::int)^nat(((p - 1) div 2)*((q - 1) div 2))"
paulson@13871
   620
  by (auto simp add: QRTEMP.QR_short zprime_zOdd_eq_grt_2 [THEN sym] 
paulson@13871
   621
                     QRTEMP_def)
paulson@13871
   622
paulson@13871
   623
end