src/HOL/Old_Number_Theory/Euler.thy
 author haftmann Tue Sep 01 15:39:33 2009 +0200 (2009-09-01) changeset 32479 521cc9bf2958 parent 30042 src/HOL/NumberTheory/Euler.thy@31039ee583fa child 35544 342a448ae141 permissions -rw-r--r--
some reorganization of number theory
```     1 (*  Title:      HOL/Quadratic_Reciprocity/Euler.thy
```
```     2     ID:         \$Id\$
```
```     3     Authors:    Jeremy Avigad, David Gray, and Adam Kramer
```
```     4 *)
```
```     5
```
```     6 header {* Euler's criterion *}
```
```     7
```
```     8 theory Euler imports Residues EvenOdd begin
```
```     9
```
```    10 definition
```
```    11   MultInvPair :: "int => int => int => int set" where
```
```    12   "MultInvPair a p j = {StandardRes p j, StandardRes p (a * (MultInv p j))}"
```
```    13
```
```    14 definition
```
```    15   SetS        :: "int => int => int set set" where
```
```    16   "SetS        a p   =  (MultInvPair a p ` SRStar p)"
```
```    17
```
```    18
```
```    19 subsection {* Property for MultInvPair *}
```
```    20
```
```    21 lemma MultInvPair_prop1a:
```
```    22   "[| zprime p; 2 < p; ~([a = 0](mod p));
```
```    23       X \<in> (SetS a p); Y \<in> (SetS a p);
```
```    24       ~((X \<inter> Y) = {}) |] ==> X = Y"
```
```    25   apply (auto simp add: SetS_def)
```
```    26   apply (drule StandardRes_SRStar_prop1a)+ defer 1
```
```    27   apply (drule StandardRes_SRStar_prop1a)+
```
```    28   apply (auto simp add: MultInvPair_def StandardRes_prop2 zcong_sym)
```
```    29   apply (drule notE, rule MultInv_zcong_prop1, auto)[]
```
```    30   apply (drule notE, rule MultInv_zcong_prop2, auto simp add: zcong_sym)[]
```
```    31   apply (drule MultInv_zcong_prop2, auto simp add: zcong_sym)[]
```
```    32   apply (drule MultInv_zcong_prop3, auto simp add: zcong_sym)[]
```
```    33   apply (drule MultInv_zcong_prop1, auto)[]
```
```    34   apply (drule MultInv_zcong_prop2, auto simp add: zcong_sym)[]
```
```    35   apply (drule MultInv_zcong_prop2, auto simp add: zcong_sym)[]
```
```    36   apply (drule MultInv_zcong_prop3, auto simp add: zcong_sym)[]
```
```    37   done
```
```    38
```
```    39 lemma MultInvPair_prop1b:
```
```    40   "[| zprime p; 2 < p; ~([a = 0](mod p));
```
```    41       X \<in> (SetS a p); Y \<in> (SetS a p);
```
```    42       X \<noteq> Y |] ==> X \<inter> Y = {}"
```
```    43   apply (rule notnotD)
```
```    44   apply (rule notI)
```
```    45   apply (drule MultInvPair_prop1a, auto)
```
```    46   done
```
```    47
```
```    48 lemma MultInvPair_prop1c: "[| zprime p; 2 < p; ~([a = 0](mod p)) |] ==>
```
```    49     \<forall>X \<in> SetS a p. \<forall>Y \<in> SetS a p. X \<noteq> Y --> X\<inter>Y = {}"
```
```    50   by (auto simp add: MultInvPair_prop1b)
```
```    51
```
```    52 lemma MultInvPair_prop2: "[| zprime p; 2 < p; ~([a = 0](mod p)) |] ==>
```
```    53                           Union ( SetS a p) = SRStar p"
```
```    54   apply (auto simp add: SetS_def MultInvPair_def StandardRes_SRStar_prop4
```
```    55     SRStar_mult_prop2)
```
```    56   apply (frule StandardRes_SRStar_prop3)
```
```    57   apply (rule bexI, auto)
```
```    58   done
```
```    59
```
```    60 lemma MultInvPair_distinct: "[| zprime p; 2 < p; ~([a = 0] (mod p));
```
```    61                                 ~([j = 0] (mod p));
```
```    62                                 ~(QuadRes p a) |]  ==>
```
```    63                              ~([j = a * MultInv p j] (mod p))"
```
```    64 proof
```
```    65   assume "zprime p" and "2 < p" and "~([a = 0] (mod p))" and
```
```    66     "~([j = 0] (mod p))" and "~(QuadRes p a)"
```
```    67   assume "[j = a * MultInv p j] (mod p)"
```
```    68   then have "[j * j = (a * MultInv p j) * j] (mod p)"
```
```    69     by (auto simp add: zcong_scalar)
```
```    70   then have a:"[j * j = a * (MultInv p j * j)] (mod p)"
```
```    71     by (auto simp add: zmult_ac)
```
```    72   have "[j * j = a] (mod p)"
```
```    73     proof -
```
```    74       from prems have b: "[MultInv p j * j = 1] (mod p)"
```
```    75         by (simp add: MultInv_prop2a)
```
```    76       from b a show ?thesis
```
```    77         by (auto simp add: zcong_zmult_prop2)
```
```    78     qed
```
```    79   then have "[j^2 = a] (mod p)"
```
```    80     by (metis  number_of_is_id power2_eq_square succ_bin_simps)
```
```    81   with prems show False
```
```    82     by (simp add: QuadRes_def)
```
```    83 qed
```
```    84
```
```    85 lemma MultInvPair_card_two: "[| zprime p; 2 < p; ~([a = 0] (mod p));
```
```    86                                 ~(QuadRes p a); ~([j = 0] (mod p)) |]  ==>
```
```    87                              card (MultInvPair a p j) = 2"
```
```    88   apply (auto simp add: MultInvPair_def)
```
```    89   apply (subgoal_tac "~ (StandardRes p j = StandardRes p (a * MultInv p j))")
```
```    90   apply auto
```
```    91   apply (metis MultInvPair_distinct Pls_def StandardRes_def aux number_of_is_id one_is_num_one)
```
```    92   done
```
```    93
```
```    94
```
```    95 subsection {* Properties of SetS *}
```
```    96
```
```    97 lemma SetS_finite: "2 < p ==> finite (SetS a p)"
```
```    98   by (auto simp add: SetS_def SRStar_finite [of p] finite_imageI)
```
```    99
```
```   100 lemma SetS_elems_finite: "\<forall>X \<in> SetS a p. finite X"
```
```   101   by (auto simp add: SetS_def MultInvPair_def)
```
```   102
```
```   103 lemma SetS_elems_card: "[| zprime p; 2 < p; ~([a = 0] (mod p));
```
```   104                         ~(QuadRes p a) |]  ==>
```
```   105                         \<forall>X \<in> SetS a p. card X = 2"
```
```   106   apply (auto simp add: SetS_def)
```
```   107   apply (frule StandardRes_SRStar_prop1a)
```
```   108   apply (rule MultInvPair_card_two, auto)
```
```   109   done
```
```   110
```
```   111 lemma Union_SetS_finite: "2 < p ==> finite (Union (SetS a p))"
```
```   112   by (auto simp add: SetS_finite SetS_elems_finite finite_Union)
```
```   113
```
```   114 lemma card_setsum_aux: "[| finite S; \<forall>X \<in> S. finite (X::int set);
```
```   115     \<forall>X \<in> S. card X = n |] ==> setsum card S = setsum (%x. n) S"
```
```   116   by (induct set: finite) auto
```
```   117
```
```   118 lemma SetS_card: "[| zprime p; 2 < p; ~([a = 0] (mod p)); ~(QuadRes p a) |] ==>
```
```   119                   int(card(SetS a p)) = (p - 1) div 2"
```
```   120 proof -
```
```   121   assume "zprime p" and "2 < p" and  "~([a = 0] (mod p))" and "~(QuadRes p a)"
```
```   122   then have "(p - 1) = 2 * int(card(SetS a p))"
```
```   123   proof -
```
```   124     have "p - 1 = int(card(Union (SetS a p)))"
```
```   125       by (auto simp add: prems MultInvPair_prop2 SRStar_card)
```
```   126     also have "... = int (setsum card (SetS a p))"
```
```   127       by (auto simp add: prems SetS_finite SetS_elems_finite
```
```   128                          MultInvPair_prop1c [of p a] card_Union_disjoint)
```
```   129     also have "... = int(setsum (%x.2) (SetS a p))"
```
```   130       using prems
```
```   131       by (auto simp add: SetS_elems_card SetS_finite SetS_elems_finite
```
```   132         card_setsum_aux simp del: setsum_constant)
```
```   133     also have "... = 2 * int(card( SetS a p))"
```
```   134       by (auto simp add: prems SetS_finite setsum_const2)
```
```   135     finally show ?thesis .
```
```   136   qed
```
```   137   from this show ?thesis
```
```   138     by auto
```
```   139 qed
```
```   140
```
```   141 lemma SetS_setprod_prop: "[| zprime p; 2 < p; ~([a = 0] (mod p));
```
```   142                               ~(QuadRes p a); x \<in> (SetS a p) |] ==>
```
```   143                           [\<Prod>x = a] (mod p)"
```
```   144   apply (auto simp add: SetS_def MultInvPair_def)
```
```   145   apply (frule StandardRes_SRStar_prop1a)
```
```   146   apply (subgoal_tac "StandardRes p x \<noteq> StandardRes p (a * MultInv p x)")
```
```   147   apply (auto simp add: StandardRes_prop2 MultInvPair_distinct)
```
```   148   apply (frule_tac m = p and x = x and y = "(a * MultInv p x)" in
```
```   149     StandardRes_prop4)
```
```   150   apply (subgoal_tac "[x * (a * MultInv p x) = a * (x * MultInv p x)] (mod p)")
```
```   151   apply (drule_tac a = "StandardRes p x * StandardRes p (a * MultInv p x)" and
```
```   152                    b = "x * (a * MultInv p x)" and
```
```   153                    c = "a * (x * MultInv p x)" in  zcong_trans, force)
```
```   154   apply (frule_tac p = p and x = x in MultInv_prop2, auto)
```
```   155 apply (metis StandardRes_SRStar_prop3 mult_1_right mult_commute zcong_sym zcong_zmult_prop1)
```
```   156   apply (auto simp add: zmult_ac)
```
```   157   done
```
```   158
```
```   159 lemma aux1: "[| 0 < x; (x::int) < a; x \<noteq> (a - 1) |] ==> x < a - 1"
```
```   160   by arith
```
```   161
```
```   162 lemma aux2: "[| (a::int) < c; b < c |] ==> (a \<le> b | b \<le> a)"
```
```   163   by auto
```
```   164
```
```   165 lemma SRStar_d22set_prop: "2 < p \<Longrightarrow> (SRStar p) = {1} \<union> (d22set (p - 1))"
```
```   166   apply (induct p rule: d22set.induct)
```
```   167   apply auto
```
```   168   apply (simp add: SRStar_def d22set.simps)
```
```   169   apply (simp add: SRStar_def d22set.simps, clarify)
```
```   170   apply (frule aux1)
```
```   171   apply (frule aux2, auto)
```
```   172   apply (simp_all add: SRStar_def)
```
```   173   apply (simp add: d22set.simps)
```
```   174   apply (frule d22set_le)
```
```   175   apply (frule d22set_g_1, auto)
```
```   176   done
```
```   177
```
```   178 lemma Union_SetS_setprod_prop1: "[| zprime p; 2 < p; ~([a = 0] (mod p)); ~(QuadRes p a) |] ==>
```
```   179                                  [\<Prod>(Union (SetS a p)) = a ^ nat ((p - 1) div 2)] (mod p)"
```
```   180 proof -
```
```   181   assume "zprime p" and "2 < p" and  "~([a = 0] (mod p))" and "~(QuadRes p a)"
```
```   182   then have "[\<Prod>(Union (SetS a p)) =
```
```   183       setprod (setprod (%x. x)) (SetS a p)] (mod p)"
```
```   184     by (auto simp add: SetS_finite SetS_elems_finite
```
```   185                        MultInvPair_prop1c setprod_Union_disjoint)
```
```   186   also have "[setprod (setprod (%x. x)) (SetS a p) =
```
```   187       setprod (%x. a) (SetS a p)] (mod p)"
```
```   188     by (rule setprod_same_function_zcong)
```
```   189       (auto simp add: prems SetS_setprod_prop SetS_finite)
```
```   190   also (zcong_trans) have "[setprod (%x. a) (SetS a p) =
```
```   191       a^(card (SetS a p))] (mod p)"
```
```   192     by (auto simp add: prems SetS_finite setprod_constant)
```
```   193   finally (zcong_trans) show ?thesis
```
```   194     apply (rule zcong_trans)
```
```   195     apply (subgoal_tac "card(SetS a p) = nat((p - 1) div 2)", auto)
```
```   196     apply (subgoal_tac "nat(int(card(SetS a p))) = nat((p - 1) div 2)", force)
```
```   197     apply (auto simp add: prems SetS_card)
```
```   198     done
```
```   199 qed
```
```   200
```
```   201 lemma Union_SetS_setprod_prop2: "[| zprime p; 2 < p; ~([a = 0](mod p)) |] ==>
```
```   202                                     \<Prod>(Union (SetS a p)) = zfact (p - 1)"
```
```   203 proof -
```
```   204   assume "zprime p" and "2 < p" and "~([a = 0](mod p))"
```
```   205   then have "\<Prod>(Union (SetS a p)) = \<Prod>(SRStar p)"
```
```   206     by (auto simp add: MultInvPair_prop2)
```
```   207   also have "... = \<Prod>({1} \<union> (d22set (p - 1)))"
```
```   208     by (auto simp add: prems SRStar_d22set_prop)
```
```   209   also have "... = zfact(p - 1)"
```
```   210   proof -
```
```   211     have "~(1 \<in> d22set (p - 1)) & finite( d22set (p - 1))"
```
```   212       by (metis d22set_fin d22set_g_1 linorder_neq_iff)
```
```   213     then have "\<Prod>({1} \<union> (d22set (p - 1))) = \<Prod>(d22set (p - 1))"
```
```   214       by auto
```
```   215     then show ?thesis
```
```   216       by (auto simp add: d22set_prod_zfact)
```
```   217   qed
```
```   218   finally show ?thesis .
```
```   219 qed
```
```   220
```
```   221 lemma zfact_prop: "[| zprime p; 2 < p; ~([a = 0] (mod p)); ~(QuadRes p a) |] ==>
```
```   222                    [zfact (p - 1) = a ^ nat ((p - 1) div 2)] (mod p)"
```
```   223   apply (frule Union_SetS_setprod_prop1)
```
```   224   apply (auto simp add: Union_SetS_setprod_prop2)
```
```   225   done
```
```   226
```
```   227 text {* \medskip Prove the first part of Euler's Criterion: *}
```
```   228
```
```   229 lemma Euler_part1: "[| 2 < p; zprime p; ~([x = 0](mod p));
```
```   230     ~(QuadRes p x) |] ==>
```
```   231       [x^(nat (((p) - 1) div 2)) = -1](mod p)"
```
```   232   by (metis Wilson_Russ number_of_is_id zcong_sym zcong_trans zfact_prop)
```
```   233
```
```   234 text {* \medskip Prove another part of Euler Criterion: *}
```
```   235
```
```   236 lemma aux_1: "0 < p ==> (a::int) ^ nat (p) = a * a ^ (nat (p) - 1)"
```
```   237 proof -
```
```   238   assume "0 < p"
```
```   239   then have "a ^ (nat p) =  a ^ (1 + (nat p - 1))"
```
```   240     by (auto simp add: diff_add_assoc)
```
```   241   also have "... = (a ^ 1) * a ^ (nat(p) - 1)"
```
```   242     by (simp only: zpower_zadd_distrib)
```
```   243   also have "... = a * a ^ (nat(p) - 1)"
```
```   244     by auto
```
```   245   finally show ?thesis .
```
```   246 qed
```
```   247
```
```   248 lemma aux_2: "[| (2::int) < p; p \<in> zOdd |] ==> 0 < ((p - 1) div 2)"
```
```   249 proof -
```
```   250   assume "2 < p" and "p \<in> zOdd"
```
```   251   then have "(p - 1):zEven"
```
```   252     by (auto simp add: zEven_def zOdd_def)
```
```   253   then have aux_1: "2 * ((p - 1) div 2) = (p - 1)"
```
```   254     by (auto simp add: even_div_2_prop2)
```
```   255   with `2 < p` have "1 < (p - 1)"
```
```   256     by auto
```
```   257   then have " 1 < (2 * ((p - 1) div 2))"
```
```   258     by (auto simp add: aux_1)
```
```   259   then have "0 < (2 * ((p - 1) div 2)) div 2"
```
```   260     by auto
```
```   261   then show ?thesis by auto
```
```   262 qed
```
```   263
```
```   264 lemma Euler_part2:
```
```   265     "[| 2 < p; zprime p; [a = 0] (mod p) |] ==> [0 = a ^ nat ((p - 1) div 2)] (mod p)"
```
```   266   apply (frule zprime_zOdd_eq_grt_2)
```
```   267   apply (frule aux_2, auto)
```
```   268   apply (frule_tac a = a in aux_1, auto)
```
```   269   apply (frule zcong_zmult_prop1, auto)
```
```   270   done
```
```   271
```
```   272 text {* \medskip Prove the final part of Euler's Criterion: *}
```
```   273
```
```   274 lemma aux__1: "[| ~([x = 0] (mod p)); [y ^ 2 = x] (mod p)|] ==> ~(p dvd y)"
```
```   275   by (metis dvdI power2_eq_square zcong_sym zcong_trans zcong_zero_equiv_div dvd_trans)
```
```   276
```
```   277 lemma aux__2: "2 * nat((p - 1) div 2) =  nat (2 * ((p - 1) div 2))"
```
```   278   by (auto simp add: nat_mult_distrib)
```
```   279
```
```   280 lemma Euler_part3: "[| 2 < p; zprime p; ~([x = 0](mod p)); QuadRes p x |] ==>
```
```   281                       [x^(nat (((p) - 1) div 2)) = 1](mod p)"
```
```   282   apply (subgoal_tac "p \<in> zOdd")
```
```   283   apply (auto simp add: QuadRes_def)
```
```   284    prefer 2
```
```   285    apply (metis number_of_is_id numeral_1_eq_1 zprime_zOdd_eq_grt_2)
```
```   286   apply (frule aux__1, auto)
```
```   287   apply (drule_tac z = "nat ((p - 1) div 2)" in zcong_zpower)
```
```   288   apply (auto simp add: zpower_zpower)
```
```   289   apply (rule zcong_trans)
```
```   290   apply (auto simp add: zcong_sym [of "x ^ nat ((p - 1) div 2)"])
```
```   291   apply (metis Little_Fermat even_div_2_prop2 mult_Bit0 number_of_is_id odd_minus_one_even one_is_num_one zmult_1 aux__2)
```
```   292   done
```
```   293
```
```   294
```
```   295 text {* \medskip Finally show Euler's Criterion: *}
```
```   296
```
```   297 theorem Euler_Criterion: "[| 2 < p; zprime p |] ==> [(Legendre a p) =
```
```   298     a^(nat (((p) - 1) div 2))] (mod p)"
```
```   299   apply (auto simp add: Legendre_def Euler_part2)
```
```   300   apply (frule Euler_part3, auto simp add: zcong_sym)[]
```
```   301   apply (frule Euler_part1, auto simp add: zcong_sym)[]
```
```   302   done
```
```   303
```
```   304 end
```