src/HOL/Old_Number_Theory/Euler.thy
 author paulson Fri Nov 13 12:27:13 2015 +0000 (2015-11-13) changeset 61649 268d88ec9087 parent 61382 efac889fccbc child 61952 546958347e05 permissions -rw-r--r--
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
```     1 (*  Title:      HOL/Old_Number_Theory/Euler.thy
```
```     2     Authors:    Jeremy Avigad, David Gray, and Adam Kramer
```
```     3 *)
```
```     4
```
```     5 section \<open>Euler's criterion\<close>
```
```     6
```
```     7 theory Euler
```
```     8 imports Residues EvenOdd
```
```     9 begin
```
```    10
```
```    11 definition MultInvPair :: "int => int => int => int set"
```
```    12   where "MultInvPair a p j = {StandardRes p j, StandardRes p (a * (MultInv p j))}"
```
```    13
```
```    14 definition SetS :: "int => int => int set set"
```
```    15   where "SetS a p = MultInvPair a p ` SRStar p"
```
```    16
```
```    17
```
```    18 subsection \<open>Property for MultInvPair\<close>
```
```    19
```
```    20 lemma MultInvPair_prop1a:
```
```    21   "[| zprime p; 2 < p; ~([a = 0](mod p));
```
```    22       X \<in> (SetS a p); Y \<in> (SetS a p);
```
```    23       ~((X \<inter> Y) = {}) |] ==> X = Y"
```
```    24   apply (auto simp add: SetS_def)
```
```    25   apply (drule StandardRes_SRStar_prop1a)+ defer 1
```
```    26   apply (drule StandardRes_SRStar_prop1a)+
```
```    27   apply (auto simp add: MultInvPair_def StandardRes_prop2 zcong_sym)
```
```    28   apply (drule notE, rule MultInv_zcong_prop1, auto)[]
```
```    29   apply (drule notE, rule MultInv_zcong_prop2, auto simp add: zcong_sym)[]
```
```    30   apply (drule MultInv_zcong_prop2, auto simp add: zcong_sym)[]
```
```    31   apply (drule MultInv_zcong_prop3, auto simp add: zcong_sym)[]
```
```    32   apply (drule MultInv_zcong_prop1, auto)[]
```
```    33   apply (drule MultInv_zcong_prop2, auto simp add: zcong_sym)[]
```
```    34   apply (drule MultInv_zcong_prop2, auto simp add: zcong_sym)[]
```
```    35   apply (drule MultInv_zcong_prop3, auto simp add: zcong_sym)[]
```
```    36   done
```
```    37
```
```    38 lemma MultInvPair_prop1b:
```
```    39   "[| zprime p; 2 < p; ~([a = 0](mod p));
```
```    40       X \<in> (SetS a p); Y \<in> (SetS a p);
```
```    41       X \<noteq> Y |] ==> X \<inter> Y = {}"
```
```    42   apply (rule notnotD)
```
```    43   apply (rule notI)
```
```    44   apply (drule MultInvPair_prop1a, auto)
```
```    45   done
```
```    46
```
```    47 lemma MultInvPair_prop1c: "[| zprime p; 2 < p; ~([a = 0](mod p)) |] ==>
```
```    48     \<forall>X \<in> SetS a p. \<forall>Y \<in> SetS a p. X \<noteq> Y --> X\<inter>Y = {}"
```
```    49   by (auto simp add: MultInvPair_prop1b)
```
```    50
```
```    51 lemma MultInvPair_prop2: "[| zprime p; 2 < p; ~([a = 0](mod p)) |] ==>
```
```    52                           Union ( SetS a p) = SRStar p"
```
```    53   apply (auto simp add: SetS_def MultInvPair_def StandardRes_SRStar_prop4
```
```    54     SRStar_mult_prop2)
```
```    55   apply (frule StandardRes_SRStar_prop3)
```
```    56   apply (rule bexI, auto)
```
```    57   done
```
```    58
```
```    59 lemma MultInvPair_distinct:
```
```    60   assumes "zprime p" and "2 < p" and
```
```    61     "~([a = 0] (mod p))" and
```
```    62     "~([j = 0] (mod p))" and
```
```    63     "~(QuadRes p a)"
```
```    64   shows "~([j = a * MultInv p j] (mod p))"
```
```    65 proof
```
```    66   assume "[j = a * MultInv p j] (mod p)"
```
```    67   then have "[j * j = (a * MultInv p j) * j] (mod p)"
```
```    68     by (auto simp add: zcong_scalar)
```
```    69   then have a:"[j * j = a * (MultInv p j * j)] (mod p)"
```
```    70     by (auto simp add: ac_simps)
```
```    71   have "[j * j = a] (mod p)"
```
```    72   proof -
```
```    73     from assms(1,2,4) have "[MultInv p j * j = 1] (mod p)"
```
```    74       by (simp add: MultInv_prop2a)
```
```    75     from this and a show ?thesis
```
```    76       by (auto simp add: zcong_zmult_prop2)
```
```    77   qed
```
```    78   then have "[j\<^sup>2 = a] (mod p)" by (simp add: power2_eq_square)
```
```    79   with assms show False by (simp add: QuadRes_def)
```
```    80 qed
```
```    81
```
```    82 lemma MultInvPair_card_two: "[| zprime p; 2 < p; ~([a = 0] (mod p));
```
```    83                                 ~(QuadRes p a); ~([j = 0] (mod p)) |]  ==>
```
```    84                              card (MultInvPair a p j) = 2"
```
```    85   apply (auto simp add: MultInvPair_def)
```
```    86   apply (subgoal_tac "~ (StandardRes p j = StandardRes p (a * MultInv p j))")
```
```    87   apply auto
```
```    88   apply (metis MultInvPair_distinct StandardRes_def aux)
```
```    89   done
```
```    90
```
```    91
```
```    92 subsection \<open>Properties of SetS\<close>
```
```    93
```
```    94 lemma SetS_finite: "2 < p ==> finite (SetS a p)"
```
```    95   by (auto simp add: SetS_def SRStar_finite [of p])
```
```    96
```
```    97 lemma SetS_elems_finite: "\<forall>X \<in> SetS a p. finite X"
```
```    98   by (auto simp add: SetS_def MultInvPair_def)
```
```    99
```
```   100 lemma SetS_elems_card: "[| zprime p; 2 < p; ~([a = 0] (mod p));
```
```   101                         ~(QuadRes p a) |]  ==>
```
```   102                         \<forall>X \<in> SetS a p. card X = 2"
```
```   103   apply (auto simp add: SetS_def)
```
```   104   apply (frule StandardRes_SRStar_prop1a)
```
```   105   apply (rule MultInvPair_card_two, auto)
```
```   106   done
```
```   107
```
```   108 lemma Union_SetS_finite: "2 < p ==> finite (Union (SetS a p))"
```
```   109   by (auto simp add: SetS_finite SetS_elems_finite)
```
```   110
```
```   111 lemma card_setsum_aux: "[| finite S; \<forall>X \<in> S. finite (X::int set);
```
```   112     \<forall>X \<in> S. card X = n |] ==> setsum card S = setsum (%x. n) S"
```
```   113   by (induct set: finite) auto
```
```   114
```
```   115 lemma SetS_card:
```
```   116   assumes "zprime p" and "2 < p" and "~([a = 0] (mod p))" and "~(QuadRes p a)"
```
```   117   shows "int(card(SetS a p)) = (p - 1) div 2"
```
```   118 proof -
```
```   119   have "(p - 1) = 2 * int(card(SetS a p))"
```
```   120   proof -
```
```   121     have "p - 1 = int(card(Union (SetS a p)))"
```
```   122       by (auto simp add: assms MultInvPair_prop2 SRStar_card)
```
```   123     also have "... = int (setsum card (SetS a p))"
```
```   124       by (auto simp add: assms SetS_finite SetS_elems_finite
```
```   125         MultInvPair_prop1c [of p a] card_Union_disjoint)
```
```   126     also have "... = int(setsum (%x.2) (SetS a p))"
```
```   127       using assms by (auto simp add: SetS_elems_card SetS_finite SetS_elems_finite
```
```   128         card_setsum_aux simp del: setsum_constant)
```
```   129     also have "... = 2 * int(card( SetS a p))"
```
```   130       by (auto simp add: assms SetS_finite setsum_const2)
```
```   131     finally show ?thesis .
```
```   132   qed
```
```   133   then show ?thesis by auto
```
```   134 qed
```
```   135
```
```   136 lemma SetS_setprod_prop: "[| zprime p; 2 < p; ~([a = 0] (mod p));
```
```   137                               ~(QuadRes p a); x \<in> (SetS a p) |] ==>
```
```   138                           [\<Prod>x = a] (mod p)"
```
```   139   apply (auto simp add: SetS_def MultInvPair_def)
```
```   140   apply (frule StandardRes_SRStar_prop1a)
```
```   141   apply hypsubst_thin
```
```   142   apply (subgoal_tac "StandardRes p x \<noteq> StandardRes p (a * MultInv p x)")
```
```   143   apply (auto simp add: StandardRes_prop2 MultInvPair_distinct)
```
```   144   apply (frule_tac m = p and x = x and y = "(a * MultInv p x)" in
```
```   145     StandardRes_prop4)
```
```   146   apply (subgoal_tac "[x * (a * MultInv p x) = a * (x * MultInv p x)] (mod p)")
```
```   147   apply (drule_tac a = "StandardRes p x * StandardRes p (a * MultInv p x)" and
```
```   148                    b = "x * (a * MultInv p x)" and
```
```   149                    c = "a * (x * MultInv p x)" in  zcong_trans, force)
```
```   150   apply (frule_tac p = p and x = x in MultInv_prop2, auto)
```
```   151 apply (metis StandardRes_SRStar_prop3 mult_1_right mult.commute zcong_sym zcong_zmult_prop1)
```
```   152   apply (auto simp add: ac_simps)
```
```   153   done
```
```   154
```
```   155 lemma aux1: "[| 0 < x; (x::int) < a; x \<noteq> (a - 1) |] ==> x < a - 1"
```
```   156   by arith
```
```   157
```
```   158 lemma aux2: "[| (a::int) < c; b < c |] ==> (a \<le> b | b \<le> a)"
```
```   159   by auto
```
```   160
```
```   161 lemma d22set_induct_old: "(\<And>a::int. 1 < a \<longrightarrow> P (a - 1) \<Longrightarrow> P a) \<Longrightarrow> P x"
```
```   162 using d22set.induct by blast
```
```   163
```
```   164 lemma SRStar_d22set_prop: "2 < p \<Longrightarrow> (SRStar p) = {1} \<union> (d22set (p - 1))"
```
```   165   apply (induct p rule: d22set_induct_old)
```
```   166   apply auto
```
```   167   apply (simp add: SRStar_def d22set.simps)
```
```   168   apply (simp add: SRStar_def d22set.simps, clarify)
```
```   169   apply (frule aux1)
```
```   170   apply (frule aux2, auto)
```
```   171   apply (simp_all add: SRStar_def)
```
```   172   apply (simp add: d22set.simps)
```
```   173   apply (frule d22set_le)
```
```   174   apply (frule d22set_g_1, auto)
```
```   175   done
```
```   176
```
```   177 lemma Union_SetS_setprod_prop1:
```
```   178   assumes "zprime p" and "2 < p" and "~([a = 0] (mod p))" and
```
```   179     "~(QuadRes p a)"
```
```   180   shows "[\<Prod>(Union (SetS a p)) = a ^ nat ((p - 1) div 2)] (mod p)"
```
```   181 proof -
```
```   182   from assms have "[\<Prod>(Union (SetS a p)) = setprod (setprod (%x. x)) (SetS a p)] (mod p)"
```
```   183     by (auto simp add: SetS_finite SetS_elems_finite
```
```   184       MultInvPair_prop1c setprod.Union_disjoint)
```
```   185   also have "[setprod (setprod (%x. x)) (SetS a p) =
```
```   186       setprod (%x. a) (SetS a p)] (mod p)"
```
```   187     by (rule setprod_same_function_zcong)
```
```   188       (auto simp add: assms SetS_setprod_prop SetS_finite)
```
```   189   also (zcong_trans) have "[setprod (%x. a) (SetS a p) =
```
```   190       a^(card (SetS a p))] (mod p)"
```
```   191     by (auto simp add: assms SetS_finite setprod_constant)
```
```   192   finally (zcong_trans) show ?thesis
```
```   193     apply (rule zcong_trans)
```
```   194     apply (subgoal_tac "card(SetS a p) = nat((p - 1) div 2)", auto)
```
```   195     apply (subgoal_tac "nat(int(card(SetS a p))) = nat((p - 1) div 2)", force)
```
```   196     apply (auto simp add: assms SetS_card)
```
```   197     done
```
```   198 qed
```
```   199
```
```   200 lemma Union_SetS_setprod_prop2:
```
```   201   assumes "zprime p" and "2 < p" and "~([a = 0](mod p))"
```
```   202   shows "\<Prod>(Union (SetS a p)) = zfact (p - 1)"
```
```   203 proof -
```
```   204   from assms have "\<Prod>(Union (SetS a p)) = \<Prod>(SRStar p)"
```
```   205     by (auto simp add: MultInvPair_prop2)
```
```   206   also have "... = \<Prod>({1} \<union> (d22set (p - 1)))"
```
```   207     by (auto simp add: assms SRStar_d22set_prop)
```
```   208   also have "... = zfact(p - 1)"
```
```   209   proof -
```
```   210     have "~(1 \<in> d22set (p - 1)) & finite( d22set (p - 1))"
```
```   211       by (metis d22set_fin d22set_g_1 linorder_neq_iff)
```
```   212     then have "\<Prod>({1} \<union> (d22set (p - 1))) = \<Prod>(d22set (p - 1))"
```
```   213       by auto
```
```   214     then show ?thesis
```
```   215       by (auto simp add: d22set_prod_zfact)
```
```   216   qed
```
```   217   finally show ?thesis .
```
```   218 qed
```
```   219
```
```   220 lemma zfact_prop: "[| zprime p; 2 < p; ~([a = 0] (mod p)); ~(QuadRes p a) |] ==>
```
```   221                    [zfact (p - 1) = a ^ nat ((p - 1) div 2)] (mod p)"
```
```   222   apply (frule Union_SetS_setprod_prop1)
```
```   223   apply (auto simp add: Union_SetS_setprod_prop2)
```
```   224   done
```
```   225
```
```   226 text \<open>\medskip Prove the first part of Euler's Criterion:\<close>
```
```   227
```
```   228 lemma Euler_part1: "[| 2 < p; zprime p; ~([x = 0](mod p));
```
```   229     ~(QuadRes p x) |] ==>
```
```   230       [x^(nat (((p) - 1) div 2)) = -1](mod p)"
```
```   231   by (metis Wilson_Russ zcong_sym zcong_trans zfact_prop)
```
```   232
```
```   233 text \<open>\medskip Prove another part of Euler Criterion:\<close>
```
```   234
```
```   235 lemma aux_1: "0 < p ==> (a::int) ^ nat (p) = a * a ^ (nat (p) - 1)"
```
```   236 proof -
```
```   237   assume "0 < p"
```
```   238   then have "a ^ (nat p) =  a ^ (1 + (nat p - 1))"
```
```   239     by (auto simp add: diff_add_assoc)
```
```   240   also have "... = (a ^ 1) * a ^ (nat(p) - 1)"
```
```   241     by (simp only: power_add)
```
```   242   also have "... = a * a ^ (nat(p) - 1)"
```
```   243     by auto
```
```   244   finally show ?thesis .
```
```   245 qed
```
```   246
```
```   247 lemma aux_2: "[| (2::int) < p; p \<in> zOdd |] ==> 0 < ((p - 1) div 2)"
```
```   248 proof -
```
```   249   assume "2 < p" and "p \<in> zOdd"
```
```   250   then have "(p - 1):zEven"
```
```   251     by (auto simp add: zEven_def zOdd_def)
```
```   252   then have aux_1: "2 * ((p - 1) div 2) = (p - 1)"
```
```   253     by (auto simp add: even_div_2_prop2)
```
```   254   with \<open>2 < p\<close> have "1 < (p - 1)"
```
```   255     by auto
```
```   256   then have " 1 < (2 * ((p - 1) div 2))"
```
```   257     by (auto simp add: aux_1)
```
```   258   then have "0 < (2 * ((p - 1) div 2)) div 2"
```
```   259     by auto
```
```   260   then show ?thesis by auto
```
```   261 qed
```
```   262
```
```   263 lemma Euler_part2:
```
```   264     "[| 2 < p; zprime p; [a = 0] (mod p) |] ==> [0 = a ^ nat ((p - 1) div 2)] (mod p)"
```
```   265   apply (frule zprime_zOdd_eq_grt_2)
```
```   266   apply (frule aux_2, auto)
```
```   267   apply (frule_tac a = a in aux_1, auto)
```
```   268   apply (frule zcong_zmult_prop1, auto)
```
```   269   done
```
```   270
```
```   271 text \<open>\medskip Prove the final part of Euler's Criterion:\<close>
```
```   272
```
```   273 lemma aux__1: "[| ~([x = 0] (mod p)); [y\<^sup>2 = x] (mod p)|] ==> ~(p dvd y)"
```
```   274   by (metis dvdI power2_eq_square zcong_sym zcong_trans zcong_zero_equiv_div dvd_trans)
```
```   275
```
```   276 lemma aux__2: "2 * nat((p - 1) div 2) =  nat (2 * ((p - 1) div 2))"
```
```   277   by (auto simp add: nat_mult_distrib)
```
```   278
```
```   279 lemma Euler_part3: "[| 2 < p; zprime p; ~([x = 0](mod p)); QuadRes p x |] ==>
```
```   280                       [x^(nat (((p) - 1) div 2)) = 1](mod p)"
```
```   281   apply (subgoal_tac "p \<in> zOdd")
```
```   282   apply (auto simp add: QuadRes_def)
```
```   283    prefer 2
```
```   284    apply (metis zprime_zOdd_eq_grt_2)
```
```   285   apply (frule aux__1, auto)
```
```   286   apply (drule_tac z = "nat ((p - 1) div 2)" in zcong_zpower)
```
```   287   apply (auto simp add: power_mult [symmetric])
```
```   288   apply (rule zcong_trans)
```
```   289   apply (auto simp add: zcong_sym [of "x ^ nat ((p - 1) div 2)"])
```
```   290   apply (metis Little_Fermat even_div_2_prop2 odd_minus_one_even mult_1 aux__2)
```
```   291   done
```
```   292
```
```   293
```
```   294 text \<open>\medskip Finally show Euler's Criterion:\<close>
```
```   295
```
```   296 theorem Euler_Criterion: "[| 2 < p; zprime p |] ==> [(Legendre a p) =
```
```   297     a^(nat (((p) - 1) div 2))] (mod p)"
```
```   298   apply (auto simp add: Legendre_def Euler_part2)
```
```   299   apply (frule Euler_part3, auto simp add: zcong_sym)[]
```
```   300   apply (frule Euler_part1, auto simp add: zcong_sym)[]
```
```   301   done
```
```   302
```
```   303 end
```