src/HOL/Number_Theory/Gauss.thy
 author haftmann Sun Oct 08 22:28:22 2017 +0200 (19 months ago) changeset 66817 0b12755ccbb2 parent 65435 378175f44328 child 66888 930abfdf8727 permissions -rw-r--r--
euclidean rings need no normalization
```     1 (*  Title:      HOL/Number_Theory/Gauss.thy
```
```     2     Authors:    Jeremy Avigad, David Gray, and Adam Kramer
```
```     3
```
```     4 Ported by lcp but unfinished.
```
```     5 *)
```
```     6
```
```     7 section \<open>Gauss' Lemma\<close>
```
```     8
```
```     9 theory Gauss
```
```    10   imports Euler_Criterion
```
```    11 begin
```
```    12
```
```    13 lemma cong_prime_prod_zero_nat:
```
```    14   "[a * b = 0] (mod p) \<Longrightarrow> prime p \<Longrightarrow> [a = 0] (mod p) \<or> [b = 0] (mod p)"
```
```    15   for a :: nat
```
```    16   by (auto simp add: cong_altdef_nat prime_dvd_mult_iff)
```
```    17
```
```    18 lemma cong_prime_prod_zero_int:
```
```    19   "[a * b = 0] (mod p) \<Longrightarrow> prime p \<Longrightarrow> [a = 0] (mod p) \<or> [b = 0] (mod p)"
```
```    20   for a :: int
```
```    21   by (auto simp add: cong_altdef_int prime_dvd_mult_iff)
```
```    22
```
```    23
```
```    24 locale GAUSS =
```
```    25   fixes p :: "nat"
```
```    26   fixes a :: "int"
```
```    27   assumes p_prime: "prime p"
```
```    28   assumes p_ge_2: "2 < p"
```
```    29   assumes p_a_relprime: "[a \<noteq> 0](mod p)"
```
```    30   assumes a_nonzero: "0 < a"
```
```    31 begin
```
```    32
```
```    33 definition "A = {0::int <.. ((int p - 1) div 2)}"
```
```    34 definition "B = (\<lambda>x. x * a) ` A"
```
```    35 definition "C = (\<lambda>x. x mod p) ` B"
```
```    36 definition "D = C \<inter> {.. (int p - 1) div 2}"
```
```    37 definition "E = C \<inter> {(int p - 1) div 2 <..}"
```
```    38 definition "F = (\<lambda>x. (int p - x)) ` E"
```
```    39
```
```    40
```
```    41 subsection \<open>Basic properties of p\<close>
```
```    42
```
```    43 lemma odd_p: "odd p"
```
```    44   by (metis p_prime p_ge_2 prime_odd_nat)
```
```    45
```
```    46 lemma p_minus_one_l: "(int p - 1) div 2 < p"
```
```    47 proof -
```
```    48   have "(p - 1) div 2 \<le> (p - 1) div 1"
```
```    49     by (metis div_by_1 div_le_dividend)
```
```    50   also have "\<dots> = p - 1" by simp
```
```    51   finally show ?thesis
```
```    52     using p_ge_2 by arith
```
```    53 qed
```
```    54
```
```    55 lemma p_eq2: "int p = (2 * ((int p - 1) div 2)) + 1"
```
```    56   using odd_p p_ge_2 nonzero_mult_div_cancel_left [of 2 "p - 1"] by simp
```
```    57
```
```    58 lemma p_odd_int: obtains z :: int where "int p = 2 * z + 1" "0 < z"
```
```    59 proof
```
```    60   let ?z = "(int p - 1) div 2"
```
```    61   show "int p = 2 * ?z + 1" by (rule p_eq2)
```
```    62   show "0 < ?z"
```
```    63     using p_ge_2 by linarith
```
```    64 qed
```
```    65
```
```    66
```
```    67 subsection \<open>Basic Properties of the Gauss Sets\<close>
```
```    68
```
```    69 lemma finite_A: "finite A"
```
```    70   by (auto simp add: A_def)
```
```    71
```
```    72 lemma finite_B: "finite B"
```
```    73   by (auto simp add: B_def finite_A)
```
```    74
```
```    75 lemma finite_C: "finite C"
```
```    76   by (auto simp add: C_def finite_B)
```
```    77
```
```    78 lemma finite_D: "finite D"
```
```    79   by (auto simp add: D_def finite_C)
```
```    80
```
```    81 lemma finite_E: "finite E"
```
```    82   by (auto simp add: E_def finite_C)
```
```    83
```
```    84 lemma finite_F: "finite F"
```
```    85   by (auto simp add: F_def finite_E)
```
```    86
```
```    87 lemma C_eq: "C = D \<union> E"
```
```    88   by (auto simp add: C_def D_def E_def)
```
```    89
```
```    90 lemma A_card_eq: "card A = nat ((int p - 1) div 2)"
```
```    91   by (auto simp add: A_def)
```
```    92
```
```    93 lemma inj_on_xa_A: "inj_on (\<lambda>x. x * a) A"
```
```    94   using a_nonzero by (simp add: A_def inj_on_def)
```
```    95
```
```    96 definition ResSet :: "int \<Rightarrow> int set \<Rightarrow> bool"
```
```    97   where "ResSet m X \<longleftrightarrow> (\<forall>y1 y2. y1 \<in> X \<and> y2 \<in> X \<and> [y1 = y2] (mod m) \<longrightarrow> y1 = y2)"
```
```    98
```
```    99 lemma ResSet_image:
```
```   100   "0 < m \<Longrightarrow> ResSet m A \<Longrightarrow> \<forall>x \<in> A. \<forall>y \<in> A. ([f x = f y](mod m) \<longrightarrow> x = y) \<Longrightarrow> ResSet m (f ` A)"
```
```   101   by (auto simp add: ResSet_def)
```
```   102
```
```   103 lemma A_res: "ResSet p A"
```
```   104   using p_ge_2 by (auto simp add: A_def ResSet_def intro!: cong_less_imp_eq_int)
```
```   105
```
```   106 lemma B_res: "ResSet p B"
```
```   107 proof -
```
```   108   have *: "x = y"
```
```   109     if a: "[x * a = y * a] (mod p)"
```
```   110     and b: "0 < x"
```
```   111     and c: "x \<le> (int p - 1) div 2"
```
```   112     and d: "0 < y"
```
```   113     and e: "y \<le> (int p - 1) div 2"
```
```   114     for x y
```
```   115   proof -
```
```   116     from p_a_relprime have "\<not> p dvd a"
```
```   117       by (simp add: cong_altdef_int)
```
```   118     with p_prime have "coprime a (int p)"
```
```   119       by (subst gcd.commute, intro prime_imp_coprime) auto
```
```   120     with a cong_mult_rcancel_int [of a "int p" x y] have "[x = y] (mod p)"
```
```   121       by simp
```
```   122     with cong_less_imp_eq_int [of x y p] p_minus_one_l
```
```   123       order_le_less_trans [of x "(int p - 1) div 2" p]
```
```   124       order_le_less_trans [of y "(int p - 1) div 2" p]
```
```   125     show ?thesis
```
```   126       by (metis b c cong_less_imp_eq_int d e zero_less_imp_eq_int of_nat_0_le_iff)
```
```   127   qed
```
```   128   show ?thesis
```
```   129     apply (insert p_ge_2 p_a_relprime p_minus_one_l)
```
```   130     apply (auto simp add: B_def)
```
```   131     apply (rule ResSet_image)
```
```   132       apply (auto simp add: A_res)
```
```   133     apply (auto simp add: A_def *)
```
```   134     done
```
```   135 qed
```
```   136
```
```   137 lemma SR_B_inj: "inj_on (\<lambda>x. x mod p) B"
```
```   138 proof -
```
```   139   have False
```
```   140     if a: "x * a mod p = y * a mod p"
```
```   141     and b: "0 < x"
```
```   142     and c: "x \<le> (int p - 1) div 2"
```
```   143     and d: "0 < y"
```
```   144     and e: "y \<le> (int p - 1) div 2"
```
```   145     and f: "x \<noteq> y"
```
```   146     for x y
```
```   147   proof -
```
```   148     from a have a': "[x * a = y * a](mod p)"
```
```   149       by (metis cong_int_def)
```
```   150     from p_a_relprime have "\<not>p dvd a"
```
```   151       by (simp add: cong_altdef_int)
```
```   152     with p_prime have "coprime a (int p)"
```
```   153       by (subst gcd.commute, intro prime_imp_coprime) auto
```
```   154     with a' cong_mult_rcancel_int [of a "int p" x y]
```
```   155     have "[x = y] (mod p)" by simp
```
```   156     with cong_less_imp_eq_int [of x y p] p_minus_one_l
```
```   157       order_le_less_trans [of x "(int p - 1) div 2" p]
```
```   158       order_le_less_trans [of y "(int p - 1) div 2" p]
```
```   159     have "x = y"
```
```   160       by (metis b c cong_less_imp_eq_int d e zero_less_imp_eq_int of_nat_0_le_iff)
```
```   161     then show ?thesis
```
```   162       by (simp add: f)
```
```   163   qed
```
```   164   then show ?thesis
```
```   165     by (auto simp add: B_def inj_on_def A_def) metis
```
```   166 qed
```
```   167
```
```   168 lemma inj_on_pminusx_E: "inj_on (\<lambda>x. p - x) E"
```
```   169   apply (auto simp add: E_def C_def B_def A_def)
```
```   170   apply (rule inj_on_inverseI [where g = "op - (int p)"])
```
```   171   apply auto
```
```   172   done
```
```   173
```
```   174 lemma nonzero_mod_p: "0 < x \<Longrightarrow> x < int p \<Longrightarrow> [x \<noteq> 0](mod p)"
```
```   175   for x :: int
```
```   176   by (simp add: cong_int_def)
```
```   177
```
```   178 lemma A_ncong_p: "x \<in> A \<Longrightarrow> [x \<noteq> 0](mod p)"
```
```   179   by (rule nonzero_mod_p) (auto simp add: A_def)
```
```   180
```
```   181 lemma A_greater_zero: "x \<in> A \<Longrightarrow> 0 < x"
```
```   182   by (auto simp add: A_def)
```
```   183
```
```   184 lemma B_ncong_p: "x \<in> B \<Longrightarrow> [x \<noteq> 0](mod p)"
```
```   185   by (auto simp: B_def p_prime p_a_relprime A_ncong_p dest: cong_prime_prod_zero_int)
```
```   186
```
```   187 lemma B_greater_zero: "x \<in> B \<Longrightarrow> 0 < x"
```
```   188   using a_nonzero by (auto simp add: B_def A_greater_zero)
```
```   189
```
```   190 lemma C_greater_zero: "y \<in> C \<Longrightarrow> 0 < y"
```
```   191 proof (auto simp add: C_def)
```
```   192   fix x :: int
```
```   193   assume x: "x \<in> B"
```
```   194   moreover from x have "x mod int p \<noteq> 0"
```
```   195     using B_ncong_p cong_int_def by simp
```
```   196   moreover have "int y = 0 \<or> 0 < int y" for y
```
```   197     by linarith
```
```   198   ultimately show "0 < x mod int p"
```
```   199     using B_greater_zero [of x]
```
```   200     by (auto simp add: mod_int_pos_iff intro: neq_le_trans)
```
```   201 qed
```
```   202
```
```   203 lemma F_subset: "F \<subseteq> {x. 0 < x \<and> x \<le> ((int p - 1) div 2)}"
```
```   204   apply (auto simp add: F_def E_def C_def)
```
```   205    apply (metis p_ge_2 Divides.pos_mod_bound nat_int zless_nat_conj)
```
```   206   apply (auto intro: p_odd_int)
```
```   207   done
```
```   208
```
```   209 lemma D_subset: "D \<subseteq> {x. 0 < x \<and> x \<le> ((p - 1) div 2)}"
```
```   210   by (auto simp add: D_def C_greater_zero)
```
```   211
```
```   212 lemma F_eq: "F = {x. \<exists>y \<in> A. (x = p - ((y * a) mod p) \<and> (int p - 1) div 2 < (y * a) mod p)}"
```
```   213   by (auto simp add: F_def E_def D_def C_def B_def A_def)
```
```   214
```
```   215 lemma D_eq: "D = {x. \<exists>y \<in> A. (x = (y * a) mod p \<and> (y * a) mod p \<le> (int p - 1) div 2)}"
```
```   216   by (auto simp add: D_def C_def B_def A_def)
```
```   217
```
```   218 lemma all_A_relprime:
```
```   219   assumes "x \<in> A"
```
```   220   shows "gcd x p = 1"
```
```   221   using p_prime A_ncong_p [OF assms]
```
```   222   by (auto simp: cong_altdef_int gcd.commute[of _ "int p"] intro!: prime_imp_coprime)
```
```   223
```
```   224 lemma A_prod_relprime: "gcd (prod id A) p = 1"
```
```   225   by (metis id_def all_A_relprime prod_coprime)
```
```   226
```
```   227
```
```   228 subsection \<open>Relationships Between Gauss Sets\<close>
```
```   229
```
```   230 lemma StandardRes_inj_on_ResSet: "ResSet m X \<Longrightarrow> inj_on (\<lambda>b. b mod m) X"
```
```   231   by (auto simp add: ResSet_def inj_on_def cong_int_def)
```
```   232
```
```   233 lemma B_card_eq_A: "card B = card A"
```
```   234   using finite_A by (simp add: finite_A B_def inj_on_xa_A card_image)
```
```   235
```
```   236 lemma B_card_eq: "card B = nat ((int p - 1) div 2)"
```
```   237   by (simp add: B_card_eq_A A_card_eq)
```
```   238
```
```   239 lemma F_card_eq_E: "card F = card E"
```
```   240   using finite_E by (simp add: F_def inj_on_pminusx_E card_image)
```
```   241
```
```   242 lemma C_card_eq_B: "card C = card B"
```
```   243 proof -
```
```   244   have "inj_on (\<lambda>x. x mod p) B"
```
```   245     by (metis SR_B_inj)
```
```   246   then show ?thesis
```
```   247     by (metis C_def card_image)
```
```   248 qed
```
```   249
```
```   250 lemma D_E_disj: "D \<inter> E = {}"
```
```   251   by (auto simp add: D_def E_def)
```
```   252
```
```   253 lemma C_card_eq_D_plus_E: "card C = card D + card E"
```
```   254   by (auto simp add: C_eq card_Un_disjoint D_E_disj finite_D finite_E)
```
```   255
```
```   256 lemma C_prod_eq_D_times_E: "prod id E * prod id D = prod id C"
```
```   257   by (metis C_eq D_E_disj finite_D finite_E inf_commute prod.union_disjoint sup_commute)
```
```   258
```
```   259 lemma C_B_zcong_prod: "[prod id C = prod id B] (mod p)"
```
```   260   apply (auto simp add: C_def)
```
```   261   apply (insert finite_B SR_B_inj)
```
```   262   apply (drule prod.reindex [of "\<lambda>x. x mod int p" B id])
```
```   263   apply auto
```
```   264   apply (rule cong_prod_int)
```
```   265   apply (auto simp add: cong_int_def)
```
```   266   done
```
```   267
```
```   268 lemma F_Un_D_subset: "(F \<union> D) \<subseteq> A"
```
```   269   by (intro Un_least subset_trans [OF F_subset] subset_trans [OF D_subset]) (auto simp: A_def)
```
```   270
```
```   271 lemma F_D_disj: "(F \<inter> D) = {}"
```
```   272 proof (auto simp add: F_eq D_eq)
```
```   273   fix y z :: int
```
```   274   assume "p - (y * a) mod p = (z * a) mod p"
```
```   275   then have "[(y * a) mod p + (z * a) mod p = 0] (mod p)"
```
```   276     by (metis add.commute diff_eq_eq dvd_refl cong_int_def dvd_eq_mod_eq_0 mod_0)
```
```   277   moreover have "[y * a = (y * a) mod p] (mod p)"
```
```   278     by (metis cong_int_def mod_mod_trivial)
```
```   279   ultimately have "[a * (y + z) = 0] (mod p)"
```
```   280     by (metis cong_int_def mod_add_left_eq mod_add_right_eq mult.commute ring_class.ring_distribs(1))
```
```   281   with p_prime a_nonzero p_a_relprime have a: "[y + z = 0] (mod p)"
```
```   282     by (auto dest!: cong_prime_prod_zero_int)
```
```   283   assume b: "y \<in> A" and c: "z \<in> A"
```
```   284   then have "0 < y + z"
```
```   285     by (auto simp: A_def)
```
```   286   moreover from b c p_eq2 have "y + z < p"
```
```   287     by (auto simp: A_def)
```
```   288   ultimately show False
```
```   289     by (metis a nonzero_mod_p)
```
```   290 qed
```
```   291
```
```   292 lemma F_Un_D_card: "card (F \<union> D) = nat ((p - 1) div 2)"
```
```   293 proof -
```
```   294   have "card (F \<union> D) = card E + card D"
```
```   295     by (auto simp add: finite_F finite_D F_D_disj card_Un_disjoint F_card_eq_E)
```
```   296   then have "card (F \<union> D) = card C"
```
```   297     by (simp add: C_card_eq_D_plus_E)
```
```   298   then show "card (F \<union> D) = nat ((p - 1) div 2)"
```
```   299     by (simp add: C_card_eq_B B_card_eq)
```
```   300 qed
```
```   301
```
```   302 lemma F_Un_D_eq_A: "F \<union> D = A"
```
```   303   using finite_A F_Un_D_subset A_card_eq F_Un_D_card by (auto simp add: card_seteq)
```
```   304
```
```   305 lemma prod_D_F_eq_prod_A: "prod id D * prod id F = prod id A"
```
```   306   by (metis F_D_disj F_Un_D_eq_A Int_commute Un_commute finite_D finite_F prod.union_disjoint)
```
```   307
```
```   308 lemma prod_F_zcong: "[prod id F = ((-1) ^ (card E)) * prod id E] (mod p)"
```
```   309 proof -
```
```   310   have FE: "prod id F = prod (op - p) E"
```
```   311     apply (auto simp add: F_def)
```
```   312     apply (insert finite_E inj_on_pminusx_E)
```
```   313     apply (drule prod.reindex)
```
```   314     apply auto
```
```   315     done
```
```   316   then have "\<forall>x \<in> E. [(p-x) mod p = - x](mod p)"
```
```   317     by (metis cong_int_def minus_mod_self1 mod_mod_trivial)
```
```   318   then have "[prod ((\<lambda>x. x mod p) o (op - p)) E = prod (uminus) E](mod p)"
```
```   319     using finite_E p_ge_2 cong_prod_int [of E "(\<lambda>x. x mod p) o (op - p)" uminus p]
```
```   320     by auto
```
```   321   then have two: "[prod id F = prod (uminus) E](mod p)"
```
```   322     by (metis FE cong_cong_mod_int cong_refl_int cong_prod_int minus_mod_self1)
```
```   323   have "prod uminus E = (-1) ^ card E * prod id E"
```
```   324     using finite_E by (induct set: finite) auto
```
```   325   with two show ?thesis
```
```   326     by simp
```
```   327 qed
```
```   328
```
```   329
```
```   330 subsection \<open>Gauss' Lemma\<close>
```
```   331
```
```   332 lemma aux: "prod id A * (- 1) ^ card E * a ^ card A * (- 1) ^ card E = prod id A * a ^ card A"
```
```   333   by (metis (no_types) minus_minus mult.commute mult.left_commute power_minus power_one)
```
```   334
```
```   335 theorem pre_gauss_lemma: "[a ^ nat((int p - 1) div 2) = (-1) ^ (card E)] (mod p)"
```
```   336 proof -
```
```   337   have "[prod id A = prod id F * prod id D](mod p)"
```
```   338     by (auto simp: prod_D_F_eq_prod_A mult.commute cong del: prod.strong_cong)
```
```   339   then have "[prod id A = ((-1)^(card E) * prod id E) * prod id D] (mod p)"
```
```   340     by (rule cong_trans_int) (metis cong_scalar_int prod_F_zcong)
```
```   341   then have "[prod id A = ((-1)^(card E) * prod id C)] (mod p)"
```
```   342     by (metis C_prod_eq_D_times_E mult.commute mult.left_commute)
```
```   343   then have "[prod id A = ((-1)^(card E) * prod id B)] (mod p)"
```
```   344     by (rule cong_trans_int) (metis C_B_zcong_prod cong_scalar2_int)
```
```   345   then have "[prod id A = ((-1)^(card E) * prod id ((\<lambda>x. x * a) ` A))] (mod p)"
```
```   346     by (simp add: B_def)
```
```   347   then have "[prod id A = ((-1)^(card E) * prod (\<lambda>x. x * a) A)] (mod p)"
```
```   348     by (simp add: inj_on_xa_A prod.reindex)
```
```   349   moreover have "prod (\<lambda>x. x * a) A = prod (\<lambda>x. a) A * prod id A"
```
```   350     using finite_A by (induct set: finite) auto
```
```   351   ultimately have "[prod id A = ((-1)^(card E) * (prod (\<lambda>x. a) A * prod id A))] (mod p)"
```
```   352     by simp
```
```   353   then have "[prod id A = ((-1)^(card E) * a^(card A) * prod id A)](mod p)"
```
```   354     by (rule cong_trans_int)
```
```   355       (simp add: cong_scalar2_int cong_scalar_int finite_A prod_constant mult.assoc)
```
```   356   then have a: "[prod id A * (-1)^(card E) =
```
```   357       ((-1)^(card E) * a^(card A) * prod id A * (-1)^(card E))](mod p)"
```
```   358     by (rule cong_scalar_int)
```
```   359   then have "[prod id A * (-1)^(card E) = prod id A *
```
```   360       (-1)^(card E) * a^(card A) * (-1)^(card E)](mod p)"
```
```   361     by (rule cong_trans_int) (simp add: a mult.commute mult.left_commute)
```
```   362   then have "[prod id A * (-1)^(card E) = prod id A * a^(card A)](mod p)"
```
```   363     by (rule cong_trans_int) (simp add: aux cong del: prod.strong_cong)
```
```   364   with A_prod_relprime have "[(- 1) ^ card E = a ^ card A](mod p)"
```
```   365     by (metis cong_mult_lcancel_int)
```
```   366   then show ?thesis
```
```   367     by (simp add: A_card_eq cong_sym_int)
```
```   368 qed
```
```   369
```
```   370 theorem gauss_lemma: "Legendre a p = (-1) ^ (card E)"
```
```   371 proof -
```
```   372   from euler_criterion p_prime p_ge_2 have "[Legendre a p = a^(nat (((p) - 1) div 2))] (mod p)"
```
```   373     by auto
```
```   374   moreover have "int ((p - 1) div 2) = (int p - 1) div 2"
```
```   375     using p_eq2 by linarith
```
```   376   then have "[a ^ nat (int ((p - 1) div 2)) = a ^ nat ((int p - 1) div 2)] (mod int p)"
```
```   377     by force
```
```   378   ultimately have "[Legendre a p = (-1) ^ (card E)] (mod p)"
```
```   379     using pre_gauss_lemma cong_trans_int by blast
```
```   380   moreover from p_a_relprime have "Legendre a p = 1 \<or> Legendre a p = -1"
```
```   381     by (auto simp add: Legendre_def)
```
```   382   moreover have "(-1::int) ^ (card E) = 1 \<or> (-1::int) ^ (card E) = -1"
```
```   383     using neg_one_even_power neg_one_odd_power by blast
```
```   384   moreover have "[1 \<noteq> - 1] (mod int p)"
```
```   385     using cong_altdef_int nonzero_mod_p[of 2] p_odd_int by fastforce
```
```   386   ultimately show ?thesis
```
```   387     by (auto simp add: cong_sym_int)
```
```   388 qed
```
```   389
```
```   390 end
```
```   391
```
```   392 end
```