src/HOL/Number_Theory/Residues.thy
 author paulson Thu Jun 14 15:20:10 2018 +0100 (12 months ago) changeset 68447 0beb927eed89 parent 67341 df79ef3b3a41 child 68458 023b353911c5 permissions -rw-r--r--
```     1 (*  Title:      HOL/Number_Theory/Residues.thy
```
```     2     Author:     Jeremy Avigad
```
```     3
```
```     4 An algebraic treatment of residue rings, and resulting proofs of
```
```     5 Euler's theorem and Wilson's theorem.
```
```     6 *)
```
```     7
```
```     8 section \<open>Residue rings\<close>
```
```     9
```
```    10 theory Residues
```
```    11 imports
```
```    12   Cong
```
```    13   "HOL-Algebra.Multiplicative_Group"
```
```    14   Totient
```
```    15 begin
```
```    16
```
```    17 definition QuadRes :: "int \<Rightarrow> int \<Rightarrow> bool"
```
```    18   where "QuadRes p a = (\<exists>y. ([y^2 = a] (mod p)))"
```
```    19
```
```    20 definition Legendre :: "int \<Rightarrow> int \<Rightarrow> int"
```
```    21   where "Legendre a p =
```
```    22     (if ([a = 0] (mod p)) then 0
```
```    23      else if QuadRes p a then 1
```
```    24      else -1)"
```
```    25
```
```    26
```
```    27 subsection \<open>A locale for residue rings\<close>
```
```    28
```
```    29 definition residue_ring :: "int \<Rightarrow> int ring"
```
```    30   where
```
```    31     "residue_ring m =
```
```    32       \<lparr>carrier = {0..m - 1},
```
```    33        monoid.mult = \<lambda>x y. (x * y) mod m,
```
```    34        one = 1,
```
```    35        zero = 0,
```
```    36        add = \<lambda>x y. (x + y) mod m\<rparr>"
```
```    37
```
```    38 locale residues =
```
```    39   fixes m :: int and R (structure)
```
```    40   assumes m_gt_one: "m > 1"
```
```    41   defines "R \<equiv> residue_ring m"
```
```    42 begin
```
```    43
```
```    44 lemma abelian_group: "abelian_group R"
```
```    45 proof -
```
```    46   have "\<exists>y\<in>{0..m - 1}. (x + y) mod m = 0" if "0 \<le> x" "x < m" for x
```
```    47   proof (cases "x = 0")
```
```    48     case True
```
```    49     with m_gt_one show ?thesis by simp
```
```    50   next
```
```    51     case False
```
```    52     then have "(x + (m - x)) mod m = 0"
```
```    53       by simp
```
```    54     with m_gt_one that show ?thesis
```
```    55       by (metis False atLeastAtMost_iff diff_ge_0_iff_ge diff_left_mono int_one_le_iff_zero_less less_le)
```
```    56   qed
```
```    57   with m_gt_one show ?thesis
```
```    58     by (fastforce simp add: R_def residue_ring_def mod_add_right_eq ac_simps  intro!: abelian_groupI)
```
```    59 qed
```
```    60
```
```    61 lemma comm_monoid: "comm_monoid R"
```
```    62   unfolding R_def residue_ring_def
```
```    63   apply (rule comm_monoidI)
```
```    64     using m_gt_one  apply auto
```
```    65   apply (metis mod_mult_right_eq mult.assoc mult.commute)
```
```    66   apply (metis mult.commute)
```
```    67   done
```
```    68
```
```    69 lemma cring: "cring R"
```
```    70   apply (intro cringI abelian_group comm_monoid)
```
```    71   unfolding R_def residue_ring_def
```
```    72   apply (auto simp add: comm_semiring_class.distrib mod_add_eq mod_mult_left_eq)
```
```    73   done
```
```    74
```
```    75 end
```
```    76
```
```    77 sublocale residues < cring
```
```    78   by (rule cring)
```
```    79
```
```    80
```
```    81 context residues
```
```    82 begin
```
```    83
```
```    84 text \<open>
```
```    85   These lemmas translate back and forth between internal and
```
```    86   external concepts.
```
```    87 \<close>
```
```    88
```
```    89 lemma res_carrier_eq: "carrier R = {0..m - 1}"
```
```    90   by (auto simp: R_def residue_ring_def)
```
```    91
```
```    92 lemma res_add_eq: "x \<oplus> y = (x + y) mod m"
```
```    93   by (auto simp: R_def residue_ring_def)
```
```    94
```
```    95 lemma res_mult_eq: "x \<otimes> y = (x * y) mod m"
```
```    96   by (auto simp: R_def residue_ring_def)
```
```    97
```
```    98 lemma res_zero_eq: "\<zero> = 0"
```
```    99   by (auto simp: R_def residue_ring_def)
```
```   100
```
```   101 lemma res_one_eq: "\<one> = 1"
```
```   102   by (auto simp: R_def residue_ring_def units_of_def)
```
```   103
```
```   104 lemma res_units_eq: "Units R = {x. 0 < x \<and> x < m \<and> coprime x m}"
```
```   105   using m_gt_one
```
```   106   apply (auto simp add: Units_def R_def residue_ring_def ac_simps invertible_coprime intro: ccontr)
```
```   107   apply (subst (asm) coprime_iff_invertible'_int)
```
```   108    apply (auto simp add: cong_def)
```
```   109   done
```
```   110
```
```   111 lemma res_neg_eq: "\<ominus> x = (- x) mod m"
```
```   112   using m_gt_one unfolding R_def a_inv_def m_inv_def residue_ring_def
```
```   113   apply simp
```
```   114   apply (rule the_equality)
```
```   115    apply (simp add: mod_add_right_eq)
```
```   116    apply (simp add: add.commute mod_add_right_eq)
```
```   117   apply (metis add.right_neutral minus_add_cancel mod_add_right_eq mod_pos_pos_trivial)
```
```   118   done
```
```   119
```
```   120 lemma finite [iff]: "finite (carrier R)"
```
```   121   by (simp add: res_carrier_eq)
```
```   122
```
```   123 lemma finite_Units [iff]: "finite (Units R)"
```
```   124   by (simp add: finite_ring_finite_units)
```
```   125
```
```   126 text \<open>
```
```   127   The function \<open>a \<mapsto> a mod m\<close> maps the integers to the
```
```   128   residue classes. The following lemmas show that this mapping
```
```   129   respects addition and multiplication on the integers.
```
```   130 \<close>
```
```   131
```
```   132 lemma mod_in_carrier [iff]: "a mod m \<in> carrier R"
```
```   133   unfolding res_carrier_eq
```
```   134   using insert m_gt_one by auto
```
```   135
```
```   136 lemma add_cong: "(x mod m) \<oplus> (y mod m) = (x + y) mod m"
```
```   137   by (auto simp: R_def residue_ring_def mod_simps)
```
```   138
```
```   139 lemma mult_cong: "(x mod m) \<otimes> (y mod m) = (x * y) mod m"
```
```   140   by (auto simp: R_def residue_ring_def mod_simps)
```
```   141
```
```   142 lemma zero_cong: "\<zero> = 0"
```
```   143   by (auto simp: R_def residue_ring_def)
```
```   144
```
```   145 lemma one_cong: "\<one> = 1 mod m"
```
```   146   using m_gt_one by (auto simp: R_def residue_ring_def)
```
```   147
```
```   148 (* FIXME revise algebra library to use 1? *)
```
```   149 lemma pow_cong: "(x mod m) [^] n = x^n mod m"
```
```   150   using m_gt_one
```
```   151   apply (induct n)
```
```   152   apply (auto simp add: nat_pow_def one_cong)
```
```   153   apply (metis mult.commute mult_cong)
```
```   154   done
```
```   155
```
```   156 lemma neg_cong: "\<ominus> (x mod m) = (- x) mod m"
```
```   157   by (metis mod_minus_eq res_neg_eq)
```
```   158
```
```   159 lemma (in residues) prod_cong: "finite A \<Longrightarrow> (\<Otimes>i\<in>A. (f i) mod m) = (\<Prod>i\<in>A. f i) mod m"
```
```   160   by (induct set: finite) (auto simp: one_cong mult_cong)
```
```   161
```
```   162 lemma (in residues) sum_cong: "finite A \<Longrightarrow> (\<Oplus>i\<in>A. (f i) mod m) = (\<Sum>i\<in>A. f i) mod m"
```
```   163   by (induct set: finite) (auto simp: zero_cong add_cong)
```
```   164
```
```   165 lemma mod_in_res_units [simp]:
```
```   166   assumes "1 < m" and "coprime a m"
```
```   167   shows "a mod m \<in> Units R"
```
```   168 proof (cases "a mod m = 0")
```
```   169   case True
```
```   170   with assms show ?thesis
```
```   171     by (auto simp add: res_units_eq gcd_red_int [symmetric])
```
```   172 next
```
```   173   case False
```
```   174   from assms have "0 < m" by simp
```
```   175   then have "0 \<le> a mod m" by (rule pos_mod_sign [of m a])
```
```   176   with False have "0 < a mod m" by simp
```
```   177   with assms show ?thesis
```
```   178     by (auto simp add: res_units_eq gcd_red_int [symmetric] ac_simps)
```
```   179 qed
```
```   180
```
```   181 lemma res_eq_to_cong: "(a mod m) = (b mod m) \<longleftrightarrow> [a = b] (mod m)"
```
```   182   by (auto simp: cong_def)
```
```   183
```
```   184
```
```   185 text \<open>Simplifying with these will translate a ring equation in R to a congruence.\<close>
```
```   186 lemmas res_to_cong_simps =
```
```   187   add_cong mult_cong pow_cong one_cong
```
```   188   prod_cong sum_cong neg_cong res_eq_to_cong
```
```   189
```
```   190 text \<open>Other useful facts about the residue ring.\<close>
```
```   191 lemma one_eq_neg_one: "\<one> = \<ominus> \<one> \<Longrightarrow> m = 2"
```
```   192   apply (simp add: res_one_eq res_neg_eq)
```
```   193   apply (metis add.commute add_diff_cancel mod_mod_trivial one_add_one uminus_add_conv_diff
```
```   194     zero_neq_one zmod_zminus1_eq_if)
```
```   195   done
```
```   196
```
```   197 end
```
```   198
```
```   199
```
```   200 subsection \<open>Prime residues\<close>
```
```   201
```
```   202 locale residues_prime =
```
```   203   fixes p :: nat and R (structure)
```
```   204   assumes p_prime [intro]: "prime p"
```
```   205   defines "R \<equiv> residue_ring (int p)"
```
```   206
```
```   207 sublocale residues_prime < residues p
```
```   208   unfolding R_def residues_def
```
```   209   using p_prime apply auto
```
```   210   apply (metis (full_types) of_nat_1 of_nat_less_iff prime_gt_1_nat)
```
```   211   done
```
```   212
```
```   213 context residues_prime
```
```   214 begin
```
```   215
```
```   216 lemma p_coprime_left:
```
```   217   "coprime p a \<longleftrightarrow> \<not> p dvd a"
```
```   218   using p_prime by (auto intro: prime_imp_coprime dest: coprime_common_divisor)
```
```   219
```
```   220 lemma p_coprime_right:
```
```   221   "coprime a p  \<longleftrightarrow> \<not> p dvd a"
```
```   222   using p_coprime_left [of a] by (simp add: ac_simps)
```
```   223
```
```   224 lemma p_coprime_left_int:
```
```   225   "coprime (int p) a \<longleftrightarrow> \<not> int p dvd a"
```
```   226   using p_prime by (auto intro: prime_imp_coprime dest: coprime_common_divisor)
```
```   227
```
```   228 lemma p_coprime_right_int:
```
```   229   "coprime a (int p) \<longleftrightarrow> \<not> int p dvd a"
```
```   230   using p_coprime_left_int [of a] by (simp add: ac_simps)
```
```   231
```
```   232 lemma is_field: "field R"
```
```   233 proof -
```
```   234   have "0 < x \<Longrightarrow> x < int p \<Longrightarrow> coprime (int p) x" for x
```
```   235     by (rule prime_imp_coprime) (auto simp add: zdvd_not_zless)
```
```   236   then show ?thesis
```
```   237     by (intro cring.field_intro2 cring)
```
```   238       (auto simp add: res_carrier_eq res_one_eq res_zero_eq res_units_eq ac_simps)
```
```   239 qed
```
```   240
```
```   241 lemma res_prime_units_eq: "Units R = {1..p - 1}"
```
```   242   apply (subst res_units_eq)
```
```   243   apply (auto simp add: p_coprime_right_int zdvd_not_zless)
```
```   244   done
```
```   245
```
```   246 end
```
```   247
```
```   248 sublocale residues_prime < field
```
```   249   by (rule is_field)
```
```   250
```
```   251
```
```   252 section \<open>Test cases: Euler's theorem and Wilson's theorem\<close>
```
```   253
```
```   254 subsection \<open>Euler's theorem\<close>
```
```   255
```
```   256 lemma (in residues) totatives_eq:
```
```   257   "totatives (nat m) = nat ` Units R"
```
```   258 proof -
```
```   259   from m_gt_one have "\<bar>m\<bar> > 1"
```
```   260     by simp
```
```   261   then have "totatives (nat \<bar>m\<bar>) = nat ` abs ` Units R"
```
```   262     by (auto simp add: totatives_def res_units_eq image_iff le_less)
```
```   263       (use m_gt_one zless_nat_eq_int_zless in force)
```
```   264   moreover have "\<bar>m\<bar> = m" "abs ` Units R = Units R"
```
```   265     using m_gt_one by (auto simp add: res_units_eq image_iff)
```
```   266   ultimately show ?thesis
```
```   267     by simp
```
```   268 qed
```
```   269
```
```   270 lemma (in residues) totient_eq:
```
```   271   "totient (nat m) = card (Units R)"
```
```   272 proof  -
```
```   273   have *: "inj_on nat (Units R)"
```
```   274     by (rule inj_onI) (auto simp add: res_units_eq)
```
```   275   then show ?thesis
```
```   276     by (simp add: totient_def totatives_eq card_image)
```
```   277 qed
```
```   278
```
```   279 lemma (in residues_prime) totient_eq: "totient p = p - 1"
```
```   280   using totient_eq by (simp add: res_prime_units_eq)
```
```   281
```
```   282 lemma (in residues) euler_theorem:
```
```   283   assumes "coprime a m"
```
```   284   shows "[a ^ totient (nat m) = 1] (mod m)"
```
```   285 proof -
```
```   286   have "a ^ totient (nat m) mod m = 1 mod m"
```
```   287     by (metis assms finite_Units m_gt_one mod_in_res_units one_cong totient_eq pow_cong units_power_order_eq_one)
```
```   288   then show ?thesis
```
```   289     using res_eq_to_cong by blast
```
```   290 qed
```
```   291
```
```   292 lemma euler_theorem:
```
```   293   fixes a m :: nat
```
```   294   assumes "coprime a m"
```
```   295   shows "[a ^ totient m = 1] (mod m)"
```
```   296 proof (cases "m = 0 \<or> m = 1")
```
```   297   case True
```
```   298   then show ?thesis by auto
```
```   299 next
```
```   300   case False
```
```   301   with assms show ?thesis
```
```   302     using residues.euler_theorem [of "int m" "int a"] cong_int_iff
```
```   303     by (auto simp add: residues_def gcd_int_def) fastforce
```
```   304 qed
```
```   305
```
```   306 lemma fermat_theorem:
```
```   307   fixes p a :: nat
```
```   308   assumes "prime p" and "\<not> p dvd a"
```
```   309   shows "[a ^ (p - 1) = 1] (mod p)"
```
```   310 proof -
```
```   311   from assms prime_imp_coprime [of p a] have "coprime a p"
```
```   312     by (auto simp add: ac_simps)
```
```   313   then have "[a ^ totient p = 1] (mod p)"
```
```   314      by (rule euler_theorem)
```
```   315   also have "totient p = p - 1"
```
```   316     by (rule totient_prime) (rule assms)
```
```   317   finally show ?thesis .
```
```   318 qed
```
```   319
```
```   320
```
```   321 subsection \<open>Wilson's theorem\<close>
```
```   322
```
```   323 lemma (in field) inv_pair_lemma: "x \<in> Units R \<Longrightarrow> y \<in> Units R \<Longrightarrow>
```
```   324     {x, inv x} \<noteq> {y, inv y} \<Longrightarrow> {x, inv x} \<inter> {y, inv y} = {}"
```
```   325   apply auto
```
```   326   apply (metis Units_inv_inv)+
```
```   327   done
```
```   328
```
```   329 lemma (in residues_prime) wilson_theorem1:
```
```   330   assumes a: "p > 2"
```
```   331   shows "[fact (p - 1) = (-1::int)] (mod p)"
```
```   332 proof -
```
```   333   let ?Inverse_Pairs = "{{x, inv x}| x. x \<in> Units R - {\<one>, \<ominus> \<one>}}"
```
```   334   have UR: "Units R = {\<one>, \<ominus> \<one>} \<union> \<Union>?Inverse_Pairs"
```
```   335     by auto
```
```   336   have "(\<Otimes>i\<in>Units R. i) = (\<Otimes>i\<in>{\<one>, \<ominus> \<one>}. i) \<otimes> (\<Otimes>i\<in>\<Union>?Inverse_Pairs. i)"
```
```   337     apply (subst UR)
```
```   338     apply (subst finprod_Un_disjoint)
```
```   339          apply (auto intro: funcsetI)
```
```   340     using inv_one apply auto[1]
```
```   341     using inv_eq_neg_one_eq apply auto
```
```   342     done
```
```   343   also have "(\<Otimes>i\<in>{\<one>, \<ominus> \<one>}. i) = \<ominus> \<one>"
```
```   344     apply (subst finprod_insert)
```
```   345         apply auto
```
```   346     apply (frule one_eq_neg_one)
```
```   347     using a apply force
```
```   348     done
```
```   349   also have "(\<Otimes>i\<in>(\<Union>?Inverse_Pairs). i) = (\<Otimes>A\<in>?Inverse_Pairs. (\<Otimes>y\<in>A. y))"
```
```   350     apply (subst finprod_Union_disjoint)
```
```   351        apply auto
```
```   352      apply (metis Units_inv_inv)+
```
```   353     done
```
```   354   also have "\<dots> = \<one>"
```
```   355     apply (rule finprod_one_eqI)
```
```   356      apply auto
```
```   357     apply (subst finprod_insert)
```
```   358         apply auto
```
```   359     apply (metis inv_eq_self)
```
```   360     done
```
```   361   finally have "(\<Otimes>i\<in>Units R. i) = \<ominus> \<one>"
```
```   362     by simp
```
```   363   also have "(\<Otimes>i\<in>Units R. i) = (\<Otimes>i\<in>Units R. i mod p)"
```
```   364     by (rule finprod_cong') (auto simp: res_units_eq)
```
```   365   also have "\<dots> = (\<Prod>i\<in>Units R. i) mod p"
```
```   366     by (rule prod_cong) auto
```
```   367   also have "\<dots> = fact (p - 1) mod p"
```
```   368     apply (simp add: fact_prod)
```
```   369     using assms
```
```   370     apply (subst res_prime_units_eq)
```
```   371     apply (simp add: int_prod zmod_int prod_int_eq)
```
```   372     done
```
```   373   finally have "fact (p - 1) mod p = \<ominus> \<one>" .
```
```   374   then show ?thesis
```
```   375     by (simp add: cong_def res_neg_eq res_one_eq zmod_int)
```
```   376 qed
```
```   377
```
```   378 lemma wilson_theorem:
```
```   379   assumes "prime p"
```
```   380   shows "[fact (p - 1) = - 1] (mod p)"
```
```   381 proof (cases "p = 2")
```
```   382   case True
```
```   383   then show ?thesis
```
```   384     by (simp add: cong_def fact_prod)
```
```   385 next
```
```   386   case False
```
```   387   then show ?thesis
```
```   388     using assms prime_ge_2_nat
```
```   389     by (metis residues_prime.wilson_theorem1 residues_prime.intro le_eq_less_or_eq)
```
```   390 qed
```
```   391
```
```   392 text \<open>
```
```   393   This result can be transferred to the multiplicative group of
```
```   394   \<open>\<int>/p\<int>\<close> for \<open>p\<close> prime.\<close>
```
```   395
```
```   396 lemma mod_nat_int_pow_eq:
```
```   397   fixes n :: nat and p a :: int
```
```   398   shows "a \<ge> 0 \<Longrightarrow> p \<ge> 0 \<Longrightarrow> (nat a ^ n) mod (nat p) = nat ((a ^ n) mod p)"
```
```   399   by (simp add: int_one_le_iff_zero_less nat_mod_distrib order_less_imp_le nat_power_eq[symmetric])
```
```   400
```
```   401 theorem residue_prime_mult_group_has_gen :
```
```   402  fixes p :: nat
```
```   403  assumes prime_p : "prime p"
```
```   404  shows "\<exists>a \<in> {1 .. p - 1}. {1 .. p - 1} = {a^i mod p|i . i \<in> UNIV}"
```
```   405 proof -
```
```   406   have "p \<ge> 2"
```
```   407     using prime_gt_1_nat[OF prime_p] by simp
```
```   408   interpret R: residues_prime p "residue_ring p"
```
```   409     by (simp add: residues_prime_def prime_p)
```
```   410   have car: "carrier (residue_ring (int p)) - {\<zero>\<^bsub>residue_ring (int p)\<^esub>} = {1 .. int p - 1}"
```
```   411     by (auto simp add: R.zero_cong R.res_carrier_eq)
```
```   412
```
```   413   have "x [^]\<^bsub>residue_ring (int p)\<^esub> i = x ^ i mod (int p)"
```
```   414     if "x \<in> {1 .. int p - 1}" for x and i :: nat
```
```   415     using that R.pow_cong[of x i] by auto
```
```   416   moreover
```
```   417   obtain a where a: "a \<in> {1 .. int p - 1}"
```
```   418     and a_gen: "{1 .. int p - 1} = {a[^]\<^bsub>residue_ring (int p)\<^esub>i|i::nat . i \<in> UNIV}"
```
```   419     using field.finite_field_mult_group_has_gen[OF R.is_field]
```
```   420     by (auto simp add: car[symmetric] carrier_mult_of)
```
```   421   moreover
```
```   422   have "nat ` {1 .. int p - 1} = {1 .. p - 1}" (is "?L = ?R")
```
```   423   proof
```
```   424     have "n \<in> ?R" if "n \<in> ?L" for n
```
```   425       using that \<open>p\<ge>2\<close> by force
```
```   426     then show "?L \<subseteq> ?R" by blast
```
```   427     have "n \<in> ?L" if "n \<in> ?R" for n
```
```   428       using that \<open>p\<ge>2\<close> by (auto intro: rev_image_eqI [of "int n"])
```
```   429     then show "?R \<subseteq> ?L" by blast
```
```   430   qed
```
```   431   moreover
```
```   432   have "nat ` {a^i mod (int p) | i::nat. i \<in> UNIV} = {nat a^i mod p | i . i \<in> UNIV}" (is "?L = ?R")
```
```   433   proof
```
```   434     have "x \<in> ?R" if "x \<in> ?L" for x
```
```   435     proof -
```
```   436       from that obtain i where i: "x = nat (a^i mod (int p))"
```
```   437         by blast
```
```   438       then have "x = nat a ^ i mod p"
```
```   439         using mod_nat_int_pow_eq[of a "int p" i] a \<open>p\<ge>2\<close> by auto
```
```   440       with i show ?thesis by blast
```
```   441     qed
```
```   442     then show "?L \<subseteq> ?R" by blast
```
```   443     have "x \<in> ?L" if "x \<in> ?R" for x
```
```   444     proof -
```
```   445       from that obtain i where i: "x = nat a^i mod p"
```
```   446         by blast
```
```   447       with mod_nat_int_pow_eq[of a "int p" i] a \<open>p\<ge>2\<close> show ?thesis
```
```   448         by auto
```
```   449     qed
```
```   450     then show "?R \<subseteq> ?L" by blast
```
```   451   qed
```
```   452   ultimately have "{1 .. p - 1} = {nat a^i mod p | i. i \<in> UNIV}"
```
```   453     by presburger
```
```   454   moreover from a have "nat a \<in> {1 .. p - 1}" by force
```
```   455   ultimately show ?thesis ..
```
```   456 qed
```
```   457
```
```   458 end
```