src/HOL/Number_Theory/Residues.thy
 author haftmann Sat Jul 09 13:26:16 2016 +0200 (2016-07-09) changeset 63417 c184ec919c70 parent 63167 0909deb8059b child 63534 523b488b15c9 permissions -rw-r--r--
more lemmas to emphasize {0::nat..(<)n} as canonical representation of intervals on nat
* * *
more rules for setsum, setprod on intervals
```     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 UniqueFactorization MiscAlgebra
```
```    12 begin
```
```    13
```
```    14 subsection \<open>A locale for residue rings\<close>
```
```    15
```
```    16 definition residue_ring :: "int \<Rightarrow> int ring"
```
```    17 where
```
```    18   "residue_ring m =
```
```    19     \<lparr>carrier = {0..m - 1},
```
```    20      mult = \<lambda>x y. (x * y) mod m,
```
```    21      one = 1,
```
```    22      zero = 0,
```
```    23      add = \<lambda>x y. (x + y) mod m\<rparr>"
```
```    24
```
```    25 locale residues =
```
```    26   fixes m :: int and R (structure)
```
```    27   assumes m_gt_one: "m > 1"
```
```    28   defines "R \<equiv> residue_ring m"
```
```    29 begin
```
```    30
```
```    31 lemma abelian_group: "abelian_group R"
```
```    32   apply (insert m_gt_one)
```
```    33   apply (rule abelian_groupI)
```
```    34   apply (unfold R_def residue_ring_def)
```
```    35   apply (auto simp add: mod_add_right_eq [symmetric] ac_simps)
```
```    36   apply (case_tac "x = 0")
```
```    37   apply force
```
```    38   apply (subgoal_tac "(x + (m - x)) mod m = 0")
```
```    39   apply (erule bexI)
```
```    40   apply auto
```
```    41   done
```
```    42
```
```    43 lemma comm_monoid: "comm_monoid R"
```
```    44   apply (insert m_gt_one)
```
```    45   apply (unfold R_def residue_ring_def)
```
```    46   apply (rule comm_monoidI)
```
```    47   apply auto
```
```    48   apply (subgoal_tac "x * y mod m * z mod m = z * (x * y mod m) mod m")
```
```    49   apply (erule ssubst)
```
```    50   apply (subst mod_mult_right_eq [symmetric])+
```
```    51   apply (simp_all only: ac_simps)
```
```    52   done
```
```    53
```
```    54 lemma cring: "cring R"
```
```    55   apply (rule cringI)
```
```    56   apply (rule abelian_group)
```
```    57   apply (rule comm_monoid)
```
```    58   apply (unfold R_def residue_ring_def, auto)
```
```    59   apply (subst mod_add_eq [symmetric])
```
```    60   apply (subst mult.commute)
```
```    61   apply (subst mod_mult_right_eq [symmetric])
```
```    62   apply (simp add: field_simps)
```
```    63   done
```
```    64
```
```    65 end
```
```    66
```
```    67 sublocale residues < cring
```
```    68   by (rule cring)
```
```    69
```
```    70
```
```    71 context residues
```
```    72 begin
```
```    73
```
```    74 text \<open>
```
```    75   These lemmas translate back and forth between internal and
```
```    76   external concepts.
```
```    77 \<close>
```
```    78
```
```    79 lemma res_carrier_eq: "carrier R = {0..m - 1}"
```
```    80   unfolding R_def residue_ring_def by auto
```
```    81
```
```    82 lemma res_add_eq: "x \<oplus> y = (x + y) mod m"
```
```    83   unfolding R_def residue_ring_def by auto
```
```    84
```
```    85 lemma res_mult_eq: "x \<otimes> y = (x * y) mod m"
```
```    86   unfolding R_def residue_ring_def by auto
```
```    87
```
```    88 lemma res_zero_eq: "\<zero> = 0"
```
```    89   unfolding R_def residue_ring_def by auto
```
```    90
```
```    91 lemma res_one_eq: "\<one> = 1"
```
```    92   unfolding R_def residue_ring_def units_of_def by auto
```
```    93
```
```    94 lemma res_units_eq: "Units R = {x. 0 < x \<and> x < m \<and> coprime x m}"
```
```    95   apply (insert m_gt_one)
```
```    96   apply (unfold Units_def R_def residue_ring_def)
```
```    97   apply auto
```
```    98   apply (subgoal_tac "x \<noteq> 0")
```
```    99   apply auto
```
```   100   apply (metis invertible_coprime_int)
```
```   101   apply (subst (asm) coprime_iff_invertible'_int)
```
```   102   apply (auto simp add: cong_int_def mult.commute)
```
```   103   done
```
```   104
```
```   105 lemma res_neg_eq: "\<ominus> x = (- x) mod m"
```
```   106   apply (insert m_gt_one)
```
```   107   apply (unfold R_def a_inv_def m_inv_def residue_ring_def)
```
```   108   apply auto
```
```   109   apply (rule the_equality)
```
```   110   apply auto
```
```   111   apply (subst mod_add_right_eq [symmetric])
```
```   112   apply auto
```
```   113   apply (subst mod_add_left_eq [symmetric])
```
```   114   apply auto
```
```   115   apply (subgoal_tac "y mod m = - x mod m")
```
```   116   apply simp
```
```   117   apply (metis minus_add_cancel mod_mult_self1 mult.commute)
```
```   118   done
```
```   119
```
```   120 lemma finite [iff]: "finite (carrier R)"
```
```   121   by (subst res_carrier_eq) auto
```
```   122
```
```   123 lemma finite_Units [iff]: "finite (Units R)"
```
```   124   by (subst res_units_eq) auto
```
```   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   unfolding R_def residue_ring_def
```
```   138   apply auto
```
```   139   apply presburger
```
```   140   done
```
```   141
```
```   142 lemma mult_cong: "(x mod m) \<otimes> (y mod m) = (x * y) mod m"
```
```   143   unfolding R_def residue_ring_def
```
```   144   by auto (metis mod_mult_eq)
```
```   145
```
```   146 lemma zero_cong: "\<zero> = 0"
```
```   147   unfolding R_def residue_ring_def by auto
```
```   148
```
```   149 lemma one_cong: "\<one> = 1 mod m"
```
```   150   using m_gt_one unfolding R_def residue_ring_def by auto
```
```   151
```
```   152 (* FIXME revise algebra library to use 1? *)
```
```   153 lemma pow_cong: "(x mod m) (^) n = x^n mod m"
```
```   154   apply (insert m_gt_one)
```
```   155   apply (induct n)
```
```   156   apply (auto simp add: nat_pow_def one_cong)
```
```   157   apply (metis mult.commute mult_cong)
```
```   158   done
```
```   159
```
```   160 lemma neg_cong: "\<ominus> (x mod m) = (- x) mod m"
```
```   161   by (metis mod_minus_eq res_neg_eq)
```
```   162
```
```   163 lemma (in residues) prod_cong: "finite A \<Longrightarrow> (\<Otimes>i\<in>A. (f i) mod m) = (\<Prod>i\<in>A. f i) mod m"
```
```   164   by (induct set: finite) (auto simp: one_cong mult_cong)
```
```   165
```
```   166 lemma (in residues) sum_cong: "finite A \<Longrightarrow> (\<Oplus>i\<in>A. (f i) mod m) = (\<Sum>i\<in>A. f i) mod m"
```
```   167   by (induct set: finite) (auto simp: zero_cong add_cong)
```
```   168
```
```   169 lemma mod_in_res_units [simp]:
```
```   170   assumes "1 < m" and "coprime a m"
```
```   171   shows "a mod m \<in> Units R"
```
```   172 proof (cases "a mod m = 0")
```
```   173   case True with assms show ?thesis
```
```   174     by (auto simp add: res_units_eq gcd_red_int [symmetric])
```
```   175 next
```
```   176   case False
```
```   177   from assms have "0 < m" by simp
```
```   178   with pos_mod_sign [of m a] have "0 \<le> a mod m" .
```
```   179   with False have "0 < a mod m" by simp
```
```   180   with assms show ?thesis
```
```   181     by (auto simp add: res_units_eq gcd_red_int [symmetric] ac_simps)
```
```   182 qed
```
```   183
```
```   184 lemma res_eq_to_cong: "(a mod m) = (b mod m) \<longleftrightarrow> [a = b] (mod m)"
```
```   185   unfolding cong_int_def by auto
```
```   186
```
```   187
```
```   188 text \<open>Simplifying with these will translate a ring equation in R to a congruence.\<close>
```
```   189 lemmas res_to_cong_simps = add_cong mult_cong pow_cong one_cong
```
```   190     prod_cong sum_cong neg_cong res_eq_to_cong
```
```   191
```
```   192 text \<open>Other useful facts about the residue ring.\<close>
```
```   193 lemma one_eq_neg_one: "\<one> = \<ominus> \<one> \<Longrightarrow> m = 2"
```
```   194   apply (simp add: res_one_eq res_neg_eq)
```
```   195   apply (metis add.commute add_diff_cancel mod_mod_trivial one_add_one uminus_add_conv_diff
```
```   196     zero_neq_one zmod_zminus1_eq_if)
```
```   197   done
```
```   198
```
```   199 end
```
```   200
```
```   201
```
```   202 subsection \<open>Prime residues\<close>
```
```   203
```
```   204 locale residues_prime =
```
```   205   fixes p and R (structure)
```
```   206   assumes p_prime [intro]: "prime p"
```
```   207   defines "R \<equiv> residue_ring p"
```
```   208
```
```   209 sublocale residues_prime < residues p
```
```   210   apply (unfold R_def residues_def)
```
```   211   using p_prime apply auto
```
```   212   apply (metis (full_types) of_nat_1 of_nat_less_iff prime_gt_1_nat)
```
```   213   done
```
```   214
```
```   215 context residues_prime
```
```   216 begin
```
```   217
```
```   218 lemma is_field: "field R"
```
```   219   apply (rule cring.field_intro2)
```
```   220   apply (rule cring)
```
```   221   apply (auto simp add: res_carrier_eq res_one_eq res_zero_eq res_units_eq)
```
```   222   apply (rule classical)
```
```   223   apply (erule notE)
```
```   224   apply (subst gcd.commute)
```
```   225   apply (rule prime_imp_coprime_int)
```
```   226   apply (rule p_prime)
```
```   227   apply (rule notI)
```
```   228   apply (frule zdvd_imp_le)
```
```   229   apply auto
```
```   230   done
```
```   231
```
```   232 lemma res_prime_units_eq: "Units R = {1..p - 1}"
```
```   233   apply (subst res_units_eq)
```
```   234   apply auto
```
```   235   apply (subst gcd.commute)
```
```   236   apply (auto simp add: p_prime prime_imp_coprime_int zdvd_not_zless)
```
```   237   done
```
```   238
```
```   239 end
```
```   240
```
```   241 sublocale residues_prime < field
```
```   242   by (rule is_field)
```
```   243
```
```   244
```
```   245 section \<open>Test cases: Euler's theorem and Wilson's theorem\<close>
```
```   246
```
```   247 subsection \<open>Euler's theorem\<close>
```
```   248
```
```   249 text \<open>The definition of the phi function.\<close>
```
```   250
```
```   251 definition phi :: "int \<Rightarrow> nat"
```
```   252   where "phi m = card {x. 0 < x \<and> x < m \<and> gcd x m = 1}"
```
```   253
```
```   254 lemma phi_def_nat: "phi m = card {x. 0 < x \<and> x < nat m \<and> gcd x (nat m) = 1}"
```
```   255   apply (simp add: phi_def)
```
```   256   apply (rule bij_betw_same_card [of nat])
```
```   257   apply (auto simp add: inj_on_def bij_betw_def image_def)
```
```   258   apply (metis dual_order.irrefl dual_order.strict_trans leI nat_1 transfer_nat_int_gcd(1))
```
```   259   apply (metis One_nat_def of_nat_0 of_nat_1 of_nat_less_0_iff int_nat_eq nat_int
```
```   260     transfer_int_nat_gcd(1) of_nat_less_iff)
```
```   261   done
```
```   262
```
```   263 lemma prime_phi:
```
```   264   assumes "2 \<le> p" "phi p = p - 1"
```
```   265   shows "prime p"
```
```   266 proof -
```
```   267   have *: "{x. 0 < x \<and> x < p \<and> coprime x p} = {1..p - 1}"
```
```   268     using assms unfolding phi_def_nat
```
```   269     by (intro card_seteq) fastforce+
```
```   270   have False if **: "1 < x" "x < p" and "x dvd p" for x :: nat
```
```   271   proof -
```
```   272     from * have cop: "x \<in> {1..p - 1} \<Longrightarrow> coprime x p"
```
```   273       by blast
```
```   274     have "coprime x p"
```
```   275       apply (rule cop)
```
```   276       using ** apply auto
```
```   277       done
```
```   278     with \<open>x dvd p\<close> \<open>1 < x\<close> show ?thesis
```
```   279       by auto
```
```   280   qed
```
```   281   then show ?thesis
```
```   282     using \<open>2 \<le> p\<close>
```
```   283     by (simp add: prime_def)
```
```   284        (metis One_nat_def dvd_pos_nat nat_dvd_not_less nat_neq_iff not_gr0
```
```   285               not_numeral_le_zero one_dvd)
```
```   286 qed
```
```   287
```
```   288 lemma phi_zero [simp]: "phi 0 = 0"
```
```   289   unfolding phi_def
```
```   290 (* Auto hangs here. Once again, where is the simplification rule
```
```   291    1 \<equiv> Suc 0 coming from? *)
```
```   292   apply (auto simp add: card_eq_0_iff)
```
```   293 (* Add card_eq_0_iff as a simp rule? delete card_empty_imp? *)
```
```   294   done
```
```   295
```
```   296 lemma phi_one [simp]: "phi 1 = 0"
```
```   297   by (auto simp add: phi_def card_eq_0_iff)
```
```   298
```
```   299 lemma (in residues) phi_eq: "phi m = card (Units R)"
```
```   300   by (simp add: phi_def res_units_eq)
```
```   301
```
```   302 lemma (in residues) euler_theorem1:
```
```   303   assumes a: "gcd a m = 1"
```
```   304   shows "[a^phi m = 1] (mod m)"
```
```   305 proof -
```
```   306   from a m_gt_one have [simp]: "a mod m \<in> Units R"
```
```   307     by (intro mod_in_res_units)
```
```   308   from phi_eq have "(a mod m) (^) (phi m) = (a mod m) (^) (card (Units R))"
```
```   309     by simp
```
```   310   also have "\<dots> = \<one>"
```
```   311     by (intro units_power_order_eq_one) auto
```
```   312   finally show ?thesis
```
```   313     by (simp add: res_to_cong_simps)
```
```   314 qed
```
```   315
```
```   316 (* In fact, there is a two line proof!
```
```   317
```
```   318 lemma (in residues) euler_theorem1:
```
```   319   assumes a: "gcd a m = 1"
```
```   320   shows "[a^phi m = 1] (mod m)"
```
```   321 proof -
```
```   322   have "(a mod m) (^) (phi m) = \<one>"
```
```   323     by (simp add: phi_eq units_power_order_eq_one a m_gt_one)
```
```   324   then show ?thesis
```
```   325     by (simp add: res_to_cong_simps)
```
```   326 qed
```
```   327
```
```   328 *)
```
```   329
```
```   330 text \<open>Outside the locale, we can relax the restriction \<open>m > 1\<close>.\<close>
```
```   331 lemma euler_theorem:
```
```   332   assumes "m \<ge> 0"
```
```   333     and "gcd a m = 1"
```
```   334   shows "[a^phi m = 1] (mod m)"
```
```   335 proof (cases "m = 0 | m = 1")
```
```   336   case True
```
```   337   then show ?thesis by auto
```
```   338 next
```
```   339   case False
```
```   340   with assms show ?thesis
```
```   341     by (intro residues.euler_theorem1, unfold residues_def, auto)
```
```   342 qed
```
```   343
```
```   344 lemma (in residues_prime) phi_prime: "phi p = nat p - 1"
```
```   345   apply (subst phi_eq)
```
```   346   apply (subst res_prime_units_eq)
```
```   347   apply auto
```
```   348   done
```
```   349
```
```   350 lemma phi_prime: "prime p \<Longrightarrow> phi p = nat p - 1"
```
```   351   apply (rule residues_prime.phi_prime)
```
```   352   apply (erule residues_prime.intro)
```
```   353   done
```
```   354
```
```   355 lemma fermat_theorem:
```
```   356   fixes a :: int
```
```   357   assumes "prime p"
```
```   358     and "\<not> p dvd a"
```
```   359   shows "[a^(p - 1) = 1] (mod p)"
```
```   360 proof -
```
```   361   from assms have "[a ^ phi p = 1] (mod p)"
```
```   362     by (auto intro!: euler_theorem dest!: prime_imp_coprime_int simp add: ac_simps)
```
```   363   also have "phi p = nat p - 1"
```
```   364     by (rule phi_prime) (rule assms)
```
```   365   finally show ?thesis
```
```   366     by (metis nat_int)
```
```   367 qed
```
```   368
```
```   369 lemma fermat_theorem_nat:
```
```   370   assumes "prime p" and "\<not> p dvd a"
```
```   371   shows "[a ^ (p - 1) = 1] (mod p)"
```
```   372   using fermat_theorem [of p a] assms
```
```   373   by (metis of_nat_1 of_nat_power transfer_int_nat_cong zdvd_int)
```
```   374
```
```   375
```
```   376 subsection \<open>Wilson's theorem\<close>
```
```   377
```
```   378 lemma (in field) inv_pair_lemma: "x \<in> Units R \<Longrightarrow> y \<in> Units R \<Longrightarrow>
```
```   379     {x, inv x} \<noteq> {y, inv y} \<Longrightarrow> {x, inv x} \<inter> {y, inv y} = {}"
```
```   380   apply auto
```
```   381   apply (metis Units_inv_inv)+
```
```   382   done
```
```   383
```
```   384 lemma (in residues_prime) wilson_theorem1:
```
```   385   assumes a: "p > 2"
```
```   386   shows "[fact (p - 1) = (-1::int)] (mod p)"
```
```   387 proof -
```
```   388   let ?Inverse_Pairs = "{{x, inv x}| x. x \<in> Units R - {\<one>, \<ominus> \<one>}}"
```
```   389   have UR: "Units R = {\<one>, \<ominus> \<one>} \<union> \<Union>?Inverse_Pairs"
```
```   390     by auto
```
```   391   have "(\<Otimes>i\<in>Units R. i) = (\<Otimes>i\<in>{\<one>, \<ominus> \<one>}. i) \<otimes> (\<Otimes>i\<in>\<Union>?Inverse_Pairs. i)"
```
```   392     apply (subst UR)
```
```   393     apply (subst finprod_Un_disjoint)
```
```   394     apply (auto intro: funcsetI)
```
```   395     using inv_one apply auto[1]
```
```   396     using inv_eq_neg_one_eq apply auto
```
```   397     done
```
```   398   also have "(\<Otimes>i\<in>{\<one>, \<ominus> \<one>}. i) = \<ominus> \<one>"
```
```   399     apply (subst finprod_insert)
```
```   400     apply auto
```
```   401     apply (frule one_eq_neg_one)
```
```   402     using a apply force
```
```   403     done
```
```   404   also have "(\<Otimes>i\<in>(\<Union>?Inverse_Pairs). i) = (\<Otimes>A\<in>?Inverse_Pairs. (\<Otimes>y\<in>A. y))"
```
```   405     apply (subst finprod_Union_disjoint)
```
```   406     apply auto
```
```   407     apply (metis Units_inv_inv)+
```
```   408     done
```
```   409   also have "\<dots> = \<one>"
```
```   410     apply (rule finprod_one)
```
```   411     apply auto
```
```   412     apply (subst finprod_insert)
```
```   413     apply auto
```
```   414     apply (metis inv_eq_self)
```
```   415     done
```
```   416   finally have "(\<Otimes>i\<in>Units R. i) = \<ominus> \<one>"
```
```   417     by simp
```
```   418   also have "(\<Otimes>i\<in>Units R. i) = (\<Otimes>i\<in>Units R. i mod p)"
```
```   419     apply (rule finprod_cong')
```
```   420     apply auto
```
```   421     apply (subst (asm) res_prime_units_eq)
```
```   422     apply auto
```
```   423     done
```
```   424   also have "\<dots> = (\<Prod>i\<in>Units R. i) mod p"
```
```   425     apply (rule prod_cong)
```
```   426     apply auto
```
```   427     done
```
```   428   also have "\<dots> = fact (p - 1) mod p"
```
```   429     apply (simp add: fact_setprod)
```
```   430     apply (insert assms)
```
```   431     apply (subst res_prime_units_eq)
```
```   432     apply (simp add: int_setprod zmod_int setprod_int_eq)
```
```   433     done
```
```   434   finally have "fact (p - 1) mod p = \<ominus> \<one>" .
```
```   435   then show ?thesis
```
```   436     by (metis of_nat_fact Divides.transfer_int_nat_functions(2)
```
```   437       cong_int_def res_neg_eq res_one_eq)
```
```   438 qed
```
```   439
```
```   440 lemma wilson_theorem:
```
```   441   assumes "prime p"
```
```   442   shows "[fact (p - 1) = - 1] (mod p)"
```
```   443 proof (cases "p = 2")
```
```   444   case True
```
```   445   then show ?thesis
```
```   446     by (simp add: cong_int_def fact_setprod)
```
```   447 next
```
```   448   case False
```
```   449   then show ?thesis
```
```   450     using assms prime_ge_2_nat
```
```   451     by (metis residues_prime.wilson_theorem1 residues_prime.intro le_eq_less_or_eq)
```
```   452 qed
```
```   453
```
```   454 end
```