src/HOL/Number_Theory/Residues.thy
author wenzelm
Fri Jun 19 23:40:46 2015 +0200 (2015-06-19)
changeset 60527 eb431a5651fe
parent 60526 fad653acf58f
child 60528 190b4a7d8b87
permissions -rw-r--r--
tuned proofs;
     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 @{text "a \<mapsto> a mod m"} 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: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: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]: "1 < m \<Longrightarrow> coprime a m \<Longrightarrow> a mod m \<in> Units R"
   170   apply (subst res_units_eq)
   171   apply auto
   172   apply (insert pos_mod_sign [of m a])
   173   apply (subgoal_tac "a mod m \<noteq> 0")
   174   apply arith
   175   apply auto
   176   apply (metis gcd_int.commute gcd_red_int)
   177   done
   178 
   179 lemma res_eq_to_cong: "((a mod m) = (b mod m)) = [a = b] (mod (m::int))"
   180   unfolding cong_int_def by auto
   181 
   182 
   183 text \<open>Simplifying with these will translate a ring equation in R to a
   184    congruence.\<close>
   185 lemmas res_to_cong_simps = add_cong mult_cong pow_cong one_cong
   186     prod_cong sum_cong neg_cong res_eq_to_cong
   187 
   188 text \<open>Other useful facts about the residue ring.\<close>
   189 
   190 lemma one_eq_neg_one: "\<one> = \<ominus> \<one> \<Longrightarrow> m = 2"
   191   apply (simp add: res_one_eq res_neg_eq)
   192   apply (metis add.commute add_diff_cancel mod_mod_trivial one_add_one uminus_add_conv_diff
   193             zero_neq_one zmod_zminus1_eq_if)
   194   done
   195 
   196 end
   197 
   198 
   199 subsection \<open>Prime residues\<close>
   200 
   201 locale residues_prime =
   202   fixes p and R (structure)
   203   assumes p_prime [intro]: "prime p"
   204   defines "R \<equiv> residue_ring p"
   205 
   206 sublocale residues_prime < residues p
   207   apply (unfold R_def residues_def)
   208   using p_prime apply auto
   209   apply (metis (full_types) int_1 of_nat_less_iff prime_gt_1_nat)
   210   done
   211 
   212 context residues_prime
   213 begin
   214 
   215 lemma is_field: "field R"
   216   apply (rule cring.field_intro2)
   217   apply (rule cring)
   218   apply (auto simp add: res_carrier_eq res_one_eq res_zero_eq res_units_eq)
   219   apply (rule classical)
   220   apply (erule notE)
   221   apply (subst gcd_commute_int)
   222   apply (rule prime_imp_coprime_int)
   223   apply (rule p_prime)
   224   apply (rule notI)
   225   apply (frule zdvd_imp_le)
   226   apply auto
   227   done
   228 
   229 lemma res_prime_units_eq: "Units R = {1..p - 1}"
   230   apply (subst res_units_eq)
   231   apply auto
   232   apply (subst gcd_commute_int)
   233   apply (auto simp add: p_prime prime_imp_coprime_int zdvd_not_zless)
   234   done
   235 
   236 end
   237 
   238 sublocale residues_prime < field
   239   by (rule is_field)
   240 
   241 
   242 section \<open>Test cases: Euler's theorem and Wilson's theorem\<close>
   243 
   244 subsection \<open>Euler's theorem\<close>
   245 
   246 text \<open>The definition of the phi function.\<close>
   247 
   248 definition phi :: "int \<Rightarrow> nat"
   249   where "phi m = card {x. 0 < x \<and> x < m \<and> gcd x m = 1}"
   250 
   251 lemma phi_def_nat: "phi m = card {x. 0 < x \<and> x < nat m \<and> gcd x (nat m) = 1}"
   252   apply (simp add: phi_def)
   253   apply (rule bij_betw_same_card [of nat])
   254   apply (auto simp add: inj_on_def bij_betw_def image_def)
   255   apply (metis dual_order.irrefl dual_order.strict_trans leI nat_1 transfer_nat_int_gcd(1))
   256   apply (metis One_nat_def int_0 int_1 int_less_0_conv int_nat_eq nat_int
   257     transfer_int_nat_gcd(1) zless_int)
   258   done
   259 
   260 lemma prime_phi:
   261   assumes "2 \<le> p" "phi p = p - 1"
   262   shows "prime p"
   263 proof -
   264   have "{x. 0 < x \<and> x < p \<and> coprime x p} = {1..p - 1}"
   265     using assms unfolding phi_def_nat
   266     by (intro card_seteq) fastforce+
   267   then have cop: "\<And>x::nat. x \<in> {1..p - 1} \<Longrightarrow> coprime x p"
   268     by blast
   269   have False if *: "1 < x" "x < p" and "x dvd p" for x :: nat
   270   proof -
   271     have "coprime x p"
   272       apply (rule cop)
   273       using * apply auto
   274       done
   275     with \<open>x dvd p\<close> \<open>1 < x\<close> show ?thesis
   276       by auto
   277   qed
   278   then show ?thesis
   279     using \<open>2 \<le> p\<close>
   280     by (simp add: prime_def)
   281        (metis One_nat_def dvd_pos_nat nat_dvd_not_less nat_neq_iff not_gr0
   282               not_numeral_le_zero one_dvd)
   283 qed
   284 
   285 lemma phi_zero [simp]: "phi 0 = 0"
   286   unfolding phi_def
   287 (* Auto hangs here. Once again, where is the simplification rule
   288    1 \<equiv> Suc 0 coming from? *)
   289   apply (auto simp add: card_eq_0_iff)
   290 (* Add card_eq_0_iff as a simp rule? delete card_empty_imp? *)
   291   done
   292 
   293 lemma phi_one [simp]: "phi 1 = 0"
   294   by (auto simp add: phi_def card_eq_0_iff)
   295 
   296 lemma (in residues) phi_eq: "phi m = card (Units R)"
   297   by (simp add: phi_def res_units_eq)
   298 
   299 lemma (in residues) euler_theorem1:
   300   assumes a: "gcd a m = 1"
   301   shows "[a^phi m = 1] (mod m)"
   302 proof -
   303   from a m_gt_one have [simp]: "a mod m \<in> Units R"
   304     by (intro mod_in_res_units)
   305   from phi_eq have "(a mod m) (^) (phi m) = (a mod m) (^) (card (Units R))"
   306     by simp
   307   also have "\<dots> = \<one>"
   308     by (intro units_power_order_eq_one) auto
   309   finally show ?thesis
   310     by (simp add: res_to_cong_simps)
   311 qed
   312 
   313 (* In fact, there is a two line proof!
   314 
   315 lemma (in residues) euler_theorem1:
   316   assumes a: "gcd a m = 1"
   317   shows "[a^phi m = 1] (mod m)"
   318 proof -
   319   have "(a mod m) (^) (phi m) = \<one>"
   320     by (simp add: phi_eq units_power_order_eq_one a m_gt_one)
   321   then show ?thesis
   322     by (simp add: res_to_cong_simps)
   323 qed
   324 
   325 *)
   326 
   327 (* outside the locale, we can relax the restriction m > 1 *)
   328 
   329 lemma euler_theorem:
   330   assumes "m \<ge> 0"
   331     and "gcd a m = 1"
   332   shows "[a^phi m = 1] (mod m)"
   333 proof (cases "m = 0 | m = 1")
   334   case True
   335   then show ?thesis by auto
   336 next
   337   case False
   338   with assms show ?thesis
   339     by (intro residues.euler_theorem1, unfold residues_def, auto)
   340 qed
   341 
   342 lemma (in residues_prime) phi_prime: "phi p = nat p - 1"
   343   apply (subst phi_eq)
   344   apply (subst res_prime_units_eq)
   345   apply auto
   346   done
   347 
   348 lemma phi_prime: "prime p \<Longrightarrow> phi p = nat p - 1"
   349   apply (rule residues_prime.phi_prime)
   350   apply (erule residues_prime.intro)
   351   done
   352 
   353 lemma fermat_theorem:
   354   fixes a :: int
   355   assumes "prime p"
   356     and "\<not> p dvd a"
   357   shows "[a^(p - 1) = 1] (mod p)"
   358 proof -
   359   from assms have "[a ^ phi p = 1] (mod p)"
   360     apply (intro euler_theorem)
   361     apply (metis of_nat_0_le_iff)
   362     apply (metis gcd_int.commute prime_imp_coprime_int)
   363     done
   364   also have "phi p = nat p - 1"
   365     by (rule phi_prime) (rule assms)
   366   finally show ?thesis
   367     by (metis nat_int)
   368 qed
   369 
   370 lemma fermat_theorem_nat:
   371   assumes "prime p" and "\<not> p dvd a"
   372   shows "[a ^ (p - 1) = 1] (mod p)"
   373   using fermat_theorem [of p a] assms
   374   by (metis int_1 of_nat_power transfer_int_nat_cong zdvd_int)
   375 
   376 
   377 subsection \<open>Wilson's theorem\<close>
   378 
   379 lemma (in field) inv_pair_lemma: "x \<in> Units R \<Longrightarrow> y \<in> Units R \<Longrightarrow>
   380     {x, inv x} \<noteq> {y, inv y} \<Longrightarrow> {x, inv x} \<inter> {y, inv y} = {}"
   381   apply auto
   382   apply (metis Units_inv_inv)+
   383   done
   384 
   385 lemma (in residues_prime) wilson_theorem1:
   386   assumes a: "p > 2"
   387   shows "[fact (p - 1) = (-1::int)] (mod p)"
   388 proof -
   389   let ?Inverse_Pairs = "{{x, inv x}| x. x \<in> Units R - {\<one>, \<ominus> \<one>}}"
   390   have UR: "Units R = {\<one>, \<ominus> \<one>} \<union> \<Union>?Inverse_Pairs"
   391     by auto
   392   have "(\<Otimes>i\<in>Units R. i) = (\<Otimes>i\<in>{\<one>, \<ominus> \<one>}. i) \<otimes> (\<Otimes>i\<in>\<Union>?Inverse_Pairs. i)"
   393     apply (subst UR)
   394     apply (subst finprod_Un_disjoint)
   395     apply (auto intro: funcsetI)
   396     using inv_one apply auto[1]
   397     using inv_eq_neg_one_eq apply auto
   398     done
   399   also have "(\<Otimes>i\<in>{\<one>, \<ominus> \<one>}. i) = \<ominus> \<one>"
   400     apply (subst finprod_insert)
   401     apply auto
   402     apply (frule one_eq_neg_one)
   403     using a apply force
   404     done
   405   also have "(\<Otimes>i\<in>(\<Union>?Inverse_Pairs). i) = (\<Otimes>A\<in>?Inverse_Pairs. (\<Otimes>y\<in>A. y))"
   406     apply (subst finprod_Union_disjoint)
   407     apply auto
   408     apply (metis Units_inv_inv)+
   409     done
   410   also have "\<dots> = \<one>"
   411     apply (rule finprod_one)
   412     apply auto
   413     apply (subst finprod_insert)
   414     apply auto
   415     apply (metis inv_eq_self)
   416     done
   417   finally have "(\<Otimes>i\<in>Units R. i) = \<ominus> \<one>"
   418     by simp
   419   also have "(\<Otimes>i\<in>Units R. i) = (\<Otimes>i\<in>Units R. i mod p)"
   420     apply (rule finprod_cong')
   421     apply auto
   422     apply (subst (asm) res_prime_units_eq)
   423     apply auto
   424     done
   425   also have "\<dots> = (\<Prod>i\<in>Units R. i) mod p"
   426     apply (rule prod_cong)
   427     apply auto
   428     done
   429   also have "\<dots> = fact (p - 1) mod p"
   430     apply (subst fact_altdef_nat)
   431     apply (insert assms)
   432     apply (subst res_prime_units_eq)
   433     apply (simp add: int_setprod zmod_int setprod_int_eq)
   434     done
   435   finally have "fact (p - 1) mod p = \<ominus> \<one>" .
   436   then show ?thesis
   437     by (metis of_nat_fact Divides.transfer_int_nat_functions(2) 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_altdef_nat)
   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