src/HOL/Number_Theory/Residues.thy
author paulson <lp15@cam.ac.uk>
Mon Feb 03 00:22:48 2014 +0000 (2014-02-03)
changeset 55262 16724746ad89
parent 55261 ad3604df6bc6
child 55352 1d2852dfc4a7
permissions -rw-r--r--
fixed indentation
     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 header {* Residue rings *}
     9 
    10 theory Residues
    11 imports
    12   UniqueFactorization
    13   Binomial
    14   MiscAlgebra
    15 begin
    16 
    17 (*
    18 
    19   A locale for residue rings
    20 
    21 *)
    22 
    23 definition residue_ring :: "int => int ring" where
    24   "residue_ring m == (|
    25     carrier =       {0..m - 1},
    26     mult =          (%x y. (x * y) mod m),
    27     one =           1,
    28     zero =          0,
    29     add =           (%x y. (x + y) mod m) |)"
    30 
    31 locale residues =
    32   fixes m :: int and R (structure)
    33   assumes m_gt_one: "m > 1"
    34   defines "R == residue_ring m"
    35 
    36 context residues
    37 begin
    38 
    39 lemma abelian_group: "abelian_group R"
    40   apply (insert m_gt_one)
    41   apply (rule abelian_groupI)
    42   apply (unfold R_def residue_ring_def)
    43   apply (auto simp add: mod_add_right_eq [symmetric] add_ac)
    44   apply (case_tac "x = 0")
    45   apply force
    46   apply (subgoal_tac "(x + (m - x)) mod m = 0")
    47   apply (erule bexI)
    48   apply auto
    49   done
    50 
    51 lemma comm_monoid: "comm_monoid R"
    52   apply (insert m_gt_one)
    53   apply (unfold R_def residue_ring_def)
    54   apply (rule comm_monoidI)
    55   apply auto
    56   apply (subgoal_tac "x * y mod m * z mod m = z * (x * y mod m) mod m")
    57   apply (erule ssubst)
    58   apply (subst mod_mult_right_eq [symmetric])+
    59   apply (simp_all only: mult_ac)
    60   done
    61 
    62 lemma cring: "cring R"
    63   apply (rule cringI)
    64   apply (rule abelian_group)
    65   apply (rule comm_monoid)
    66   apply (unfold R_def residue_ring_def, auto)
    67   apply (subst mod_add_eq [symmetric])
    68   apply (subst mult_commute)
    69   apply (subst mod_mult_right_eq [symmetric])
    70   apply (simp add: field_simps)
    71   done
    72 
    73 end
    74 
    75 sublocale residues < cring
    76   by (rule cring)
    77 
    78 
    79 context residues
    80 begin
    81 
    82 (* These lemmas translate back and forth between internal and
    83    external concepts *)
    84 
    85 lemma res_carrier_eq: "carrier R = {0..m - 1}"
    86   unfolding R_def residue_ring_def by auto
    87 
    88 lemma res_add_eq: "x \<oplus> y = (x + y) mod m"
    89   unfolding R_def residue_ring_def by auto
    90 
    91 lemma res_mult_eq: "x \<otimes> y = (x * y) mod m"
    92   unfolding R_def residue_ring_def by auto
    93 
    94 lemma res_zero_eq: "\<zero> = 0"
    95   unfolding R_def residue_ring_def by auto
    96 
    97 lemma res_one_eq: "\<one> = 1"
    98   unfolding R_def residue_ring_def units_of_def by auto
    99 
   100 lemma res_units_eq: "Units R = { x. 0 < x & x < m & coprime x m}"
   101   apply (insert m_gt_one)
   102   apply (unfold Units_def R_def residue_ring_def)
   103   apply auto
   104   apply (subgoal_tac "x ~= 0")
   105   apply auto
   106   apply (rule invertible_coprime_int)
   107   apply (subgoal_tac "x ~= 0")
   108   apply auto
   109   apply (subst (asm) coprime_iff_invertible'_int)
   110   apply arith
   111   apply (auto simp add: cong_int_def mult_commute)
   112   done
   113 
   114 lemma res_neg_eq: "\<ominus> x = (- x) mod m"
   115   apply (insert m_gt_one)
   116   apply (unfold R_def a_inv_def m_inv_def residue_ring_def)
   117   apply auto
   118   apply (rule the_equality)
   119   apply auto
   120   apply (subst mod_add_right_eq [symmetric])
   121   apply auto
   122   apply (subst mod_add_left_eq [symmetric])
   123   apply auto
   124   apply (subgoal_tac "y mod m = - x mod m")
   125   apply simp
   126   apply (subst zmod_eq_dvd_iff)
   127   apply auto
   128   done
   129 
   130 lemma finite [iff]: "finite (carrier R)"
   131   by (subst res_carrier_eq, auto)
   132 
   133 lemma finite_Units [iff]: "finite (Units R)"
   134   by (subst res_units_eq) auto
   135 
   136 (* The function a -> a mod m maps the integers to the
   137    residue classes. The following lemmas show that this mapping
   138    respects addition and multiplication on the integers. *)
   139 
   140 lemma mod_in_carrier [iff]: "a mod m : carrier R"
   141   apply (unfold res_carrier_eq)
   142   apply (insert m_gt_one, auto)
   143   done
   144 
   145 lemma add_cong: "(x mod m) \<oplus> (y mod m) = (x + y) mod m"
   146   unfolding R_def residue_ring_def
   147   apply auto
   148   apply presburger
   149   done
   150 
   151 lemma mult_cong: "(x mod m) \<otimes> (y mod m) = (x * y) mod m"
   152   apply (unfold R_def residue_ring_def, auto)
   153   apply (subst mod_mult_right_eq [symmetric])
   154   apply (subst mult_commute)
   155   apply (subst mod_mult_right_eq [symmetric])
   156   apply (subst mult_commute)
   157   apply auto
   158   done
   159 
   160 lemma zero_cong: "\<zero> = 0"
   161   unfolding R_def residue_ring_def by auto
   162 
   163 lemma one_cong: "\<one> = 1 mod m"
   164   using m_gt_one unfolding R_def residue_ring_def by auto
   165 
   166 (* revise algebra library to use 1? *)
   167 lemma pow_cong: "(x mod m) (^) n = x^n mod m"
   168   apply (insert m_gt_one)
   169   apply (induct n)
   170   apply (auto simp add: nat_pow_def one_cong)
   171   apply (subst mult_commute)
   172   apply (rule mult_cong)
   173   done
   174 
   175 lemma neg_cong: "\<ominus> (x mod m) = (- x) mod m"
   176   apply (rule sym)
   177   apply (rule sum_zero_eq_neg)
   178   apply auto
   179   apply (subst add_cong)
   180   apply (subst zero_cong)
   181   apply auto
   182   done
   183 
   184 lemma (in residues) prod_cong:
   185     "finite A \<Longrightarrow> (\<Otimes> i:A. (f i) mod m) = (PROD i:A. f i) mod m"
   186   apply (induct set: finite)
   187   apply (auto simp: one_cong mult_cong)
   188   done
   189 
   190 lemma (in residues) sum_cong:
   191     "finite A \<Longrightarrow> (\<Oplus> i:A. (f i) mod m) = (SUM i: A. f i) mod m"
   192   apply (induct set: finite)
   193   apply (auto simp: zero_cong add_cong)
   194   done
   195 
   196 lemma mod_in_res_units [simp]: "1 < m \<Longrightarrow> coprime a m \<Longrightarrow>
   197     a mod m : Units R"
   198   apply (subst res_units_eq, auto)
   199   apply (insert pos_mod_sign [of m a])
   200   apply (subgoal_tac "a mod m ~= 0")
   201   apply arith
   202   apply auto
   203   apply (subst (asm) gcd_red_int)
   204   apply (subst gcd_commute_int, assumption)
   205   done
   206 
   207 lemma res_eq_to_cong: "((a mod m) = (b mod m)) = [a = b] (mod (m::int))"
   208   unfolding cong_int_def by auto
   209 
   210 (* Simplifying with these will translate a ring equation in R to a
   211    congruence. *)
   212 
   213 lemmas res_to_cong_simps = add_cong mult_cong pow_cong one_cong
   214     prod_cong sum_cong neg_cong res_eq_to_cong
   215 
   216 (* Other useful facts about the residue ring *)
   217 
   218 lemma one_eq_neg_one: "\<one> = \<ominus> \<one> \<Longrightarrow> m = 2"
   219   apply (simp add: res_one_eq res_neg_eq)
   220   apply (insert m_gt_one)
   221   apply (subgoal_tac "~(m > 2)")
   222   apply arith
   223   apply (rule notI)
   224   apply (subgoal_tac "-1 mod m = m - 1")
   225   apply force
   226   apply (subst mod_add_self2 [symmetric])
   227   apply (subst mod_pos_pos_trivial)
   228   apply auto
   229   done
   230 
   231 end
   232 
   233 
   234 (* prime residues *)
   235 
   236 locale residues_prime =
   237   fixes p and R (structure)
   238   assumes p_prime [intro]: "prime p"
   239   defines "R == residue_ring p"
   240 
   241 sublocale residues_prime < residues p
   242   apply (unfold R_def residues_def)
   243   using p_prime apply auto
   244   apply (metis (full_types) int_1 of_nat_less_iff prime_gt_1_nat)
   245   done
   246 
   247 context residues_prime
   248 begin
   249 
   250 lemma is_field: "field R"
   251   apply (rule cring.field_intro2)
   252   apply (rule cring)
   253   apply (auto simp add: res_carrier_eq res_one_eq res_zero_eq res_units_eq)
   254   apply (rule classical)
   255   apply (erule notE)
   256   apply (subst gcd_commute_int)
   257   apply (rule prime_imp_coprime_int)
   258   apply (rule p_prime)
   259   apply (rule notI)
   260   apply (frule zdvd_imp_le)
   261   apply auto
   262   done
   263 
   264 lemma res_prime_units_eq: "Units R = {1..p - 1}"
   265   apply (subst res_units_eq)
   266   apply auto
   267   apply (subst gcd_commute_int)
   268   apply (rule prime_imp_coprime_int)
   269   apply (rule p_prime)
   270   apply (rule zdvd_not_zless)
   271   apply auto
   272   done
   273 
   274 end
   275 
   276 sublocale residues_prime < field
   277   by (rule is_field)
   278 
   279 
   280 (*
   281   Test cases: Euler's theorem and Wilson's theorem.
   282 *)
   283 
   284 
   285 subsection{* Euler's theorem *}
   286 
   287 (* the definition of the phi function *)
   288 
   289 definition phi :: "int => nat"
   290   where "phi m = card({ x. 0 < x & x < m & gcd x m = 1})"
   291 
   292 lemma phi_def_nat: "phi m = card({ x. 0 < x & x < nat m & gcd x (nat m) = 1})"
   293   apply (simp add: phi_def)
   294   apply (rule bij_betw_same_card [of nat])
   295   apply (auto simp add: inj_on_def bij_betw_def image_def)
   296   apply (metis dual_order.irrefl dual_order.strict_trans leI nat_1 transfer_nat_int_gcd(1))
   297   apply (metis One_nat_def int_0 int_1 int_less_0_conv int_nat_eq nat_int transfer_int_nat_gcd(1) zless_int)
   298   done
   299 
   300 lemma prime_phi:
   301   assumes  "2 \<le> p" "phi p = p - 1" shows "prime p"
   302 proof -
   303   have "{x. 0 < x \<and> x < p \<and> coprime x p} = {1..p - 1}"
   304     using assms unfolding phi_def_nat
   305     by (intro card_seteq) fastforce+
   306   then have cop: "\<And>x. x \<in> {1::nat..p - 1} \<Longrightarrow> coprime x p"
   307     by blast
   308   { fix x::nat assume *: "1 < x" "x < p" and "x dvd p"
   309     have "coprime x p" 
   310       apply (rule cop)
   311       using * apply auto
   312       done
   313     with `x dvd p` `1 < x` have "False" by auto }
   314   then show ?thesis 
   315     using `2 \<le> p` 
   316     by (simp add: prime_def)
   317        (metis One_nat_def dvd_pos_nat nat_dvd_not_less nat_neq_iff not_gr0 not_numeral_le_zero one_dvd)
   318 qed
   319 
   320 lemma phi_zero [simp]: "phi 0 = 0"
   321   apply (subst phi_def)
   322 (* Auto hangs here. Once again, where is the simplification rule
   323    1 == Suc 0 coming from? *)
   324   apply (auto simp add: card_eq_0_iff)
   325 (* Add card_eq_0_iff as a simp rule? delete card_empty_imp? *)
   326   done
   327 
   328 lemma phi_one [simp]: "phi 1 = 0"
   329   by (auto simp add: phi_def card_eq_0_iff)
   330 
   331 lemma (in residues) phi_eq: "phi m = card(Units R)"
   332   by (simp add: phi_def res_units_eq)
   333 
   334 lemma (in residues) euler_theorem1:
   335   assumes a: "gcd a m = 1"
   336   shows "[a^phi m = 1] (mod m)"
   337 proof -
   338   from a m_gt_one have [simp]: "a mod m : Units R"
   339     by (intro mod_in_res_units)
   340   from phi_eq have "(a mod m) (^) (phi m) = (a mod m) (^) (card (Units R))"
   341     by simp
   342   also have "\<dots> = \<one>"
   343     by (intro units_power_order_eq_one, auto)
   344   finally show ?thesis
   345     by (simp add: res_to_cong_simps)
   346 qed
   347 
   348 (* In fact, there is a two line proof!
   349 
   350 lemma (in residues) euler_theorem1:
   351   assumes a: "gcd a m = 1"
   352   shows "[a^phi m = 1] (mod m)"
   353 proof -
   354   have "(a mod m) (^) (phi m) = \<one>"
   355     by (simp add: phi_eq units_power_order_eq_one a m_gt_one)
   356   then show ?thesis
   357     by (simp add: res_to_cong_simps)
   358 qed
   359 
   360 *)
   361 
   362 (* outside the locale, we can relax the restriction m > 1 *)
   363 
   364 lemma euler_theorem:
   365   assumes "m >= 0" and "gcd a m = 1"
   366   shows "[a^phi m = 1] (mod m)"
   367 proof (cases)
   368   assume "m = 0 | m = 1"
   369   then show ?thesis by auto
   370 next
   371   assume "~(m = 0 | m = 1)"
   372   with assms show ?thesis
   373     by (intro residues.euler_theorem1, unfold residues_def, auto)
   374 qed
   375 
   376 lemma (in residues_prime) phi_prime: "phi p = (nat p - 1)"
   377   apply (subst phi_eq)
   378   apply (subst res_prime_units_eq)
   379   apply auto
   380   done
   381 
   382 lemma phi_prime: "prime p \<Longrightarrow> phi p = (nat p - 1)"
   383   apply (rule residues_prime.phi_prime)
   384   apply (erule residues_prime.intro)
   385   done
   386 
   387 lemma fermat_theorem:
   388   fixes a::int
   389   assumes "prime p" and "~ (p dvd a)"
   390   shows "[a^(p - 1) = 1] (mod p)"
   391 proof -
   392   from assms have "[a^phi p = 1] (mod p)"
   393     apply (intro euler_theorem)
   394     apply (metis of_nat_0_le_iff)
   395     apply (metis gcd_int.commute prime_imp_coprime_int)
   396     done
   397   also have "phi p = nat p - 1"
   398     by (rule phi_prime, rule assms)
   399   finally show ?thesis
   400     by (metis nat_int) 
   401 qed
   402 
   403 lemma fermat_theorem_nat:
   404   assumes "prime p" and "~ (p dvd a)"
   405   shows "[a^(p - 1) = 1] (mod p)"
   406 using fermat_theorem [of p a] assms
   407 by (metis int_1 of_nat_power transfer_int_nat_cong zdvd_int)
   408 
   409 
   410 subsection {* Wilson's theorem *}
   411 
   412 lemma (in field) inv_pair_lemma: "x : Units R \<Longrightarrow> y : Units R \<Longrightarrow>
   413     {x, inv x} ~= {y, inv y} \<Longrightarrow> {x, inv x} Int {y, inv y} = {}"
   414   apply auto
   415   apply (erule notE)
   416   apply (erule inv_eq_imp_eq)
   417   apply auto
   418   apply (erule notE)
   419   apply (erule inv_eq_imp_eq)
   420   apply auto
   421   done
   422 
   423 lemma (in residues_prime) wilson_theorem1:
   424   assumes a: "p > 2"
   425   shows "[fact (p - 1) = - 1] (mod p)"
   426 proof -
   427   let ?InversePairs = "{ {x, inv x} | x. x : Units R - {\<one>, \<ominus> \<one>}}"
   428   have UR: "Units R = {\<one>, \<ominus> \<one>} Un (Union ?InversePairs)"
   429     by auto
   430   have "(\<Otimes>i: Units R. i) =
   431     (\<Otimes>i: {\<one>, \<ominus> \<one>}. i) \<otimes> (\<Otimes>i: Union ?InversePairs. i)"
   432     apply (subst UR)
   433     apply (subst finprod_Un_disjoint)
   434     apply (auto intro:funcsetI)
   435     apply (drule sym, subst (asm) inv_eq_one_eq)
   436     apply auto
   437     apply (drule sym, subst (asm) inv_eq_neg_one_eq)
   438     apply auto
   439     done
   440   also have "(\<Otimes>i: {\<one>, \<ominus> \<one>}. i) = \<ominus> \<one>"
   441     apply (subst finprod_insert)
   442     apply auto
   443     apply (frule one_eq_neg_one)
   444     apply (insert a, force)
   445     done
   446   also have "(\<Otimes>i:(Union ?InversePairs). i) =
   447       (\<Otimes>A: ?InversePairs. (\<Otimes>y:A. y))"
   448     apply (subst finprod_Union_disjoint)
   449     apply force
   450     apply force
   451     apply clarify
   452     apply (rule inv_pair_lemma)
   453     apply auto
   454     done
   455   also have "\<dots> = \<one>"
   456     apply (rule finprod_one)
   457     apply auto
   458     apply (subst finprod_insert)
   459     apply auto
   460     apply (frule inv_eq_self)
   461     apply (auto)
   462     done
   463   finally have "(\<Otimes>i: Units R. i) = \<ominus> \<one>"
   464     by simp
   465   also have "(\<Otimes>i: Units R. i) = (\<Otimes>i: Units R. i mod p)"
   466     apply (rule finprod_cong')
   467     apply (auto)
   468     apply (subst (asm) res_prime_units_eq)
   469     apply auto
   470     done
   471   also have "\<dots> = (PROD i: Units R. i) mod p"
   472     apply (rule prod_cong)
   473     apply auto
   474     done
   475   also have "\<dots> = fact (p - 1) mod p"
   476     apply (subst fact_altdef_nat)
   477     apply (insert assms)
   478     apply (subst res_prime_units_eq)
   479     apply (simp add: int_setprod zmod_int setprod_int_eq)
   480     done
   481   finally have "fact (p - 1) mod p = \<ominus> \<one>".
   482   then show ?thesis
   483     by (metis Divides.transfer_int_nat_functions(2) cong_int_def res_neg_eq res_one_eq)
   484 qed
   485 
   486 lemma wilson_theorem: "prime p \<Longrightarrow> [fact (p - 1) = - 1] (mod p)"
   487   apply (frule prime_gt_1_nat)
   488   apply (case_tac "p = 2")
   489   apply (subst fact_altdef_nat, simp)
   490   apply (subst cong_int_def)
   491   apply simp
   492   apply (rule residues_prime.wilson_theorem1)
   493   apply (rule residues_prime.intro)
   494   apply auto
   495   done
   496 
   497 end