src/HOL/Number_Theory/Residues.thy
author haftmann
Tue Sep 01 15:39:33 2009 +0200 (2009-09-01)
changeset 32479 521cc9bf2958
parent 31952 src/HOL/NewNumberTheory/Residues.thy@40501bb2d57c
child 35416 d8d7d1b785af
permissions -rw-r--r--
some reorganization of number theory
     1 (*  Title:      HOL/Library/Residues.thy
     2     ID:         
     3     Author:     Jeremy Avigad
     4 
     5     An algebraic treatment of residue rings, and resulting proofs of
     6     Euler's theorem and Wilson's theorem. 
     7 *)
     8 
     9 header {* Residue rings *}
    10 
    11 theory Residues
    12 imports
    13    UniqueFactorization
    14    Binomial
    15    MiscAlgebra
    16 begin
    17 
    18 
    19 (*
    20  
    21   A locale for residue rings
    22 
    23 *)
    24 
    25 constdefs 
    26   residue_ring :: "int => int ring"
    27   "residue_ring m == (| 
    28     carrier =       {0..m - 1}, 
    29     mult =          (%x y. (x * y) mod m),
    30     one =           1,
    31     zero =          0,
    32     add =           (%x y. (x + y) mod m) |)"
    33 
    34 locale residues =
    35   fixes m :: int and R (structure)
    36   assumes m_gt_one: "m > 1"
    37   defines "R == residue_ring m"
    38 
    39 context residues begin
    40 
    41 lemma abelian_group: "abelian_group R"
    42   apply (insert m_gt_one)
    43   apply (rule abelian_groupI)
    44   apply (unfold R_def residue_ring_def)
    45   apply (auto simp add: mod_pos_pos_trivial mod_add_right_eq [symmetric]
    46     add_ac)
    47   apply (case_tac "x = 0")
    48   apply force
    49   apply (subgoal_tac "(x + (m - x)) mod m = 0")
    50   apply (erule bexI)
    51   apply auto
    52 done
    53 
    54 lemma comm_monoid: "comm_monoid R"
    55   apply (insert m_gt_one)
    56   apply (unfold R_def residue_ring_def)
    57   apply (rule comm_monoidI)
    58   apply auto
    59   apply (subgoal_tac "x * y mod m * z mod m = z * (x * y mod m) mod m")
    60   apply (erule ssubst)
    61   apply (subst zmod_zmult1_eq [symmetric])+
    62   apply (simp_all only: mult_ac)
    63 done
    64 
    65 lemma cring: "cring R"
    66   apply (rule cringI)
    67   apply (rule abelian_group)
    68   apply (rule comm_monoid)
    69   apply (unfold R_def residue_ring_def, auto)
    70   apply (subst mod_add_eq [symmetric])
    71   apply (subst mult_commute)
    72   apply (subst zmod_zmult1_eq [symmetric])
    73   apply (simp add: ring_simps)
    74 done
    75 
    76 end
    77 
    78 sublocale residues < cring
    79   by (rule cring)
    80 
    81 
    82 context residues begin
    83 
    84 (* These lemmas translate back and forth between internal and 
    85    external concepts *)
    86 
    87 lemma res_carrier_eq: "carrier R = {0..m - 1}"
    88   by (unfold R_def residue_ring_def, auto)
    89 
    90 lemma res_add_eq: "x \<oplus> y = (x + y) mod m"
    91   by (unfold R_def residue_ring_def, auto)
    92 
    93 lemma res_mult_eq: "x \<otimes> y = (x * y) mod m"
    94   by (unfold R_def residue_ring_def, auto)
    95 
    96 lemma res_zero_eq: "\<zero> = 0"
    97   by (unfold R_def residue_ring_def, auto)
    98 
    99 lemma res_one_eq: "\<one> = 1"
   100   by (unfold R_def residue_ring_def units_of_def residue_ring_def, auto)
   101 
   102 lemma res_units_eq: "Units R = { x. 0 < x & x < m & coprime x m}"
   103   apply (insert m_gt_one)
   104   apply (unfold Units_def R_def residue_ring_def)
   105   apply auto
   106   apply (subgoal_tac "x ~= 0")
   107   apply auto
   108   apply (rule invertible_coprime_int)
   109   apply (subgoal_tac "x ~= 0")
   110   apply auto
   111   apply (subst (asm) coprime_iff_invertible'_int)
   112   apply (rule m_gt_one)
   113   apply (auto simp add: cong_int_def mult_commute)
   114 done
   115 
   116 lemma res_neg_eq: "\<ominus> x = (- x) mod m"
   117   apply (insert m_gt_one)
   118   apply (unfold R_def a_inv_def m_inv_def residue_ring_def)
   119   apply auto
   120   apply (rule the_equality)
   121   apply auto
   122   apply (subst mod_add_right_eq [symmetric])
   123   apply auto
   124   apply (subst mod_add_left_eq [symmetric])
   125   apply auto
   126   apply (subgoal_tac "y mod m = - x mod m")
   127   apply simp
   128   apply (subst zmod_eq_dvd_iff)
   129   apply auto
   130 done
   131 
   132 lemma finite [iff]: "finite(carrier R)"
   133   by (subst res_carrier_eq, auto)
   134 
   135 lemma finite_Units [iff]: "finite(Units R)"
   136   by (subst res_units_eq, auto)
   137 
   138 (* The function a -> a mod m maps the integers to the 
   139    residue classes. The following lemmas show that this mapping 
   140    respects addition and multiplication on the integers. *)
   141 
   142 lemma mod_in_carrier [iff]: "a mod m : carrier R"
   143   apply (unfold res_carrier_eq)
   144   apply (insert m_gt_one, auto)
   145 done
   146 
   147 lemma add_cong: "(x mod m) \<oplus> (y mod m) = (x + y) mod m"
   148   by (unfold R_def residue_ring_def, auto, arith)
   149 
   150 lemma mult_cong: "(x mod m) \<otimes> (y mod m) = (x * y) mod m"
   151   apply (unfold R_def residue_ring_def, auto)
   152   apply (subst zmod_zmult1_eq [symmetric])
   153   apply (subst mult_commute)
   154   apply (subst zmod_zmult1_eq [symmetric])
   155   apply (subst mult_commute)
   156   apply auto
   157 done  
   158 
   159 lemma zero_cong: "\<zero> = 0"
   160   apply (unfold R_def residue_ring_def, auto)
   161 done
   162 
   163 lemma one_cong: "\<one> = 1 mod m"
   164   apply (insert m_gt_one)
   165   apply (unfold R_def residue_ring_def, auto)
   166 done
   167 
   168 (* revise algebra library to use 1? *)
   169 lemma pow_cong: "(x mod m) (^) n = x^n mod m"
   170   apply (insert m_gt_one)
   171   apply (induct n)
   172   apply (auto simp add: nat_pow_def one_cong One_nat_def)
   173   apply (subst mult_commute)
   174   apply (rule mult_cong)
   175 done
   176 
   177 lemma neg_cong: "\<ominus> (x mod m) = (- x) mod m"
   178   apply (rule sym)
   179   apply (rule sum_zero_eq_neg)
   180   apply auto
   181   apply (subst add_cong)
   182   apply (subst zero_cong)
   183   apply auto
   184 done
   185 
   186 lemma (in residues) prod_cong: 
   187   "finite A \<Longrightarrow> (\<Otimes> i:A. (f i) mod m) = (PROD i:A. f i) mod m"
   188   apply (induct set: finite)
   189   apply (auto simp: one_cong mult_cong)
   190 done
   191 
   192 lemma (in residues) sum_cong:
   193   "finite A \<Longrightarrow> (\<Oplus> i:A. (f i) mod m) = (SUM i: A. f i) mod m"
   194   apply (induct set: finite)
   195   apply (auto simp: zero_cong add_cong)
   196 done
   197 
   198 lemma mod_in_res_units [simp]: "1 < m \<Longrightarrow> coprime a m \<Longrightarrow> 
   199     a mod m : Units R"
   200   apply (subst res_units_eq, auto)
   201   apply (insert pos_mod_sign [of m a])
   202   apply (subgoal_tac "a mod m ~= 0")
   203   apply arith
   204   apply auto
   205   apply (subst (asm) gcd_red_int)
   206   apply (subst gcd_commute_int, assumption)
   207 done
   208 
   209 lemma res_eq_to_cong: "((a mod m) = (b mod m)) = [a = b] (mod (m::int))" 
   210   unfolding cong_int_def by auto
   211 
   212 (* Simplifying with these will translate a ring equation in R to a 
   213    congruence. *)
   214 
   215 lemmas res_to_cong_simps = add_cong mult_cong pow_cong one_cong
   216     prod_cong sum_cong neg_cong res_eq_to_cong
   217 
   218 (* Other useful facts about the residue ring *)
   219 
   220 lemma one_eq_neg_one: "\<one> = \<ominus> \<one> \<Longrightarrow> m = 2"
   221   apply (simp add: res_one_eq res_neg_eq)
   222   apply (insert m_gt_one)
   223   apply (subgoal_tac "~(m > 2)")
   224   apply arith
   225   apply (rule notI)
   226   apply (subgoal_tac "-1 mod m = m - 1")
   227   apply force
   228   apply (subst mod_add_self2 [symmetric])
   229   apply (subst mod_pos_pos_trivial)
   230   apply auto
   231 done
   232 
   233 end
   234 
   235 
   236 (* prime residues *)
   237 
   238 locale residues_prime =
   239   fixes p :: int and R (structure)
   240   assumes p_prime [intro]: "prime p"
   241   defines "R == residue_ring p"
   242 
   243 sublocale residues_prime < residues p
   244   apply (unfold R_def residues_def)
   245   using p_prime apply auto
   246 done
   247 
   248 context residues_prime 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
   254     res_units_eq)
   255   apply (rule classical)
   256   apply (erule notE)
   257   apply (subst gcd_commute_int)
   258   apply (rule prime_imp_coprime_int)
   259   apply (rule p_prime)
   260   apply (rule notI)
   261   apply (frule zdvd_imp_le)
   262   apply auto
   263 done
   264 
   265 lemma res_prime_units_eq: "Units R = {1..p - 1}"
   266   apply (subst res_units_eq)
   267   apply auto
   268   apply (subst gcd_commute_int)
   269   apply (rule prime_imp_coprime_int)
   270   apply (rule p_prime)
   271   apply (rule zdvd_not_zless)
   272   apply auto
   273 done
   274 
   275 end
   276 
   277 sublocale residues_prime < field
   278   by (rule is_field)
   279 
   280 
   281 (*
   282   Test cases: Euler's theorem and Wilson's theorem.
   283 *)
   284 
   285 
   286 subsection{* Euler's theorem *}
   287 
   288 (* the definition of the phi function *)
   289 
   290 constdefs
   291   phi :: "int => nat"
   292   "phi m == card({ x. 0 < x & x < m & gcd x m = 1})" 
   293 
   294 lemma phi_zero [simp]: "phi 0 = 0"
   295   apply (subst phi_def)
   296 (* Auto hangs here. Once again, where is the simplification rule 
   297    1 == Suc 0 coming from? *)
   298   apply (auto simp add: card_eq_0_iff)
   299 (* Add card_eq_0_iff as a simp rule? delete card_empty_imp? *)
   300 done
   301 
   302 lemma phi_one [simp]: "phi 1 = 0"
   303   apply (auto simp add: phi_def card_eq_0_iff)
   304 done
   305 
   306 lemma (in residues) phi_eq: "phi m = card(Units R)"
   307   by (simp add: phi_def res_units_eq)
   308 
   309 lemma (in residues) euler_theorem1: 
   310   assumes a: "gcd a m = 1"
   311   shows "[a^phi m = 1] (mod m)"
   312 proof -
   313   from a m_gt_one have [simp]: "a mod m : Units R"
   314     by (intro mod_in_res_units)
   315   from phi_eq have "(a mod m) (^) (phi m) = (a mod m) (^) (card (Units R))"
   316     by simp
   317   also have "\<dots> = \<one>" 
   318     by (intro units_power_order_eq_one, auto)
   319   finally show ?thesis
   320     by (simp add: res_to_cong_simps)
   321 qed
   322 
   323 (* In fact, there is a two line proof!
   324 
   325 lemma (in residues) euler_theorem1: 
   326   assumes a: "gcd a m = 1"
   327   shows "[a^phi m = 1] (mod m)"
   328 proof -
   329   have "(a mod m) (^) (phi m) = \<one>"
   330     by (simp add: phi_eq units_power_order_eq_one a m_gt_one)
   331   thus ?thesis
   332     by (simp add: res_to_cong_simps)
   333 qed
   334 
   335 *)
   336 
   337 (* outside the locale, we can relax the restriction m > 1 *)
   338 
   339 lemma euler_theorem:
   340   assumes "m >= 0" and "gcd a m = 1"
   341   shows "[a^phi m = 1] (mod m)"
   342 proof (cases)
   343   assume "m = 0 | m = 1"
   344   thus ?thesis by auto
   345 next
   346   assume "~(m = 0 | m = 1)"
   347   with prems show ?thesis
   348     by (intro residues.euler_theorem1, unfold residues_def, auto)
   349 qed
   350 
   351 lemma (in residues_prime) phi_prime: "phi p = (nat p - 1)"
   352   apply (subst phi_eq)
   353   apply (subst res_prime_units_eq)
   354   apply auto
   355 done
   356 
   357 lemma phi_prime: "prime p \<Longrightarrow> phi p = (nat p - 1)"
   358   apply (rule residues_prime.phi_prime)
   359   apply (erule residues_prime.intro)
   360 done
   361 
   362 lemma fermat_theorem:
   363   assumes "prime p" and "~ (p dvd a)"
   364   shows "[a^(nat p - 1) = 1] (mod p)"
   365 proof -
   366   from prems have "[a^phi p = 1] (mod p)"
   367     apply (intro euler_theorem)
   368     (* auto should get this next part. matching across
   369        substitutions is needed. *)
   370     apply (frule prime_gt_1_int, arith)
   371     apply (subst gcd_commute_int, erule prime_imp_coprime_int, assumption)
   372     done
   373   also have "phi p = nat p - 1"
   374     by (rule phi_prime, rule prems)
   375   finally show ?thesis .
   376 qed
   377 
   378 
   379 subsection {* Wilson's theorem *}
   380 
   381 lemma (in field) inv_pair_lemma: "x : Units R \<Longrightarrow> y : Units R \<Longrightarrow> 
   382   {x, inv x} ~= {y, inv y} \<Longrightarrow> {x, inv x} Int {y, inv y} = {}" 
   383   apply auto
   384   apply (erule notE)
   385   apply (erule inv_eq_imp_eq)
   386   apply auto
   387   apply (erule notE)
   388   apply (erule inv_eq_imp_eq)
   389   apply auto
   390 done
   391 
   392 lemma (in residues_prime) wilson_theorem1:
   393   assumes a: "p > 2"
   394   shows "[fact (p - 1) = - 1] (mod p)"
   395 proof -
   396   let ?InversePairs = "{ {x, inv x} | x. x : Units R - {\<one>, \<ominus> \<one>}}" 
   397   have UR: "Units R = {\<one>, \<ominus> \<one>} Un (Union ?InversePairs)"
   398     by auto
   399   have "(\<Otimes>i: Units R. i) = 
   400     (\<Otimes>i: {\<one>, \<ominus> \<one>}. i) \<otimes> (\<Otimes>i: Union ?InversePairs. i)"
   401     apply (subst UR)
   402     apply (subst finprod_Un_disjoint)
   403     apply (auto intro:funcsetI)
   404     apply (drule sym, subst (asm) inv_eq_one_eq)
   405     apply auto
   406     apply (drule sym, subst (asm) inv_eq_neg_one_eq)
   407     apply auto
   408     done
   409   also have "(\<Otimes>i: {\<one>, \<ominus> \<one>}. i) = \<ominus> \<one>"
   410     apply (subst finprod_insert)
   411     apply auto
   412     apply (frule one_eq_neg_one)
   413     apply (insert a, force)
   414     done
   415   also have "(\<Otimes>i:(Union ?InversePairs). i) = 
   416       (\<Otimes> A: ?InversePairs. (\<Otimes> y:A. y))"
   417     apply (subst finprod_Union_disjoint)
   418     apply force
   419     apply force
   420     apply clarify
   421     apply (rule inv_pair_lemma)
   422     apply auto
   423     done
   424   also have "\<dots> = \<one>"
   425     apply (rule finprod_one)
   426     apply auto
   427     apply (subst finprod_insert)
   428     apply auto
   429     apply (frule inv_eq_self)
   430     apply (auto)
   431     done
   432   finally have "(\<Otimes>i: Units R. i) = \<ominus> \<one>"
   433     by simp
   434   also have "(\<Otimes>i: Units R. i) = (\<Otimes>i: Units R. i mod p)"
   435     apply (rule finprod_cong')
   436     apply (auto)
   437     apply (subst (asm) res_prime_units_eq)
   438     apply auto
   439     done
   440   also have "\<dots> = (PROD i: Units R. i) mod p"
   441     apply (rule prod_cong)
   442     apply auto
   443     done
   444   also have "\<dots> = fact (p - 1) mod p"
   445     apply (subst fact_altdef_int)
   446     apply (insert prems, force)
   447     apply (subst res_prime_units_eq, rule refl)
   448     done
   449   finally have "fact (p - 1) mod p = \<ominus> \<one>".
   450   thus ?thesis
   451     by (simp add: res_to_cong_simps)
   452 qed
   453 
   454 lemma wilson_theorem: "prime (p::int) \<Longrightarrow> [fact (p - 1) = - 1] (mod p)"
   455   apply (frule prime_gt_1_int)
   456   apply (case_tac "p = 2")
   457   apply (subst fact_altdef_int, simp)
   458   apply (subst cong_int_def)
   459   apply simp
   460   apply (rule residues_prime.wilson_theorem1)
   461   apply (rule residues_prime.intro)
   462   apply auto
   463 done
   464 
   465 
   466 end