src/HOL/Number_Theory/Residues.thy
changeset 32479 521cc9bf2958
parent 31952 40501bb2d57c
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32478:87201c60ae7d 32479:521cc9bf2958
       
     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