src/HOL/Old_Number_Theory/WilsonBij.thy
author huffman
Tue Sep 06 19:03:41 2011 -0700 (2011-09-06)
changeset 44766 d4d33a4d7548
parent 39159 0dec18004e75
child 45605 a89b4bc311a5
permissions -rw-r--r--
avoid using legacy theorem names
     1 (*  Title:      HOL/Old_Number_Theory/WilsonBij.thy
     2     Author:     Thomas M. Rasmussen
     3     Copyright   2000  University of Cambridge
     4 *)
     5 
     6 header {* Wilson's Theorem using a more abstract approach *}
     7 
     8 theory WilsonBij
     9 imports BijectionRel IntFact
    10 begin
    11 
    12 text {*
    13   Wilson's Theorem using a more ``abstract'' approach based on
    14   bijections between sets.  Does not use Fermat's Little Theorem
    15   (unlike Russinoff).
    16 *}
    17 
    18 
    19 subsection {* Definitions and lemmas *}
    20 
    21 definition reciR :: "int => int => int => bool"
    22   where "reciR p = (\<lambda>a b. zcong (a * b) 1 p \<and> 1 < a \<and> a < p - 1 \<and> 1 < b \<and> b < p - 1)"
    23 
    24 definition inv :: "int => int => int" where
    25   "inv p a =
    26     (if zprime p \<and> 0 < a \<and> a < p then
    27       (SOME x. 0 \<le> x \<and> x < p \<and> zcong (a * x) 1 p)
    28      else 0)"
    29 
    30 
    31 text {* \medskip Inverse *}
    32 
    33 lemma inv_correct:
    34   "zprime p ==> 0 < a ==> a < p
    35     ==> 0 \<le> inv p a \<and> inv p a < p \<and> [a * inv p a = 1] (mod p)"
    36   apply (unfold inv_def)
    37   apply (simp (no_asm_simp))
    38   apply (rule zcong_lineq_unique [THEN ex1_implies_ex, THEN someI_ex])
    39    apply (erule_tac [2] zless_zprime_imp_zrelprime)
    40     apply (unfold zprime_def)
    41     apply auto
    42   done
    43 
    44 lemmas inv_ge = inv_correct [THEN conjunct1, standard]
    45 lemmas inv_less = inv_correct [THEN conjunct2, THEN conjunct1, standard]
    46 lemmas inv_is_inv = inv_correct [THEN conjunct2, THEN conjunct2, standard]
    47 
    48 lemma inv_not_0:
    49   "zprime p ==> 1 < a ==> a < p - 1 ==> inv p a \<noteq> 0"
    50   -- {* same as @{text WilsonRuss} *}
    51   apply safe
    52   apply (cut_tac a = a and p = p in inv_is_inv)
    53      apply (unfold zcong_def)
    54      apply auto
    55   apply (subgoal_tac "\<not> p dvd 1")
    56    apply (rule_tac [2] zdvd_not_zless)
    57     apply (subgoal_tac "p dvd 1")
    58      prefer 2
    59      apply (subst dvd_minus_iff [symmetric])
    60      apply auto
    61   done
    62 
    63 lemma inv_not_1:
    64   "zprime p ==> 1 < a ==> a < p - 1 ==> inv p a \<noteq> 1"
    65   -- {* same as @{text WilsonRuss} *}
    66   apply safe
    67   apply (cut_tac a = a and p = p in inv_is_inv)
    68      prefer 4
    69      apply simp
    70      apply (subgoal_tac "a = 1")
    71       apply (rule_tac [2] zcong_zless_imp_eq)
    72           apply auto
    73   done
    74 
    75 lemma aux: "[a * (p - 1) = 1] (mod p) = [a = p - 1] (mod p)"
    76   -- {* same as @{text WilsonRuss} *}
    77   apply (unfold zcong_def)
    78   apply (simp add: diff_diff_eq diff_diff_eq2 right_diff_distrib)
    79   apply (rule_tac s = "p dvd -((a + 1) + (p * -a))" in trans)
    80    apply (simp add: algebra_simps)
    81   apply (subst dvd_minus_iff)
    82   apply (subst zdvd_reduce)
    83   apply (rule_tac s = "p dvd (a + 1) + (p * -1)" in trans)
    84    apply (subst zdvd_reduce)
    85    apply auto
    86   done
    87 
    88 lemma inv_not_p_minus_1:
    89   "zprime p ==> 1 < a ==> a < p - 1 ==> inv p a \<noteq> p - 1"
    90   -- {* same as @{text WilsonRuss} *}
    91   apply safe
    92   apply (cut_tac a = a and p = p in inv_is_inv)
    93      apply auto
    94   apply (simp add: aux)
    95   apply (subgoal_tac "a = p - 1")
    96    apply (rule_tac [2] zcong_zless_imp_eq)
    97        apply auto
    98   done
    99 
   100 text {*
   101   Below is slightly different as we don't expand @{term [source] inv}
   102   but use ``@{text correct}'' theorems.
   103 *}
   104 
   105 lemma inv_g_1: "zprime p ==> 1 < a ==> a < p - 1 ==> 1 < inv p a"
   106   apply (subgoal_tac "inv p a \<noteq> 1")
   107    apply (subgoal_tac "inv p a \<noteq> 0")
   108     apply (subst order_less_le)
   109     apply (subst zle_add1_eq_le [symmetric])
   110     apply (subst order_less_le)
   111     apply (rule_tac [2] inv_not_0)
   112       apply (rule_tac [5] inv_not_1)
   113         apply auto
   114   apply (rule inv_ge)
   115     apply auto
   116   done
   117 
   118 lemma inv_less_p_minus_1:
   119   "zprime p ==> 1 < a ==> a < p - 1 ==> inv p a < p - 1"
   120   -- {* ditto *}
   121   apply (subst order_less_le)
   122   apply (simp add: inv_not_p_minus_1 inv_less)
   123   done
   124 
   125 
   126 text {* \medskip Bijection *}
   127 
   128 lemma aux1: "1 < x ==> 0 \<le> (x::int)"
   129   apply auto
   130   done
   131 
   132 lemma aux2: "1 < x ==> 0 < (x::int)"
   133   apply auto
   134   done
   135 
   136 lemma aux3: "x \<le> p - 2 ==> x < (p::int)"
   137   apply auto
   138   done
   139 
   140 lemma aux4: "x \<le> p - 2 ==> x < (p::int) - 1"
   141   apply auto
   142   done
   143 
   144 lemma inv_inj: "zprime p ==> inj_on (inv p) (d22set (p - 2))"
   145   apply (unfold inj_on_def)
   146   apply auto
   147   apply (rule zcong_zless_imp_eq)
   148       apply (tactic {* stac (@{thm zcong_cancel} RS sym) 5 *})
   149         apply (rule_tac [7] zcong_trans)
   150          apply (tactic {* stac @{thm zcong_sym} 8 *})
   151          apply (erule_tac [7] inv_is_inv)
   152           apply (tactic "asm_simp_tac @{simpset} 9")
   153           apply (erule_tac [9] inv_is_inv)
   154            apply (rule_tac [6] zless_zprime_imp_zrelprime)
   155              apply (rule_tac [8] inv_less)
   156                apply (rule_tac [7] inv_g_1 [THEN aux2])
   157                  apply (unfold zprime_def)
   158                  apply (auto intro: d22set_g_1 d22set_le
   159                    aux1 aux2 aux3 aux4)
   160   done
   161 
   162 lemma inv_d22set_d22set:
   163     "zprime p ==> inv p ` d22set (p - 2) = d22set (p - 2)"
   164   apply (rule endo_inj_surj)
   165     apply (rule d22set_fin)
   166    apply (erule_tac [2] inv_inj)
   167   apply auto
   168   apply (rule d22set_mem)
   169    apply (erule inv_g_1)
   170     apply (subgoal_tac [3] "inv p xa < p - 1")
   171      apply (erule_tac [4] inv_less_p_minus_1)
   172       apply (auto intro: d22set_g_1 d22set_le aux4)
   173   done
   174 
   175 lemma d22set_d22set_bij:
   176     "zprime p ==> (d22set (p - 2), d22set (p - 2)) \<in> bijR (reciR p)"
   177   apply (unfold reciR_def)
   178   apply (rule_tac s = "(d22set (p - 2), inv p ` d22set (p - 2))" in subst)
   179    apply (simp add: inv_d22set_d22set)
   180   apply (rule inj_func_bijR)
   181     apply (rule_tac [3] d22set_fin)
   182    apply (erule_tac [2] inv_inj)
   183   apply auto
   184       apply (erule inv_is_inv)
   185        apply (erule_tac [5] inv_g_1)
   186         apply (erule_tac [7] inv_less_p_minus_1)
   187          apply (auto intro: d22set_g_1 d22set_le aux2 aux3 aux4)
   188   done
   189 
   190 lemma reciP_bijP: "zprime p ==> bijP (reciR p) (d22set (p - 2))"
   191   apply (unfold reciR_def bijP_def)
   192   apply auto
   193   apply (rule d22set_mem)
   194    apply auto
   195   done
   196 
   197 lemma reciP_uniq: "zprime p ==> uniqP (reciR p)"
   198   apply (unfold reciR_def uniqP_def)
   199   apply auto
   200    apply (rule zcong_zless_imp_eq)
   201        apply (tactic {* stac (@{thm zcong_cancel2} RS sym) 5 *})
   202          apply (rule_tac [7] zcong_trans)
   203           apply (tactic {* stac @{thm zcong_sym} 8 *})
   204           apply (rule_tac [6] zless_zprime_imp_zrelprime)
   205             apply auto
   206   apply (rule zcong_zless_imp_eq)
   207       apply (tactic {* stac (@{thm zcong_cancel} RS sym) 5 *})
   208         apply (rule_tac [7] zcong_trans)
   209          apply (tactic {* stac @{thm zcong_sym} 8 *})
   210          apply (rule_tac [6] zless_zprime_imp_zrelprime)
   211            apply auto
   212   done
   213 
   214 lemma reciP_sym: "zprime p ==> symP (reciR p)"
   215   apply (unfold reciR_def symP_def)
   216   apply (simp add: mult_commute)
   217   apply auto
   218   done
   219 
   220 lemma bijER_d22set: "zprime p ==> d22set (p - 2) \<in> bijER (reciR p)"
   221   apply (rule bijR_bijER)
   222      apply (erule d22set_d22set_bij)
   223     apply (erule reciP_bijP)
   224    apply (erule reciP_uniq)
   225   apply (erule reciP_sym)
   226   done
   227 
   228 
   229 subsection {* Wilson *}
   230 
   231 lemma bijER_zcong_prod_1:
   232     "zprime p ==> A \<in> bijER (reciR p) ==> [\<Prod>A = 1] (mod p)"
   233   apply (unfold reciR_def)
   234   apply (erule bijER.induct)
   235     apply (subgoal_tac [2] "a = 1 \<or> a = p - 1")
   236      apply (rule_tac [3] zcong_square_zless)
   237         apply auto
   238   apply (subst setprod_insert)
   239     prefer 3
   240     apply (subst setprod_insert)
   241       apply (auto simp add: fin_bijER)
   242   apply (subgoal_tac "zcong ((a * b) * \<Prod>A) (1 * 1) p")
   243    apply (simp add: mult_assoc)
   244   apply (rule zcong_zmult)
   245    apply auto
   246   done
   247 
   248 theorem Wilson_Bij: "zprime p ==> [zfact (p - 1) = -1] (mod p)"
   249   apply (subgoal_tac "zcong ((p - 1) * zfact (p - 2)) (-1 * 1) p")
   250    apply (rule_tac [2] zcong_zmult)
   251     apply (simp add: zprime_def)
   252     apply (subst zfact.simps)
   253     apply (rule_tac t = "p - 1 - 1" and s = "p - 2" in subst)
   254      apply auto
   255    apply (simp add: zcong_def)
   256   apply (subst d22set_prod_zfact [symmetric])
   257   apply (rule bijER_zcong_prod_1)
   258    apply (rule_tac [2] bijER_d22set)
   259    apply auto
   260   done
   261 
   262 end