src/HOL/Old_Number_Theory/WilsonBij.thy
 author wenzelm Mon Sep 06 19:13:10 2010 +0200 (2010-09-06) changeset 39159 0dec18004e75 parent 38159 e9b4835a54ee child 44766 d4d33a4d7548 permissions -rw-r--r--
more antiquotations;
```     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 zdiff_zmult_distrib2)
```
```    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: zmult_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: zmult_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
```