src/HOL/NumberTheory/WilsonBij.thy
 author haftmann Wed Sep 26 20:27:55 2007 +0200 (2007-09-26) changeset 24728 e2b3a1065676 parent 23894 1a4167d761ac child 30042 31039ee583fa permissions -rw-r--r--
moved Finite_Set before Datatype
```     1 (*  Title:      HOL/NumberTheory/WilsonBij.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Thomas M. Rasmussen
```
```     4     Copyright   2000  University of Cambridge
```
```     5 *)
```
```     6
```
```     7 header {* Wilson's Theorem using a more abstract approach *}
```
```     8
```
```     9 theory WilsonBij imports BijectionRel IntFact begin
```
```    10
```
```    11 text {*
```
```    12   Wilson's Theorem using a more ``abstract'' approach based on
```
```    13   bijections between sets.  Does not use Fermat's Little Theorem
```
```    14   (unlike Russinoff).
```
```    15 *}
```
```    16
```
```    17
```
```    18 subsection {* Definitions and lemmas *}
```
```    19
```
```    20 definition
```
```    21   reciR :: "int => int => int => bool" where
```
```    22   "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
```
```    25   inv :: "int => int => int" where
```
```    26   "inv p a =
```
```    27     (if zprime p \<and> 0 < a \<and> a < p then
```
```    28       (SOME x. 0 \<le> x \<and> x < p \<and> zcong (a * x) 1 p)
```
```    29      else 0)"
```
```    30
```
```    31
```
```    32 text {* \medskip Inverse *}
```
```    33
```
```    34 lemma inv_correct:
```
```    35   "zprime p ==> 0 < a ==> a < p
```
```    36     ==> 0 \<le> inv p a \<and> inv p a < p \<and> [a * inv p a = 1] (mod p)"
```
```    37   apply (unfold inv_def)
```
```    38   apply (simp (no_asm_simp))
```
```    39   apply (rule zcong_lineq_unique [THEN ex1_implies_ex, THEN someI_ex])
```
```    40    apply (erule_tac [2] zless_zprime_imp_zrelprime)
```
```    41     apply (unfold zprime_def)
```
```    42     apply auto
```
```    43   done
```
```    44
```
```    45 lemmas inv_ge = inv_correct [THEN conjunct1, standard]
```
```    46 lemmas inv_less = inv_correct [THEN conjunct2, THEN conjunct1, standard]
```
```    47 lemmas inv_is_inv = inv_correct [THEN conjunct2, THEN conjunct2, standard]
```
```    48
```
```    49 lemma inv_not_0:
```
```    50   "zprime p ==> 1 < a ==> a < p - 1 ==> inv p a \<noteq> 0"
```
```    51   -- {* same as @{text WilsonRuss} *}
```
```    52   apply safe
```
```    53   apply (cut_tac a = a and p = p in inv_is_inv)
```
```    54      apply (unfold zcong_def)
```
```    55      apply auto
```
```    56   apply (subgoal_tac "\<not> p dvd 1")
```
```    57    apply (rule_tac [2] zdvd_not_zless)
```
```    58     apply (subgoal_tac "p dvd 1")
```
```    59      prefer 2
```
```    60      apply (subst zdvd_zminus_iff [symmetric])
```
```    61      apply auto
```
```    62   done
```
```    63
```
```    64 lemma inv_not_1:
```
```    65   "zprime p ==> 1 < a ==> a < p - 1 ==> inv p a \<noteq> 1"
```
```    66   -- {* same as @{text WilsonRuss} *}
```
```    67   apply safe
```
```    68   apply (cut_tac a = a and p = p in inv_is_inv)
```
```    69      prefer 4
```
```    70      apply simp
```
```    71      apply (subgoal_tac "a = 1")
```
```    72       apply (rule_tac [2] zcong_zless_imp_eq)
```
```    73           apply auto
```
```    74   done
```
```    75
```
```    76 lemma aux: "[a * (p - 1) = 1] (mod p) = [a = p - 1] (mod p)"
```
```    77   -- {* same as @{text WilsonRuss} *}
```
```    78   apply (unfold zcong_def)
```
```    79   apply (simp add: OrderedGroup.diff_diff_eq diff_diff_eq2 zdiff_zmult_distrib2)
```
```    80   apply (rule_tac s = "p dvd -((a + 1) + (p * -a))" in trans)
```
```    81    apply (simp add: mult_commute)
```
```    82   apply (subst zdvd_zminus_iff)
```
```    83   apply (subst zdvd_reduce)
```
```    84   apply (rule_tac s = "p dvd (a + 1) + (p * -1)" in trans)
```
```    85    apply (subst zdvd_reduce)
```
```    86    apply auto
```
```    87   done
```
```    88
```
```    89 lemma inv_not_p_minus_1:
```
```    90   "zprime p ==> 1 < a ==> a < p - 1 ==> inv p a \<noteq> p - 1"
```
```    91   -- {* same as @{text WilsonRuss} *}
```
```    92   apply safe
```
```    93   apply (cut_tac a = a and p = p in inv_is_inv)
```
```    94      apply auto
```
```    95   apply (simp add: aux)
```
```    96   apply (subgoal_tac "a = p - 1")
```
```    97    apply (rule_tac [2] zcong_zless_imp_eq)
```
```    98        apply auto
```
```    99   done
```
```   100
```
```   101 text {*
```
```   102   Below is slightly different as we don't expand @{term [source] inv}
```
```   103   but use ``@{text correct}'' theorems.
```
```   104 *}
```
```   105
```
```   106 lemma inv_g_1: "zprime p ==> 1 < a ==> a < p - 1 ==> 1 < inv p a"
```
```   107   apply (subgoal_tac "inv p a \<noteq> 1")
```
```   108    apply (subgoal_tac "inv p a \<noteq> 0")
```
```   109     apply (subst order_less_le)
```
```   110     apply (subst zle_add1_eq_le [symmetric])
```
```   111     apply (subst order_less_le)
```
```   112     apply (rule_tac [2] inv_not_0)
```
```   113       apply (rule_tac [5] inv_not_1)
```
```   114         apply auto
```
```   115   apply (rule inv_ge)
```
```   116     apply auto
```
```   117   done
```
```   118
```
```   119 lemma inv_less_p_minus_1:
```
```   120   "zprime p ==> 1 < a ==> a < p - 1 ==> inv p a < p - 1"
```
```   121   -- {* ditto *}
```
```   122   apply (subst order_less_le)
```
```   123   apply (simp add: inv_not_p_minus_1 inv_less)
```
```   124   done
```
```   125
```
```   126
```
```   127 text {* \medskip Bijection *}
```
```   128
```
```   129 lemma aux1: "1 < x ==> 0 \<le> (x::int)"
```
```   130   apply auto
```
```   131   done
```
```   132
```
```   133 lemma aux2: "1 < x ==> 0 < (x::int)"
```
```   134   apply auto
```
```   135   done
```
```   136
```
```   137 lemma aux3: "x \<le> p - 2 ==> x < (p::int)"
```
```   138   apply auto
```
```   139   done
```
```   140
```
```   141 lemma aux4: "x \<le> p - 2 ==> x < (p::int) - 1"
```
```   142   apply auto
```
```   143   done
```
```   144
```
```   145 lemma inv_inj: "zprime p ==> inj_on (inv p) (d22set (p - 2))"
```
```   146   apply (unfold inj_on_def)
```
```   147   apply auto
```
```   148   apply (rule zcong_zless_imp_eq)
```
```   149       apply (tactic {* stac (thm "zcong_cancel" RS sym) 5 *})
```
```   150         apply (rule_tac [7] zcong_trans)
```
```   151          apply (tactic {* stac (thm "zcong_sym") 8 *})
```
```   152          apply (erule_tac [7] inv_is_inv)
```
```   153           apply (tactic "asm_simp_tac @{simpset} 9")
```
```   154           apply (erule_tac [9] inv_is_inv)
```
```   155            apply (rule_tac [6] zless_zprime_imp_zrelprime)
```
```   156              apply (rule_tac [8] inv_less)
```
```   157                apply (rule_tac [7] inv_g_1 [THEN aux2])
```
```   158                  apply (unfold zprime_def)
```
```   159                  apply (auto intro: d22set_g_1 d22set_le
```
```   160 		   aux1 aux2 aux3 aux4)
```
```   161   done
```
```   162
```
```   163 lemma inv_d22set_d22set:
```
```   164     "zprime p ==> inv p ` d22set (p - 2) = d22set (p - 2)"
```
```   165   apply (rule endo_inj_surj)
```
```   166     apply (rule d22set_fin)
```
```   167    apply (erule_tac [2] inv_inj)
```
```   168   apply auto
```
```   169   apply (rule d22set_mem)
```
```   170    apply (erule inv_g_1)
```
```   171     apply (subgoal_tac [3] "inv p xa < p - 1")
```
```   172      apply (erule_tac [4] inv_less_p_minus_1)
```
```   173       apply (auto intro: d22set_g_1 d22set_le aux4)
```
```   174   done
```
```   175
```
```   176 lemma d22set_d22set_bij:
```
```   177     "zprime p ==> (d22set (p - 2), d22set (p - 2)) \<in> bijR (reciR p)"
```
```   178   apply (unfold reciR_def)
```
```   179   apply (rule_tac s = "(d22set (p - 2), inv p ` d22set (p - 2))" in subst)
```
```   180    apply (simp add: inv_d22set_d22set)
```
```   181   apply (rule inj_func_bijR)
```
```   182     apply (rule_tac [3] d22set_fin)
```
```   183    apply (erule_tac [2] inv_inj)
```
```   184   apply auto
```
```   185       apply (erule inv_is_inv)
```
```   186        apply (erule_tac [5] inv_g_1)
```
```   187         apply (erule_tac [7] inv_less_p_minus_1)
```
```   188          apply (auto intro: d22set_g_1 d22set_le aux2 aux3 aux4)
```
```   189   done
```
```   190
```
```   191 lemma reciP_bijP: "zprime p ==> bijP (reciR p) (d22set (p - 2))"
```
```   192   apply (unfold reciR_def bijP_def)
```
```   193   apply auto
```
```   194   apply (rule d22set_mem)
```
```   195    apply auto
```
```   196   done
```
```   197
```
```   198 lemma reciP_uniq: "zprime p ==> uniqP (reciR p)"
```
```   199   apply (unfold reciR_def uniqP_def)
```
```   200   apply auto
```
```   201    apply (rule zcong_zless_imp_eq)
```
```   202        apply (tactic {* stac (thm "zcong_cancel2" RS sym) 5 *})
```
```   203          apply (rule_tac [7] zcong_trans)
```
```   204           apply (tactic {* stac (thm "zcong_sym") 8 *})
```
```   205           apply (rule_tac [6] zless_zprime_imp_zrelprime)
```
```   206             apply auto
```
```   207   apply (rule zcong_zless_imp_eq)
```
```   208       apply (tactic {* stac (thm "zcong_cancel" RS sym) 5 *})
```
```   209         apply (rule_tac [7] zcong_trans)
```
```   210          apply (tactic {* stac (thm "zcong_sym") 8 *})
```
```   211          apply (rule_tac [6] zless_zprime_imp_zrelprime)
```
```   212            apply auto
```
```   213   done
```
```   214
```
```   215 lemma reciP_sym: "zprime p ==> symP (reciR p)"
```
```   216   apply (unfold reciR_def symP_def)
```
```   217   apply (simp add: zmult_commute)
```
```   218   apply auto
```
```   219   done
```
```   220
```
```   221 lemma bijER_d22set: "zprime p ==> d22set (p - 2) \<in> bijER (reciR p)"
```
```   222   apply (rule bijR_bijER)
```
```   223      apply (erule d22set_d22set_bij)
```
```   224     apply (erule reciP_bijP)
```
```   225    apply (erule reciP_uniq)
```
```   226   apply (erule reciP_sym)
```
```   227   done
```
```   228
```
```   229
```
```   230 subsection {* Wilson *}
```
```   231
```
```   232 lemma bijER_zcong_prod_1:
```
```   233     "zprime p ==> A \<in> bijER (reciR p) ==> [\<Prod>A = 1] (mod p)"
```
```   234   apply (unfold reciR_def)
```
```   235   apply (erule bijER.induct)
```
```   236     apply (subgoal_tac [2] "a = 1 \<or> a = p - 1")
```
```   237      apply (rule_tac [3] zcong_square_zless)
```
```   238         apply auto
```
```   239   apply (subst setprod_insert)
```
```   240     prefer 3
```
```   241     apply (subst setprod_insert)
```
```   242       apply (auto simp add: fin_bijER)
```
```   243   apply (subgoal_tac "zcong ((a * b) * \<Prod>A) (1 * 1) p")
```
```   244    apply (simp add: zmult_assoc)
```
```   245   apply (rule zcong_zmult)
```
```   246    apply auto
```
```   247   done
```
```   248
```
```   249 theorem Wilson_Bij: "zprime p ==> [zfact (p - 1) = -1] (mod p)"
```
```   250   apply (subgoal_tac "zcong ((p - 1) * zfact (p - 2)) (-1 * 1) p")
```
```   251    apply (rule_tac [2] zcong_zmult)
```
```   252     apply (simp add: zprime_def)
```
```   253     apply (subst zfact.simps)
```
```   254     apply (rule_tac t = "p - 1 - 1" and s = "p - 2" in subst)
```
```   255      apply auto
```
```   256    apply (simp add: zcong_def)
```
```   257   apply (subst d22set_prod_zfact [symmetric])
```
```   258   apply (rule bijER_zcong_prod_1)
```
```   259    apply (rule_tac [2] bijER_d22set)
```
```   260    apply auto
```
```   261   done
```
```   262
```
```   263 end
```