src/HOL/Old_Number_Theory/WilsonRuss.thy
 author krauss Tue Mar 02 12:26:50 2010 +0100 (2010-03-02) changeset 35440 bdf8ad377877 parent 35048 82ab78fff970 child 38159 e9b4835a54ee permissions -rw-r--r--
killed more recdefs
```     1 (*  Author:     Thomas M. Rasmussen
```
```     2     Copyright   2000  University of Cambridge
```
```     3 *)
```
```     4
```
```     5 header {* Wilson's Theorem according to Russinoff *}
```
```     6
```
```     7 theory WilsonRuss imports EulerFermat begin
```
```     8
```
```     9 text {*
```
```    10   Wilson's Theorem following quite closely Russinoff's approach
```
```    11   using Boyer-Moore (using finite sets instead of lists, though).
```
```    12 *}
```
```    13
```
```    14 subsection {* Definitions and lemmas *}
```
```    15
```
```    16 definition
```
```    17   inv :: "int => int => int" where
```
```    18   "inv p a = (a^(nat (p - 2))) mod p"
```
```    19
```
```    20 fun
```
```    21   wset :: "int \<Rightarrow> int => int set"
```
```    22 where
```
```    23   "wset a p =
```
```    24     (if 1 < a then
```
```    25       let ws = wset (a - 1) p
```
```    26       in (if a \<in> ws then ws else insert a (insert (inv p a) ws)) else {})"
```
```    27
```
```    28
```
```    29 text {* \medskip @{term [source] inv} *}
```
```    30
```
```    31 lemma inv_is_inv_aux: "1 < m ==> Suc (nat (m - 2)) = nat (m - 1)"
```
```    32 by (subst int_int_eq [symmetric], auto)
```
```    33
```
```    34 lemma inv_is_inv:
```
```    35     "zprime p \<Longrightarrow> 0 < a \<Longrightarrow> a < p ==> [a * inv p a = 1] (mod p)"
```
```    36   apply (unfold inv_def)
```
```    37   apply (subst zcong_zmod)
```
```    38   apply (subst zmod_zmult1_eq [symmetric])
```
```    39   apply (subst zcong_zmod [symmetric])
```
```    40   apply (subst power_Suc [symmetric])
```
```    41   apply (subst inv_is_inv_aux)
```
```    42    apply (erule_tac [2] Little_Fermat)
```
```    43    apply (erule_tac [2] zdvd_not_zless)
```
```    44    apply (unfold zprime_def, auto)
```
```    45   done
```
```    46
```
```    47 lemma inv_distinct:
```
```    48     "zprime p \<Longrightarrow> 1 < a \<Longrightarrow> a < p - 1 ==> a \<noteq> inv p a"
```
```    49   apply safe
```
```    50   apply (cut_tac a = a and p = p in zcong_square)
```
```    51      apply (cut_tac [3] a = a and p = p in inv_is_inv, auto)
```
```    52    apply (subgoal_tac "a = 1")
```
```    53     apply (rule_tac [2] m = p in zcong_zless_imp_eq)
```
```    54         apply (subgoal_tac [7] "a = p - 1")
```
```    55          apply (rule_tac [8] m = p in zcong_zless_imp_eq, auto)
```
```    56   done
```
```    57
```
```    58 lemma inv_not_0:
```
```    59     "zprime p \<Longrightarrow> 1 < a \<Longrightarrow> a < p - 1 ==> inv p a \<noteq> 0"
```
```    60   apply safe
```
```    61   apply (cut_tac a = a and p = p in inv_is_inv)
```
```    62      apply (unfold zcong_def, auto)
```
```    63   apply (subgoal_tac "\<not> p dvd 1")
```
```    64    apply (rule_tac [2] zdvd_not_zless)
```
```    65     apply (subgoal_tac "p dvd 1")
```
```    66      prefer 2
```
```    67      apply (subst dvd_minus_iff [symmetric], auto)
```
```    68   done
```
```    69
```
```    70 lemma inv_not_1:
```
```    71     "zprime p \<Longrightarrow> 1 < a \<Longrightarrow> a < p - 1 ==> inv p a \<noteq> 1"
```
```    72   apply safe
```
```    73   apply (cut_tac a = a and p = p in inv_is_inv)
```
```    74      prefer 4
```
```    75      apply simp
```
```    76      apply (subgoal_tac "a = 1")
```
```    77       apply (rule_tac [2] zcong_zless_imp_eq, auto)
```
```    78   done
```
```    79
```
```    80 lemma inv_not_p_minus_1_aux:
```
```    81     "[a * (p - 1) = 1] (mod p) = [a = p - 1] (mod p)"
```
```    82   apply (unfold zcong_def)
```
```    83   apply (simp add: diff_diff_eq diff_diff_eq2 zdiff_zmult_distrib2)
```
```    84   apply (rule_tac s = "p dvd -((a + 1) + (p * -a))" in trans)
```
```    85    apply (simp add: algebra_simps)
```
```    86   apply (subst dvd_minus_iff)
```
```    87   apply (subst zdvd_reduce)
```
```    88   apply (rule_tac s = "p dvd (a + 1) + (p * -1)" in trans)
```
```    89    apply (subst zdvd_reduce, auto)
```
```    90   done
```
```    91
```
```    92 lemma inv_not_p_minus_1:
```
```    93     "zprime p \<Longrightarrow> 1 < a \<Longrightarrow> a < p - 1 ==> inv p a \<noteq> p - 1"
```
```    94   apply safe
```
```    95   apply (cut_tac a = a and p = p in inv_is_inv, auto)
```
```    96   apply (simp add: inv_not_p_minus_1_aux)
```
```    97   apply (subgoal_tac "a = p - 1")
```
```    98    apply (rule_tac [2] zcong_zless_imp_eq, auto)
```
```    99   done
```
```   100
```
```   101 lemma inv_g_1:
```
```   102     "zprime p \<Longrightarrow> 1 < a \<Longrightarrow> a < p - 1 ==> 1 < inv p a"
```
```   103   apply (case_tac "0\<le> inv p a")
```
```   104    apply (subgoal_tac "inv p a \<noteq> 1")
```
```   105     apply (subgoal_tac "inv p a \<noteq> 0")
```
```   106      apply (subst order_less_le)
```
```   107      apply (subst zle_add1_eq_le [symmetric])
```
```   108      apply (subst order_less_le)
```
```   109      apply (rule_tac [2] inv_not_0)
```
```   110        apply (rule_tac [5] inv_not_1, auto)
```
```   111   apply (unfold inv_def zprime_def, simp)
```
```   112   done
```
```   113
```
```   114 lemma inv_less_p_minus_1:
```
```   115     "zprime p \<Longrightarrow> 1 < a \<Longrightarrow> a < p - 1 ==> inv p a < p - 1"
```
```   116   apply (case_tac "inv p a < p")
```
```   117    apply (subst order_less_le)
```
```   118    apply (simp add: inv_not_p_minus_1, auto)
```
```   119   apply (unfold inv_def zprime_def, simp)
```
```   120   done
```
```   121
```
```   122 lemma inv_inv_aux: "5 \<le> p ==>
```
```   123     nat (p - 2) * nat (p - 2) = Suc (nat (p - 1) * nat (p - 3))"
```
```   124   apply (subst int_int_eq [symmetric])
```
```   125   apply (simp add: zmult_int [symmetric])
```
```   126   apply (simp add: zdiff_zmult_distrib zdiff_zmult_distrib2)
```
```   127   done
```
```   128
```
```   129 lemma zcong_zpower_zmult:
```
```   130     "[x^y = 1] (mod p) \<Longrightarrow> [x^(y * z) = 1] (mod p)"
```
```   131   apply (induct z)
```
```   132    apply (auto simp add: zpower_zadd_distrib)
```
```   133   apply (subgoal_tac "zcong (x^y * x^(y * z)) (1 * 1) p")
```
```   134    apply (rule_tac [2] zcong_zmult, simp_all)
```
```   135   done
```
```   136
```
```   137 lemma inv_inv: "zprime p \<Longrightarrow>
```
```   138     5 \<le> p \<Longrightarrow> 0 < a \<Longrightarrow> a < p ==> inv p (inv p a) = a"
```
```   139   apply (unfold inv_def)
```
```   140   apply (subst zpower_zmod)
```
```   141   apply (subst zpower_zpower)
```
```   142   apply (rule zcong_zless_imp_eq)
```
```   143       prefer 5
```
```   144       apply (subst zcong_zmod)
```
```   145       apply (subst mod_mod_trivial)
```
```   146       apply (subst zcong_zmod [symmetric])
```
```   147       apply (subst inv_inv_aux)
```
```   148        apply (subgoal_tac [2]
```
```   149          "zcong (a * a^(nat (p - 1) * nat (p - 3))) (a * 1) p")
```
```   150         apply (rule_tac [3] zcong_zmult)
```
```   151          apply (rule_tac [4] zcong_zpower_zmult)
```
```   152          apply (erule_tac [4] Little_Fermat)
```
```   153          apply (rule_tac [4] zdvd_not_zless, simp_all)
```
```   154   done
```
```   155
```
```   156
```
```   157 text {* \medskip @{term wset} *}
```
```   158
```
```   159 declare wset.simps [simp del]
```
```   160
```
```   161 lemma wset_induct:
```
```   162   assumes "!!a p. P {} a p"
```
```   163     and "!!a p. 1 < (a::int) \<Longrightarrow>
```
```   164       P (wset (a - 1) p) (a - 1) p ==> P (wset a p) a p"
```
```   165   shows "P (wset u v) u v"
```
```   166   apply (rule wset.induct)
```
```   167   apply (case_tac "1 < a")
```
```   168    apply (rule assms)
```
```   169     apply (simp_all add: wset.simps assms)
```
```   170   done
```
```   171
```
```   172 lemma wset_mem_imp_or [rule_format]:
```
```   173   "1 < a \<Longrightarrow> b \<notin> wset (a - 1) p
```
```   174     ==> b \<in> wset a p --> b = a \<or> b = inv p a"
```
```   175   apply (subst wset.simps)
```
```   176   apply (unfold Let_def, simp)
```
```   177   done
```
```   178
```
```   179 lemma wset_mem_mem [simp]: "1 < a ==> a \<in> wset a p"
```
```   180   apply (subst wset.simps)
```
```   181   apply (unfold Let_def, simp)
```
```   182   done
```
```   183
```
```   184 lemma wset_subset: "1 < a \<Longrightarrow> b \<in> wset (a - 1) p ==> b \<in> wset a p"
```
```   185   apply (subst wset.simps)
```
```   186   apply (unfold Let_def, auto)
```
```   187   done
```
```   188
```
```   189 lemma wset_g_1 [rule_format]:
```
```   190     "zprime p --> a < p - 1 --> b \<in> wset a p --> 1 < b"
```
```   191   apply (induct a p rule: wset_induct, auto)
```
```   192   apply (case_tac "b = a")
```
```   193    apply (case_tac [2] "b = inv p a")
```
```   194     apply (subgoal_tac [3] "b = a \<or> b = inv p a")
```
```   195      apply (rule_tac [4] wset_mem_imp_or)
```
```   196        prefer 2
```
```   197        apply simp
```
```   198        apply (rule inv_g_1, auto)
```
```   199   done
```
```   200
```
```   201 lemma wset_less [rule_format]:
```
```   202     "zprime p --> a < p - 1 --> b \<in> wset a p --> b < p - 1"
```
```   203   apply (induct a p rule: wset_induct, auto)
```
```   204   apply (case_tac "b = a")
```
```   205    apply (case_tac [2] "b = inv p a")
```
```   206     apply (subgoal_tac [3] "b = a \<or> b = inv p a")
```
```   207      apply (rule_tac [4] wset_mem_imp_or)
```
```   208        prefer 2
```
```   209        apply simp
```
```   210        apply (rule inv_less_p_minus_1, auto)
```
```   211   done
```
```   212
```
```   213 lemma wset_mem [rule_format]:
```
```   214   "zprime p -->
```
```   215     a < p - 1 --> 1 < b --> b \<le> a --> b \<in> wset a p"
```
```   216   apply (induct a p rule: wset.induct, auto)
```
```   217   apply (rule_tac wset_subset)
```
```   218   apply (simp (no_asm_simp))
```
```   219   apply auto
```
```   220   done
```
```   221
```
```   222 lemma wset_mem_inv_mem [rule_format]:
```
```   223   "zprime p --> 5 \<le> p --> a < p - 1 --> b \<in> wset a p
```
```   224     --> inv p b \<in> wset a p"
```
```   225   apply (induct a p rule: wset_induct, auto)
```
```   226    apply (case_tac "b = a")
```
```   227     apply (subst wset.simps)
```
```   228     apply (unfold Let_def)
```
```   229     apply (rule_tac [3] wset_subset, auto)
```
```   230   apply (case_tac "b = inv p a")
```
```   231    apply (simp (no_asm_simp))
```
```   232    apply (subst inv_inv)
```
```   233        apply (subgoal_tac [6] "b = a \<or> b = inv p a")
```
```   234         apply (rule_tac [7] wset_mem_imp_or, auto)
```
```   235   done
```
```   236
```
```   237 lemma wset_inv_mem_mem:
```
```   238   "zprime p \<Longrightarrow> 5 \<le> p \<Longrightarrow> a < p - 1 \<Longrightarrow> 1 < b \<Longrightarrow> b < p - 1
```
```   239     \<Longrightarrow> inv p b \<in> wset a p \<Longrightarrow> b \<in> wset a p"
```
```   240   apply (rule_tac s = "inv p (inv p b)" and t = b in subst)
```
```   241    apply (rule_tac [2] wset_mem_inv_mem)
```
```   242       apply (rule inv_inv, simp_all)
```
```   243   done
```
```   244
```
```   245 lemma wset_fin: "finite (wset a p)"
```
```   246   apply (induct a p rule: wset_induct)
```
```   247    prefer 2
```
```   248    apply (subst wset.simps)
```
```   249    apply (unfold Let_def, auto)
```
```   250   done
```
```   251
```
```   252 lemma wset_zcong_prod_1 [rule_format]:
```
```   253   "zprime p -->
```
```   254     5 \<le> p --> a < p - 1 --> [(\<Prod>x\<in>wset a p. x) = 1] (mod p)"
```
```   255   apply (induct a p rule: wset_induct)
```
```   256    prefer 2
```
```   257    apply (subst wset.simps)
```
```   258    apply (auto, unfold Let_def, auto)
```
```   259   apply (subst setprod_insert)
```
```   260     apply (tactic {* stac (thm "setprod_insert") 3 *})
```
```   261       apply (subgoal_tac [5]
```
```   262         "zcong (a * inv p a * (\<Prod>x\<in>wset (a - 1) p. x)) (1 * 1) p")
```
```   263        prefer 5
```
```   264        apply (simp add: zmult_assoc)
```
```   265       apply (rule_tac [5] zcong_zmult)
```
```   266        apply (rule_tac [5] inv_is_inv)
```
```   267          apply (tactic "clarify_tac @{claset} 4")
```
```   268          apply (subgoal_tac [4] "a \<in> wset (a - 1) p")
```
```   269           apply (rule_tac [5] wset_inv_mem_mem)
```
```   270                apply (simp_all add: wset_fin)
```
```   271   apply (rule inv_distinct, auto)
```
```   272   done
```
```   273
```
```   274 lemma d22set_eq_wset: "zprime p ==> d22set (p - 2) = wset (p - 2) p"
```
```   275   apply safe
```
```   276    apply (erule wset_mem)
```
```   277      apply (rule_tac [2] d22set_g_1)
```
```   278      apply (rule_tac [3] d22set_le)
```
```   279      apply (rule_tac [4] d22set_mem)
```
```   280       apply (erule_tac [4] wset_g_1)
```
```   281        prefer 6
```
```   282        apply (subst zle_add1_eq_le [symmetric])
```
```   283        apply (subgoal_tac "p - 2 + 1 = p - 1")
```
```   284         apply (simp (no_asm_simp))
```
```   285         apply (erule wset_less, auto)
```
```   286   done
```
```   287
```
```   288
```
```   289 subsection {* Wilson *}
```
```   290
```
```   291 lemma prime_g_5: "zprime p \<Longrightarrow> p \<noteq> 2 \<Longrightarrow> p \<noteq> 3 ==> 5 \<le> p"
```
```   292   apply (unfold zprime_def dvd_def)
```
```   293   apply (case_tac "p = 4", auto)
```
```   294    apply (rule notE)
```
```   295     prefer 2
```
```   296     apply assumption
```
```   297    apply (simp (no_asm))
```
```   298    apply (rule_tac x = 2 in exI)
```
```   299    apply (safe, arith)
```
```   300      apply (rule_tac x = 2 in exI, auto)
```
```   301   done
```
```   302
```
```   303 theorem Wilson_Russ:
```
```   304     "zprime p ==> [zfact (p - 1) = -1] (mod p)"
```
```   305   apply (subgoal_tac "[(p - 1) * zfact (p - 2) = -1 * 1] (mod p)")
```
```   306    apply (rule_tac [2] zcong_zmult)
```
```   307     apply (simp only: zprime_def)
```
```   308     apply (subst zfact.simps)
```
```   309     apply (rule_tac t = "p - 1 - 1" and s = "p - 2" in subst, auto)
```
```   310    apply (simp only: zcong_def)
```
```   311    apply (simp (no_asm_simp))
```
```   312   apply (case_tac "p = 2")
```
```   313    apply (simp add: zfact.simps)
```
```   314   apply (case_tac "p = 3")
```
```   315    apply (simp add: zfact.simps)
```
```   316   apply (subgoal_tac "5 \<le> p")
```
```   317    apply (erule_tac [2] prime_g_5)
```
```   318     apply (subst d22set_prod_zfact [symmetric])
```
```   319     apply (subst d22set_eq_wset)
```
```   320      apply (rule_tac [2] wset_zcong_prod_1, auto)
```
```   321   done
```
```   322
```
```   323 end
```