src/HOL/Old_Number_Theory/WilsonRuss.thy
 author paulson Fri Nov 13 12:27:13 2015 +0000 (2015-11-13) changeset 61649 268d88ec9087 parent 61382 efac889fccbc child 61694 6571c78c9667 permissions -rw-r--r--
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
```     1 (*  Title:      HOL/Old_Number_Theory/WilsonRuss.thy
```
```     2     Author:     Thomas M. Rasmussen
```
```     3     Copyright   2000  University of Cambridge
```
```     4 *)
```
```     5
```
```     6 section \<open>Wilson's Theorem according to Russinoff\<close>
```
```     7
```
```     8 theory WilsonRuss
```
```     9 imports EulerFermat
```
```    10 begin
```
```    11
```
```    12 text \<open>
```
```    13   Wilson's Theorem following quite closely Russinoff's approach
```
```    14   using Boyer-Moore (using finite sets instead of lists, though).
```
```    15 \<close>
```
```    16
```
```    17 subsection \<open>Definitions and lemmas\<close>
```
```    18
```
```    19 definition inv :: "int => int => int"
```
```    20   where "inv p a = (a^(nat (p - 2))) mod p"
```
```    21
```
```    22 fun wset :: "int \<Rightarrow> int => int set" 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 \<open>\medskip @{term [source] inv}\<close>
```
```    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 mod_mult_right_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   done
```
```    64
```
```    65 lemma inv_not_1:
```
```    66     "zprime p \<Longrightarrow> 1 < a \<Longrightarrow> a < p - 1 ==> inv p a \<noteq> 1"
```
```    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, auto)
```
```    73   done
```
```    74
```
```    75 lemma inv_not_p_minus_1_aux:
```
```    76     "[a * (p - 1) = 1] (mod p) = [a = p - 1] (mod p)"
```
```    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, auto)
```
```    85   done
```
```    86
```
```    87 lemma inv_not_p_minus_1:
```
```    88     "zprime p \<Longrightarrow> 1 < a \<Longrightarrow> a < p - 1 ==> inv p a \<noteq> p - 1"
```
```    89   apply safe
```
```    90   apply (cut_tac a = a and p = p in inv_is_inv, auto)
```
```    91   apply (simp add: inv_not_p_minus_1_aux)
```
```    92   apply (subgoal_tac "a = p - 1")
```
```    93    apply (rule_tac [2] zcong_zless_imp_eq, auto)
```
```    94   done
```
```    95
```
```    96 lemma inv_g_1:
```
```    97     "zprime p \<Longrightarrow> 1 < a \<Longrightarrow> a < p - 1 ==> 1 < inv p a"
```
```    98   apply (case_tac "0\<le> inv p a")
```
```    99    apply (subgoal_tac "inv p a \<noteq> 1")
```
```   100     apply (subgoal_tac "inv p a \<noteq> 0")
```
```   101      apply (subst order_less_le)
```
```   102      apply (subst zle_add1_eq_le [symmetric])
```
```   103      apply (subst order_less_le)
```
```   104      apply (rule_tac [2] inv_not_0)
```
```   105        apply (rule_tac [5] inv_not_1, auto)
```
```   106   apply (unfold inv_def zprime_def, simp)
```
```   107   done
```
```   108
```
```   109 lemma inv_less_p_minus_1:
```
```   110     "zprime p \<Longrightarrow> 1 < a \<Longrightarrow> a < p - 1 ==> inv p a < p - 1"
```
```   111   apply (case_tac "inv p a < p")
```
```   112    apply (subst order_less_le)
```
```   113    apply (simp add: inv_not_p_minus_1, auto)
```
```   114   apply (unfold inv_def zprime_def, simp)
```
```   115   done
```
```   116
```
```   117 lemma inv_inv_aux: "5 \<le> p ==>
```
```   118     nat (p - 2) * nat (p - 2) = Suc (nat (p - 1) * nat (p - 3))"
```
```   119   apply (subst int_int_eq [symmetric])
```
```   120   apply (simp add: of_nat_mult)
```
```   121   apply (simp add: left_diff_distrib right_diff_distrib)
```
```   122   done
```
```   123
```
```   124 lemma zcong_zpower_zmult:
```
```   125     "[x^y = 1] (mod p) \<Longrightarrow> [x^(y * z) = 1] (mod p)"
```
```   126   apply (induct z)
```
```   127    apply (auto simp add: power_add)
```
```   128   apply (subgoal_tac "zcong (x^y * x^(y * z)) (1 * 1) p")
```
```   129    apply (rule_tac [2] zcong_zmult, simp_all)
```
```   130   done
```
```   131
```
```   132 lemma inv_inv: "zprime p \<Longrightarrow>
```
```   133     5 \<le> p \<Longrightarrow> 0 < a \<Longrightarrow> a < p ==> inv p (inv p a) = a"
```
```   134   apply (unfold inv_def)
```
```   135   apply (subst power_mod)
```
```   136   apply (subst power_mult [symmetric])
```
```   137   apply (rule zcong_zless_imp_eq)
```
```   138       prefer 5
```
```   139       apply (subst zcong_zmod)
```
```   140       apply (subst mod_mod_trivial)
```
```   141       apply (subst zcong_zmod [symmetric])
```
```   142       apply (subst inv_inv_aux)
```
```   143        apply (subgoal_tac [2]
```
```   144          "zcong (a * a^(nat (p - 1) * nat (p - 3))) (a * 1) p")
```
```   145         apply (rule_tac [3] zcong_zmult)
```
```   146          apply (rule_tac [4] zcong_zpower_zmult)
```
```   147          apply (erule_tac [4] Little_Fermat)
```
```   148          apply (rule_tac [4] zdvd_not_zless, simp_all)
```
```   149   done
```
```   150
```
```   151
```
```   152 text \<open>\medskip @{term wset}\<close>
```
```   153
```
```   154 declare wset.simps [simp del]
```
```   155
```
```   156 lemma wset_induct:
```
```   157   assumes "!!a p. P {} a p"
```
```   158     and "!!a p. 1 < (a::int) \<Longrightarrow>
```
```   159       P (wset (a - 1) p) (a - 1) p ==> P (wset a p) a p"
```
```   160   shows "P (wset u v) u v"
```
```   161   apply (rule wset.induct)
```
```   162   apply (case_tac "1 < a")
```
```   163    apply (rule assms)
```
```   164     apply (simp_all add: wset.simps assms)
```
```   165   done
```
```   166
```
```   167 lemma wset_mem_imp_or [rule_format]:
```
```   168   "1 < a \<Longrightarrow> b \<notin> wset (a - 1) p
```
```   169     ==> b \<in> wset a p --> b = a \<or> b = inv p a"
```
```   170   apply (subst wset.simps)
```
```   171   apply (unfold Let_def, simp)
```
```   172   done
```
```   173
```
```   174 lemma wset_mem_mem [simp]: "1 < a ==> a \<in> wset a p"
```
```   175   apply (subst wset.simps)
```
```   176   apply (unfold Let_def, simp)
```
```   177   done
```
```   178
```
```   179 lemma wset_subset: "1 < a \<Longrightarrow> b \<in> wset (a - 1) p ==> b \<in> wset a p"
```
```   180   apply (subst wset.simps)
```
```   181   apply (unfold Let_def, auto)
```
```   182   done
```
```   183
```
```   184 lemma wset_g_1 [rule_format]:
```
```   185     "zprime p --> a < p - 1 --> b \<in> wset a p --> 1 < b"
```
```   186   apply (induct a p rule: wset_induct, auto)
```
```   187   apply (case_tac "b = a")
```
```   188    apply (case_tac [2] "b = inv p a")
```
```   189     apply (subgoal_tac [3] "b = a \<or> b = inv p a")
```
```   190      apply (rule_tac [4] wset_mem_imp_or)
```
```   191        prefer 2
```
```   192        apply simp
```
```   193        apply (rule inv_g_1, auto)
```
```   194   done
```
```   195
```
```   196 lemma wset_less [rule_format]:
```
```   197     "zprime p --> a < p - 1 --> b \<in> wset a p --> b < p - 1"
```
```   198   apply (induct a p rule: wset_induct, auto)
```
```   199   apply (case_tac "b = a")
```
```   200    apply (case_tac [2] "b = inv p a")
```
```   201     apply (subgoal_tac [3] "b = a \<or> b = inv p a")
```
```   202      apply (rule_tac [4] wset_mem_imp_or)
```
```   203        prefer 2
```
```   204        apply simp
```
```   205        apply (rule inv_less_p_minus_1, auto)
```
```   206   done
```
```   207
```
```   208 lemma wset_mem [rule_format]:
```
```   209   "zprime p -->
```
```   210     a < p - 1 --> 1 < b --> b \<le> a --> b \<in> wset a p"
```
```   211   apply (induct a p rule: wset.induct, auto)
```
```   212   apply (rule_tac wset_subset)
```
```   213   apply (simp (no_asm_simp))
```
```   214   apply auto
```
```   215   done
```
```   216
```
```   217 lemma wset_mem_inv_mem [rule_format]:
```
```   218   "zprime p --> 5 \<le> p --> a < p - 1 --> b \<in> wset a p
```
```   219     --> inv p b \<in> wset a p"
```
```   220   apply (induct a p rule: wset_induct, auto)
```
```   221    apply (case_tac "b = a")
```
```   222     apply (subst wset.simps)
```
```   223     apply (unfold Let_def)
```
```   224     apply (rule_tac [3] wset_subset, auto)
```
```   225   apply (case_tac "b = inv p a")
```
```   226    apply (simp (no_asm_simp))
```
```   227    apply (subst inv_inv)
```
```   228        apply (subgoal_tac [6] "b = a \<or> b = inv p a")
```
```   229         apply (rule_tac [7] wset_mem_imp_or, auto)
```
```   230   done
```
```   231
```
```   232 lemma wset_inv_mem_mem:
```
```   233   "zprime p \<Longrightarrow> 5 \<le> p \<Longrightarrow> a < p - 1 \<Longrightarrow> 1 < b \<Longrightarrow> b < p - 1
```
```   234     \<Longrightarrow> inv p b \<in> wset a p \<Longrightarrow> b \<in> wset a p"
```
```   235   apply (rule_tac s = "inv p (inv p b)" and t = b in subst)
```
```   236    apply (rule_tac [2] wset_mem_inv_mem)
```
```   237       apply (rule inv_inv, simp_all)
```
```   238   done
```
```   239
```
```   240 lemma wset_fin: "finite (wset a p)"
```
```   241   apply (induct a p rule: wset_induct)
```
```   242    prefer 2
```
```   243    apply (subst wset.simps)
```
```   244    apply (unfold Let_def, auto)
```
```   245   done
```
```   246
```
```   247 lemma wset_zcong_prod_1 [rule_format]:
```
```   248   "zprime p -->
```
```   249     5 \<le> p --> a < p - 1 --> [(\<Prod>x\<in>wset a p. x) = 1] (mod p)"
```
```   250   apply (induct a p rule: wset_induct)
```
```   251    prefer 2
```
```   252    apply (subst wset.simps)
```
```   253    apply (auto, unfold Let_def, auto)
```
```   254   apply (subst setprod.insert)
```
```   255     apply (tactic \<open>stac @{context} @{thm setprod.insert} 3\<close>)
```
```   256       apply (subgoal_tac [5]
```
```   257         "zcong (a * inv p a * (\<Prod>x\<in>wset (a - 1) p. x)) (1 * 1) p")
```
```   258        prefer 5
```
```   259        apply (simp add: mult.assoc)
```
```   260       apply (rule_tac [5] zcong_zmult)
```
```   261        apply (rule_tac [5] inv_is_inv)
```
```   262          apply (tactic "clarify_tac @{context} 4")
```
```   263          apply (subgoal_tac [4] "a \<in> wset (a - 1) p")
```
```   264           apply (rule_tac [5] wset_inv_mem_mem)
```
```   265                apply (simp_all add: wset_fin)
```
```   266   apply (rule inv_distinct, auto)
```
```   267   done
```
```   268
```
```   269 lemma d22set_eq_wset: "zprime p ==> d22set (p - 2) = wset (p - 2) p"
```
```   270   apply safe
```
```   271    apply (erule wset_mem)
```
```   272      apply (rule_tac [2] d22set_g_1)
```
```   273      apply (rule_tac [3] d22set_le)
```
```   274      apply (rule_tac [4] d22set_mem)
```
```   275       apply (erule_tac [4] wset_g_1)
```
```   276        prefer 6
```
```   277        apply (subst zle_add1_eq_le [symmetric])
```
```   278        apply (subgoal_tac "p - 2 + 1 = p - 1")
```
```   279         apply (simp (no_asm_simp))
```
```   280         apply (erule wset_less, auto)
```
```   281   done
```
```   282
```
```   283
```
```   284 subsection \<open>Wilson\<close>
```
```   285
```
```   286 lemma prime_g_5: "zprime p \<Longrightarrow> p \<noteq> 2 \<Longrightarrow> p \<noteq> 3 ==> 5 \<le> p"
```
```   287   apply (unfold zprime_def dvd_def)
```
```   288   apply (case_tac "p = 4", auto)
```
```   289    apply (rule notE)
```
```   290     prefer 2
```
```   291     apply assumption
```
```   292    apply (simp (no_asm))
```
```   293    apply (rule_tac x = 2 in exI)
```
```   294    apply (safe, arith)
```
```   295      apply (rule_tac x = 2 in exI, auto)
```
```   296   done
```
```   297
```
```   298 theorem Wilson_Russ:
```
```   299     "zprime p ==> [zfact (p - 1) = -1] (mod p)"
```
```   300   apply (subgoal_tac "[(p - 1) * zfact (p - 2) = -1 * 1] (mod p)")
```
```   301    apply (rule_tac [2] zcong_zmult)
```
```   302     apply (simp only: zprime_def)
```
```   303     apply (subst zfact.simps)
```
```   304     apply (rule_tac t = "p - 1 - 1" and s = "p - 2" in subst, auto)
```
```   305    apply (simp only: zcong_def)
```
```   306    apply (simp (no_asm_simp))
```
```   307   apply (case_tac "p = 2")
```
```   308    apply (simp add: zfact.simps)
```
```   309   apply (case_tac "p = 3")
```
```   310    apply (simp add: zfact.simps)
```
```   311   apply (subgoal_tac "5 \<le> p")
```
```   312    apply (erule_tac [2] prime_g_5)
```
```   313     apply (subst d22set_prod_zfact [symmetric])
```
```   314     apply (subst d22set_eq_wset)
```
```   315      apply (rule_tac [2] wset_zcong_prod_1, auto)
```
```   316   done
```
```   317
```
```   318 end
```