src/HOL/Old_Number_Theory/WilsonRuss.thy
author paulson <lp15@cam.ac.uk>
Tue Nov 17 12:32:08 2015 +0000 (2015-11-17)
changeset 61694 6571c78c9667
parent 61649 268d88ec9087
child 64272 f76b6dda2e56
permissions -rw-r--r--
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
     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 simp
    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   using Little_Fermat inv_is_inv_aux zdvd_not_zless apply auto
    42   done
    43 
    44 lemma inv_distinct:
    45     "zprime p \<Longrightarrow> 1 < a \<Longrightarrow> a < p - 1 ==> a \<noteq> inv p a"
    46   apply safe
    47   apply (cut_tac a = a and p = p in zcong_square)
    48      apply (cut_tac [3] a = a and p = p in inv_is_inv, auto)
    49    apply (subgoal_tac "a = 1")
    50     apply (rule_tac [2] m = p in zcong_zless_imp_eq)
    51         apply (subgoal_tac [7] "a = p - 1")
    52          apply (rule_tac [8] m = p in zcong_zless_imp_eq, auto)
    53   done
    54 
    55 lemma inv_not_0:
    56     "zprime p \<Longrightarrow> 1 < a \<Longrightarrow> a < p - 1 ==> inv p a \<noteq> 0"
    57   apply safe
    58   apply (cut_tac a = a and p = p in inv_is_inv)
    59      apply (unfold zcong_def, auto)
    60   done
    61 
    62 lemma inv_not_1:
    63     "zprime p \<Longrightarrow> 1 < a \<Longrightarrow> a < p - 1 ==> inv p a \<noteq> 1"
    64   apply safe
    65   apply (cut_tac a = a and p = p in inv_is_inv)
    66      prefer 4
    67      apply simp
    68      apply (subgoal_tac "a = 1")
    69       apply (rule_tac [2] zcong_zless_imp_eq, auto)
    70   done
    71 
    72 lemma inv_not_p_minus_1_aux:
    73     "[a * (p - 1) = 1] (mod p) = [a = p - 1] (mod p)"
    74   apply (unfold zcong_def)
    75   apply (simp add: diff_diff_eq diff_diff_eq2 right_diff_distrib)
    76   apply (rule_tac s = "p dvd -((a + 1) + (p * -a))" in trans)
    77    apply (simp add: algebra_simps)
    78   apply (subst dvd_minus_iff)
    79   apply (subst zdvd_reduce)
    80   apply (rule_tac s = "p dvd (a + 1) + (p * -1)" in trans)
    81    apply (subst zdvd_reduce, auto)
    82   done
    83 
    84 lemma inv_not_p_minus_1:
    85     "zprime p \<Longrightarrow> 1 < a \<Longrightarrow> a < p - 1 ==> inv p a \<noteq> p - 1"
    86   apply safe
    87   apply (cut_tac a = a and p = p in inv_is_inv, auto)
    88   apply (simp add: inv_not_p_minus_1_aux)
    89   apply (subgoal_tac "a = p - 1")
    90    apply (rule_tac [2] zcong_zless_imp_eq, auto)
    91   done
    92 
    93 lemma inv_g_1:
    94     "zprime p \<Longrightarrow> 1 < a \<Longrightarrow> a < p - 1 ==> 1 < inv p a"
    95   apply (case_tac "0\<le> inv p a")
    96    apply (subgoal_tac "inv p a \<noteq> 1")
    97     apply (subgoal_tac "inv p a \<noteq> 0")
    98      apply (subst order_less_le)
    99      apply (subst zle_add1_eq_le [symmetric])
   100      apply (subst order_less_le)
   101      apply (rule_tac [2] inv_not_0)
   102        apply (rule_tac [5] inv_not_1, auto)
   103   apply (unfold inv_def zprime_def, simp)
   104   done
   105 
   106 lemma inv_less_p_minus_1:
   107     "zprime p \<Longrightarrow> 1 < a \<Longrightarrow> a < p - 1 ==> inv p a < p - 1"
   108   apply (case_tac "inv p a < p")
   109    apply (subst order_less_le)
   110    apply (simp add: inv_not_p_minus_1, auto)
   111   apply (unfold inv_def zprime_def, simp)
   112   done
   113 
   114 lemma inv_inv_aux: "5 \<le> p ==>
   115     nat (p - 2) * nat (p - 2) = Suc (nat (p - 1) * nat (p - 3))"
   116   apply (subst of_nat_eq_iff [where 'a = int, symmetric])
   117   apply (simp add: left_diff_distrib right_diff_distrib)
   118   done
   119 
   120 lemma zcong_zpower_zmult:
   121     "[x^y = 1] (mod p) \<Longrightarrow> [x^(y * z) = 1] (mod p)"
   122   apply (induct z)
   123    apply (auto simp add: power_add)
   124   apply (subgoal_tac "zcong (x^y * x^(y * z)) (1 * 1) p")
   125    apply (rule_tac [2] zcong_zmult, simp_all)
   126   done
   127 
   128 lemma inv_inv: "zprime p \<Longrightarrow>
   129     5 \<le> p \<Longrightarrow> 0 < a \<Longrightarrow> a < p ==> inv p (inv p a) = a"
   130   apply (unfold inv_def)
   131   apply (subst power_mod)
   132   apply (subst power_mult [symmetric])
   133   apply (rule zcong_zless_imp_eq)
   134       prefer 5
   135       apply (subst zcong_zmod)
   136       apply (subst mod_mod_trivial)
   137       apply (subst zcong_zmod [symmetric])
   138       apply (subst inv_inv_aux)
   139        apply (subgoal_tac [2]
   140          "zcong (a * a^(nat (p - 1) * nat (p - 3))) (a * 1) p")
   141         apply (rule_tac [3] zcong_zmult)
   142          apply (rule_tac [4] zcong_zpower_zmult)
   143          apply (erule_tac [4] Little_Fermat)
   144          apply (rule_tac [4] zdvd_not_zless, simp_all)
   145   done
   146 
   147 
   148 text \<open>\medskip @{term wset}\<close>
   149 
   150 declare wset.simps [simp del]
   151 
   152 lemma wset_induct:
   153   assumes "!!a p. P {} a p"
   154     and "!!a p. 1 < (a::int) \<Longrightarrow>
   155       P (wset (a - 1) p) (a - 1) p ==> P (wset a p) a p"
   156   shows "P (wset u v) u v"
   157   apply (rule wset.induct)
   158   apply (case_tac "1 < a")
   159    apply (rule assms)
   160     apply (simp_all add: wset.simps assms)
   161   done
   162 
   163 lemma wset_mem_imp_or [rule_format]:
   164   "1 < a \<Longrightarrow> b \<notin> wset (a - 1) p
   165     ==> b \<in> wset a p --> b = a \<or> b = inv p a"
   166   apply (subst wset.simps)
   167   apply (unfold Let_def, simp)
   168   done
   169 
   170 lemma wset_mem_mem [simp]: "1 < a ==> a \<in> wset a p"
   171   apply (subst wset.simps)
   172   apply (unfold Let_def, simp)
   173   done
   174 
   175 lemma wset_subset: "1 < a \<Longrightarrow> b \<in> wset (a - 1) p ==> b \<in> wset a p"
   176   apply (subst wset.simps)
   177   apply (unfold Let_def, auto)
   178   done
   179 
   180 lemma wset_g_1 [rule_format]:
   181     "zprime p --> a < p - 1 --> b \<in> wset a p --> 1 < b"
   182   apply (induct a p rule: wset_induct, auto)
   183   apply (case_tac "b = a")
   184    apply (case_tac [2] "b = inv p a")
   185     apply (subgoal_tac [3] "b = a \<or> b = inv p a")
   186      apply (rule_tac [4] wset_mem_imp_or)
   187        prefer 2
   188        apply simp
   189        apply (rule inv_g_1, auto)
   190   done
   191 
   192 lemma wset_less [rule_format]:
   193     "zprime p --> a < p - 1 --> b \<in> wset a p --> b < p - 1"
   194   apply (induct a p rule: wset_induct, auto)
   195   apply (case_tac "b = a")
   196    apply (case_tac [2] "b = inv p a")
   197     apply (subgoal_tac [3] "b = a \<or> b = inv p a")
   198      apply (rule_tac [4] wset_mem_imp_or)
   199        prefer 2
   200        apply simp
   201        apply (rule inv_less_p_minus_1, auto)
   202   done
   203 
   204 lemma wset_mem [rule_format]:
   205   "zprime p -->
   206     a < p - 1 --> 1 < b --> b \<le> a --> b \<in> wset a p"
   207   apply (induct a p rule: wset.induct, auto)
   208   apply (rule_tac wset_subset)
   209   apply (simp (no_asm_simp))
   210   apply auto
   211   done
   212 
   213 lemma wset_mem_inv_mem [rule_format]:
   214   "zprime p --> 5 \<le> p --> a < p - 1 --> b \<in> wset a p
   215     --> inv p b \<in> wset a p"
   216   apply (induct a p rule: wset_induct, auto)
   217    apply (case_tac "b = a")
   218     apply (subst wset.simps)
   219     apply (unfold Let_def)
   220     apply (rule_tac [3] wset_subset, auto)
   221   apply (case_tac "b = inv p a")
   222    apply (simp (no_asm_simp))
   223    apply (subst inv_inv)
   224        apply (subgoal_tac [6] "b = a \<or> b = inv p a")
   225         apply (rule_tac [7] wset_mem_imp_or, auto)
   226   done
   227 
   228 lemma wset_inv_mem_mem:
   229   "zprime p \<Longrightarrow> 5 \<le> p \<Longrightarrow> a < p - 1 \<Longrightarrow> 1 < b \<Longrightarrow> b < p - 1
   230     \<Longrightarrow> inv p b \<in> wset a p \<Longrightarrow> b \<in> wset a p"
   231   apply (rule_tac s = "inv p (inv p b)" and t = b in subst)
   232    apply (rule_tac [2] wset_mem_inv_mem)
   233       apply (rule inv_inv, simp_all)
   234   done
   235 
   236 lemma wset_fin: "finite (wset a p)"
   237   apply (induct a p rule: wset_induct)
   238    prefer 2
   239    apply (subst wset.simps)
   240    apply (unfold Let_def, auto)
   241   done
   242 
   243 lemma wset_zcong_prod_1 [rule_format]:
   244   "zprime p -->
   245     5 \<le> p --> a < p - 1 --> [(\<Prod>x\<in>wset a p. x) = 1] (mod p)"
   246   apply (induct a p rule: wset_induct)
   247    prefer 2
   248    apply (subst wset.simps)
   249    apply (auto, unfold Let_def, auto)
   250   apply (subst setprod.insert)
   251     apply (tactic \<open>stac @{context} @{thm setprod.insert} 3\<close>)
   252       apply (subgoal_tac [5]
   253         "zcong (a * inv p a * (\<Prod>x\<in>wset (a - 1) p. x)) (1 * 1) p")
   254        prefer 5
   255        apply (simp add: mult.assoc)
   256       apply (rule_tac [5] zcong_zmult)
   257        apply (rule_tac [5] inv_is_inv)
   258          apply (tactic "clarify_tac @{context} 4")
   259          apply (subgoal_tac [4] "a \<in> wset (a - 1) p")
   260           apply (rule_tac [5] wset_inv_mem_mem)
   261                apply (simp_all add: wset_fin)
   262   apply (rule inv_distinct, auto)
   263   done
   264 
   265 lemma d22set_eq_wset: "zprime p ==> d22set (p - 2) = wset (p - 2) p"
   266   apply safe
   267    apply (erule wset_mem)
   268      apply (rule_tac [2] d22set_g_1)
   269      apply (rule_tac [3] d22set_le)
   270      apply (rule_tac [4] d22set_mem)
   271       apply (erule_tac [4] wset_g_1)
   272        prefer 6
   273        apply (subst zle_add1_eq_le [symmetric])
   274        apply (subgoal_tac "p - 2 + 1 = p - 1")
   275         apply (simp (no_asm_simp))
   276         apply (erule wset_less, auto)
   277   done
   278 
   279 
   280 subsection \<open>Wilson\<close>
   281 
   282 lemma prime_g_5: "zprime p \<Longrightarrow> p \<noteq> 2 \<Longrightarrow> p \<noteq> 3 ==> 5 \<le> p"
   283   apply (unfold zprime_def dvd_def)
   284   apply (case_tac "p = 4", auto)
   285    apply (rule notE)
   286     prefer 2
   287     apply assumption
   288    apply (simp (no_asm))
   289    apply (rule_tac x = 2 in exI)
   290    apply (safe, arith)
   291      apply (rule_tac x = 2 in exI, auto)
   292   done
   293 
   294 theorem Wilson_Russ:
   295     "zprime p ==> [zfact (p - 1) = -1] (mod p)"
   296   apply (subgoal_tac "[(p - 1) * zfact (p - 2) = -1 * 1] (mod p)")
   297    apply (rule_tac [2] zcong_zmult)
   298     apply (simp only: zprime_def)
   299     apply (subst zfact.simps)
   300     apply (rule_tac t = "p - 1 - 1" and s = "p - 2" in subst, auto)
   301    apply (simp only: zcong_def)
   302    apply (simp (no_asm_simp))
   303   apply (case_tac "p = 2")
   304    apply (simp add: zfact.simps)
   305   apply (case_tac "p = 3")
   306    apply (simp add: zfact.simps)
   307   apply (subgoal_tac "5 \<le> p")
   308    apply (erule_tac [2] prime_g_5)
   309     apply (subst d22set_prod_zfact [symmetric])
   310     apply (subst d22set_eq_wset)
   311      apply (rule_tac [2] wset_zcong_prod_1, auto)
   312   done
   313 
   314 end