src/HOL/Old_Number_Theory/WilsonRuss.thy
 author haftmann Fri Nov 27 08:41:10 2009 +0100 (2009-11-27) changeset 33963 977b94b64905 parent 32960 69916a850301 child 35048 82ab78fff970 permissions -rw-r--r--
renamed former datatype.ML to datatype_data.ML; datatype.ML provides uniform view on datatype.ML and datatype_rep_proofs.ML
```     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 consts
```
```    21   wset :: "int * int => int set"
```
```    22
```
```    23 recdef wset
```
```    24   "measure ((\<lambda>(a, p). nat a) :: int * int => nat)"
```
```    25   "wset (a, p) =
```
```    26     (if 1 < a then
```
```    27       let ws = wset (a - 1, p)
```
```    28       in (if a \<in> ws then ws else insert a (insert (inv p a) ws)) else {})"
```
```    29
```
```    30
```
```    31 text {* \medskip @{term [source] inv} *}
```
```    32
```
```    33 lemma inv_is_inv_aux: "1 < m ==> Suc (nat (m - 2)) = nat (m - 1)"
```
```    34 by (subst int_int_eq [symmetric], auto)
```
```    35
```
```    36 lemma inv_is_inv:
```
```    37     "zprime p \<Longrightarrow> 0 < a \<Longrightarrow> a < p ==> [a * inv p a = 1] (mod p)"
```
```    38   apply (unfold inv_def)
```
```    39   apply (subst zcong_zmod)
```
```    40   apply (subst zmod_zmult1_eq [symmetric])
```
```    41   apply (subst zcong_zmod [symmetric])
```
```    42   apply (subst power_Suc [symmetric])
```
```    43   apply (subst inv_is_inv_aux)
```
```    44    apply (erule_tac [2] Little_Fermat)
```
```    45    apply (erule_tac [2] zdvd_not_zless)
```
```    46    apply (unfold zprime_def, auto)
```
```    47   done
```
```    48
```
```    49 lemma inv_distinct:
```
```    50     "zprime p \<Longrightarrow> 1 < a \<Longrightarrow> a < p - 1 ==> a \<noteq> inv p a"
```
```    51   apply safe
```
```    52   apply (cut_tac a = a and p = p in zcong_square)
```
```    53      apply (cut_tac [3] a = a and p = p in inv_is_inv, auto)
```
```    54    apply (subgoal_tac "a = 1")
```
```    55     apply (rule_tac [2] m = p in zcong_zless_imp_eq)
```
```    56         apply (subgoal_tac [7] "a = p - 1")
```
```    57          apply (rule_tac [8] m = p in zcong_zless_imp_eq, auto)
```
```    58   done
```
```    59
```
```    60 lemma inv_not_0:
```
```    61     "zprime p \<Longrightarrow> 1 < a \<Longrightarrow> a < p - 1 ==> inv p a \<noteq> 0"
```
```    62   apply safe
```
```    63   apply (cut_tac a = a and p = p in inv_is_inv)
```
```    64      apply (unfold zcong_def, auto)
```
```    65   apply (subgoal_tac "\<not> p dvd 1")
```
```    66    apply (rule_tac [2] zdvd_not_zless)
```
```    67     apply (subgoal_tac "p dvd 1")
```
```    68      prefer 2
```
```    69      apply (subst dvd_minus_iff [symmetric], auto)
```
```    70   done
```
```    71
```
```    72 lemma inv_not_1:
```
```    73     "zprime p \<Longrightarrow> 1 < a \<Longrightarrow> a < p - 1 ==> inv p a \<noteq> 1"
```
```    74   apply safe
```
```    75   apply (cut_tac a = a and p = p in inv_is_inv)
```
```    76      prefer 4
```
```    77      apply simp
```
```    78      apply (subgoal_tac "a = 1")
```
```    79       apply (rule_tac [2] zcong_zless_imp_eq, auto)
```
```    80   done
```
```    81
```
```    82 lemma inv_not_p_minus_1_aux:
```
```    83     "[a * (p - 1) = 1] (mod p) = [a = p - 1] (mod p)"
```
```    84   apply (unfold zcong_def)
```
```    85   apply (simp add: OrderedGroup.diff_diff_eq diff_diff_eq2 zdiff_zmult_distrib2)
```
```    86   apply (rule_tac s = "p dvd -((a + 1) + (p * -a))" in trans)
```
```    87    apply (simp add: mult_commute)
```
```    88   apply (subst dvd_minus_iff)
```
```    89   apply (subst zdvd_reduce)
```
```    90   apply (rule_tac s = "p dvd (a + 1) + (p * -1)" in trans)
```
```    91    apply (subst zdvd_reduce, auto)
```
```    92   done
```
```    93
```
```    94 lemma inv_not_p_minus_1:
```
```    95     "zprime p \<Longrightarrow> 1 < a \<Longrightarrow> a < p - 1 ==> inv p a \<noteq> p - 1"
```
```    96   apply safe
```
```    97   apply (cut_tac a = a and p = p in inv_is_inv, auto)
```
```    98   apply (simp add: inv_not_p_minus_1_aux)
```
```    99   apply (subgoal_tac "a = p - 1")
```
```   100    apply (rule_tac [2] zcong_zless_imp_eq, auto)
```
```   101   done
```
```   102
```
```   103 lemma inv_g_1:
```
```   104     "zprime p \<Longrightarrow> 1 < a \<Longrightarrow> a < p - 1 ==> 1 < inv p a"
```
```   105   apply (case_tac "0\<le> 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, auto)
```
```   113   apply (unfold inv_def zprime_def, simp)
```
```   114   done
```
```   115
```
```   116 lemma inv_less_p_minus_1:
```
```   117     "zprime p \<Longrightarrow> 1 < a \<Longrightarrow> a < p - 1 ==> inv p a < p - 1"
```
```   118   apply (case_tac "inv p a < p")
```
```   119    apply (subst order_less_le)
```
```   120    apply (simp add: inv_not_p_minus_1, auto)
```
```   121   apply (unfold inv_def zprime_def, simp)
```
```   122   done
```
```   123
```
```   124 lemma inv_inv_aux: "5 \<le> p ==>
```
```   125     nat (p - 2) * nat (p - 2) = Suc (nat (p - 1) * nat (p - 3))"
```
```   126   apply (subst int_int_eq [symmetric])
```
```   127   apply (simp add: zmult_int [symmetric])
```
```   128   apply (simp add: zdiff_zmult_distrib zdiff_zmult_distrib2)
```
```   129   done
```
```   130
```
```   131 lemma zcong_zpower_zmult:
```
```   132     "[x^y = 1] (mod p) \<Longrightarrow> [x^(y * z) = 1] (mod p)"
```
```   133   apply (induct z)
```
```   134    apply (auto simp add: zpower_zadd_distrib)
```
```   135   apply (subgoal_tac "zcong (x^y * x^(y * z)) (1 * 1) p")
```
```   136    apply (rule_tac [2] zcong_zmult, simp_all)
```
```   137   done
```
```   138
```
```   139 lemma inv_inv: "zprime p \<Longrightarrow>
```
```   140     5 \<le> p \<Longrightarrow> 0 < a \<Longrightarrow> a < p ==> inv p (inv p a) = a"
```
```   141   apply (unfold inv_def)
```
```   142   apply (subst zpower_zmod)
```
```   143   apply (subst zpower_zpower)
```
```   144   apply (rule zcong_zless_imp_eq)
```
```   145       prefer 5
```
```   146       apply (subst zcong_zmod)
```
```   147       apply (subst mod_mod_trivial)
```
```   148       apply (subst zcong_zmod [symmetric])
```
```   149       apply (subst inv_inv_aux)
```
```   150        apply (subgoal_tac [2]
```
```   151          "zcong (a * a^(nat (p - 1) * nat (p - 3))) (a * 1) p")
```
```   152         apply (rule_tac [3] zcong_zmult)
```
```   153          apply (rule_tac [4] zcong_zpower_zmult)
```
```   154          apply (erule_tac [4] Little_Fermat)
```
```   155          apply (rule_tac [4] zdvd_not_zless, simp_all)
```
```   156   done
```
```   157
```
```   158
```
```   159 text {* \medskip @{term wset} *}
```
```   160
```
```   161 declare wset.simps [simp del]
```
```   162
```
```   163 lemma wset_induct:
```
```   164   assumes "!!a p. P {} a p"
```
```   165     and "!!a p. 1 < (a::int) \<Longrightarrow>
```
```   166       P (wset (a - 1, p)) (a - 1) p ==> P (wset (a, p)) a p"
```
```   167   shows "P (wset (u, v)) u v"
```
```   168   apply (rule wset.induct, safe)
```
```   169    prefer 2
```
```   170    apply (case_tac "1 < a")
```
```   171     apply (rule prems)
```
```   172      apply simp_all
```
```   173    apply (simp_all add: wset.simps prems)
```
```   174   done
```
```   175
```
```   176 lemma wset_mem_imp_or [rule_format]:
```
```   177   "1 < a \<Longrightarrow> b \<notin> wset (a - 1, p)
```
```   178     ==> b \<in> wset (a, p) --> b = a \<or> b = inv p a"
```
```   179   apply (subst wset.simps)
```
```   180   apply (unfold Let_def, simp)
```
```   181   done
```
```   182
```
```   183 lemma wset_mem_mem [simp]: "1 < a ==> a \<in> wset (a, p)"
```
```   184   apply (subst wset.simps)
```
```   185   apply (unfold Let_def, simp)
```
```   186   done
```
```   187
```
```   188 lemma wset_subset: "1 < a \<Longrightarrow> b \<in> wset (a - 1, p) ==> b \<in> wset (a, p)"
```
```   189   apply (subst wset.simps)
```
```   190   apply (unfold Let_def, auto)
```
```   191   done
```
```   192
```
```   193 lemma wset_g_1 [rule_format]:
```
```   194     "zprime p --> a < p - 1 --> b \<in> wset (a, p) --> 1 < b"
```
```   195   apply (induct a p rule: wset_induct, auto)
```
```   196   apply (case_tac "b = a")
```
```   197    apply (case_tac [2] "b = inv p a")
```
```   198     apply (subgoal_tac [3] "b = a \<or> b = inv p a")
```
```   199      apply (rule_tac [4] wset_mem_imp_or)
```
```   200        prefer 2
```
```   201        apply simp
```
```   202        apply (rule inv_g_1, auto)
```
```   203   done
```
```   204
```
```   205 lemma wset_less [rule_format]:
```
```   206     "zprime p --> a < p - 1 --> b \<in> wset (a, p) --> b < p - 1"
```
```   207   apply (induct a p rule: wset_induct, auto)
```
```   208   apply (case_tac "b = a")
```
```   209    apply (case_tac [2] "b = inv p a")
```
```   210     apply (subgoal_tac [3] "b = a \<or> b = inv p a")
```
```   211      apply (rule_tac [4] wset_mem_imp_or)
```
```   212        prefer 2
```
```   213        apply simp
```
```   214        apply (rule inv_less_p_minus_1, auto)
```
```   215   done
```
```   216
```
```   217 lemma wset_mem [rule_format]:
```
```   218   "zprime p -->
```
```   219     a < p - 1 --> 1 < b --> b \<le> a --> b \<in> wset (a, p)"
```
```   220   apply (induct a p rule: wset.induct, auto)
```
```   221   apply (rule_tac wset_subset)
```
```   222   apply (simp (no_asm_simp))
```
```   223   apply auto
```
```   224   done
```
```   225
```
```   226 lemma wset_mem_inv_mem [rule_format]:
```
```   227   "zprime p --> 5 \<le> p --> a < p - 1 --> b \<in> wset (a, p)
```
```   228     --> inv p b \<in> wset (a, p)"
```
```   229   apply (induct a p rule: wset_induct, auto)
```
```   230    apply (case_tac "b = a")
```
```   231     apply (subst wset.simps)
```
```   232     apply (unfold Let_def)
```
```   233     apply (rule_tac [3] wset_subset, auto)
```
```   234   apply (case_tac "b = inv p a")
```
```   235    apply (simp (no_asm_simp))
```
```   236    apply (subst inv_inv)
```
```   237        apply (subgoal_tac [6] "b = a \<or> b = inv p a")
```
```   238         apply (rule_tac [7] wset_mem_imp_or, auto)
```
```   239   done
```
```   240
```
```   241 lemma wset_inv_mem_mem:
```
```   242   "zprime p \<Longrightarrow> 5 \<le> p \<Longrightarrow> a < p - 1 \<Longrightarrow> 1 < b \<Longrightarrow> b < p - 1
```
```   243     \<Longrightarrow> inv p b \<in> wset (a, p) \<Longrightarrow> b \<in> wset (a, p)"
```
```   244   apply (rule_tac s = "inv p (inv p b)" and t = b in subst)
```
```   245    apply (rule_tac [2] wset_mem_inv_mem)
```
```   246       apply (rule inv_inv, simp_all)
```
```   247   done
```
```   248
```
```   249 lemma wset_fin: "finite (wset (a, p))"
```
```   250   apply (induct a p rule: wset_induct)
```
```   251    prefer 2
```
```   252    apply (subst wset.simps)
```
```   253    apply (unfold Let_def, auto)
```
```   254   done
```
```   255
```
```   256 lemma wset_zcong_prod_1 [rule_format]:
```
```   257   "zprime p -->
```
```   258     5 \<le> p --> a < p - 1 --> [(\<Prod>x\<in>wset(a, p). x) = 1] (mod p)"
```
```   259   apply (induct a p rule: wset_induct)
```
```   260    prefer 2
```
```   261    apply (subst wset.simps)
```
```   262    apply (unfold Let_def, auto)
```
```   263   apply (subst setprod_insert)
```
```   264     apply (tactic {* stac (thm "setprod_insert") 3 *})
```
```   265       apply (subgoal_tac [5]
```
```   266         "zcong (a * inv p a * (\<Prod>x\<in> wset(a - 1, p). x)) (1 * 1) p")
```
```   267        prefer 5
```
```   268        apply (simp add: zmult_assoc)
```
```   269       apply (rule_tac [5] zcong_zmult)
```
```   270        apply (rule_tac [5] inv_is_inv)
```
```   271          apply (tactic "clarify_tac @{claset} 4")
```
```   272          apply (subgoal_tac [4] "a \<in> wset (a - 1, p)")
```
```   273           apply (rule_tac [5] wset_inv_mem_mem)
```
```   274                apply (simp_all add: wset_fin)
```
```   275   apply (rule inv_distinct, auto)
```
```   276   done
```
```   277
```
```   278 lemma d22set_eq_wset: "zprime p ==> d22set (p - 2) = wset (p - 2, p)"
```
```   279   apply safe
```
```   280    apply (erule wset_mem)
```
```   281      apply (rule_tac [2] d22set_g_1)
```
```   282      apply (rule_tac [3] d22set_le)
```
```   283      apply (rule_tac [4] d22set_mem)
```
```   284       apply (erule_tac [4] wset_g_1)
```
```   285        prefer 6
```
```   286        apply (subst zle_add1_eq_le [symmetric])
```
```   287        apply (subgoal_tac "p - 2 + 1 = p - 1")
```
```   288         apply (simp (no_asm_simp))
```
```   289         apply (erule wset_less, auto)
```
```   290   done
```
```   291
```
```   292
```
```   293 subsection {* Wilson *}
```
```   294
```
```   295 lemma prime_g_5: "zprime p \<Longrightarrow> p \<noteq> 2 \<Longrightarrow> p \<noteq> 3 ==> 5 \<le> p"
```
```   296   apply (unfold zprime_def dvd_def)
```
```   297   apply (case_tac "p = 4", auto)
```
```   298    apply (rule notE)
```
```   299     prefer 2
```
```   300     apply assumption
```
```   301    apply (simp (no_asm))
```
```   302    apply (rule_tac x = 2 in exI)
```
```   303    apply (safe, arith)
```
```   304      apply (rule_tac x = 2 in exI, auto)
```
```   305   done
```
```   306
```
```   307 theorem Wilson_Russ:
```
```   308     "zprime p ==> [zfact (p - 1) = -1] (mod p)"
```
```   309   apply (subgoal_tac "[(p - 1) * zfact (p - 2) = -1 * 1] (mod p)")
```
```   310    apply (rule_tac [2] zcong_zmult)
```
```   311     apply (simp only: zprime_def)
```
```   312     apply (subst zfact.simps)
```
```   313     apply (rule_tac t = "p - 1 - 1" and s = "p - 2" in subst, auto)
```
```   314    apply (simp only: zcong_def)
```
```   315    apply (simp (no_asm_simp))
```
```   316   apply (case_tac "p = 2")
```
```   317    apply (simp add: zfact.simps)
```
```   318   apply (case_tac "p = 3")
```
```   319    apply (simp add: zfact.simps)
```
```   320   apply (subgoal_tac "5 \<le> p")
```
```   321    apply (erule_tac [2] prime_g_5)
```
```   322     apply (subst d22set_prod_zfact [symmetric])
```
```   323     apply (subst d22set_eq_wset)
```
```   324      apply (rule_tac [2] wset_zcong_prod_1, auto)
```
```   325   done
```
```   326
```
```   327 end
```