src/HOL/Old_Number_Theory/WilsonRuss.thy
 author huffman Tue Mar 27 16:04:51 2012 +0200 (2012-03-27) changeset 47164 6a4c479ba94f parent 47163 248376f8881d child 47268 262d96552e50 permissions -rw-r--r--
generalized lemma zpower_zmod
1 (*  Title:      HOL/Old_Number_Theory/WilsonRuss.thy
2     Author:     Thomas M. Rasmussen
3     Copyright   2000  University of Cambridge
4 *)
6 header {* Wilson's Theorem according to Russinoff *}
8 theory WilsonRuss
9 imports EulerFermat
10 begin
12 text {*
13   Wilson's Theorem following quite closely Russinoff's approach
14   using Boyer-Moore (using finite sets instead of lists, though).
15 *}
17 subsection {* Definitions and lemmas *}
19 definition inv :: "int => int => int"
20   where "inv p a = (a^(nat (p - 2))) mod p"
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 {})"
29 text {* \medskip @{term [source] inv} *}
31 lemma inv_is_inv_aux: "1 < m ==> Suc (nat (m - 2)) = nat (m - 1)"
32   by (subst int_int_eq [symmetric]) auto
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
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
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
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
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 right_diff_distrib)
84   apply (rule_tac s = "p dvd -((a + 1) + (p * -a))" in trans)
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
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)
97   apply (subgoal_tac "a = p - 1")
98    apply (rule_tac [2] zcong_zless_imp_eq, auto)
99   done
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)
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
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
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])
126   apply (simp add: left_diff_distrib right_diff_distrib)
127   done
129 lemma zcong_zpower_zmult:
130     "[x^y = 1] (mod p) \<Longrightarrow> [x^(y * z) = 1] (mod p)"
131   apply (induct z)
133   apply (subgoal_tac "zcong (x^y * x^(y * z)) (1 * 1) p")
134    apply (rule_tac [2] zcong_zmult, simp_all)
135   done
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 power_mod)
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
157 text {* \medskip @{term wset} *}
159 declare wset.simps [simp del]
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
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
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
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
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
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
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
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
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
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
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
265       apply (rule_tac [5] zcong_zmult)
266        apply (rule_tac [5] inv_is_inv)
267          apply (tactic "clarify_tac @{context} 4")
268          apply (subgoal_tac [4] "a \<in> wset (a - 1) p")
269           apply (rule_tac [5] wset_inv_mem_mem)
271   apply (rule inv_distinct, auto)
272   done
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
283        apply (subgoal_tac "p - 2 + 1 = p - 1")
284         apply (simp (no_asm_simp))
285         apply (erule wset_less, auto)
286   done
289 subsection {* Wilson *}
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
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")