src/HOL/Old_Number_Theory/Euler.thy
 author haftmann Fri Nov 27 08:41:10 2009 +0100 (2009-11-27) changeset 33963 977b94b64905 parent 32479 521cc9bf2958 child 35544 342a448ae141 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 (*  Title:      HOL/Quadratic_Reciprocity/Euler.thy
2     ID:         \$Id\$
3     Authors:    Jeremy Avigad, David Gray, and Adam Kramer
4 *)
6 header {* Euler's criterion *}
8 theory Euler imports Residues EvenOdd begin
10 definition
11   MultInvPair :: "int => int => int => int set" where
12   "MultInvPair a p j = {StandardRes p j, StandardRes p (a * (MultInv p j))}"
14 definition
15   SetS        :: "int => int => int set set" where
16   "SetS        a p   =  (MultInvPair a p ` SRStar p)"
19 subsection {* Property for MultInvPair *}
21 lemma MultInvPair_prop1a:
22   "[| zprime p; 2 < p; ~([a = 0](mod p));
23       X \<in> (SetS a p); Y \<in> (SetS a p);
24       ~((X \<inter> Y) = {}) |] ==> X = Y"
25   apply (auto simp add: SetS_def)
26   apply (drule StandardRes_SRStar_prop1a)+ defer 1
27   apply (drule StandardRes_SRStar_prop1a)+
28   apply (auto simp add: MultInvPair_def StandardRes_prop2 zcong_sym)
29   apply (drule notE, rule MultInv_zcong_prop1, auto)[]
30   apply (drule notE, rule MultInv_zcong_prop2, auto simp add: zcong_sym)[]
31   apply (drule MultInv_zcong_prop2, auto simp add: zcong_sym)[]
32   apply (drule MultInv_zcong_prop3, auto simp add: zcong_sym)[]
33   apply (drule MultInv_zcong_prop1, auto)[]
34   apply (drule MultInv_zcong_prop2, auto simp add: zcong_sym)[]
35   apply (drule MultInv_zcong_prop2, auto simp add: zcong_sym)[]
36   apply (drule MultInv_zcong_prop3, auto simp add: zcong_sym)[]
37   done
39 lemma MultInvPair_prop1b:
40   "[| zprime p; 2 < p; ~([a = 0](mod p));
41       X \<in> (SetS a p); Y \<in> (SetS a p);
42       X \<noteq> Y |] ==> X \<inter> Y = {}"
43   apply (rule notnotD)
44   apply (rule notI)
45   apply (drule MultInvPair_prop1a, auto)
46   done
48 lemma MultInvPair_prop1c: "[| zprime p; 2 < p; ~([a = 0](mod p)) |] ==>
49     \<forall>X \<in> SetS a p. \<forall>Y \<in> SetS a p. X \<noteq> Y --> X\<inter>Y = {}"
50   by (auto simp add: MultInvPair_prop1b)
52 lemma MultInvPair_prop2: "[| zprime p; 2 < p; ~([a = 0](mod p)) |] ==>
53                           Union ( SetS a p) = SRStar p"
54   apply (auto simp add: SetS_def MultInvPair_def StandardRes_SRStar_prop4
55     SRStar_mult_prop2)
56   apply (frule StandardRes_SRStar_prop3)
57   apply (rule bexI, auto)
58   done
60 lemma MultInvPair_distinct: "[| zprime p; 2 < p; ~([a = 0] (mod p));
61                                 ~([j = 0] (mod p));
62                                 ~(QuadRes p a) |]  ==>
63                              ~([j = a * MultInv p j] (mod p))"
64 proof
65   assume "zprime p" and "2 < p" and "~([a = 0] (mod p))" and
66     "~([j = 0] (mod p))" and "~(QuadRes p a)"
67   assume "[j = a * MultInv p j] (mod p)"
68   then have "[j * j = (a * MultInv p j) * j] (mod p)"
69     by (auto simp add: zcong_scalar)
70   then have a:"[j * j = a * (MultInv p j * j)] (mod p)"
71     by (auto simp add: zmult_ac)
72   have "[j * j = a] (mod p)"
73     proof -
74       from prems have b: "[MultInv p j * j = 1] (mod p)"
75         by (simp add: MultInv_prop2a)
76       from b a show ?thesis
77         by (auto simp add: zcong_zmult_prop2)
78     qed
79   then have "[j^2 = a] (mod p)"
80     by (metis  number_of_is_id power2_eq_square succ_bin_simps)
81   with prems show False
83 qed
85 lemma MultInvPair_card_two: "[| zprime p; 2 < p; ~([a = 0] (mod p));
86                                 ~(QuadRes p a); ~([j = 0] (mod p)) |]  ==>
87                              card (MultInvPair a p j) = 2"
88   apply (auto simp add: MultInvPair_def)
89   apply (subgoal_tac "~ (StandardRes p j = StandardRes p (a * MultInv p j))")
90   apply auto
91   apply (metis MultInvPair_distinct Pls_def StandardRes_def aux number_of_is_id one_is_num_one)
92   done
95 subsection {* Properties of SetS *}
97 lemma SetS_finite: "2 < p ==> finite (SetS a p)"
98   by (auto simp add: SetS_def SRStar_finite [of p] finite_imageI)
100 lemma SetS_elems_finite: "\<forall>X \<in> SetS a p. finite X"
101   by (auto simp add: SetS_def MultInvPair_def)
103 lemma SetS_elems_card: "[| zprime p; 2 < p; ~([a = 0] (mod p));
104                         ~(QuadRes p a) |]  ==>
105                         \<forall>X \<in> SetS a p. card X = 2"
106   apply (auto simp add: SetS_def)
107   apply (frule StandardRes_SRStar_prop1a)
108   apply (rule MultInvPair_card_two, auto)
109   done
111 lemma Union_SetS_finite: "2 < p ==> finite (Union (SetS a p))"
112   by (auto simp add: SetS_finite SetS_elems_finite finite_Union)
114 lemma card_setsum_aux: "[| finite S; \<forall>X \<in> S. finite (X::int set);
115     \<forall>X \<in> S. card X = n |] ==> setsum card S = setsum (%x. n) S"
116   by (induct set: finite) auto
118 lemma SetS_card: "[| zprime p; 2 < p; ~([a = 0] (mod p)); ~(QuadRes p a) |] ==>
119                   int(card(SetS a p)) = (p - 1) div 2"
120 proof -
121   assume "zprime p" and "2 < p" and  "~([a = 0] (mod p))" and "~(QuadRes p a)"
122   then have "(p - 1) = 2 * int(card(SetS a p))"
123   proof -
124     have "p - 1 = int(card(Union (SetS a p)))"
125       by (auto simp add: prems MultInvPair_prop2 SRStar_card)
126     also have "... = int (setsum card (SetS a p))"
127       by (auto simp add: prems SetS_finite SetS_elems_finite
128                          MultInvPair_prop1c [of p a] card_Union_disjoint)
129     also have "... = int(setsum (%x.2) (SetS a p))"
130       using prems
131       by (auto simp add: SetS_elems_card SetS_finite SetS_elems_finite
132         card_setsum_aux simp del: setsum_constant)
133     also have "... = 2 * int(card( SetS a p))"
134       by (auto simp add: prems SetS_finite setsum_const2)
135     finally show ?thesis .
136   qed
137   from this show ?thesis
138     by auto
139 qed
141 lemma SetS_setprod_prop: "[| zprime p; 2 < p; ~([a = 0] (mod p));
142                               ~(QuadRes p a); x \<in> (SetS a p) |] ==>
143                           [\<Prod>x = a] (mod p)"
144   apply (auto simp add: SetS_def MultInvPair_def)
145   apply (frule StandardRes_SRStar_prop1a)
146   apply (subgoal_tac "StandardRes p x \<noteq> StandardRes p (a * MultInv p x)")
147   apply (auto simp add: StandardRes_prop2 MultInvPair_distinct)
148   apply (frule_tac m = p and x = x and y = "(a * MultInv p x)" in
149     StandardRes_prop4)
150   apply (subgoal_tac "[x * (a * MultInv p x) = a * (x * MultInv p x)] (mod p)")
151   apply (drule_tac a = "StandardRes p x * StandardRes p (a * MultInv p x)" and
152                    b = "x * (a * MultInv p x)" and
153                    c = "a * (x * MultInv p x)" in  zcong_trans, force)
154   apply (frule_tac p = p and x = x in MultInv_prop2, auto)
155 apply (metis StandardRes_SRStar_prop3 mult_1_right mult_commute zcong_sym zcong_zmult_prop1)
156   apply (auto simp add: zmult_ac)
157   done
159 lemma aux1: "[| 0 < x; (x::int) < a; x \<noteq> (a - 1) |] ==> x < a - 1"
160   by arith
162 lemma aux2: "[| (a::int) < c; b < c |] ==> (a \<le> b | b \<le> a)"
163   by auto
165 lemma SRStar_d22set_prop: "2 < p \<Longrightarrow> (SRStar p) = {1} \<union> (d22set (p - 1))"
166   apply (induct p rule: d22set.induct)
167   apply auto
168   apply (simp add: SRStar_def d22set.simps)
169   apply (simp add: SRStar_def d22set.simps, clarify)
170   apply (frule aux1)
171   apply (frule aux2, auto)
172   apply (simp_all add: SRStar_def)
173   apply (simp add: d22set.simps)
174   apply (frule d22set_le)
175   apply (frule d22set_g_1, auto)
176   done
178 lemma Union_SetS_setprod_prop1: "[| zprime p; 2 < p; ~([a = 0] (mod p)); ~(QuadRes p a) |] ==>
179                                  [\<Prod>(Union (SetS a p)) = a ^ nat ((p - 1) div 2)] (mod p)"
180 proof -
181   assume "zprime p" and "2 < p" and  "~([a = 0] (mod p))" and "~(QuadRes p a)"
182   then have "[\<Prod>(Union (SetS a p)) =
183       setprod (setprod (%x. x)) (SetS a p)] (mod p)"
184     by (auto simp add: SetS_finite SetS_elems_finite
185                        MultInvPair_prop1c setprod_Union_disjoint)
186   also have "[setprod (setprod (%x. x)) (SetS a p) =
187       setprod (%x. a) (SetS a p)] (mod p)"
188     by (rule setprod_same_function_zcong)
189       (auto simp add: prems SetS_setprod_prop SetS_finite)
190   also (zcong_trans) have "[setprod (%x. a) (SetS a p) =
191       a^(card (SetS a p))] (mod p)"
192     by (auto simp add: prems SetS_finite setprod_constant)
193   finally (zcong_trans) show ?thesis
194     apply (rule zcong_trans)
195     apply (subgoal_tac "card(SetS a p) = nat((p - 1) div 2)", auto)
196     apply (subgoal_tac "nat(int(card(SetS a p))) = nat((p - 1) div 2)", force)
197     apply (auto simp add: prems SetS_card)
198     done
199 qed
201 lemma Union_SetS_setprod_prop2: "[| zprime p; 2 < p; ~([a = 0](mod p)) |] ==>
202                                     \<Prod>(Union (SetS a p)) = zfact (p - 1)"
203 proof -
204   assume "zprime p" and "2 < p" and "~([a = 0](mod p))"
205   then have "\<Prod>(Union (SetS a p)) = \<Prod>(SRStar p)"
206     by (auto simp add: MultInvPair_prop2)
207   also have "... = \<Prod>({1} \<union> (d22set (p - 1)))"
208     by (auto simp add: prems SRStar_d22set_prop)
209   also have "... = zfact(p - 1)"
210   proof -
211     have "~(1 \<in> d22set (p - 1)) & finite( d22set (p - 1))"
212       by (metis d22set_fin d22set_g_1 linorder_neq_iff)
213     then have "\<Prod>({1} \<union> (d22set (p - 1))) = \<Prod>(d22set (p - 1))"
214       by auto
215     then show ?thesis
216       by (auto simp add: d22set_prod_zfact)
217   qed
218   finally show ?thesis .
219 qed
221 lemma zfact_prop: "[| zprime p; 2 < p; ~([a = 0] (mod p)); ~(QuadRes p a) |] ==>
222                    [zfact (p - 1) = a ^ nat ((p - 1) div 2)] (mod p)"
223   apply (frule Union_SetS_setprod_prop1)
224   apply (auto simp add: Union_SetS_setprod_prop2)
225   done
227 text {* \medskip Prove the first part of Euler's Criterion: *}
229 lemma Euler_part1: "[| 2 < p; zprime p; ~([x = 0](mod p));
230     ~(QuadRes p x) |] ==>
231       [x^(nat (((p) - 1) div 2)) = -1](mod p)"
232   by (metis Wilson_Russ number_of_is_id zcong_sym zcong_trans zfact_prop)
234 text {* \medskip Prove another part of Euler Criterion: *}
236 lemma aux_1: "0 < p ==> (a::int) ^ nat (p) = a * a ^ (nat (p) - 1)"
237 proof -
238   assume "0 < p"
239   then have "a ^ (nat p) =  a ^ (1 + (nat p - 1))"
241   also have "... = (a ^ 1) * a ^ (nat(p) - 1)"
242     by (simp only: zpower_zadd_distrib)
243   also have "... = a * a ^ (nat(p) - 1)"
244     by auto
245   finally show ?thesis .
246 qed
248 lemma aux_2: "[| (2::int) < p; p \<in> zOdd |] ==> 0 < ((p - 1) div 2)"
249 proof -
250   assume "2 < p" and "p \<in> zOdd"
251   then have "(p - 1):zEven"
252     by (auto simp add: zEven_def zOdd_def)
253   then have aux_1: "2 * ((p - 1) div 2) = (p - 1)"
254     by (auto simp add: even_div_2_prop2)
255   with `2 < p` have "1 < (p - 1)"
256     by auto
257   then have " 1 < (2 * ((p - 1) div 2))"
258     by (auto simp add: aux_1)
259   then have "0 < (2 * ((p - 1) div 2)) div 2"
260     by auto
261   then show ?thesis by auto
262 qed
264 lemma Euler_part2:
265     "[| 2 < p; zprime p; [a = 0] (mod p) |] ==> [0 = a ^ nat ((p - 1) div 2)] (mod p)"
266   apply (frule zprime_zOdd_eq_grt_2)
267   apply (frule aux_2, auto)
268   apply (frule_tac a = a in aux_1, auto)
269   apply (frule zcong_zmult_prop1, auto)
270   done
272 text {* \medskip Prove the final part of Euler's Criterion: *}
274 lemma aux__1: "[| ~([x = 0] (mod p)); [y ^ 2 = x] (mod p)|] ==> ~(p dvd y)"
275   by (metis dvdI power2_eq_square zcong_sym zcong_trans zcong_zero_equiv_div dvd_trans)
277 lemma aux__2: "2 * nat((p - 1) div 2) =  nat (2 * ((p - 1) div 2))"
278   by (auto simp add: nat_mult_distrib)
280 lemma Euler_part3: "[| 2 < p; zprime p; ~([x = 0](mod p)); QuadRes p x |] ==>
281                       [x^(nat (((p) - 1) div 2)) = 1](mod p)"
282   apply (subgoal_tac "p \<in> zOdd")
284    prefer 2
285    apply (metis number_of_is_id numeral_1_eq_1 zprime_zOdd_eq_grt_2)
286   apply (frule aux__1, auto)
287   apply (drule_tac z = "nat ((p - 1) div 2)" in zcong_zpower)
288   apply (auto simp add: zpower_zpower)
289   apply (rule zcong_trans)
290   apply (auto simp add: zcong_sym [of "x ^ nat ((p - 1) div 2)"])
291   apply (metis Little_Fermat even_div_2_prop2 mult_Bit0 number_of_is_id odd_minus_one_even one_is_num_one zmult_1 aux__2)
292   done
295 text {* \medskip Finally show Euler's Criterion: *}
297 theorem Euler_Criterion: "[| 2 < p; zprime p |] ==> [(Legendre a p) =
298     a^(nat (((p) - 1) div 2))] (mod p)"
299   apply (auto simp add: Legendre_def Euler_part2)
300   apply (frule Euler_part3, auto simp add: zcong_sym)[]
301   apply (frule Euler_part1, auto simp add: zcong_sym)[]
302   done
304 end