author nipkow Wed Aug 18 11:09:40 2004 +0200 (2004-08-18) changeset 15140 322485b816ac parent 14981 e73f8140af78 child 15392 290bc97038c7 permissions -rw-r--r--
import -> imports
2     ID:         \$Id\$
3     Authors:    Jeremy Avigad, David Gray, and Adam Kramer
4 *)
6 header {* The law of Quadratic reciprocity *}
8 theory Quadratic_Reciprocity = Gauss:;
10 (***************************************************************)
11 (*                                                             *)
12 (*  Lemmas leading up to the proof of theorem 3.3 in           *)
13 (*  Niven and Zuckerman's presentation                         *)
14 (*                                                             *)
15 (***************************************************************)
17 lemma (in GAUSS) QRLemma1: "a * setsum id A =
18   p * setsum (%x. ((x * a) div p)) A + setsum id D + setsum id E";
19 proof -;
20   from finite_A have "a * setsum id A = setsum (%x. a * x) A";
21     by (auto simp add: setsum_const_mult id_def)
22   also have "setsum (%x. a * x) = setsum (%x. x * a)";
23     by (auto simp add: zmult_commute)
24   also; have "setsum (%x. x * a) A = setsum id B";
25     by (auto simp add: B_def sum_prop_id finite_A inj_on_xa_A)
26   also have "... = setsum (%x. p * (x div p) + StandardRes p x) B";
27     apply (rule setsum_same_function)
28     by (auto simp add: finite_B StandardRes_def zmod_zdiv_equality)
29   also have "... = setsum (%x. p * (x div p)) B + setsum (StandardRes p) B";
30     by (rule setsum_addf)
31   also; have "setsum (StandardRes p) B = setsum id C";
32     by (auto simp add: C_def sum_prop_id [THEN sym] finite_B
33       SR_B_inj)
34   also; from C_eq have "... = setsum id (D \<union> E)";
35     by auto
36   also; from finite_D finite_E have "... = setsum id D + setsum id E";
37     apply (rule setsum_Un_disjoint)
38     by (auto simp add: D_def E_def)
39   also have "setsum (%x. p * (x div p)) B =
40       setsum ((%x. p * (x div p)) o (%x. (x * a))) A";
41     by (auto simp add: B_def sum_prop finite_A inj_on_xa_A)
42   also have "... = setsum (%x. p * ((x * a) div p)) A";
43     by (auto simp add: o_def)
44   also from finite_A have "setsum (%x. p * ((x * a) div p)) A =
45     p * setsum (%x. ((x * a) div p)) A";
46     by (auto simp add: setsum_const_mult)
47   finally show ?thesis by arith
48 qed;
50 lemma (in GAUSS) QRLemma2: "setsum id A = p * int (card E) - setsum id E +
51   setsum id D";
52 proof -;
53   from F_Un_D_eq_A have "setsum id A = setsum id (D \<union> F)";
54     by (simp add: Un_commute)
55   also from F_D_disj finite_D finite_F have
56       "... = setsum id D + setsum id F";
57     apply (simp add: Int_commute)
58     by (intro setsum_Un_disjoint)
59   also from F_def have "F = (%x. (p - x)) ` E";
60     by auto
61   also from finite_E inj_on_pminusx_E have "setsum id ((%x. (p - x)) ` E) =
62       setsum (%x. (p - x)) E";
63     by (auto simp add: sum_prop)
64   also from finite_E have "setsum (op - p) E = setsum (%x. p) E - setsum id E";
65     by (auto simp add: setsum_minus id_def)
66   also from finite_E have "setsum (%x. p) E = p * int(card E)";
67     by (intro setsum_const)
68   finally show ?thesis;
69     by arith
70 qed;
72 lemma (in GAUSS) QRLemma3: "(a - 1) * setsum id A =
73     p * (setsum (%x. ((x * a) div p)) A - int(card E)) + 2 * setsum id E";
74 proof -;
75   have "(a - 1) * setsum id A = a * setsum id A - setsum id A";
76     by (auto simp add: zdiff_zmult_distrib)
77   also note QRLemma1;
78   also; from QRLemma2 have "p * (\<Sum>x \<in> A. x * a div p) + setsum id D +
79      setsum id E - setsum id A =
80       p * (\<Sum>x \<in> A. x * a div p) + setsum id D +
81       setsum id E - (p * int (card E) - setsum id E + setsum id D)";
82     by auto
83   also; have "... = p * (\<Sum>x \<in> A. x * a div p) -
84       p * int (card E) + 2 * setsum id E";
85     by arith
86   finally show ?thesis;
87     by (auto simp only: zdiff_zmult_distrib2)
88 qed;
90 lemma (in GAUSS) QRLemma4: "a \<in> zOdd ==>
91     (setsum (%x. ((x * a) div p)) A \<in> zEven) = (int(card E): zEven)";
92 proof -;
93   assume a_odd: "a \<in> zOdd";
94   from QRLemma3 have a: "p * (setsum (%x. ((x * a) div p)) A - int(card E)) =
95       (a - 1) * setsum id A - 2 * setsum id E";
96     by arith
97   from a_odd have "a - 1 \<in> zEven"
98     by (rule odd_minus_one_even)
99   hence "(a - 1) * setsum id A \<in> zEven";
100     by (rule even_times_either)
101   moreover have "2 * setsum id E \<in> zEven";
102     by (auto simp add: zEven_def)
103   ultimately have "(a - 1) * setsum id A - 2 * setsum id E \<in> zEven"
104     by (rule even_minus_even)
105   with a have "p * (setsum (%x. ((x * a) div p)) A - int(card E)): zEven";
106     by simp
107   hence "p \<in> zEven | (setsum (%x. ((x * a) div p)) A - int(card E)): zEven";
108     by (rule EvenOdd.even_product)
109   with p_odd have "(setsum (%x. ((x * a) div p)) A - int(card E)): zEven";
110     by (auto simp add: odd_iff_not_even)
111   thus ?thesis;
112     by (auto simp only: even_diff [THEN sym])
113 qed;
115 lemma (in GAUSS) QRLemma5: "a \<in> zOdd ==>
116    (-1::int)^(card E) = (-1::int)^(nat(setsum (%x. ((x * a) div p)) A))";
117 proof -;
118   assume "a \<in> zOdd";
119   from QRLemma4 have
120     "(int(card E): zEven) = (setsum (%x. ((x * a) div p)) A \<in> zEven)";..;
121   moreover have "0 \<le> int(card E)";
122     by auto
123   moreover have "0 \<le> setsum (%x. ((x * a) div p)) A";
124     proof (intro setsum_non_neg);
125       from finite_A show "finite A";.;
126       next show "\<forall>x \<in> A. 0 \<le> x * a div p";
127       proof;
128         fix x;
129         assume "x \<in> A";
130         then have "0 \<le> x";
131           by (auto simp add: A_def)
132         with a_nonzero have "0 \<le> x * a";
133           by (auto simp add: zero_le_mult_iff)
134         with p_g_2 show "0 \<le> x * a div p";
135           by (auto simp add: pos_imp_zdiv_nonneg_iff)
136       qed;
137     qed;
138   ultimately have "(-1::int)^nat((int (card E))) =
139       (-1)^nat(((\<Sum>x \<in> A. x * a div p)))";
140     by (intro neg_one_power_parity, auto)
141   also have "nat (int(card E)) = card E";
142     by auto
143   finally show ?thesis;.;
144 qed;
146 lemma MainQRLemma: "[| a \<in> zOdd; 0 < a; ~([a = 0] (mod p));p \<in> zprime; 2 < p;
147   A = {x. 0 < x & x \<le> (p - 1) div 2} |] ==>
148   (Legendre a p) = (-1::int)^(nat(setsum (%x. ((x * a) div p)) A))";
149   apply (subst GAUSS.gauss_lemma)
150   apply (auto simp add: GAUSS_def)
151   apply (subst GAUSS.QRLemma5)
152 by (auto simp add: GAUSS_def)
154 (******************************************************************)
155 (*                                                                *)
156 (* Stuff about S, S1 and S2...                                    *)
157 (*                                                                *)
158 (******************************************************************)
160 locale QRTEMP =
161   fixes p     :: "int"
162   fixes q     :: "int"
163   fixes P_set :: "int set"
164   fixes Q_set :: "int set"
165   fixes S     :: "(int * int) set"
166   fixes S1    :: "(int * int) set"
167   fixes S2    :: "(int * int) set"
168   fixes f1    :: "int => (int * int) set"
169   fixes f2    :: "int => (int * int) set"
171   assumes p_prime: "p \<in> zprime"
172   assumes p_g_2: "2 < p"
173   assumes q_prime: "q \<in> zprime"
174   assumes q_g_2: "2 < q"
175   assumes p_neq_q:      "p \<noteq> q"
177   defines P_set_def: "P_set == {x. 0 < x & x \<le> ((p - 1) div 2) }"
178   defines Q_set_def: "Q_set == {x. 0 < x & x \<le> ((q - 1) div 2) }"
179   defines S_def:     "S     == P_set <*> Q_set"
180   defines S1_def:    "S1    == { (x, y). (x, y):S & ((p * y) < (q * x)) }"
181   defines S2_def:    "S2    == { (x, y). (x, y):S & ((q * x) < (p * y)) }"
182   defines f1_def:    "f1 j  == { (j1, y). (j1, y):S & j1 = j &
183                                  (y \<le> (q * j) div p) }"
184   defines f2_def:    "f2 j  == { (x, j1). (x, j1):S & j1 = j &
185                                  (x \<le> (p * j) div q) }";
187 lemma (in QRTEMP) p_fact: "0 < (p - 1) div 2";
188 proof -;
189   from prems have "2 < p" by (simp add: QRTEMP_def)
190   then have "2 \<le> p - 1" by arith
191   then have "2 div 2 \<le> (p - 1) div 2" by (rule zdiv_mono1, auto)
192   then show ?thesis by auto
193 qed;
195 lemma (in QRTEMP) q_fact: "0 < (q - 1) div 2";
196 proof -;
197   from prems have "2 < q" by (simp add: QRTEMP_def)
198   then have "2 \<le> q - 1" by arith
199   then have "2 div 2 \<le> (q - 1) div 2" by (rule zdiv_mono1, auto)
200   then show ?thesis by auto
201 qed;
203 lemma (in QRTEMP) pb_neq_qa: "[|1 \<le> b; b \<le> (q - 1) div 2 |] ==>
204     (p * b \<noteq> q * a)";
205 proof;
206   assume "p * b = q * a" and "1 \<le> b" and "b \<le> (q - 1) div 2";
207   then have "q dvd (p * b)" by (auto simp add: dvd_def)
208   with q_prime p_g_2 have "q dvd p | q dvd b";
209     by (auto simp add: zprime_zdvd_zmult)
210   moreover have "~ (q dvd p)";
211   proof;
212     assume "q dvd p";
213     with p_prime have "q = 1 | q = p"
214       apply (auto simp add: zprime_def QRTEMP_def)
215       apply (drule_tac x = q and R = False in allE)
216       apply (simp add: QRTEMP_def)
217       apply (subgoal_tac "0 \<le> q", simp add: QRTEMP_def)
218       apply (insert prems)
219     by (auto simp add: QRTEMP_def)
220     with q_g_2 p_neq_q show False by auto
221   qed;
222   ultimately have "q dvd b" by auto
223   then have "q \<le> b";
224   proof -;
225     assume "q dvd b";
226     moreover from prems have "0 < b" by auto
227     ultimately show ?thesis by (insert zdvd_bounds [of q b], auto)
228   qed;
229   with prems have "q \<le> (q - 1) div 2" by auto
230   then have "2 * q \<le> 2 * ((q - 1) div 2)" by arith
231   then have "2 * q \<le> q - 1";
232   proof -;
233     assume "2 * q \<le> 2 * ((q - 1) div 2)";
234     with prems have "q \<in> zOdd" by (auto simp add: QRTEMP_def zprime_zOdd_eq_grt_2)
235     with odd_minus_one_even have "(q - 1):zEven" by auto
236     with even_div_2_prop2 have "(q - 1) = 2 * ((q - 1) div 2)" by auto
237     with prems show ?thesis by auto
238   qed;
239   then have p1: "q \<le> -1" by arith
240   with q_g_2 show False by auto
241 qed;
243 lemma (in QRTEMP) P_set_finite: "finite (P_set)";
244   by (insert p_fact, auto simp add: P_set_def bdd_int_set_l_le_finite)
246 lemma (in QRTEMP) Q_set_finite: "finite (Q_set)";
247   by (insert q_fact, auto simp add: Q_set_def bdd_int_set_l_le_finite)
249 lemma (in QRTEMP) S_finite: "finite S";
250   by (auto simp add: S_def  P_set_finite Q_set_finite cartesian_product_finite)
252 lemma (in QRTEMP) S1_finite: "finite S1";
253 proof -;
254   have "finite S" by (auto simp add: S_finite)
255   moreover have "S1 \<subseteq> S" by (auto simp add: S1_def S_def)
256   ultimately show ?thesis by (auto simp add: finite_subset)
257 qed;
259 lemma (in QRTEMP) S2_finite: "finite S2";
260 proof -;
261   have "finite S" by (auto simp add: S_finite)
262   moreover have "S2 \<subseteq> S" by (auto simp add: S2_def S_def)
263   ultimately show ?thesis by (auto simp add: finite_subset)
264 qed;
266 lemma (in QRTEMP) P_set_card: "(p - 1) div 2 = int (card (P_set))";
267   by (insert p_fact, auto simp add: P_set_def card_bdd_int_set_l_le)
269 lemma (in QRTEMP) Q_set_card: "(q - 1) div 2 = int (card (Q_set))";
270   by (insert q_fact, auto simp add: Q_set_def card_bdd_int_set_l_le)
272 lemma (in QRTEMP) S_card: "((p - 1) div 2) * ((q - 1) div 2) = int (card(S))";
273   apply (insert P_set_card Q_set_card P_set_finite Q_set_finite)
274   apply (auto simp add: S_def zmult_int setsum_constant_nat)
275 done
277 lemma (in QRTEMP) S1_Int_S2_prop: "S1 \<inter> S2 = {}";
278   by (auto simp add: S1_def S2_def)
280 lemma (in QRTEMP) S1_Union_S2_prop: "S = S1 \<union> S2";
281   apply (auto simp add: S_def P_set_def Q_set_def S1_def S2_def)
282   proof -;
283     fix a and b;
284     assume "~ q * a < p * b" and b1: "0 < b" and b2: "b \<le> (q - 1) div 2";
285     with zless_linear have "(p * b < q * a) | (p * b = q * a)" by auto
286     moreover from pb_neq_qa b1 b2 have "(p * b \<noteq> q * a)" by auto
287     ultimately show "p * b < q * a" by auto
288   qed;
290 lemma (in QRTEMP) card_sum_S1_S2: "((p - 1) div 2) * ((q - 1) div 2) =
291     int(card(S1)) + int(card(S2))";
292 proof-;
293   have "((p - 1) div 2) * ((q - 1) div 2) = int (card(S))";
294     by (auto simp add: S_card)
295   also have "... = int( card(S1) + card(S2))";
296     apply (insert S1_finite S2_finite S1_Int_S2_prop S1_Union_S2_prop)
297     apply (drule card_Un_disjoint, auto)
298   done
299   also have "... = int(card(S1)) + int(card(S2))" by auto
300   finally show ?thesis .;
301 qed;
303 lemma (in QRTEMP) aux1a: "[| 0 < a; a \<le> (p - 1) div 2;
304                              0 < b; b \<le> (q - 1) div 2 |] ==>
305                           (p * b < q * a) = (b \<le> q * a div p)";
306 proof -;
307   assume "0 < a" and "a \<le> (p - 1) div 2" and "0 < b" and "b \<le> (q - 1) div 2";
308   have "p * b < q * a ==> b \<le> q * a div p";
309   proof -;
310     assume "p * b < q * a";
311     then have "p * b \<le> q * a" by auto
312     then have "(p * b) div p \<le> (q * a) div p";
313       by (rule zdiv_mono1, insert p_g_2, auto)
314     then show "b \<le> (q * a) div p";
315       apply (subgoal_tac "p \<noteq> 0")
316       apply (frule zdiv_zmult_self2, force)
317       by (insert p_g_2, auto)
318   qed;
319   moreover have "b \<le> q * a div p ==> p * b < q * a";
320   proof -;
321     assume "b \<le> q * a div p";
322     then have "p * b \<le> p * ((q * a) div p)";
323       by (insert p_g_2, auto simp add: mult_le_cancel_left)
324     also have "... \<le> q * a";
325       by (rule zdiv_leq_prop, insert p_g_2, auto)
326     finally have "p * b \<le> q * a" .;
327     then have "p * b < q * a | p * b = q * a";
328       by (simp only: order_le_imp_less_or_eq)
329     moreover have "p * b \<noteq> q * a";
330       by (rule  pb_neq_qa, insert prems, auto)
331     ultimately show ?thesis by auto
332   qed;
333   ultimately show ?thesis ..;
334 qed;
336 lemma (in QRTEMP) aux1b: "[| 0 < a; a \<le> (p - 1) div 2;
337                              0 < b; b \<le> (q - 1) div 2 |] ==>
338                           (q * a < p * b) = (a \<le> p * b div q)";
339 proof -;
340   assume "0 < a" and "a \<le> (p - 1) div 2" and "0 < b" and "b \<le> (q - 1) div 2";
341   have "q * a < p * b ==> a \<le> p * b div q";
342   proof -;
343     assume "q * a < p * b";
344     then have "q * a \<le> p * b" by auto
345     then have "(q * a) div q \<le> (p * b) div q";
346       by (rule zdiv_mono1, insert q_g_2, auto)
347     then show "a \<le> (p * b) div q";
348       apply (subgoal_tac "q \<noteq> 0")
349       apply (frule zdiv_zmult_self2, force)
350       by (insert q_g_2, auto)
351   qed;
352   moreover have "a \<le> p * b div q ==> q * a < p * b";
353   proof -;
354     assume "a \<le> p * b div q";
355     then have "q * a \<le> q * ((p * b) div q)";
356       by (insert q_g_2, auto simp add: mult_le_cancel_left)
357     also have "... \<le> p * b";
358       by (rule zdiv_leq_prop, insert q_g_2, auto)
359     finally have "q * a \<le> p * b" .;
360     then have "q * a < p * b | q * a = p * b";
361       by (simp only: order_le_imp_less_or_eq)
362     moreover have "p * b \<noteq> q * a";
363       by (rule  pb_neq_qa, insert prems, auto)
364     ultimately show ?thesis by auto
365   qed;
366   ultimately show ?thesis ..;
367 qed;
369 lemma aux2: "[| p \<in> zprime; q \<in> zprime; 2 < p; 2 < q |] ==>
370              (q * ((p - 1) div 2)) div p \<le> (q - 1) div 2";
371 proof-;
372   assume "p \<in> zprime" and "q \<in> zprime" and "2 < p" and "2 < q";
373   (* Set up what's even and odd *)
374   then have "p \<in> zOdd & q \<in> zOdd";
375     by (auto simp add:  zprime_zOdd_eq_grt_2)
376   then have even1: "(p - 1):zEven & (q - 1):zEven";
377     by (auto simp add: odd_minus_one_even)
378   then have even2: "(2 * p):zEven & ((q - 1) * p):zEven";
379     by (auto simp add: zEven_def)
380   then have even3: "(((q - 1) * p) + (2 * p)):zEven";
381     by (auto simp: EvenOdd.even_plus_even)
382   (* using these prove it *)
383   from prems have "q * (p - 1) < ((q - 1) * p) + (2 * p)";
384     by (auto simp add: int_distrib)
385   then have "((p - 1) * q) div 2 < (((q - 1) * p) + (2 * p)) div 2";
386     apply (rule_tac x = "((p - 1) * q)" in even_div_2_l);
387     by (auto simp add: even3, auto simp add: zmult_ac)
388   also have "((p - 1) * q) div 2 = q * ((p - 1) div 2)";
389     by (auto simp add: even1 even_prod_div_2)
390   also have "(((q - 1) * p) + (2 * p)) div 2 = (((q - 1) div 2) * p) + p";
391     by (auto simp add: even1 even2 even_prod_div_2 even_sum_div_2)
392   finally show ?thesis
393     apply (rule_tac x = " q * ((p - 1) div 2)" and
394                     y = "(q - 1) div 2" in div_prop2);
395     by (insert prems, auto)
396 qed;
398 lemma (in QRTEMP) aux3a: "\<forall>j \<in> P_set. int (card (f1 j)) = (q * j) div p";
399 proof;
400   fix j;
401   assume j_fact: "j \<in> P_set";
402   have "int (card (f1 j)) = int (card {y. y \<in> Q_set & y \<le> (q * j) div p})";
403   proof -;
404     have "finite (f1 j)";
405     proof -;
406       have "(f1 j) \<subseteq> S" by (auto simp add: f1_def)
407       with S_finite show ?thesis by (auto simp add: finite_subset)
408     qed;
409     moreover have "inj_on (%(x,y). y) (f1 j)";
410       by (auto simp add: f1_def inj_on_def)
411     ultimately have "card ((%(x,y). y) ` (f1 j)) = card  (f1 j)";
412       by (auto simp add: f1_def card_image)
413     moreover have "((%(x,y). y) ` (f1 j)) = {y. y \<in> Q_set & y \<le> (q * j) div p}";
414       by (insert prems, auto simp add: f1_def S_def Q_set_def P_set_def
415         image_def)
416     ultimately show ?thesis by (auto simp add: f1_def)
417   qed;
418   also have "... = int (card {y. 0 < y & y \<le> (q * j) div p})";
419   proof -;
420     have "{y. y \<in> Q_set & y \<le> (q * j) div p} =
421         {y. 0 < y & y \<le> (q * j) div p}";
422       apply (auto simp add: Q_set_def)
423       proof -;
424         fix x;
425         assume "0 < x" and "x \<le> q * j div p";
426         with j_fact P_set_def  have "j \<le> (p - 1) div 2"; by auto
427         with q_g_2; have "q * j \<le> q * ((p - 1) div 2)";
428           by (auto simp add: mult_le_cancel_left)
429         with p_g_2 have "q * j div p \<le> q * ((p - 1) div 2) div p";
430           by (auto simp add: zdiv_mono1)
431         also from prems have "... \<le> (q - 1) div 2";
432           apply simp apply (insert aux2) by (simp add: QRTEMP_def)
433         finally show "x \<le> (q - 1) div 2" by (insert prems, auto)
434       qed;
435     then show ?thesis by auto
436   qed;
437   also have "... = (q * j) div p";
438   proof -;
439     from j_fact P_set_def have "0 \<le> j" by auto
440     with q_g_2 have "q * 0 \<le> q * j" by (auto simp only: mult_left_mono)
441     then have "0 \<le> q * j" by auto
442     then have "0 div p \<le> (q * j) div p";
443       apply (rule_tac a = 0 in zdiv_mono1)
444       by (insert p_g_2, auto)
445     also have "0 div p = 0" by auto
446     finally show ?thesis by (auto simp add: card_bdd_int_set_l_le)
447   qed;
448   finally show "int (card (f1 j)) = q * j div p" .;
449 qed;
451 lemma (in QRTEMP) aux3b: "\<forall>j \<in> Q_set. int (card (f2 j)) = (p * j) div q";
452 proof;
453   fix j;
454   assume j_fact: "j \<in> Q_set";
455   have "int (card (f2 j)) = int (card {y. y \<in> P_set & y \<le> (p * j) div q})";
456   proof -;
457     have "finite (f2 j)";
458     proof -;
459       have "(f2 j) \<subseteq> S" by (auto simp add: f2_def)
460       with S_finite show ?thesis by (auto simp add: finite_subset)
461     qed;
462     moreover have "inj_on (%(x,y). x) (f2 j)";
463       by (auto simp add: f2_def inj_on_def)
464     ultimately have "card ((%(x,y). x) ` (f2 j)) = card  (f2 j)";
465       by (auto simp add: f2_def card_image)
466     moreover have "((%(x,y). x) ` (f2 j)) = {y. y \<in> P_set & y \<le> (p * j) div q}";
467       by (insert prems, auto simp add: f2_def S_def Q_set_def
468         P_set_def image_def)
469     ultimately show ?thesis by (auto simp add: f2_def)
470   qed;
471   also have "... = int (card {y. 0 < y & y \<le> (p * j) div q})";
472   proof -;
473     have "{y. y \<in> P_set & y \<le> (p * j) div q} =
474         {y. 0 < y & y \<le> (p * j) div q}";
475       apply (auto simp add: P_set_def)
476       proof -;
477         fix x;
478         assume "0 < x" and "x \<le> p * j div q";
479         with j_fact Q_set_def  have "j \<le> (q - 1) div 2"; by auto
480         with p_g_2; have "p * j \<le> p * ((q - 1) div 2)";
481           by (auto simp add: mult_le_cancel_left)
482         with q_g_2 have "p * j div q \<le> p * ((q - 1) div 2) div q";
483           by (auto simp add: zdiv_mono1)
484         also from prems have "... \<le> (p - 1) div 2";
485           by (auto simp add: aux2 QRTEMP_def)
486         finally show "x \<le> (p - 1) div 2" by (insert prems, auto)
487       qed;
488     then show ?thesis by auto
489   qed;
490   also have "... = (p * j) div q";
491   proof -;
492     from j_fact Q_set_def have "0 \<le> j" by auto
493     with p_g_2 have "p * 0 \<le> p * j" by (auto simp only: mult_left_mono)
494     then have "0 \<le> p * j" by auto
495     then have "0 div q \<le> (p * j) div q";
496       apply (rule_tac a = 0 in zdiv_mono1)
497       by (insert q_g_2, auto)
498     also have "0 div q = 0" by auto
499     finally show ?thesis by (auto simp add: card_bdd_int_set_l_le)
500   qed;
501   finally show "int (card (f2 j)) = p * j div q" .;
502 qed;
504 lemma (in QRTEMP) S1_card: "int (card(S1)) = setsum (%j. (q * j) div p) P_set";
505 proof -;
506   have "\<forall>x \<in> P_set. finite (f1 x)";
507   proof;
508     fix x;
509     have "f1 x \<subseteq> S" by (auto simp add: f1_def)
510     with S_finite show "finite (f1 x)" by (auto simp add: finite_subset)
511   qed;
512   moreover have "(\<forall>x \<in> P_set. \<forall>y \<in> P_set. x \<noteq> y --> (f1 x) \<inter> (f1 y) = {})";
513     by (auto simp add: f1_def)
514   moreover note P_set_finite;
515   ultimately have "int(card (UNION P_set f1)) =
516       setsum (%x. int(card (f1 x))) P_set";
517     by (rule_tac A = P_set in int_card_indexed_union_disjoint_sets, auto)
518   moreover have "S1 = UNION P_set f1";
519     by (auto simp add: f1_def S_def S1_def S2_def P_set_def Q_set_def aux1a)
520   ultimately have "int(card (S1)) = setsum (%j. int(card (f1 j))) P_set"
521     by auto
522   also have "... = setsum (%j. q * j div p) P_set";
523   proof -;
524     note aux3a
525     with  P_set_finite show ?thesis by (rule setsum_same_function)
526   qed;
527   finally show ?thesis .;
528 qed;
530 lemma (in QRTEMP) S2_card: "int (card(S2)) = setsum (%j. (p * j) div q) Q_set";
531 proof -;
532   have "\<forall>x \<in> Q_set. finite (f2 x)";
533   proof;
534     fix x;
535     have "f2 x \<subseteq> S" by (auto simp add: f2_def)
536     with S_finite show "finite (f2 x)" by (auto simp add: finite_subset)
537   qed;
538   moreover have "(\<forall>x \<in> Q_set. \<forall>y \<in> Q_set. x \<noteq> y -->
539       (f2 x) \<inter> (f2 y) = {})";
540     by (auto simp add: f2_def)
541   moreover note Q_set_finite;
542   ultimately have "int(card (UNION Q_set f2)) =
543       setsum (%x. int(card (f2 x))) Q_set";
544     by (rule_tac A = Q_set in int_card_indexed_union_disjoint_sets, auto)
545   moreover have "S2 = UNION Q_set f2";
546     by (auto simp add: f2_def S_def S1_def S2_def P_set_def Q_set_def aux1b)
547   ultimately have "int(card (S2)) = setsum (%j. int(card (f2 j))) Q_set"
548     by auto
549   also have "... = setsum (%j. p * j div q) Q_set";
550   proof -;
551     note aux3b;
552     with Q_set_finite show ?thesis by (rule setsum_same_function)
553   qed;
554   finally show ?thesis .;
555 qed;
557 lemma (in QRTEMP) S1_carda: "int (card(S1)) =
558     setsum (%j. (j * q) div p) P_set";
559   by (auto simp add: S1_card zmult_ac)
561 lemma (in QRTEMP) S2_carda: "int (card(S2)) =
562     setsum (%j. (j * p) div q) Q_set";
563   by (auto simp add: S2_card zmult_ac)
565 lemma (in QRTEMP) pq_sum_prop: "(setsum (%j. (j * p) div q) Q_set) +
566     (setsum (%j. (j * q) div p) P_set) = ((p - 1) div 2) * ((q - 1) div 2)";
567 proof -;
568   have "(setsum (%j. (j * p) div q) Q_set) +
569       (setsum (%j. (j * q) div p) P_set) = int (card S2) + int (card S1)";
570     by (auto simp add: S1_carda S2_carda)
571   also have "... = int (card S1) + int (card S2)";
572     by auto
573   also have "... = ((p - 1) div 2) * ((q - 1) div 2)";
574     by (auto simp add: card_sum_S1_S2)
575   finally show ?thesis .;
576 qed;
578 lemma pq_prime_neq: "[| p \<in> zprime; q \<in> zprime; p \<noteq> q |] ==> (~[p = 0] (mod q))";
579   apply (auto simp add: zcong_eq_zdvd_prop zprime_def)
580   apply (drule_tac x = q in allE)
581   apply (drule_tac x = p in allE)
582 by auto
584 lemma (in QRTEMP) QR_short: "(Legendre p q) * (Legendre q p) =
585     (-1::int)^nat(((p - 1) div 2)*((q - 1) div 2))";
586 proof -;
587   from prems have "~([p = 0] (mod q))";
588     by (auto simp add: pq_prime_neq QRTEMP_def)
589   with prems have a1: "(Legendre p q) = (-1::int) ^
590       nat(setsum (%x. ((x * p) div q)) Q_set)";
591     apply (rule_tac p = q in  MainQRLemma)
592     by (auto simp add: zprime_zOdd_eq_grt_2 QRTEMP_def)
593   from prems have "~([q = 0] (mod p))";
594     apply (rule_tac p = q and q = p in pq_prime_neq)
595     apply (simp add: QRTEMP_def)+;
596     by arith
597   with prems have a2: "(Legendre q p) =
598       (-1::int) ^ nat(setsum (%x. ((x * q) div p)) P_set)";
599     apply (rule_tac p = p in  MainQRLemma)
600     by (auto simp add: zprime_zOdd_eq_grt_2 QRTEMP_def)
601   from a1 a2 have "(Legendre p q) * (Legendre q p) =
602       (-1::int) ^ nat(setsum (%x. ((x * p) div q)) Q_set) *
603         (-1::int) ^ nat(setsum (%x. ((x * q) div p)) P_set)";
604     by auto
605   also have "... = (-1::int) ^ (nat(setsum (%x. ((x * p) div q)) Q_set) +
606                    nat(setsum (%x. ((x * q) div p)) P_set))";
608   also have "nat(setsum (%x. ((x * p) div q)) Q_set) +
609       nat(setsum (%x. ((x * q) div p)) P_set) =
610         nat((setsum (%x. ((x * p) div q)) Q_set) +
611           (setsum (%x. ((x * q) div p)) P_set))";
612     apply (rule_tac z1 = "setsum (%x. ((x * p) div q)) Q_set" in
613       nat_add_distrib [THEN sym]);
614     by (auto simp add: S1_carda [THEN sym] S2_carda [THEN sym])
615   also have "... = nat(((p - 1) div 2) * ((q - 1) div 2))";
616     by (auto simp add: pq_sum_prop)
617   finally show ?thesis .;
618 qed;