|
1 (* Author: Thomas M. Rasmussen |
|
2 Copyright 2000 University of Cambridge |
|
3 *) |
|
4 |
|
5 header {* The Chinese Remainder Theorem *} |
|
6 |
|
7 theory Chinese |
|
8 imports IntPrimes |
|
9 begin |
|
10 |
|
11 text {* |
|
12 The Chinese Remainder Theorem for an arbitrary finite number of |
|
13 equations. (The one-equation case is included in theory @{text |
|
14 IntPrimes}. Uses functions for indexing.\footnote{Maybe @{term |
|
15 funprod} and @{term funsum} should be based on general @{term fold} |
|
16 on indices?} |
|
17 *} |
|
18 |
|
19 |
|
20 subsection {* Definitions *} |
|
21 |
|
22 consts |
|
23 funprod :: "(nat => int) => nat => nat => int" |
|
24 funsum :: "(nat => int) => nat => nat => int" |
|
25 |
|
26 primrec |
|
27 "funprod f i 0 = f i" |
|
28 "funprod f i (Suc n) = f (Suc (i + n)) * funprod f i n" |
|
29 |
|
30 primrec |
|
31 "funsum f i 0 = f i" |
|
32 "funsum f i (Suc n) = f (Suc (i + n)) + funsum f i n" |
|
33 |
|
34 definition |
|
35 m_cond :: "nat => (nat => int) => bool" where |
|
36 "m_cond n mf = |
|
37 ((\<forall>i. i \<le> n --> 0 < mf i) \<and> |
|
38 (\<forall>i j. i \<le> n \<and> j \<le> n \<and> i \<noteq> j --> zgcd (mf i) (mf j) = 1))" |
|
39 |
|
40 definition |
|
41 km_cond :: "nat => (nat => int) => (nat => int) => bool" where |
|
42 "km_cond n kf mf = (\<forall>i. i \<le> n --> zgcd (kf i) (mf i) = 1)" |
|
43 |
|
44 definition |
|
45 lincong_sol :: |
|
46 "nat => (nat => int) => (nat => int) => (nat => int) => int => bool" where |
|
47 "lincong_sol n kf bf mf x = (\<forall>i. i \<le> n --> zcong (kf i * x) (bf i) (mf i))" |
|
48 |
|
49 definition |
|
50 mhf :: "(nat => int) => nat => nat => int" where |
|
51 "mhf mf n i = |
|
52 (if i = 0 then funprod mf (Suc 0) (n - Suc 0) |
|
53 else if i = n then funprod mf 0 (n - Suc 0) |
|
54 else funprod mf 0 (i - Suc 0) * funprod mf (Suc i) (n - Suc 0 - i))" |
|
55 |
|
56 definition |
|
57 xilin_sol :: |
|
58 "nat => nat => (nat => int) => (nat => int) => (nat => int) => int" where |
|
59 "xilin_sol i n kf bf mf = |
|
60 (if 0 < n \<and> i \<le> n \<and> m_cond n mf \<and> km_cond n kf mf then |
|
61 (SOME x. 0 \<le> x \<and> x < mf i \<and> zcong (kf i * mhf mf n i * x) (bf i) (mf i)) |
|
62 else 0)" |
|
63 |
|
64 definition |
|
65 x_sol :: "nat => (nat => int) => (nat => int) => (nat => int) => int" where |
|
66 "x_sol n kf bf mf = funsum (\<lambda>i. xilin_sol i n kf bf mf * mhf mf n i) 0 n" |
|
67 |
|
68 |
|
69 text {* \medskip @{term funprod} and @{term funsum} *} |
|
70 |
|
71 lemma funprod_pos: "(\<forall>i. i \<le> n --> 0 < mf i) ==> 0 < funprod mf 0 n" |
|
72 apply (induct n) |
|
73 apply auto |
|
74 apply (simp add: zero_less_mult_iff) |
|
75 done |
|
76 |
|
77 lemma funprod_zgcd [rule_format (no_asm)]: |
|
78 "(\<forall>i. k \<le> i \<and> i \<le> k + l --> zgcd (mf i) (mf m) = 1) --> |
|
79 zgcd (funprod mf k l) (mf m) = 1" |
|
80 apply (induct l) |
|
81 apply simp_all |
|
82 apply (rule impI)+ |
|
83 apply (subst zgcd_zmult_cancel) |
|
84 apply auto |
|
85 done |
|
86 |
|
87 lemma funprod_zdvd [rule_format]: |
|
88 "k \<le> i --> i \<le> k + l --> mf i dvd funprod mf k l" |
|
89 apply (induct l) |
|
90 apply auto |
|
91 apply (subgoal_tac "i = Suc (k + l)") |
|
92 apply (simp_all (no_asm_simp)) |
|
93 done |
|
94 |
|
95 lemma funsum_mod: |
|
96 "funsum f k l mod m = funsum (\<lambda>i. (f i) mod m) k l mod m" |
|
97 apply (induct l) |
|
98 apply auto |
|
99 apply (rule trans) |
|
100 apply (rule mod_add_eq) |
|
101 apply simp |
|
102 apply (rule mod_add_right_eq [symmetric]) |
|
103 done |
|
104 |
|
105 lemma funsum_zero [rule_format (no_asm)]: |
|
106 "(\<forall>i. k \<le> i \<and> i \<le> k + l --> f i = 0) --> (funsum f k l) = 0" |
|
107 apply (induct l) |
|
108 apply auto |
|
109 done |
|
110 |
|
111 lemma funsum_oneelem [rule_format (no_asm)]: |
|
112 "k \<le> j --> j \<le> k + l --> |
|
113 (\<forall>i. k \<le> i \<and> i \<le> k + l \<and> i \<noteq> j --> f i = 0) --> |
|
114 funsum f k l = f j" |
|
115 apply (induct l) |
|
116 prefer 2 |
|
117 apply clarify |
|
118 defer |
|
119 apply clarify |
|
120 apply (subgoal_tac "k = j") |
|
121 apply (simp_all (no_asm_simp)) |
|
122 apply (case_tac "Suc (k + l) = j") |
|
123 apply (subgoal_tac "funsum f k l = 0") |
|
124 apply (rule_tac [2] funsum_zero) |
|
125 apply (subgoal_tac [3] "f (Suc (k + l)) = 0") |
|
126 apply (subgoal_tac [3] "j \<le> k + l") |
|
127 prefer 4 |
|
128 apply arith |
|
129 apply auto |
|
130 done |
|
131 |
|
132 |
|
133 subsection {* Chinese: uniqueness *} |
|
134 |
|
135 lemma zcong_funprod_aux: |
|
136 "m_cond n mf ==> km_cond n kf mf |
|
137 ==> lincong_sol n kf bf mf x ==> lincong_sol n kf bf mf y |
|
138 ==> [x = y] (mod mf n)" |
|
139 apply (unfold m_cond_def km_cond_def lincong_sol_def) |
|
140 apply (rule iffD1) |
|
141 apply (rule_tac k = "kf n" in zcong_cancel2) |
|
142 apply (rule_tac [3] b = "bf n" in zcong_trans) |
|
143 prefer 4 |
|
144 apply (subst zcong_sym) |
|
145 defer |
|
146 apply (rule order_less_imp_le) |
|
147 apply simp_all |
|
148 done |
|
149 |
|
150 lemma zcong_funprod [rule_format]: |
|
151 "m_cond n mf --> km_cond n kf mf --> |
|
152 lincong_sol n kf bf mf x --> lincong_sol n kf bf mf y --> |
|
153 [x = y] (mod funprod mf 0 n)" |
|
154 apply (induct n) |
|
155 apply (simp_all (no_asm)) |
|
156 apply (blast intro: zcong_funprod_aux) |
|
157 apply (rule impI)+ |
|
158 apply (rule zcong_zgcd_zmult_zmod) |
|
159 apply (blast intro: zcong_funprod_aux) |
|
160 prefer 2 |
|
161 apply (subst zgcd_commute) |
|
162 apply (rule funprod_zgcd) |
|
163 apply (auto simp add: m_cond_def km_cond_def lincong_sol_def) |
|
164 done |
|
165 |
|
166 |
|
167 subsection {* Chinese: existence *} |
|
168 |
|
169 lemma unique_xi_sol: |
|
170 "0 < n ==> i \<le> n ==> m_cond n mf ==> km_cond n kf mf |
|
171 ==> \<exists>!x. 0 \<le> x \<and> x < mf i \<and> [kf i * mhf mf n i * x = bf i] (mod mf i)" |
|
172 apply (rule zcong_lineq_unique) |
|
173 apply (tactic {* stac (thm "zgcd_zmult_cancel") 2 *}) |
|
174 apply (unfold m_cond_def km_cond_def mhf_def) |
|
175 apply (simp_all (no_asm_simp)) |
|
176 apply safe |
|
177 apply (tactic {* stac (thm "zgcd_zmult_cancel") 3 *}) |
|
178 apply (rule_tac [!] funprod_zgcd) |
|
179 apply safe |
|
180 apply simp_all |
|
181 apply (subgoal_tac "i<n") |
|
182 prefer 2 |
|
183 apply arith |
|
184 apply (case_tac [2] i) |
|
185 apply simp_all |
|
186 done |
|
187 |
|
188 lemma x_sol_lin_aux: |
|
189 "0 < n ==> i \<le> n ==> j \<le> n ==> j \<noteq> i ==> mf j dvd mhf mf n i" |
|
190 apply (unfold mhf_def) |
|
191 apply (case_tac "i = 0") |
|
192 apply (case_tac [2] "i = n") |
|
193 apply (simp_all (no_asm_simp)) |
|
194 apply (case_tac [3] "j < i") |
|
195 apply (rule_tac [3] dvd_mult2) |
|
196 apply (rule_tac [4] dvd_mult) |
|
197 apply (rule_tac [!] funprod_zdvd) |
|
198 apply arith |
|
199 apply arith |
|
200 apply arith |
|
201 apply arith |
|
202 apply arith |
|
203 apply arith |
|
204 apply arith |
|
205 apply arith |
|
206 done |
|
207 |
|
208 lemma x_sol_lin: |
|
209 "0 < n ==> i \<le> n |
|
210 ==> x_sol n kf bf mf mod mf i = |
|
211 xilin_sol i n kf bf mf * mhf mf n i mod mf i" |
|
212 apply (unfold x_sol_def) |
|
213 apply (subst funsum_mod) |
|
214 apply (subst funsum_oneelem) |
|
215 apply auto |
|
216 apply (subst dvd_eq_mod_eq_0 [symmetric]) |
|
217 apply (rule dvd_mult) |
|
218 apply (rule x_sol_lin_aux) |
|
219 apply auto |
|
220 done |
|
221 |
|
222 |
|
223 subsection {* Chinese *} |
|
224 |
|
225 lemma chinese_remainder: |
|
226 "0 < n ==> m_cond n mf ==> km_cond n kf mf |
|
227 ==> \<exists>!x. 0 \<le> x \<and> x < funprod mf 0 n \<and> lincong_sol n kf bf mf x" |
|
228 apply safe |
|
229 apply (rule_tac [2] m = "funprod mf 0 n" in zcong_zless_imp_eq) |
|
230 apply (rule_tac [6] zcong_funprod) |
|
231 apply auto |
|
232 apply (rule_tac x = "x_sol n kf bf mf mod funprod mf 0 n" in exI) |
|
233 apply (unfold lincong_sol_def) |
|
234 apply safe |
|
235 apply (tactic {* stac (thm "zcong_zmod") 3 *}) |
|
236 apply (tactic {* stac (thm "mod_mult_eq") 3 *}) |
|
237 apply (tactic {* stac (thm "mod_mod_cancel") 3 *}) |
|
238 apply (tactic {* stac (thm "x_sol_lin") 4 *}) |
|
239 apply (tactic {* stac (thm "mod_mult_eq" RS sym) 6 *}) |
|
240 apply (tactic {* stac (thm "zcong_zmod" RS sym) 6 *}) |
|
241 apply (subgoal_tac [6] |
|
242 "0 \<le> xilin_sol i n kf bf mf \<and> xilin_sol i n kf bf mf < mf i |
|
243 \<and> [kf i * mhf mf n i * xilin_sol i n kf bf mf = bf i] (mod mf i)") |
|
244 prefer 6 |
|
245 apply (simp add: zmult_ac) |
|
246 apply (unfold xilin_sol_def) |
|
247 apply (tactic {* asm_simp_tac @{simpset} 6 *}) |
|
248 apply (rule_tac [6] ex1_implies_ex [THEN someI_ex]) |
|
249 apply (rule_tac [6] unique_xi_sol) |
|
250 apply (rule_tac [3] funprod_zdvd) |
|
251 apply (unfold m_cond_def) |
|
252 apply (rule funprod_pos [THEN pos_mod_sign]) |
|
253 apply (rule_tac [2] funprod_pos [THEN pos_mod_bound]) |
|
254 apply auto |
|
255 done |
|
256 |
|
257 end |