1 (* Title: HOL/Old_Number_Theory/Primes.thy |
|
2 Author: Amine Chaieb, Christophe Tabacznyj and Lawrence C Paulson |
|
3 Copyright 1996 University of Cambridge |
|
4 *) |
|
5 |
|
6 section \<open>Primality on nat\<close> |
|
7 |
|
8 theory Primes |
|
9 imports Complex_Main Legacy_GCD |
|
10 begin |
|
11 |
|
12 definition coprime :: "nat => nat => bool" |
|
13 where "coprime m n \<longleftrightarrow> gcd m n = 1" |
|
14 |
|
15 definition prime :: "nat \<Rightarrow> bool" |
|
16 where "prime p \<longleftrightarrow> (1 < p \<and> (\<forall>m. m dvd p --> m = 1 \<or> m = p))" |
|
17 |
|
18 |
|
19 lemma two_is_prime: "prime 2" |
|
20 apply (auto simp add: prime_def) |
|
21 apply (case_tac m) |
|
22 apply (auto dest!: dvd_imp_le) |
|
23 done |
|
24 |
|
25 lemma prime_imp_relprime: "prime p ==> \<not> p dvd n ==> gcd p n = 1" |
|
26 apply (auto simp add: prime_def) |
|
27 apply (metis gcd_dvd1 gcd_dvd2) |
|
28 done |
|
29 |
|
30 text \<open> |
|
31 This theorem leads immediately to a proof of the uniqueness of |
|
32 factorization. If @{term p} divides a product of primes then it is |
|
33 one of those primes. |
|
34 \<close> |
|
35 |
|
36 lemma prime_dvd_mult: "prime p ==> p dvd m * n ==> p dvd m \<or> p dvd n" |
|
37 by (blast intro: relprime_dvd_mult prime_imp_relprime) |
|
38 |
|
39 lemma prime_dvd_square: "prime p ==> p dvd m^Suc (Suc 0) ==> p dvd m" |
|
40 by (auto dest: prime_dvd_mult) |
|
41 |
|
42 lemma prime_dvd_power_two: "prime p ==> p dvd m\<^sup>2 ==> p dvd m" |
|
43 by (rule prime_dvd_square) (simp_all add: power2_eq_square) |
|
44 |
|
45 |
|
46 lemma exp_eq_1:"(x::nat)^n = 1 \<longleftrightarrow> x = 1 \<or> n = 0" |
|
47 by (induct n, auto) |
|
48 |
|
49 lemma exp_mono_lt: "(x::nat) ^ (Suc n) < y ^ (Suc n) \<longleftrightarrow> x < y" |
|
50 by(metis linorder_not_less not_less0 power_le_imp_le_base power_less_imp_less_base) |
|
51 |
|
52 lemma exp_mono_le: "(x::nat) ^ (Suc n) \<le> y ^ (Suc n) \<longleftrightarrow> x \<le> y" |
|
53 by (simp only: linorder_not_less[symmetric] exp_mono_lt) |
|
54 |
|
55 lemma exp_mono_eq: "(x::nat) ^ Suc n = y ^ Suc n \<longleftrightarrow> x = y" |
|
56 using power_inject_base[of x n y] by auto |
|
57 |
|
58 |
|
59 lemma even_square: assumes e: "even (n::nat)" shows "\<exists>x. n\<^sup>2 = 4*x" |
|
60 proof- |
|
61 from e have "2 dvd n" by presburger |
|
62 then obtain k where k: "n = 2*k" using dvd_def by auto |
|
63 hence "n\<^sup>2 = 4 * k\<^sup>2" by (simp add: power2_eq_square) |
|
64 thus ?thesis by blast |
|
65 qed |
|
66 |
|
67 lemma odd_square: assumes e: "odd (n::nat)" shows "\<exists>x. n\<^sup>2 = 4*x + 1" |
|
68 proof- |
|
69 from e have np: "n > 0" by presburger |
|
70 from e have "2 dvd (n - 1)" by presburger |
|
71 then obtain k where "n - 1 = 2 * k" .. |
|
72 hence k: "n = 2*k + 1" using e by presburger |
|
73 hence "n\<^sup>2 = 4* (k\<^sup>2 + k) + 1" by algebra |
|
74 thus ?thesis by blast |
|
75 qed |
|
76 |
|
77 lemma diff_square: "(x::nat)\<^sup>2 - y\<^sup>2 = (x+y)*(x - y)" |
|
78 proof- |
|
79 have "x \<le> y \<or> y \<le> x" by (rule nat_le_linear) |
|
80 moreover |
|
81 {assume le: "x \<le> y" |
|
82 hence "x\<^sup>2 \<le> y\<^sup>2" by (simp only: numeral_2_eq_2 exp_mono_le Let_def) |
|
83 with le have ?thesis by simp } |
|
84 moreover |
|
85 {assume le: "y \<le> x" |
|
86 hence le2: "y\<^sup>2 \<le> x\<^sup>2" by (simp only: numeral_2_eq_2 exp_mono_le Let_def) |
|
87 from le have "\<exists>z. y + z = x" by presburger |
|
88 then obtain z where z: "x = y + z" by blast |
|
89 from le2 have "\<exists>z. x\<^sup>2 = y\<^sup>2 + z" by presburger |
|
90 then obtain z2 where z2: "x\<^sup>2 = y\<^sup>2 + z2" by blast |
|
91 from z z2 have ?thesis by simp algebra } |
|
92 ultimately show ?thesis by blast |
|
93 qed |
|
94 |
|
95 text \<open>Elementary theory of divisibility\<close> |
|
96 lemma divides_ge: "(a::nat) dvd b \<Longrightarrow> b = 0 \<or> a \<le> b" unfolding dvd_def by auto |
|
97 lemma divides_antisym: "(x::nat) dvd y \<and> y dvd x \<longleftrightarrow> x = y" |
|
98 using dvd_antisym[of x y] by auto |
|
99 |
|
100 lemma divides_add_revr: assumes da: "(d::nat) dvd a" and dab:"d dvd (a + b)" |
|
101 shows "d dvd b" |
|
102 proof- |
|
103 from da obtain k where k:"a = d*k" by (auto simp add: dvd_def) |
|
104 from dab obtain k' where k': "a + b = d*k'" by (auto simp add: dvd_def) |
|
105 from k k' have "b = d *(k' - k)" by (simp add : diff_mult_distrib2) |
|
106 thus ?thesis unfolding dvd_def by blast |
|
107 qed |
|
108 |
|
109 declare nat_mult_dvd_cancel_disj[presburger] |
|
110 lemma nat_mult_dvd_cancel_disj'[presburger]: |
|
111 "(m::nat)*k dvd n*k \<longleftrightarrow> k = 0 \<or> m dvd n" unfolding mult.commute[of m k] mult.commute[of n k] by presburger |
|
112 |
|
113 lemma divides_mul_l: "(a::nat) dvd b ==> (c * a) dvd (c * b)" |
|
114 by presburger |
|
115 |
|
116 lemma divides_mul_r: "(a::nat) dvd b ==> (a * c) dvd (b * c)" by presburger |
|
117 lemma divides_cases: "(n::nat) dvd m ==> m = 0 \<or> m = n \<or> 2 * n <= m" |
|
118 by (auto simp add: dvd_def) |
|
119 |
|
120 lemma divides_div_not: "(x::nat) = (q * n) + r \<Longrightarrow> 0 < r \<Longrightarrow> r < n ==> ~(n dvd x)" |
|
121 proof(auto simp add: dvd_def) |
|
122 fix k assume H: "0 < r" "r < n" "q * n + r = n * k" |
|
123 from H(3) have r: "r = n* (k -q)" by(simp add: diff_mult_distrib2 mult.commute) |
|
124 {assume "k - q = 0" with r H(1) have False by simp} |
|
125 moreover |
|
126 {assume "k - q \<noteq> 0" with r have "r \<ge> n" by auto |
|
127 with H(2) have False by simp} |
|
128 ultimately show False by blast |
|
129 qed |
|
130 lemma divides_exp: "(x::nat) dvd y ==> x ^ n dvd y ^ n" |
|
131 by (auto simp add: power_mult_distrib dvd_def) |
|
132 |
|
133 lemma divides_exp2: "n \<noteq> 0 \<Longrightarrow> (x::nat) ^ n dvd y \<Longrightarrow> x dvd y" |
|
134 by (induct n ,auto simp add: dvd_def) |
|
135 |
|
136 fun fact :: "nat \<Rightarrow> nat" where |
|
137 "fact 0 = 1" |
|
138 | "fact (Suc n) = Suc n * fact n" |
|
139 |
|
140 lemma fact_lt: "0 < fact n" by(induct n, simp_all) |
|
141 lemma fact_le: "fact n \<ge> 1" using fact_lt[of n] by simp |
|
142 lemma fact_mono: assumes le: "m \<le> n" shows "fact m \<le> fact n" |
|
143 proof- |
|
144 from le have "\<exists>i. n = m+i" by presburger |
|
145 then obtain i where i: "n = m+i" by blast |
|
146 have "fact m \<le> fact (m + i)" |
|
147 proof(induct m) |
|
148 case 0 thus ?case using fact_le[of i] by simp |
|
149 next |
|
150 case (Suc m) |
|
151 have "fact (Suc m) = Suc m * fact m" by simp |
|
152 have th1: "Suc m \<le> Suc (m + i)" by simp |
|
153 from mult_le_mono[of "Suc m" "Suc (m+i)" "fact m" "fact (m+i)", OF th1 Suc.hyps] |
|
154 show ?case by simp |
|
155 qed |
|
156 thus ?thesis using i by simp |
|
157 qed |
|
158 |
|
159 lemma divides_fact: "1 <= p \<Longrightarrow> p <= n ==> p dvd fact n" |
|
160 proof(induct n arbitrary: p) |
|
161 case 0 thus ?case by simp |
|
162 next |
|
163 case (Suc n p) |
|
164 from Suc.prems have "p = Suc n \<or> p \<le> n" by presburger |
|
165 moreover |
|
166 {assume "p = Suc n" hence ?case by (simp only: fact.simps dvd_triv_left)} |
|
167 moreover |
|
168 {assume "p \<le> n" |
|
169 with Suc.prems(1) Suc.hyps have th: "p dvd fact n" by simp |
|
170 from dvd_mult[OF th] have ?case by (simp only: fact.simps) } |
|
171 ultimately show ?case by blast |
|
172 qed |
|
173 |
|
174 declare dvd_triv_left[presburger] |
|
175 declare dvd_triv_right[presburger] |
|
176 lemma divides_rexp: |
|
177 "x dvd y \<Longrightarrow> (x::nat) dvd (y^(Suc n))" by (simp add: dvd_mult2[of x y]) |
|
178 |
|
179 text \<open>Coprimality\<close> |
|
180 |
|
181 lemma coprime: "coprime a b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)" |
|
182 using gcd_unique[of 1 a b, simplified] by (auto simp add: coprime_def) |
|
183 lemma coprime_commute: "coprime a b \<longleftrightarrow> coprime b a" by (simp add: coprime_def gcd_commute) |
|
184 |
|
185 lemma coprime_bezout: "coprime a b \<longleftrightarrow> (\<exists>x y. a * x - b * y = 1 \<or> b * x - a * y = 1)" |
|
186 using coprime_def gcd_bezout by auto |
|
187 |
|
188 lemma coprime_divprod: "d dvd a * b \<Longrightarrow> coprime d a \<Longrightarrow> d dvd b" |
|
189 using relprime_dvd_mult_iff[of d a b] by (auto simp add: coprime_def mult.commute) |
|
190 |
|
191 lemma coprime_1[simp]: "coprime a 1" by (simp add: coprime_def) |
|
192 lemma coprime_1'[simp]: "coprime 1 a" by (simp add: coprime_def) |
|
193 lemma coprime_Suc0[simp]: "coprime a (Suc 0)" by (simp add: coprime_def) |
|
194 lemma coprime_Suc0'[simp]: "coprime (Suc 0) a" by (simp add: coprime_def) |
|
195 |
|
196 lemma gcd_coprime: |
|
197 assumes z: "gcd a b \<noteq> 0" and a: "a = a' * gcd a b" and b: "b = b' * gcd a b" |
|
198 shows "coprime a' b'" |
|
199 proof- |
|
200 let ?g = "gcd a b" |
|
201 {assume bz: "a = 0" from b bz z a have ?thesis by (simp add: gcd_zero coprime_def)} |
|
202 moreover |
|
203 {assume az: "a\<noteq> 0" |
|
204 from z have z': "?g > 0" by simp |
|
205 from bezout_gcd_strong[OF az, of b] |
|
206 obtain x y where xy: "a*x = b*y + ?g" by blast |
|
207 from xy a b have "?g * a'*x = ?g * (b'*y + 1)" by (simp add: algebra_simps) |
|
208 hence "?g * (a'*x) = ?g * (b'*y + 1)" by (simp add: mult.assoc) |
|
209 hence "a'*x = (b'*y + 1)" |
|
210 by (simp only: nat_mult_eq_cancel1[OF z']) |
|
211 hence "a'*x - b'*y = 1" by simp |
|
212 with coprime_bezout[of a' b'] have ?thesis by auto} |
|
213 ultimately show ?thesis by blast |
|
214 qed |
|
215 lemma coprime_0: "coprime d 0 \<longleftrightarrow> d = 1" by (simp add: coprime_def) |
|
216 lemma coprime_mul: assumes da: "coprime d a" and db: "coprime d b" |
|
217 shows "coprime d (a * b)" |
|
218 proof- |
|
219 from da have th: "gcd a d = 1" by (simp add: coprime_def gcd_commute) |
|
220 from gcd_mult_cancel[of a d b, OF th] db[unfolded coprime_def] have "gcd d (a*b) = 1" |
|
221 by (simp add: gcd_commute) |
|
222 thus ?thesis unfolding coprime_def . |
|
223 qed |
|
224 lemma coprime_lmul2: assumes dab: "coprime d (a * b)" shows "coprime d b" |
|
225 using dab unfolding coprime_bezout |
|
226 apply clarsimp |
|
227 apply (case_tac "d * x - a * b * y = Suc 0 ", simp_all) |
|
228 apply (rule_tac x="x" in exI) |
|
229 apply (rule_tac x="a*y" in exI) |
|
230 apply (simp add: ac_simps) |
|
231 apply (rule_tac x="a*x" in exI) |
|
232 apply (rule_tac x="y" in exI) |
|
233 apply (simp add: ac_simps) |
|
234 done |
|
235 |
|
236 lemma coprime_rmul2: "coprime d (a * b) \<Longrightarrow> coprime d a" |
|
237 unfolding coprime_bezout |
|
238 apply clarsimp |
|
239 apply (case_tac "d * x - a * b * y = Suc 0 ", simp_all) |
|
240 apply (rule_tac x="x" in exI) |
|
241 apply (rule_tac x="b*y" in exI) |
|
242 apply (simp add: ac_simps) |
|
243 apply (rule_tac x="b*x" in exI) |
|
244 apply (rule_tac x="y" in exI) |
|
245 apply (simp add: ac_simps) |
|
246 done |
|
247 lemma coprime_mul_eq: "coprime d (a * b) \<longleftrightarrow> coprime d a \<and> coprime d b" |
|
248 using coprime_rmul2[of d a b] coprime_lmul2[of d a b] coprime_mul[of d a b] |
|
249 by blast |
|
250 |
|
251 lemma gcd_coprime_exists: |
|
252 assumes nz: "gcd a b \<noteq> 0" |
|
253 shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'" |
|
254 proof- |
|
255 let ?g = "gcd a b" |
|
256 from gcd_dvd1[of a b] gcd_dvd2[of a b] |
|
257 obtain a' b' where "a = ?g*a'" "b = ?g*b'" unfolding dvd_def by blast |
|
258 hence ab': "a = a'*?g" "b = b'*?g" by algebra+ |
|
259 from ab' gcd_coprime[OF nz ab'] show ?thesis by blast |
|
260 qed |
|
261 |
|
262 lemma coprime_exp: "coprime d a ==> coprime d (a^n)" |
|
263 by(induct n, simp_all add: coprime_mul) |
|
264 |
|
265 lemma coprime_exp_imp: "coprime a b ==> coprime (a ^n) (b ^n)" |
|
266 by (induct n, simp_all add: coprime_mul_eq coprime_commute coprime_exp) |
|
267 lemma coprime_refl[simp]: "coprime n n \<longleftrightarrow> n = 1" by (simp add: coprime_def) |
|
268 lemma coprime_plus1[simp]: "coprime (n + 1) n" |
|
269 apply (simp add: coprime_bezout) |
|
270 apply (rule exI[where x=1]) |
|
271 apply (rule exI[where x=1]) |
|
272 apply simp |
|
273 done |
|
274 lemma coprime_minus1: "n \<noteq> 0 ==> coprime (n - 1) n" |
|
275 using coprime_plus1[of "n - 1"] coprime_commute[of "n - 1" n] by auto |
|
276 |
|
277 lemma bezout_gcd_pow: "\<exists>x y. a ^n * x - b ^ n * y = gcd a b ^ n \<or> b ^ n * x - a ^ n * y = gcd a b ^ n" |
|
278 proof- |
|
279 let ?g = "gcd a b" |
|
280 {assume z: "?g = 0" hence ?thesis |
|
281 apply (cases n, simp) |
|
282 apply arith |
|
283 apply (simp only: z power_0_Suc) |
|
284 apply (rule exI[where x=0]) |
|
285 apply (rule exI[where x=0]) |
|
286 apply simp |
|
287 done } |
|
288 moreover |
|
289 {assume z: "?g \<noteq> 0" |
|
290 from gcd_dvd1[of a b] gcd_dvd2[of a b] obtain a' b' where |
|
291 ab': "a = a'*?g" "b = b'*?g" unfolding dvd_def by (auto simp add: ac_simps) |
|
292 hence ab'': "?g*a' = a" "?g * b' = b" by algebra+ |
|
293 from coprime_exp_imp[OF gcd_coprime[OF z ab'], unfolded coprime_bezout, of n] |
|
294 obtain x y where "a'^n * x - b'^n * y = 1 \<or> b'^n * x - a'^n * y = 1" by blast |
|
295 hence "?g^n * (a'^n * x - b'^n * y) = ?g^n \<or> ?g^n*(b'^n * x - a'^n * y) = ?g^n" |
|
296 using z by auto |
|
297 then have "a^n * x - b^n * y = ?g^n \<or> b^n * x - a^n * y = ?g^n" |
|
298 using z ab'' by (simp only: power_mult_distrib[symmetric] |
|
299 diff_mult_distrib2 mult.assoc[symmetric]) |
|
300 hence ?thesis by blast } |
|
301 ultimately show ?thesis by blast |
|
302 qed |
|
303 |
|
304 lemma gcd_exp: "gcd (a^n) (b^n) = gcd a b^n" |
|
305 proof- |
|
306 let ?g = "gcd (a^n) (b^n)" |
|
307 let ?gn = "gcd a b^n" |
|
308 {fix e assume H: "e dvd a^n" "e dvd b^n" |
|
309 from bezout_gcd_pow[of a n b] obtain x y |
|
310 where xy: "a ^ n * x - b ^ n * y = ?gn \<or> b ^ n * x - a ^ n * y = ?gn" by blast |
|
311 from dvd_diff_nat [OF dvd_mult2[OF H(1), of x] dvd_mult2[OF H(2), of y]] |
|
312 dvd_diff_nat [OF dvd_mult2[OF H(2), of x] dvd_mult2[OF H(1), of y]] xy |
|
313 have "e dvd ?gn" by (cases "a ^ n * x - b ^ n * y = gcd a b ^ n", simp_all)} |
|
314 hence th: "\<forall>e. e dvd a^n \<and> e dvd b^n \<longrightarrow> e dvd ?gn" by blast |
|
315 from divides_exp[OF gcd_dvd1[of a b], of n] divides_exp[OF gcd_dvd2[of a b], of n] th |
|
316 gcd_unique have "?gn = ?g" by blast thus ?thesis by simp |
|
317 qed |
|
318 |
|
319 lemma coprime_exp2: "coprime (a ^ Suc n) (b^ Suc n) \<longleftrightarrow> coprime a b" |
|
320 by (simp only: coprime_def gcd_exp exp_eq_1) simp |
|
321 |
|
322 lemma division_decomp: assumes dc: "(a::nat) dvd b * c" |
|
323 shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c" |
|
324 proof- |
|
325 let ?g = "gcd a b" |
|
326 {assume "?g = 0" with dc have ?thesis apply (simp add: gcd_zero) |
|
327 apply (rule exI[where x="0"]) |
|
328 by (rule exI[where x="c"], simp)} |
|
329 moreover |
|
330 {assume z: "?g \<noteq> 0" |
|
331 from gcd_coprime_exists[OF z] |
|
332 obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'" by blast |
|
333 from gcd_dvd2[of a b] have thb: "?g dvd b" . |
|
334 from ab'(1) have "a' dvd a" unfolding dvd_def by blast |
|
335 with dc have th0: "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp |
|
336 from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto |
|
337 hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult.assoc) |
|
338 with z have th_1: "a' dvd b'*c" by simp |
|
339 from coprime_divprod[OF th_1 ab'(3)] have thc: "a' dvd c" . |
|
340 from ab' have "a = ?g*a'" by algebra |
|
341 with thb thc have ?thesis by blast } |
|
342 ultimately show ?thesis by blast |
|
343 qed |
|
344 |
|
345 lemma nat_power_eq_0_iff: "(m::nat) ^ n = 0 \<longleftrightarrow> n \<noteq> 0 \<and> m = 0" by (induct n, auto) |
|
346 |
|
347 lemma divides_rev: assumes ab: "(a::nat) ^ n dvd b ^n" and n:"n \<noteq> 0" shows "a dvd b" |
|
348 proof- |
|
349 let ?g = "gcd a b" |
|
350 from n obtain m where m: "n = Suc m" by (cases n, simp_all) |
|
351 {assume "?g = 0" with ab n have ?thesis by (simp add: gcd_zero)} |
|
352 moreover |
|
353 {assume z: "?g \<noteq> 0" |
|
354 hence zn: "?g ^ n \<noteq> 0" using n by simp |
|
355 from gcd_coprime_exists[OF z] |
|
356 obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'" by blast |
|
357 from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n" by (simp add: ab'(1,2)[symmetric]) |
|
358 hence "?g^n*a'^n dvd ?g^n *b'^n" by (simp only: power_mult_distrib mult.commute) |
|
359 with zn z n have th0:"a'^n dvd b'^n" by (auto simp add: nat_power_eq_0_iff) |
|
360 have "a' dvd a'^n" by (simp add: m) |
|
361 with th0 have "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by simp |
|
362 hence th1: "a' dvd b'^m * b'" by (simp add: m mult.commute) |
|
363 from coprime_divprod[OF th1 coprime_exp[OF ab'(3), of m]] |
|
364 have "a' dvd b'" . |
|
365 hence "a'*?g dvd b'*?g" by simp |
|
366 with ab'(1,2) have ?thesis by simp } |
|
367 ultimately show ?thesis by blast |
|
368 qed |
|
369 |
|
370 lemma divides_mul: assumes mr: "m dvd r" and nr: "n dvd r" and mn:"coprime m n" |
|
371 shows "m * n dvd r" |
|
372 proof- |
|
373 from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'" |
|
374 unfolding dvd_def by blast |
|
375 from mr n' have "m dvd n'*n" by (simp add: mult.commute) |
|
376 hence "m dvd n'" using relprime_dvd_mult_iff[OF mn[unfolded coprime_def]] by simp |
|
377 then obtain k where k: "n' = m*k" unfolding dvd_def by blast |
|
378 from n' k show ?thesis unfolding dvd_def by auto |
|
379 qed |
|
380 |
|
381 |
|
382 text \<open>A binary form of the Chinese Remainder Theorem.\<close> |
|
383 |
|
384 lemma chinese_remainder: assumes ab: "coprime a b" and a:"a \<noteq> 0" and b:"b \<noteq> 0" |
|
385 shows "\<exists>x q1 q2. x = u + q1 * a \<and> x = v + q2 * b" |
|
386 proof- |
|
387 from bezout_add_strong[OF a, of b] bezout_add_strong[OF b, of a] |
|
388 obtain d1 x1 y1 d2 x2 y2 where dxy1: "d1 dvd a" "d1 dvd b" "a * x1 = b * y1 + d1" |
|
389 and dxy2: "d2 dvd b" "d2 dvd a" "b * x2 = a * y2 + d2" by blast |
|
390 from gcd_unique[of 1 a b, simplified ab[unfolded coprime_def], simplified] |
|
391 dxy1(1,2) dxy2(1,2) have d12: "d1 = 1" "d2 =1" by auto |
|
392 let ?x = "v * a * x1 + u * b * x2" |
|
393 let ?q1 = "v * x1 + u * y2" |
|
394 let ?q2 = "v * y1 + u * x2" |
|
395 from dxy2(3)[simplified d12] dxy1(3)[simplified d12] |
|
396 have "?x = u + ?q1 * a" "?x = v + ?q2 * b" by algebra+ |
|
397 thus ?thesis by blast |
|
398 qed |
|
399 |
|
400 text \<open>Primality\<close> |
|
401 |
|
402 text \<open>A few useful theorems about primes\<close> |
|
403 |
|
404 lemma prime_0[simp]: "~prime 0" by (simp add: prime_def) |
|
405 lemma prime_1[simp]: "~ prime 1" by (simp add: prime_def) |
|
406 lemma prime_Suc0[simp]: "~ prime (Suc 0)" by (simp add: prime_def) |
|
407 |
|
408 lemma prime_ge_2: "prime p ==> p \<ge> 2" by (simp add: prime_def) |
|
409 |
|
410 lemma prime_factor: "n \<noteq> 1 \<Longrightarrow> \<exists>p. prime p \<and> p dvd n" |
|
411 proof (induct n rule: nat_less_induct) |
|
412 fix n |
|
413 assume H: "\<forall>m<n. m \<noteq> 1 \<longrightarrow> (\<exists>p. prime p \<and> p dvd m)" "n \<noteq> 1" |
|
414 show "\<exists>p. prime p \<and> p dvd n" |
|
415 proof (cases "n = 0") |
|
416 case True |
|
417 with two_is_prime show ?thesis by auto |
|
418 next |
|
419 case nz: False |
|
420 show ?thesis |
|
421 proof (cases "prime n") |
|
422 case True |
|
423 then have "prime n \<and> n dvd n" by simp |
|
424 then show ?thesis .. |
|
425 next |
|
426 case n: False |
|
427 with nz H(2) obtain k where k: "k dvd n" "k \<noteq> 1" "k \<noteq> n" |
|
428 by (auto simp: prime_def) |
|
429 from dvd_imp_le[OF k(1)] nz k(3) have kn: "k < n" by simp |
|
430 from H(1)[rule_format, OF kn k(2)] obtain p where p: "prime p" "p dvd k" by blast |
|
431 from dvd_trans[OF p(2) k(1)] p(1) show ?thesis by blast |
|
432 qed |
|
433 qed |
|
434 qed |
|
435 |
|
436 lemma prime_factor_lt: |
|
437 assumes p: "prime p" and n: "n \<noteq> 0" and npm:"n = p * m" |
|
438 shows "m < n" |
|
439 proof (cases "m = 0") |
|
440 case True |
|
441 with n show ?thesis by simp |
|
442 next |
|
443 case m: False |
|
444 from npm have mn: "m dvd n" unfolding dvd_def by auto |
|
445 from npm m have "n \<noteq> m" using p by auto |
|
446 with dvd_imp_le[OF mn] n show ?thesis by simp |
|
447 qed |
|
448 |
|
449 lemma euclid_bound: "\<exists>p. prime p \<and> n < p \<and> p <= Suc (fact n)" |
|
450 proof- |
|
451 have f1: "fact n + 1 \<noteq> 1" using fact_le[of n] by arith |
|
452 from prime_factor[OF f1] obtain p where p: "prime p" "p dvd fact n + 1" by blast |
|
453 from dvd_imp_le[OF p(2)] have pfn: "p \<le> fact n + 1" by simp |
|
454 {assume np: "p \<le> n" |
|
455 from p(1) have p1: "p \<ge> 1" by (cases p, simp_all) |
|
456 from divides_fact[OF p1 np] have pfn': "p dvd fact n" . |
|
457 from divides_add_revr[OF pfn' p(2)] p(1) have False by simp} |
|
458 hence "n < p" by arith |
|
459 with p(1) pfn show ?thesis by auto |
|
460 qed |
|
461 |
|
462 lemma euclid: "\<exists>p. prime p \<and> p > n" using euclid_bound by auto |
|
463 |
|
464 lemma primes_infinite: "\<not> (finite {p. prime p})" |
|
465 apply(simp add: finite_nat_set_iff_bounded_le) |
|
466 apply (metis euclid linorder_not_le) |
|
467 done |
|
468 |
|
469 lemma coprime_prime: assumes ab: "coprime a b" |
|
470 shows "~(prime p \<and> p dvd a \<and> p dvd b)" |
|
471 proof |
|
472 assume "prime p \<and> p dvd a \<and> p dvd b" |
|
473 thus False using ab gcd_greatest[of p a b] by (simp add: coprime_def) |
|
474 qed |
|
475 lemma coprime_prime_eq: "coprime a b \<longleftrightarrow> (\<forall>p. ~(prime p \<and> p dvd a \<and> p dvd b))" |
|
476 (is "?lhs = ?rhs") |
|
477 proof- |
|
478 {assume "?lhs" with coprime_prime have ?rhs by blast} |
|
479 moreover |
|
480 {assume r: "?rhs" and c: "\<not> ?lhs" |
|
481 then obtain g where g: "g\<noteq>1" "g dvd a" "g dvd b" unfolding coprime_def by blast |
|
482 from prime_factor[OF g(1)] obtain p where p: "prime p" "p dvd g" by blast |
|
483 from dvd_trans [OF p(2) g(2)] dvd_trans [OF p(2) g(3)] |
|
484 have "p dvd a" "p dvd b" . with p(1) r have False by blast} |
|
485 ultimately show ?thesis by blast |
|
486 qed |
|
487 |
|
488 lemma prime_coprime: assumes p: "prime p" |
|
489 shows "n = 1 \<or> p dvd n \<or> coprime p n" |
|
490 using p prime_imp_relprime[of p n] by (auto simp add: coprime_def) |
|
491 |
|
492 lemma prime_coprime_strong: "prime p \<Longrightarrow> p dvd n \<or> coprime p n" |
|
493 using prime_coprime[of p n] by auto |
|
494 |
|
495 declare coprime_0[simp] |
|
496 |
|
497 lemma coprime_0'[simp]: "coprime 0 d \<longleftrightarrow> d = 1" by (simp add: coprime_commute[of 0 d]) |
|
498 lemma coprime_bezout_strong: assumes ab: "coprime a b" and b: "b \<noteq> 1" |
|
499 shows "\<exists>x y. a * x = b * y + 1" |
|
500 proof- |
|
501 have az: "a \<noteq> 0" by (rule ccontr) (use ab b in auto) |
|
502 from bezout_gcd_strong[OF az, of b] ab[unfolded coprime_def] |
|
503 show ?thesis by auto |
|
504 qed |
|
505 |
|
506 lemma bezout_prime: assumes p: "prime p" and pa: "\<not> p dvd a" |
|
507 shows "\<exists>x y. a*x = p*y + 1" |
|
508 proof- |
|
509 from p have p1: "p \<noteq> 1" using prime_1 by blast |
|
510 from prime_coprime[OF p, of a] p1 pa have ap: "coprime a p" |
|
511 by (auto simp add: coprime_commute) |
|
512 from coprime_bezout_strong[OF ap p1] show ?thesis . |
|
513 qed |
|
514 lemma prime_divprod: assumes p: "prime p" and pab: "p dvd a*b" |
|
515 shows "p dvd a \<or> p dvd b" |
|
516 proof- |
|
517 {assume "a=1" hence ?thesis using pab by simp } |
|
518 moreover |
|
519 {assume "p dvd a" hence ?thesis by blast} |
|
520 moreover |
|
521 {assume pa: "coprime p a" from coprime_divprod[OF pab pa] have ?thesis .. } |
|
522 ultimately show ?thesis using prime_coprime[OF p, of a] by blast |
|
523 qed |
|
524 |
|
525 lemma prime_divprod_eq: assumes p: "prime p" |
|
526 shows "p dvd a*b \<longleftrightarrow> p dvd a \<or> p dvd b" |
|
527 using p prime_divprod dvd_mult dvd_mult2 by auto |
|
528 |
|
529 lemma prime_divexp: assumes p:"prime p" and px: "p dvd x^n" |
|
530 shows "p dvd x" |
|
531 using px |
|
532 proof(induct n) |
|
533 case 0 thus ?case by simp |
|
534 next |
|
535 case (Suc n) |
|
536 hence th: "p dvd x*x^n" by simp |
|
537 {assume H: "p dvd x^n" |
|
538 from Suc.hyps[OF H] have ?case .} |
|
539 with prime_divprod[OF p th] show ?case by blast |
|
540 qed |
|
541 |
|
542 lemma prime_divexp_n: "prime p \<Longrightarrow> p dvd x^n \<Longrightarrow> p^n dvd x^n" |
|
543 using prime_divexp[of p x n] divides_exp[of p x n] by blast |
|
544 |
|
545 lemma coprime_prime_dvd_ex: assumes xy: "\<not>coprime x y" |
|
546 shows "\<exists>p. prime p \<and> p dvd x \<and> p dvd y" |
|
547 proof- |
|
548 from xy[unfolded coprime_def] obtain g where g: "g \<noteq> 1" "g dvd x" "g dvd y" |
|
549 by blast |
|
550 from prime_factor[OF g(1)] obtain p where p: "prime p" "p dvd g" by blast |
|
551 from g(2,3) dvd_trans[OF p(2)] p(1) show ?thesis by auto |
|
552 qed |
|
553 lemma coprime_sos: assumes xy: "coprime x y" |
|
554 shows "coprime (x * y) (x\<^sup>2 + y\<^sup>2)" |
|
555 proof- |
|
556 {assume c: "\<not> coprime (x * y) (x\<^sup>2 + y\<^sup>2)" |
|
557 from coprime_prime_dvd_ex[OF c] obtain p |
|
558 where p: "prime p" "p dvd x*y" "p dvd x\<^sup>2 + y\<^sup>2" by blast |
|
559 {assume px: "p dvd x" |
|
560 from dvd_mult[OF px, of x] p(3) |
|
561 obtain r s where "x * x = p * r" and "x\<^sup>2 + y\<^sup>2 = p * s" |
|
562 by (auto elim!: dvdE) |
|
563 then have "y\<^sup>2 = p * (s - r)" |
|
564 by (auto simp add: power2_eq_square diff_mult_distrib2) |
|
565 then have "p dvd y\<^sup>2" .. |
|
566 with prime_divexp[OF p(1), of y 2] have py: "p dvd y" . |
|
567 from p(1) px py xy[unfolded coprime, rule_format, of p] prime_1 |
|
568 have False by simp } |
|
569 moreover |
|
570 {assume py: "p dvd y" |
|
571 from dvd_mult[OF py, of y] p(3) |
|
572 obtain r s where "y * y = p * r" and "x\<^sup>2 + y\<^sup>2 = p * s" |
|
573 by (auto elim!: dvdE) |
|
574 then have "x\<^sup>2 = p * (s - r)" |
|
575 by (auto simp add: power2_eq_square diff_mult_distrib2) |
|
576 then have "p dvd x\<^sup>2" .. |
|
577 with prime_divexp[OF p(1), of x 2] have px: "p dvd x" . |
|
578 from p(1) px py xy[unfolded coprime, rule_format, of p] prime_1 |
|
579 have False by simp } |
|
580 ultimately have False using prime_divprod[OF p(1,2)] by blast} |
|
581 thus ?thesis by blast |
|
582 qed |
|
583 |
|
584 lemma distinct_prime_coprime: "prime p \<Longrightarrow> prime q \<Longrightarrow> p \<noteq> q \<Longrightarrow> coprime p q" |
|
585 unfolding prime_def coprime_prime_eq by blast |
|
586 |
|
587 lemma prime_coprime_lt: |
|
588 assumes p: "prime p" and x: "0 < x" and xp: "x < p" |
|
589 shows "coprime x p" |
|
590 proof (rule ccontr) |
|
591 assume c: "\<not> ?thesis" |
|
592 then obtain g where g: "g \<noteq> 1" "g dvd x" "g dvd p" unfolding coprime_def by blast |
|
593 from dvd_imp_le[OF g(2)] x xp have gp: "g < p" by arith |
|
594 have "g \<noteq> 0" by (rule ccontr) (use g(2) x in simp) |
|
595 with g gp p[unfolded prime_def] show False by blast |
|
596 qed |
|
597 |
|
598 lemma prime_odd: "prime p \<Longrightarrow> p = 2 \<or> odd p" unfolding prime_def by auto |
|
599 |
|
600 |
|
601 text \<open>One property of coprimality is easier to prove via prime factors.\<close> |
|
602 |
|
603 lemma prime_divprod_pow: |
|
604 assumes p: "prime p" and ab: "coprime a b" and pab: "p^n dvd a * b" |
|
605 shows "p^n dvd a \<or> p^n dvd b" |
|
606 proof- |
|
607 {assume "n = 0 \<or> a = 1 \<or> b = 1" with pab have ?thesis |
|
608 apply (cases "n=0", simp_all) |
|
609 apply (cases "a=1", simp_all) done} |
|
610 moreover |
|
611 {assume n: "n \<noteq> 0" and a: "a\<noteq>1" and b: "b\<noteq>1" |
|
612 then obtain m where m: "n = Suc m" by (cases n, auto) |
|
613 from divides_exp2[OF n pab] have pab': "p dvd a*b" . |
|
614 from prime_divprod[OF p pab'] |
|
615 have "p dvd a \<or> p dvd b" . |
|
616 moreover |
|
617 {assume pa: "p dvd a" |
|
618 have pnba: "p^n dvd b*a" using pab by (simp add: mult.commute) |
|
619 from coprime_prime[OF ab, of p] p pa have "\<not> p dvd b" by blast |
|
620 with prime_coprime[OF p, of b] b |
|
621 have cpb: "coprime b p" using coprime_commute by blast |
|
622 from coprime_exp[OF cpb] have pnb: "coprime (p^n) b" |
|
623 by (simp add: coprime_commute) |
|
624 from coprime_divprod[OF pnba pnb] have ?thesis by blast } |
|
625 moreover |
|
626 {assume pb: "p dvd b" |
|
627 have pnba: "p^n dvd b*a" using pab by (simp add: mult.commute) |
|
628 from coprime_prime[OF ab, of p] p pb have "\<not> p dvd a" by blast |
|
629 with prime_coprime[OF p, of a] a |
|
630 have cpb: "coprime a p" using coprime_commute by blast |
|
631 from coprime_exp[OF cpb] have pnb: "coprime (p^n) a" |
|
632 by (simp add: coprime_commute) |
|
633 from coprime_divprod[OF pab pnb] have ?thesis by blast } |
|
634 ultimately have ?thesis by blast} |
|
635 ultimately show ?thesis by blast |
|
636 qed |
|
637 |
|
638 lemma nat_mult_eq_one: "(n::nat) * m = 1 \<longleftrightarrow> n = 1 \<and> m = 1" (is "?lhs \<longleftrightarrow> ?rhs") |
|
639 proof |
|
640 assume H: "?lhs" |
|
641 hence "n dvd 1" "m dvd 1" unfolding dvd_def by (auto simp add: mult.commute) |
|
642 thus ?rhs by auto |
|
643 next |
|
644 assume ?rhs then show ?lhs by auto |
|
645 qed |
|
646 |
|
647 lemma power_Suc0: "Suc 0 ^ n = Suc 0" |
|
648 unfolding One_nat_def[symmetric] power_one .. |
|
649 |
|
650 lemma coprime_pow: assumes ab: "coprime a b" and abcn: "a * b = c ^n" |
|
651 shows "\<exists>r s. a = r^n \<and> b = s ^n" |
|
652 using ab abcn |
|
653 proof(induct c arbitrary: a b rule: nat_less_induct) |
|
654 fix c a b |
|
655 assume H: "\<forall>m<c. \<forall>a b. coprime a b \<longrightarrow> a * b = m ^ n \<longrightarrow> (\<exists>r s. a = r ^ n \<and> b = s ^ n)" "coprime a b" "a * b = c ^ n" |
|
656 let ?ths = "\<exists>r s. a = r^n \<and> b = s ^n" |
|
657 {assume n: "n = 0" |
|
658 with H(3) power_one have "a*b = 1" by simp |
|
659 hence "a = 1 \<and> b = 1" by simp |
|
660 hence ?ths |
|
661 apply - |
|
662 apply (rule exI[where x=1]) |
|
663 apply (rule exI[where x=1]) |
|
664 using power_one[of n] |
|
665 by simp} |
|
666 moreover |
|
667 {assume n: "n \<noteq> 0" then obtain m where m: "n = Suc m" by (cases n, auto) |
|
668 {assume c: "c = 0" |
|
669 with H(3) m H(2) have ?ths apply simp |
|
670 apply (cases "a=0", simp_all) |
|
671 apply (rule exI[where x="0"], simp) |
|
672 apply (rule exI[where x="0"], simp) |
|
673 done} |
|
674 moreover |
|
675 {assume "c=1" with H(3) power_one have "a*b = 1" by simp |
|
676 hence "a = 1 \<and> b = 1" by simp |
|
677 hence ?ths |
|
678 apply - |
|
679 apply (rule exI[where x=1]) |
|
680 apply (rule exI[where x=1]) |
|
681 using power_one[of n] |
|
682 by simp} |
|
683 moreover |
|
684 {assume c: "c\<noteq>1" "c \<noteq> 0" |
|
685 from prime_factor[OF c(1)] obtain p where p: "prime p" "p dvd c" by blast |
|
686 from prime_divprod_pow[OF p(1) H(2), unfolded H(3), OF divides_exp[OF p(2), of n]] |
|
687 have pnab: "p ^ n dvd a \<or> p^n dvd b" . |
|
688 from p(2) obtain l where l: "c = p*l" unfolding dvd_def by blast |
|
689 have pn0: "p^n \<noteq> 0" using n prime_ge_2 [OF p(1)] by simp |
|
690 {assume pa: "p^n dvd a" |
|
691 then obtain k where k: "a = p^n * k" unfolding dvd_def by blast |
|
692 from l have "l dvd c" by auto |
|
693 with dvd_imp_le[of l c] c have "l \<le> c" by auto |
|
694 moreover {assume "l = c" with l c have "p = 1" by simp with p have False by simp} |
|
695 ultimately have lc: "l < c" by arith |
|
696 from coprime_lmul2 [OF H(2)[unfolded k coprime_commute[of "p^n*k" b]]] |
|
697 have kb: "coprime k b" by (simp add: coprime_commute) |
|
698 from H(3) l k pn0 have kbln: "k * b = l ^ n" |
|
699 by (auto simp add: power_mult_distrib) |
|
700 from H(1)[rule_format, OF lc kb kbln] |
|
701 obtain r s where rs: "k = r ^n" "b = s^n" by blast |
|
702 from k rs(1) have "a = (p*r)^n" by (simp add: power_mult_distrib) |
|
703 with rs(2) have ?ths by blast } |
|
704 moreover |
|
705 {assume pb: "p^n dvd b" |
|
706 then obtain k where k: "b = p^n * k" unfolding dvd_def by blast |
|
707 from l have "l dvd c" by auto |
|
708 with dvd_imp_le[of l c] c have "l \<le> c" by auto |
|
709 moreover {assume "l = c" with l c have "p = 1" by simp with p have False by simp} |
|
710 ultimately have lc: "l < c" by arith |
|
711 from coprime_lmul2 [OF H(2)[unfolded k coprime_commute[of "p^n*k" a]]] |
|
712 have kb: "coprime k a" by (simp add: coprime_commute) |
|
713 from H(3) l k pn0 n have kbln: "k * a = l ^ n" |
|
714 by (simp add: power_mult_distrib mult.commute) |
|
715 from H(1)[rule_format, OF lc kb kbln] |
|
716 obtain r s where rs: "k = r ^n" "a = s^n" by blast |
|
717 from k rs(1) have "b = (p*r)^n" by (simp add: power_mult_distrib) |
|
718 with rs(2) have ?ths by blast } |
|
719 ultimately have ?ths using pnab by blast} |
|
720 ultimately have ?ths by blast} |
|
721 ultimately show ?ths by blast |
|
722 qed |
|
723 |
|
724 text \<open>More useful lemmas.\<close> |
|
725 lemma prime_product: |
|
726 assumes "prime (p * q)" |
|
727 shows "p = 1 \<or> q = 1" |
|
728 proof - |
|
729 from assms have |
|
730 "1 < p * q" and P: "\<And>m. m dvd p * q \<Longrightarrow> m = 1 \<or> m = p * q" |
|
731 unfolding prime_def by auto |
|
732 from \<open>1 < p * q\<close> have "p \<noteq> 0" by (cases p) auto |
|
733 then have Q: "p = p * q \<longleftrightarrow> q = 1" by auto |
|
734 have "p dvd p * q" by simp |
|
735 then have "p = 1 \<or> p = p * q" by (rule P) |
|
736 then show ?thesis by (simp add: Q) |
|
737 qed |
|
738 |
|
739 lemma prime_exp: "prime (p^n) \<longleftrightarrow> prime p \<and> n = 1" |
|
740 proof(induct n) |
|
741 case 0 thus ?case by simp |
|
742 next |
|
743 case (Suc n) |
|
744 {assume "p = 0" hence ?case by simp} |
|
745 moreover |
|
746 {assume "p=1" hence ?case by simp} |
|
747 moreover |
|
748 {assume p: "p \<noteq> 0" "p\<noteq>1" |
|
749 {assume pp: "prime (p^Suc n)" |
|
750 hence "p = 1 \<or> p^n = 1" using prime_product[of p "p^n"] by simp |
|
751 with p have n: "n = 0" |
|
752 by (simp only: exp_eq_1 ) simp |
|
753 with pp have "prime p \<and> Suc n = 1" by simp} |
|
754 moreover |
|
755 {assume n: "prime p \<and> Suc n = 1" hence "prime (p^Suc n)" by simp} |
|
756 ultimately have ?case by blast} |
|
757 ultimately show ?case by blast |
|
758 qed |
|
759 |
|
760 lemma prime_power_mult: |
|
761 assumes p: "prime p" and xy: "x * y = p ^ k" |
|
762 shows "\<exists>i j. x = p ^i \<and> y = p^ j" |
|
763 using xy |
|
764 proof(induct k arbitrary: x y) |
|
765 case 0 |
|
766 thus ?case apply simp by (rule exI[where x="0"], simp) |
|
767 next |
|
768 case (Suc k x y) |
|
769 have p0: "p \<noteq> 0" by (rule ccontr) (use p in simp) |
|
770 from Suc.prems have pxy: "p dvd x*y" by auto |
|
771 from prime_divprod[OF p pxy] show ?case |
|
772 proof |
|
773 assume px: "p dvd x" |
|
774 then obtain d where d: "x = p*d" unfolding dvd_def by blast |
|
775 from Suc.prems d have "p*d*y = p^Suc k" by simp |
|
776 hence th: "d*y = p^k" using p0 by simp |
|
777 from Suc.hyps[OF th] obtain i j where ij: "d = p^i" "y = p^j" by blast |
|
778 with d have "x = p^Suc i" by simp |
|
779 with ij(2) show ?thesis by blast |
|
780 next |
|
781 assume px: "p dvd y" |
|
782 then obtain d where d: "y = p*d" unfolding dvd_def by blast |
|
783 from Suc.prems d have "p*d*x = p^Suc k" by (simp add: mult.commute) |
|
784 hence th: "d*x = p^k" using p0 by simp |
|
785 from Suc.hyps[OF th] obtain i j where ij: "d = p^i" "x = p^j" by blast |
|
786 with d have "y = p^Suc i" by simp |
|
787 with ij(2) show ?thesis by blast |
|
788 qed |
|
789 qed |
|
790 |
|
791 lemma prime_power_exp: assumes p: "prime p" and n:"n \<noteq> 0" |
|
792 and xn: "x^n = p^k" shows "\<exists>i. x = p^i" |
|
793 using n xn |
|
794 proof(induct n arbitrary: k) |
|
795 case 0 thus ?case by simp |
|
796 next |
|
797 case (Suc n k) hence th: "x*x^n = p^k" by simp |
|
798 {assume "n = 0" with Suc have ?case by simp (rule exI[where x="k"], simp)} |
|
799 moreover |
|
800 {assume n: "n \<noteq> 0" |
|
801 from prime_power_mult[OF p th] |
|
802 obtain i j where ij: "x = p^i" "x^n = p^j"by blast |
|
803 from Suc.hyps[OF n ij(2)] have ?case .} |
|
804 ultimately show ?case by blast |
|
805 qed |
|
806 |
|
807 lemma divides_primepow: assumes p: "prime p" |
|
808 shows "d dvd p^k \<longleftrightarrow> (\<exists> i. i \<le> k \<and> d = p ^i)" |
|
809 proof |
|
810 assume H: "d dvd p^k" then obtain e where e: "d*e = p^k" |
|
811 unfolding dvd_def apply (auto simp add: mult.commute) by blast |
|
812 from prime_power_mult[OF p e] obtain i j where ij: "d = p^i" "e=p^j" by blast |
|
813 from prime_ge_2[OF p] have p1: "p > 1" by arith |
|
814 from e ij have "p^(i + j) = p^k" by (simp add: power_add) |
|
815 hence "i + j = k" using power_inject_exp[of p "i+j" k, OF p1] by simp |
|
816 hence "i \<le> k" by arith |
|
817 with ij(1) show "\<exists>i\<le>k. d = p ^ i" by blast |
|
818 next |
|
819 {fix i assume H: "i \<le> k" "d = p^i" |
|
820 hence "\<exists>j. k = i + j" by arith |
|
821 then obtain j where j: "k = i + j" by blast |
|
822 hence "p^k = p^j*d" using H(2) by (simp add: power_add) |
|
823 hence "d dvd p^k" unfolding dvd_def by auto} |
|
824 thus "\<exists>i\<le>k. d = p ^ i \<Longrightarrow> d dvd p ^ k" by blast |
|
825 qed |
|
826 |
|
827 lemma coprime_divisors: "d dvd a \<Longrightarrow> e dvd b \<Longrightarrow> coprime a b \<Longrightarrow> coprime d e" |
|
828 by (auto simp add: dvd_def coprime) |
|
829 |
|
830 lemma mult_inj_if_coprime_nat: |
|
831 "inj_on f A \<Longrightarrow> inj_on g B \<Longrightarrow> \<forall>a\<in>A. \<forall>b\<in>B. Primes.coprime (f a) (g b) \<Longrightarrow> |
|
832 inj_on (\<lambda>(a, b). f a * g b) (A \<times> B)" |
|
833 apply (auto simp add: inj_on_def) |
|
834 apply (metis coprime_def dvd_antisym dvd_triv_left relprime_dvd_mult_iff) |
|
835 apply (metis coprime_commute coprime_divprod dvd_antisym dvd_triv_right) |
|
836 done |
|
837 |
|
838 declare power_Suc0[simp del] |
|
839 |
|
840 end |
|