src/HOL/Old_Number_Theory/Primes.thy
 author huffman Sun Apr 01 16:09:58 2012 +0200 (2012-04-01) changeset 47255 30a1692557b0 parent 41541 1fa4725c4656 child 50037 f2a32197a33a permissions -rw-r--r--
removed Nat_Numeral.thy, moving all theorems elsewhere
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 *)
6 header {* Primality on nat *}
8 theory Primes
9 imports Complex_Main Legacy_GCD
10 begin
12 definition coprime :: "nat => nat => bool"
13   where "coprime m n \<longleftrightarrow> gcd m n = 1"
15 definition prime :: "nat \<Rightarrow> bool"
16   where "prime p \<longleftrightarrow> (1 < p \<and> (\<forall>m. m dvd p --> m = 1 \<or> m = p))"
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
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 One_nat_def gcd_dvd1 gcd_dvd2)
28   done
30 text {*
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 *}
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)
39 lemma prime_dvd_square: "prime p ==> p dvd m^Suc (Suc 0) ==> p dvd m"
40   by (auto dest: prime_dvd_mult)
42 lemma prime_dvd_power_two: "prime p ==> p dvd m\<twosuperior> ==> p dvd m"
43   by (rule prime_dvd_square) (simp_all add: power2_eq_square)
46 lemma exp_eq_1:"(x::nat)^n = 1 \<longleftrightarrow> x = 1 \<or> n = 0"
47 by (induct n, auto)
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)
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)
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
59 lemma even_square: assumes e: "even (n::nat)" shows "\<exists>x. n ^ 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^2 = 4* (k^2)" by (simp add: power2_eq_square)
64   thus ?thesis by blast
65 qed
67 lemma odd_square: assumes e: "odd (n::nat)" shows "\<exists>x. n ^ 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" using dvd_def by auto
72   hence k: "n = 2*k + 1"  using e by presburger
73   hence "n^2 = 4* (k^2 + k) + 1" by algebra
74   thus ?thesis by blast
75 qed
77 lemma diff_square: "(x::nat)^2 - y^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 ^2 \<le> y^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 ^2 \<le> x^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^2 = y^2 + z" by presburger
90     then obtain z2 where z2: "x^2 = y^2 + z2"  by blast
91     from z z2 have ?thesis apply simp by algebra }
92   ultimately show ?thesis by blast
93 qed
95 text {* Elementary theory of divisibility *}
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
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
109 declare nat_mult_dvd_cancel_disj[presburger]
110 lemma nat_mult_dvd_cancel_disj'[presburger]:
111   "(m\<Colon>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
113 lemma divides_mul_l: "(a::nat) dvd b ==> (c * a) dvd (c * b)"
114   by presburger
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)
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)
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)
136 fun fact :: "nat \<Rightarrow> nat" where
137   "fact 0 = 1"
138 | "fact (Suc n) = Suc n * fact n"
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
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
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])
179 text {* Coprimality *}
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)
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
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)
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)
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: mult_ac)
231 apply (rule_tac x="a*x" in exI)
232 apply (rule_tac x="y" in exI)
233 apply (simp add: mult_ac)
234 done
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: mult_ac)
243 apply (rule_tac x="b*x" in exI)
244 apply (rule_tac x="y" in exI)
245 apply (simp add: mult_ac)
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
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
262 lemma coprime_exp: "coprime d a ==> coprime d (a^n)"
263   by(induct n, simp_all add: coprime_mul)
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
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: mult_ac)
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
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
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
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
345 lemma nat_power_eq_0_iff: "(m::nat) ^ n = 0 \<longleftrightarrow> n \<noteq> 0 \<and> m = 0" by (induct n, auto)
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
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
382 text {* A binary form of the Chinese Remainder Theorem. *}
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
400 text {* Primality *}
402 text {* A few useful theorems about primes *}
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)
408 lemma prime_ge_2: "prime p ==> p \<ge> 2" by (simp add: prime_def)
409 lemma prime_factor: assumes n: "n \<noteq> 1" shows "\<exists> p. prime p \<and> p dvd n"
410 using 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   let ?ths = "\<exists>p. prime p \<and> p dvd n"
415   {assume "n=0" hence ?ths using two_is_prime by auto}
416   moreover
417   {assume nz: "n\<noteq>0"
418     {assume "prime n" hence ?ths by - (rule exI[where x="n"], simp)}
419     moreover
420     {assume n: "\<not> prime n"
421       with nz H(2)
422       obtain k where k:"k dvd n" "k \<noteq> 1" "k \<noteq> n" by (auto simp add: prime_def)
423       from dvd_imp_le[OF k(1)] nz k(3) have kn: "k < n" by simp
424       from H(1)[rule_format, OF kn k(2)] obtain p where p: "prime p" "p dvd k" by blast
425       from dvd_trans[OF p(2) k(1)] p(1) have ?ths by blast}
426     ultimately have ?ths by blast}
427   ultimately show ?ths by blast
428 qed
430 lemma prime_factor_lt: assumes p: "prime p" and n: "n \<noteq> 0" and npm:"n = p * m"
431   shows "m < n"
432 proof-
433   {assume "m=0" with n have ?thesis by simp}
434   moreover
435   {assume m: "m \<noteq> 0"
436     from npm have mn: "m dvd n" unfolding dvd_def by auto
437     from npm m have "n \<noteq> m" using p by auto
438     with dvd_imp_le[OF mn] n have ?thesis by simp}
439   ultimately show ?thesis by blast
440 qed
442 lemma euclid_bound: "\<exists>p. prime p \<and> n < p \<and>  p <= Suc (fact n)"
443 proof-
444   have f1: "fact n + 1 \<noteq> 1" using fact_le[of n] by arith
445   from prime_factor[OF f1] obtain p where p: "prime p" "p dvd fact n + 1" by blast
446   from dvd_imp_le[OF p(2)] have pfn: "p \<le> fact n + 1" by simp
447   {assume np: "p \<le> n"
448     from p(1) have p1: "p \<ge> 1" by (cases p, simp_all)
449     from divides_fact[OF p1 np] have pfn': "p dvd fact n" .
450     from divides_add_revr[OF pfn' p(2)] p(1) have False by simp}
451   hence "n < p" by arith
452   with p(1) pfn show ?thesis by auto
453 qed
455 lemma euclid: "\<exists>p. prime p \<and> p > n" using euclid_bound by auto
457 lemma primes_infinite: "\<not> (finite {p. prime p})"
458 apply(simp add: finite_nat_set_iff_bounded_le)
459 apply (metis euclid linorder_not_le)
460 done
462 lemma coprime_prime: assumes ab: "coprime a b"
463   shows "~(prime p \<and> p dvd a \<and> p dvd b)"
464 proof
465   assume "prime p \<and> p dvd a \<and> p dvd b"
466   thus False using ab gcd_greatest[of p a b] by (simp add: coprime_def)
467 qed
468 lemma coprime_prime_eq: "coprime a b \<longleftrightarrow> (\<forall>p. ~(prime p \<and> p dvd a \<and> p dvd b))"
469   (is "?lhs = ?rhs")
470 proof-
471   {assume "?lhs" with coprime_prime  have ?rhs by blast}
472   moreover
473   {assume r: "?rhs" and c: "\<not> ?lhs"
474     then obtain g where g: "g\<noteq>1" "g dvd a" "g dvd b" unfolding coprime_def by blast
475     from prime_factor[OF g(1)] obtain p where p: "prime p" "p dvd g" by blast
476     from dvd_trans [OF p(2) g(2)] dvd_trans [OF p(2) g(3)]
477     have "p dvd a" "p dvd b" . with p(1) r have False by blast}
478   ultimately show ?thesis by blast
479 qed
481 lemma prime_coprime: assumes p: "prime p"
482   shows "n = 1 \<or> p dvd n \<or> coprime p n"
483 using p prime_imp_relprime[of p n] by (auto simp add: coprime_def)
485 lemma prime_coprime_strong: "prime p \<Longrightarrow> p dvd n \<or> coprime p n"
486   using prime_coprime[of p n] by auto
488 declare  coprime_0[simp]
490 lemma coprime_0'[simp]: "coprime 0 d \<longleftrightarrow> d = 1" by (simp add: coprime_commute[of 0 d])
491 lemma coprime_bezout_strong: assumes ab: "coprime a b" and b: "b \<noteq> 1"
492   shows "\<exists>x y. a * x = b * y + 1"
493 proof-
494   from ab b have az: "a \<noteq> 0" by - (rule ccontr, auto)
495   from bezout_gcd_strong[OF az, of b] ab[unfolded coprime_def]
496   show ?thesis by auto
497 qed
499 lemma bezout_prime: assumes p: "prime p"  and pa: "\<not> p dvd a"
500   shows "\<exists>x y. a*x = p*y + 1"
501 proof-
502   from p have p1: "p \<noteq> 1" using prime_1 by blast
503   from prime_coprime[OF p, of a] p1 pa have ap: "coprime a p"
504     by (auto simp add: coprime_commute)
505   from coprime_bezout_strong[OF ap p1] show ?thesis .
506 qed
507 lemma prime_divprod: assumes p: "prime p" and pab: "p dvd a*b"
508   shows "p dvd a \<or> p dvd b"
509 proof-
510   {assume "a=1" hence ?thesis using pab by simp }
511   moreover
512   {assume "p dvd a" hence ?thesis by blast}
513   moreover
514   {assume pa: "coprime p a" from coprime_divprod[OF pab pa]  have ?thesis .. }
515   ultimately show ?thesis using prime_coprime[OF p, of a] by blast
516 qed
518 lemma prime_divprod_eq: assumes p: "prime p"
519   shows "p dvd a*b \<longleftrightarrow> p dvd a \<or> p dvd b"
520 using p prime_divprod dvd_mult dvd_mult2 by auto
522 lemma prime_divexp: assumes p:"prime p" and px: "p dvd x^n"
523   shows "p dvd x"
524 using px
525 proof(induct n)
526   case 0 thus ?case by simp
527 next
528   case (Suc n)
529   hence th: "p dvd x*x^n" by simp
530   {assume H: "p dvd x^n"
531     from Suc.hyps[OF H] have ?case .}
532   with prime_divprod[OF p th] show ?case by blast
533 qed
535 lemma prime_divexp_n: "prime p \<Longrightarrow> p dvd x^n \<Longrightarrow> p^n dvd x^n"
536   using prime_divexp[of p x n] divides_exp[of p x n] by blast
538 lemma coprime_prime_dvd_ex: assumes xy: "\<not>coprime x y"
539   shows "\<exists>p. prime p \<and> p dvd x \<and> p dvd y"
540 proof-
541   from xy[unfolded coprime_def] obtain g where g: "g \<noteq> 1" "g dvd x" "g dvd y"
542     by blast
543   from prime_factor[OF g(1)] obtain p where p: "prime p" "p dvd g" by blast
544   from g(2,3) dvd_trans[OF p(2)] p(1) show ?thesis by auto
545 qed
546 lemma coprime_sos: assumes xy: "coprime x y"
547   shows "coprime (x * y) (x^2 + y^2)"
548 proof-
549   {assume c: "\<not> coprime (x * y) (x^2 + y^2)"
550     from coprime_prime_dvd_ex[OF c] obtain p
551       where p: "prime p" "p dvd x*y" "p dvd x^2 + y^2" by blast
552     {assume px: "p dvd x"
553       from dvd_mult[OF px, of x] p(3)
554         obtain r s where "x * x = p * r" and "x^2 + y^2 = p * s"
555           by (auto elim!: dvdE)
556         then have "y^2 = p * (s - r)"
557           by (auto simp add: power2_eq_square diff_mult_distrib2)
558         then have "p dvd y^2" ..
559       with prime_divexp[OF p(1), of y 2] have py: "p dvd y" .
560       from p(1) px py xy[unfolded coprime, rule_format, of p] prime_1
561       have False by simp }
562     moreover
563     {assume py: "p dvd y"
564       from dvd_mult[OF py, of y] p(3)
565         obtain r s where "y * y = p * r" and "x^2 + y^2 = p * s"
566           by (auto elim!: dvdE)
567         then have "x^2 = p * (s - r)"
568           by (auto simp add: power2_eq_square diff_mult_distrib2)
569         then have "p dvd x^2" ..
570       with prime_divexp[OF p(1), of x 2] have px: "p dvd x" .
571       from p(1) px py xy[unfolded coprime, rule_format, of p] prime_1
572       have False by simp }
573     ultimately have False using prime_divprod[OF p(1,2)] by blast}
574   thus ?thesis by blast
575 qed
577 lemma distinct_prime_coprime: "prime p \<Longrightarrow> prime q \<Longrightarrow> p \<noteq> q \<Longrightarrow> coprime p q"
578   unfolding prime_def coprime_prime_eq by blast
580 lemma prime_coprime_lt: assumes p: "prime p" and x: "0 < x" and xp: "x < p"
581   shows "coprime x p"
582 proof-
583   {assume c: "\<not> coprime x p"
584     then obtain g where g: "g \<noteq> 1" "g dvd x" "g dvd p" unfolding coprime_def by blast
585   from dvd_imp_le[OF g(2)] x xp have gp: "g < p" by arith
586   from g(2) x have "g \<noteq> 0" by - (rule ccontr, simp)
587   with g gp p[unfolded prime_def] have False by blast}
588 thus ?thesis by blast
589 qed
591 lemma even_dvd[simp]: "even (n::nat) \<longleftrightarrow> 2 dvd n" by presburger
592 lemma prime_odd: "prime p \<Longrightarrow> p = 2 \<or> odd p" unfolding prime_def by auto
595 text {* One property of coprimality is easier to prove via prime factors. *}
597 lemma prime_divprod_pow:
598   assumes p: "prime p" and ab: "coprime a b" and pab: "p^n dvd a * b"
599   shows "p^n dvd a \<or> p^n dvd b"
600 proof-
601   {assume "n = 0 \<or> a = 1 \<or> b = 1" with pab have ?thesis
602       apply (cases "n=0", simp_all)
603       apply (cases "a=1", simp_all) done}
604   moreover
605   {assume n: "n \<noteq> 0" and a: "a\<noteq>1" and b: "b\<noteq>1"
606     then obtain m where m: "n = Suc m" by (cases n, auto)
607     from divides_exp2[OF n pab] have pab': "p dvd a*b" .
608     from prime_divprod[OF p pab']
609     have "p dvd a \<or> p dvd b" .
610     moreover
611     {assume pa: "p dvd a"
612       have pnba: "p^n dvd b*a" using pab by (simp add: mult_commute)
613       from coprime_prime[OF ab, of p] p pa have "\<not> p dvd b" by blast
614       with prime_coprime[OF p, of b] b
615       have cpb: "coprime b p" using coprime_commute by blast
616       from coprime_exp[OF cpb] have pnb: "coprime (p^n) b"
617         by (simp add: coprime_commute)
618       from coprime_divprod[OF pnba pnb] have ?thesis by blast }
619     moreover
620     {assume pb: "p dvd b"
621       have pnba: "p^n dvd b*a" using pab by (simp add: mult_commute)
622       from coprime_prime[OF ab, of p] p pb have "\<not> p dvd a" by blast
623       with prime_coprime[OF p, of a] a
624       have cpb: "coprime a p" using coprime_commute by blast
625       from coprime_exp[OF cpb] have pnb: "coprime (p^n) a"
626         by (simp add: coprime_commute)
627       from coprime_divprod[OF pab pnb] have ?thesis by blast }
628     ultimately have ?thesis by blast}
629   ultimately show ?thesis by blast
630 qed
632 lemma nat_mult_eq_one: "(n::nat) * m = 1 \<longleftrightarrow> n = 1 \<and> m = 1" (is "?lhs \<longleftrightarrow> ?rhs")
633 proof
634   assume H: "?lhs"
635   hence "n dvd 1" "m dvd 1" unfolding dvd_def by (auto simp add: mult_commute)
636   thus ?rhs by auto
637 next
638   assume ?rhs then show ?lhs by auto
639 qed
641 lemma power_Suc0: "Suc 0 ^ n = Suc 0"
642   unfolding One_nat_def[symmetric] power_one ..
644 lemma coprime_pow: assumes ab: "coprime a b" and abcn: "a * b = c ^n"
645   shows "\<exists>r s. a = r^n  \<and> b = s ^n"
646   using ab abcn
647 proof(induct c arbitrary: a b rule: nat_less_induct)
648   fix c a b
649   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"
650   let ?ths = "\<exists>r s. a = r^n  \<and> b = s ^n"
651   {assume n: "n = 0"
652     with H(3) power_one have "a*b = 1" by simp
653     hence "a = 1 \<and> b = 1" by simp
654     hence ?ths
655       apply -
656       apply (rule exI[where x=1])
657       apply (rule exI[where x=1])
658       using power_one[of  n]
659       by simp}
660   moreover
661   {assume n: "n \<noteq> 0" then obtain m where m: "n = Suc m" by (cases n, auto)
662     {assume c: "c = 0"
663       with H(3) m H(2) have ?ths apply simp
664         apply (cases "a=0", simp_all)
665         apply (rule exI[where x="0"], simp)
666         apply (rule exI[where x="0"], simp)
667         done}
668     moreover
669     {assume "c=1" with H(3) power_one have "a*b = 1" by simp
670         hence "a = 1 \<and> b = 1" by simp
671         hence ?ths
672       apply -
673       apply (rule exI[where x=1])
674       apply (rule exI[where x=1])
675       using power_one[of  n]
676       by simp}
677   moreover
678   {assume c: "c\<noteq>1" "c \<noteq> 0"
679     from prime_factor[OF c(1)] obtain p where p: "prime p" "p dvd c" by blast
680     from prime_divprod_pow[OF p(1) H(2), unfolded H(3), OF divides_exp[OF p(2), of n]]
681     have pnab: "p ^ n dvd a \<or> p^n dvd b" .
682     from p(2) obtain l where l: "c = p*l" unfolding dvd_def by blast
683     have pn0: "p^n \<noteq> 0" using n prime_ge_2 [OF p(1)] by simp
684     {assume pa: "p^n dvd a"
685       then obtain k where k: "a = p^n * k" unfolding dvd_def by blast
686       from l have "l dvd c" by auto
687       with dvd_imp_le[of l c] c have "l \<le> c" by auto
688       moreover {assume "l = c" with l c  have "p = 1" by simp with p have False by simp}
689       ultimately have lc: "l < c" by arith
690       from coprime_lmul2 [OF H(2)[unfolded k coprime_commute[of "p^n*k" b]]]
691       have kb: "coprime k b" by (simp add: coprime_commute)
692       from H(3) l k pn0 have kbln: "k * b = l ^ n"
693         by (auto simp add: power_mult_distrib)
694       from H(1)[rule_format, OF lc kb kbln]
695       obtain r s where rs: "k = r ^n" "b = s^n" by blast
696       from k rs(1) have "a = (p*r)^n" by (simp add: power_mult_distrib)
697       with rs(2) have ?ths by blast }
698     moreover
699     {assume pb: "p^n dvd b"
700       then obtain k where k: "b = p^n * k" unfolding dvd_def by blast
701       from l have "l dvd c" by auto
702       with dvd_imp_le[of l c] c have "l \<le> c" by auto
703       moreover {assume "l = c" with l c  have "p = 1" by simp with p have False by simp}
704       ultimately have lc: "l < c" by arith
705       from coprime_lmul2 [OF H(2)[unfolded k coprime_commute[of "p^n*k" a]]]
706       have kb: "coprime k a" by (simp add: coprime_commute)
707       from H(3) l k pn0 n have kbln: "k * a = l ^ n"
708         by (simp add: power_mult_distrib mult_commute)
709       from H(1)[rule_format, OF lc kb kbln]
710       obtain r s where rs: "k = r ^n" "a = s^n" by blast
711       from k rs(1) have "b = (p*r)^n" by (simp add: power_mult_distrib)
712       with rs(2) have ?ths by blast }
713     ultimately have ?ths using pnab by blast}
714   ultimately have ?ths by blast}
715 ultimately show ?ths by blast
716 qed
718 text {* More useful lemmas. *}
719 lemma prime_product:
720   assumes "prime (p * q)"
721   shows "p = 1 \<or> q = 1"
722 proof -
723   from assms have
724     "1 < p * q" and P: "\<And>m. m dvd p * q \<Longrightarrow> m = 1 \<or> m = p * q"
725     unfolding prime_def by auto
726   from `1 < p * q` have "p \<noteq> 0" by (cases p) auto
727   then have Q: "p = p * q \<longleftrightarrow> q = 1" by auto
728   have "p dvd p * q" by simp
729   then have "p = 1 \<or> p = p * q" by (rule P)
730   then show ?thesis by (simp add: Q)
731 qed
733 lemma prime_exp: "prime (p^n) \<longleftrightarrow> prime p \<and> n = 1"
734 proof(induct n)
735   case 0 thus ?case by simp
736 next
737   case (Suc n)
738   {assume "p = 0" hence ?case by simp}
739   moreover
740   {assume "p=1" hence ?case by simp}
741   moreover
742   {assume p: "p \<noteq> 0" "p\<noteq>1"
743     {assume pp: "prime (p^Suc n)"
744       hence "p = 1 \<or> p^n = 1" using prime_product[of p "p^n"] by simp
745       with p have n: "n = 0"
746         by (simp only: exp_eq_1 ) simp
747       with pp have "prime p \<and> Suc n = 1" by simp}
748     moreover
749     {assume n: "prime p \<and> Suc n = 1" hence "prime (p^Suc n)" by simp}
750     ultimately have ?case by blast}
751   ultimately show ?case by blast
752 qed
754 lemma prime_power_mult:
755   assumes p: "prime p" and xy: "x * y = p ^ k"
756   shows "\<exists>i j. x = p ^i \<and> y = p^ j"
757   using xy
758 proof(induct k arbitrary: x y)
759   case 0 thus ?case apply simp by (rule exI[where x="0"], simp)
760 next
761   case (Suc k x y)
762   from Suc.prems have pxy: "p dvd x*y" by auto
763   from prime_divprod[OF p pxy] have pxyc: "p dvd x \<or> p dvd y" .
764   from p have p0: "p \<noteq> 0" by - (rule ccontr, simp)
765   {assume px: "p dvd x"
766     then obtain d where d: "x = p*d" unfolding dvd_def by blast
767     from Suc.prems d  have "p*d*y = p^Suc k" by simp
768     hence th: "d*y = p^k" using p0 by simp
769     from Suc.hyps[OF th] obtain i j where ij: "d = p^i" "y = p^j" by blast
770     with d have "x = p^Suc i" by simp
771     with ij(2) have ?case by blast}
772   moreover
773   {assume px: "p dvd y"
774     then obtain d where d: "y = p*d" unfolding dvd_def by blast
775     from Suc.prems d  have "p*d*x = p^Suc k" by (simp add: mult_commute)
776     hence th: "d*x = p^k" using p0 by simp
777     from Suc.hyps[OF th] obtain i j where ij: "d = p^i" "x = p^j" by blast
778     with d have "y = p^Suc i" by simp
779     with ij(2) have ?case by blast}
780   ultimately show ?case  using pxyc by blast
781 qed
783 lemma prime_power_exp: assumes p: "prime p" and n:"n \<noteq> 0"
784   and xn: "x^n = p^k" shows "\<exists>i. x = p^i"
785   using n xn
786 proof(induct n arbitrary: k)
787   case 0 thus ?case by simp
788 next
789   case (Suc n k) hence th: "x*x^n = p^k" by simp
790   {assume "n = 0" with Suc have ?case by simp (rule exI[where x="k"], simp)}
791   moreover
792   {assume n: "n \<noteq> 0"
793     from prime_power_mult[OF p th]
794     obtain i j where ij: "x = p^i" "x^n = p^j"by blast
795     from Suc.hyps[OF n ij(2)] have ?case .}
796   ultimately show ?case by blast
797 qed
799 lemma divides_primepow: assumes p: "prime p"
800   shows "d dvd p^k \<longleftrightarrow> (\<exists> i. i \<le> k \<and> d = p ^i)"
801 proof
802   assume H: "d dvd p^k" then obtain e where e: "d*e = p^k"
803     unfolding dvd_def  apply (auto simp add: mult_commute) by blast
804   from prime_power_mult[OF p e] obtain i j where ij: "d = p^i" "e=p^j" by blast
805   from prime_ge_2[OF p] have p1: "p > 1" by arith
806   from e ij have "p^(i + j) = p^k" by (simp add: power_add)
807   hence "i + j = k" using power_inject_exp[of p "i+j" k, OF p1] by simp
808   hence "i \<le> k" by arith
809   with ij(1) show "\<exists>i\<le>k. d = p ^ i" by blast
810 next
811   {fix i assume H: "i \<le> k" "d = p^i"
812     hence "\<exists>j. k = i + j" by arith
813     then obtain j where j: "k = i + j" by blast
814     hence "p^k = p^j*d" using H(2) by (simp add: power_add)
815     hence "d dvd p^k" unfolding dvd_def by auto}
816   thus "\<exists>i\<le>k. d = p ^ i \<Longrightarrow> d dvd p ^ k" by blast
817 qed
819 lemma coprime_divisors: "d dvd a \<Longrightarrow> e dvd b \<Longrightarrow> coprime a b \<Longrightarrow> coprime d e"
820   by (auto simp add: dvd_def coprime)
822 lemma mult_inj_if_coprime_nat:
823   "inj_on f A \<Longrightarrow> inj_on g B \<Longrightarrow> ALL a:A. ALL b:B. coprime (f a) (g b)
824    \<Longrightarrow> inj_on (%(a,b). f a * g b::nat) (A \<times> B)"
825 apply(auto simp add:inj_on_def)
826 apply(metis coprime_def dvd_triv_left gcd_proj2_if_dvd_nat gcd_semilattice_nat.inf_commute relprime_dvd_mult)
827 apply(metis coprime_commute coprime_divprod dvd.neq_le_trans dvd_triv_right)
828 done
830 declare power_Suc0[simp del]
831 declare even_dvd[simp del]
833 end