wenzelm@11368: (* Title: HOL/Library/Primes.thy paulson@11363: ID: $Id$ haftmann@27106: Author: Amine Chaieb, Christophe Tabacznyj and Lawrence C Paulson paulson@11363: Copyright 1996 University of Cambridge paulson@11363: *) paulson@11363: nipkow@16762: header {* Primality on nat *} paulson@11363: nipkow@15131: theory Primes haftmann@27487: imports Plain "~~/src/HOL/ATP_Linkup" GCD Parity nipkow@15131: begin paulson@11363: wenzelm@19086: definition wenzelm@21404: coprime :: "nat => nat => bool" where chaieb@27567: "coprime m n \ gcd m n = 1" paulson@11363: wenzelm@21404: definition wenzelm@21404: prime :: "nat \ bool" where haftmann@27106: [code func del]: "prime p \ (1 < p \ (\m. m dvd p --> m = 1 \ m = p))" paulson@11363: paulson@11363: nipkow@16762: lemma two_is_prime: "prime 2" nipkow@16762: apply (auto simp add: prime_def) nipkow@16762: apply (case_tac m) nipkow@16762: apply (auto dest!: dvd_imp_le) paulson@11363: done paulson@11363: haftmann@27556: lemma prime_imp_relprime: "prime p ==> \ p dvd n ==> gcd p n = 1" paulson@11363: apply (auto simp add: prime_def) paulson@23839: apply (metis One_nat_def gcd_dvd1 gcd_dvd2) paulson@11363: done paulson@11363: paulson@11363: text {* paulson@11363: This theorem leads immediately to a proof of the uniqueness of paulson@11363: factorization. If @{term p} divides a product of primes then it is paulson@11363: one of those primes. paulson@11363: *} paulson@11363: nipkow@16663: lemma prime_dvd_mult: "prime p ==> p dvd m * n ==> p dvd m \ p dvd n" wenzelm@12739: by (blast intro: relprime_dvd_mult prime_imp_relprime) paulson@11363: nipkow@16663: lemma prime_dvd_square: "prime p ==> p dvd m^Suc (Suc 0) ==> p dvd m" wenzelm@12739: by (auto dest: prime_dvd_mult) wenzelm@12739: nipkow@16663: lemma prime_dvd_power_two: "prime p ==> p dvd m\ ==> p dvd m" paulson@14353: by (rule prime_dvd_square) (simp_all add: power2_eq_square) wenzelm@11368: chaieb@26125: chaieb@26125: lemma exp_eq_1:"(x::nat)^n = 1 \ x = 1 \ n = 0" by (induct n, auto) chaieb@26125: lemma exp_mono_lt: "(x::nat) ^ (Suc n) < y ^ (Suc n) \ x < y" chaieb@26125: using power_less_imp_less_base[of x "Suc n" y] power_strict_mono[of x y "Suc n"] chaieb@26125: by auto chaieb@26125: lemma exp_mono_le: "(x::nat) ^ (Suc n) \ y ^ (Suc n) \ x \ y" chaieb@26125: by (simp only: linorder_not_less[symmetric] exp_mono_lt) chaieb@26125: chaieb@26125: lemma exp_mono_eq: "(x::nat) ^ Suc n = y ^ Suc n \ x = y" chaieb@26125: using power_inject_base[of x n y] by auto chaieb@26125: chaieb@26125: chaieb@26125: lemma even_square: assumes e: "even (n::nat)" shows "\x. n ^ 2 = 4*x" chaieb@26125: proof- chaieb@26125: from e have "2 dvd n" by presburger chaieb@26125: then obtain k where k: "n = 2*k" using dvd_def by auto chaieb@26125: hence "n^2 = 4* (k^2)" by (simp add: power2_eq_square) chaieb@26125: thus ?thesis by blast chaieb@26125: qed chaieb@26125: chaieb@26125: lemma odd_square: assumes e: "odd (n::nat)" shows "\x. n ^ 2 = 4*x + 1" chaieb@26125: proof- chaieb@26125: from e have np: "n > 0" by presburger chaieb@26125: from e have "2 dvd (n - 1)" by presburger chaieb@26125: then obtain k where "n - 1 = 2*k" using dvd_def by auto chaieb@26125: hence k: "n = 2*k + 1" using e by presburger chaieb@26125: hence "n^2 = 4* (k^2 + k) + 1" by algebra chaieb@26125: thus ?thesis by blast chaieb@26125: qed chaieb@26125: chaieb@26125: lemma diff_square: "(x::nat)^2 - y^2 = (x+y)*(x - y)" chaieb@26125: proof- chaieb@26125: have "x \ y \ y \ x" by (rule nat_le_linear) chaieb@26125: moreover chaieb@26125: {assume le: "x \ y" chaieb@26125: hence "x ^2 \ y^2" by (simp only: numeral_2_eq_2 exp_mono_le Let_def) chaieb@26125: with le have ?thesis by simp } chaieb@26125: moreover chaieb@26125: {assume le: "y \ x" chaieb@26125: hence le2: "y ^2 \ x^2" by (simp only: numeral_2_eq_2 exp_mono_le Let_def) chaieb@26125: from le have "\z. y + z = x" by presburger chaieb@26125: then obtain z where z: "x = y + z" by blast chaieb@26125: from le2 have "\z. x^2 = y^2 + z" by presburger chaieb@26125: then obtain z2 where z2: "x^2 = y^2 + z2" by blast chaieb@26125: from z z2 have ?thesis apply simp by algebra } chaieb@26125: ultimately show ?thesis by blast chaieb@26125: qed chaieb@26125: wenzelm@26144: text {* Elementary theory of divisibility *} chaieb@26125: lemma divides_ge: "(a::nat) dvd b \ b = 0 \ a \ b" unfolding dvd_def by auto chaieb@26125: lemma divides_antisym: "(x::nat) dvd y \ y dvd x \ x = y" chaieb@26125: using dvd_anti_sym[of x y] by auto chaieb@26125: chaieb@26125: lemma divides_add_revr: assumes da: "(d::nat) dvd a" and dab:"d dvd (a + b)" chaieb@26125: shows "d dvd b" chaieb@26125: proof- chaieb@26125: from da obtain k where k:"a = d*k" by (auto simp add: dvd_def) chaieb@26125: from dab obtain k' where k': "a + b = d*k'" by (auto simp add: dvd_def) chaieb@26125: from k k' have "b = d *(k' - k)" by (simp add : diff_mult_distrib2) chaieb@26125: thus ?thesis unfolding dvd_def by blast chaieb@26125: qed chaieb@26125: chaieb@26125: declare nat_mult_dvd_cancel_disj[presburger] chaieb@26125: lemma nat_mult_dvd_cancel_disj'[presburger]: chaieb@26125: "(m\nat)*k dvd n*k \ k = 0 \ m dvd n" unfolding mult_commute[of m k] mult_commute[of n k] by presburger chaieb@26125: chaieb@26125: lemma divides_mul_l: "(a::nat) dvd b ==> (c * a) dvd (c * b)" chaieb@26125: by presburger chaieb@26125: chaieb@26125: lemma divides_mul_r: "(a::nat) dvd b ==> (a * c) dvd (b * c)" by presburger chaieb@26125: lemma divides_cases: "(n::nat) dvd m ==> m = 0 \ m = n \ 2 * n <= m" chaieb@26125: by (auto simp add: dvd_def) chaieb@26125: lemma divides_le: "m dvd n ==> m <= n \ n = (0::nat)" by (auto simp add: dvd_def) chaieb@26125: chaieb@26125: lemma divides_div_not: "(x::nat) = (q * n) + r \ 0 < r \ r < n ==> ~(n dvd x)" chaieb@26125: proof(auto simp add: dvd_def) chaieb@26125: fix k assume H: "0 < r" "r < n" "q * n + r = n * k" chaieb@26125: from H(3) have r: "r = n* (k -q)" by(simp add: diff_mult_distrib2 mult_commute) chaieb@26125: {assume "k - q = 0" with r H(1) have False by simp} chaieb@26125: moreover chaieb@26125: {assume "k - q \ 0" with r have "r \ n" by auto chaieb@26125: with H(2) have False by simp} chaieb@26125: ultimately show False by blast chaieb@26125: qed chaieb@26125: lemma divides_exp: "(x::nat) dvd y ==> x ^ n dvd y ^ n" chaieb@26125: by (auto simp add: power_mult_distrib dvd_def) chaieb@26125: chaieb@26125: lemma divides_exp2: "n \ 0 \ (x::nat) ^ n dvd y \ x dvd y" chaieb@26125: by (induct n ,auto simp add: dvd_def) chaieb@26125: chaieb@26125: fun fact :: "nat \ nat" where chaieb@26125: "fact 0 = 1" chaieb@26125: | "fact (Suc n) = Suc n * fact n" chaieb@26125: chaieb@26125: lemma fact_lt: "0 < fact n" by(induct n, simp_all) chaieb@26125: lemma fact_le: "fact n \ 1" using fact_lt[of n] by simp chaieb@26125: lemma fact_mono: assumes le: "m \ n" shows "fact m \ fact n" chaieb@26125: proof- chaieb@26125: from le have "\i. n = m+i" by presburger chaieb@26125: then obtain i where i: "n = m+i" by blast chaieb@26125: have "fact m \ fact (m + i)" chaieb@26125: proof(induct m) chaieb@26125: case 0 thus ?case using fact_le[of i] by simp chaieb@26125: next chaieb@26125: case (Suc m) chaieb@26125: have "fact (Suc m) = Suc m * fact m" by simp chaieb@26125: have th1: "Suc m \ Suc (m + i)" by simp chaieb@26125: from mult_le_mono[of "Suc m" "Suc (m+i)" "fact m" "fact (m+i)", OF th1 Suc.hyps] chaieb@26125: show ?case by simp chaieb@26125: qed chaieb@26125: thus ?thesis using i by simp chaieb@26125: qed chaieb@26125: chaieb@26125: lemma divides_fact: "1 <= p \ p <= n ==> p dvd fact n" chaieb@26125: proof(induct n arbitrary: p) chaieb@26125: case 0 thus ?case by simp chaieb@26125: next chaieb@26125: case (Suc n p) chaieb@26125: from Suc.prems have "p = Suc n \ p \ n" by presburger chaieb@26125: moreover chaieb@26125: {assume "p = Suc n" hence ?case by (simp only: fact.simps dvd_triv_left)} chaieb@26125: moreover chaieb@26125: {assume "p \ n" chaieb@26125: with Suc.prems(1) Suc.hyps have th: "p dvd fact n" by simp chaieb@26125: from dvd_mult[OF th] have ?case by (simp only: fact.simps) } chaieb@26125: ultimately show ?case by blast chaieb@26125: qed chaieb@26125: chaieb@26125: declare dvd_triv_left[presburger] chaieb@26125: declare dvd_triv_right[presburger] chaieb@26125: lemma divides_rexp: chaieb@26125: "x dvd y \ (x::nat) dvd (y^(Suc n))" by (simp add: dvd_mult2[of x y]) chaieb@26125: wenzelm@26144: text {* The Bezout theorem is a bit ugly for N; it'd be easier for Z *} chaieb@26125: lemma ind_euclid: chaieb@26125: assumes c: " \a b. P (a::nat) b \ P b a" and z: "\a. P a 0" chaieb@26125: and add: "\a b. P a b \ P a (a + b)" chaieb@26125: shows "P a b" chaieb@26125: proof(induct n\"a+b" arbitrary: a b rule: nat_less_induct) chaieb@26125: fix n a b chaieb@26125: assume H: "\m < n. \a b. m = a + b \ P a b" "n = a + b" chaieb@26125: have "a = b \ a < b \ b < a" by arith chaieb@26125: moreover {assume eq: "a= b" chaieb@26125: from add[rule_format, OF z[rule_format, of a]] have "P a b" using eq by simp} chaieb@26125: moreover chaieb@26125: {assume lt: "a < b" chaieb@26125: hence "a + b - a < n \ a = 0" using H(2) by arith chaieb@26125: moreover chaieb@26125: {assume "a =0" with z c have "P a b" by blast } chaieb@26125: moreover chaieb@26125: {assume ab: "a + b - a < n" chaieb@26125: have th0: "a + b - a = a + (b - a)" using lt by arith chaieb@26125: from add[rule_format, OF H(1)[rule_format, OF ab th0]] chaieb@26125: have "P a b" by (simp add: th0[symmetric])} chaieb@26125: ultimately have "P a b" by blast} chaieb@26125: moreover chaieb@26125: {assume lt: "a > b" chaieb@26125: hence "b + a - b < n \ b = 0" using H(2) by arith chaieb@26125: moreover chaieb@26125: {assume "b =0" with z c have "P a b" by blast } chaieb@26125: moreover chaieb@26125: {assume ab: "b + a - b < n" chaieb@26125: have th0: "b + a - b = b + (a - b)" using lt by arith chaieb@26125: from add[rule_format, OF H(1)[rule_format, OF ab th0]] chaieb@26125: have "P b a" by (simp add: th0[symmetric]) chaieb@26125: hence "P a b" using c by blast } chaieb@26125: ultimately have "P a b" by blast} chaieb@26125: ultimately show "P a b" by blast chaieb@26125: qed chaieb@26125: chaieb@26125: lemma bezout_lemma: chaieb@26125: assumes ex: "\(d::nat) x y. d dvd a \ d dvd b \ (a * x = b * y + d \ b * x = a * y + d)" chaieb@26125: shows "\d x y. d dvd a \ d dvd a + b \ (a * x = (a + b) * y + d \ (a + b) * x = a * y + d)" chaieb@26125: using ex chaieb@26125: apply clarsimp chaieb@26125: apply (rule_tac x="d" in exI, simp add: dvd_add) chaieb@26125: apply (case_tac "a * x = b * y + d" , simp_all) chaieb@26125: apply (rule_tac x="x + y" in exI) chaieb@26125: apply (rule_tac x="y" in exI) chaieb@26125: apply algebra chaieb@26125: apply (rule_tac x="x" in exI) chaieb@26125: apply (rule_tac x="x + y" in exI) chaieb@26125: apply algebra chaieb@26125: done chaieb@26125: chaieb@26125: lemma bezout_add: "\(d::nat) x y. d dvd a \ d dvd b \ (a * x = b * y + d \ b * x = a * y + d)" chaieb@26125: apply(induct a b rule: ind_euclid) chaieb@26125: apply blast chaieb@26125: apply clarify chaieb@26125: apply (rule_tac x="a" in exI, simp add: dvd_add) chaieb@26125: apply clarsimp chaieb@26125: apply (rule_tac x="d" in exI) chaieb@26125: apply (case_tac "a * x = b * y + d", simp_all add: dvd_add) chaieb@26125: apply (rule_tac x="x+y" in exI) chaieb@26125: apply (rule_tac x="y" in exI) chaieb@26125: apply algebra chaieb@26125: apply (rule_tac x="x" in exI) chaieb@26125: apply (rule_tac x="x+y" in exI) chaieb@26125: apply algebra chaieb@26125: done chaieb@26125: chaieb@26125: lemma bezout: "\(d::nat) x y. d dvd a \ d dvd b \ (a * x - b * y = d \ b * x - a * y = d)" chaieb@26125: using bezout_add[of a b] chaieb@26125: apply clarsimp chaieb@26125: apply (rule_tac x="d" in exI, simp) chaieb@26125: apply (rule_tac x="x" in exI) chaieb@26125: apply (rule_tac x="y" in exI) chaieb@26125: apply auto chaieb@26125: done chaieb@26125: wenzelm@26144: text {* We can get a stronger version with a nonzeroness assumption. *} chaieb@26125: chaieb@26125: lemma bezout_add_strong: assumes nz: "a \ (0::nat)" chaieb@26125: shows "\d x y. d dvd a \ d dvd b \ a * x = b * y + d" chaieb@26125: proof- chaieb@26125: from nz have ap: "a > 0" by simp chaieb@26125: from bezout_add[of a b] chaieb@26125: have "(\d x y. d dvd a \ d dvd b \ a * x = b * y + d) \ (\d x y. d dvd a \ d dvd b \ b * x = a * y + d)" by blast chaieb@26125: moreover chaieb@26125: {fix d x y assume H: "d dvd a" "d dvd b" "a * x = b * y + d" chaieb@26125: from H have ?thesis by blast } chaieb@26125: moreover chaieb@26125: {fix d x y assume H: "d dvd a" "d dvd b" "b * x = a * y + d" chaieb@26125: {assume b0: "b = 0" with H have ?thesis by simp} chaieb@26125: moreover chaieb@26125: {assume b: "b \ 0" hence bp: "b > 0" by simp chaieb@26125: from divides_le[OF H(2)] b have "d < b \ d = b" using le_less by blast chaieb@26125: moreover chaieb@26125: {assume db: "d=b" chaieb@26125: from prems have ?thesis apply simp chaieb@26125: apply (rule exI[where x = b], simp) chaieb@26125: apply (rule exI[where x = b]) chaieb@26125: by (rule exI[where x = "a - 1"], simp add: diff_mult_distrib2)} chaieb@26125: moreover chaieb@26125: {assume db: "d < b" chaieb@26125: {assume "x=0" hence ?thesis using prems by simp } chaieb@26125: moreover chaieb@26125: {assume x0: "x \ 0" hence xp: "x > 0" by simp chaieb@26125: chaieb@26125: from db have "d \ b - 1" by simp chaieb@26125: hence "d*b \ b*(b - 1)" by simp chaieb@26125: with xp mult_mono[of "1" "x" "d*b" "b*(b - 1)"] chaieb@26125: have dble: "d*b \ x*b*(b - 1)" using bp by simp chaieb@26125: from H (3) have "d + (b - 1) * (b*x) = d + (b - 1) * (a*y + d)" by simp chaieb@26125: hence "d + (b - 1) * a * y + (b - 1) * d = d + (b - 1) * b * x" chaieb@26125: by (simp only: mult_assoc right_distrib) chaieb@26125: hence "a * ((b - 1) * y) + d * (b - 1 + 1) = d + x*b*(b - 1)" by algebra chaieb@26125: hence "a * ((b - 1) * y) = d + x*b*(b - 1) - d*b" using bp by simp chaieb@26125: hence "a * ((b - 1) * y) = d + (x*b*(b - 1) - d*b)" chaieb@26125: by (simp only: diff_add_assoc[OF dble, of d, symmetric]) chaieb@26125: hence "a * ((b - 1) * y) = b*(x*(b - 1) - d) + d" chaieb@26125: by (simp only: diff_mult_distrib2 add_commute mult_ac) chaieb@26125: hence ?thesis using H(1,2) chaieb@26125: apply - chaieb@26125: apply (rule exI[where x=d], simp) chaieb@26125: apply (rule exI[where x="(b - 1) * y"]) chaieb@26125: by (rule exI[where x="x*(b - 1) - d"], simp)} chaieb@26125: ultimately have ?thesis by blast} chaieb@26125: ultimately have ?thesis by blast} chaieb@26125: ultimately have ?thesis by blast} chaieb@26125: ultimately show ?thesis by blast chaieb@26125: qed chaieb@26125: wenzelm@26144: text {* Greatest common divisor. *} haftmann@27556: lemma gcd_unique: "d dvd a\d dvd b \ (\e. e dvd a \ e dvd b \ e dvd d) \ d = gcd a b" chaieb@26125: proof(auto) chaieb@26125: assume H: "d dvd a" "d dvd b" "\e. e dvd a \ e dvd b \ e dvd d" chaieb@26125: from H(3)[rule_format] gcd_dvd1[of a b] gcd_dvd2[of a b] haftmann@27556: have th: "gcd a b dvd d" by blast chaieb@27567: from dvd_anti_sym[OF th gcd_greatest[OF H(1,2)]] show "d = gcd a b" by blast chaieb@26125: qed chaieb@26125: chaieb@26125: lemma gcd_eq: assumes H: "\d. d dvd x \ d dvd y \ d dvd u \ d dvd v" haftmann@27556: shows "gcd x y = gcd u v" chaieb@26125: proof- haftmann@27556: from H have "\d. d dvd x \ d dvd y \ d dvd gcd u v" by simp haftmann@27556: with gcd_unique[of "gcd u v" x y] show ?thesis by auto chaieb@26125: qed chaieb@26125: haftmann@27556: lemma bezout_gcd: "\x y. a * x - b * y = gcd a b \ b * x - a * y = gcd a b" chaieb@26125: proof- haftmann@27556: let ?g = "gcd a b" chaieb@26125: from bezout[of a b] obtain d x y where d: "d dvd a" "d dvd b" "a * x - b * y = d \ b * x - a * y = d" by blast chaieb@26125: from d(1,2) have "d dvd ?g" by simp chaieb@26125: then obtain k where k: "?g = d*k" unfolding dvd_def by blast chaieb@26125: from d(3) have "(a * x - b * y)*k = d*k \ (b * x - a * y)*k = d*k" by blast chaieb@26125: hence "a * x * k - b * y*k = d*k \ b * x * k - a * y*k = d*k" chaieb@26125: by (simp only: diff_mult_distrib) chaieb@26125: hence "a * (x * k) - b * (y*k) = ?g \ b * (x * k) - a * (y*k) = ?g" chaieb@26125: by (simp add: k mult_assoc) chaieb@26125: thus ?thesis by blast chaieb@26125: qed chaieb@26125: chaieb@26125: lemma bezout_gcd_strong: assumes a: "a \ 0" haftmann@27556: shows "\x y. a * x = b * y + gcd a b" chaieb@26125: proof- haftmann@27556: let ?g = "gcd a b" chaieb@26125: from bezout_add_strong[OF a, of b] chaieb@26125: obtain d x y where d: "d dvd a" "d dvd b" "a * x = b * y + d" by blast chaieb@26125: from d(1,2) have "d dvd ?g" by simp chaieb@26125: then obtain k where k: "?g = d*k" unfolding dvd_def by blast chaieb@26125: from d(3) have "a * x * k = (b * y + d) *k " by auto chaieb@26125: hence "a * (x * k) = b * (y*k) + ?g" by (algebra add: k) chaieb@26125: thus ?thesis by blast chaieb@26125: qed chaieb@26125: chaieb@27567: lemma gcd_mult_distrib: "gcd(a * c) (b * c) = c * gcd a b" chaieb@26125: by(simp add: gcd_mult_distrib2 mult_commute) chaieb@26125: haftmann@27556: lemma gcd_bezout: "(\x y. a * x - b * y = d \ b * x - a * y = d) \ gcd a b dvd d" chaieb@26125: (is "?lhs \ ?rhs") chaieb@26125: proof- haftmann@27556: let ?g = "gcd a b" chaieb@26125: {assume H: ?rhs then obtain k where k: "d = ?g*k" unfolding dvd_def by blast chaieb@26125: from bezout_gcd[of a b] obtain x y where xy: "a * x - b * y = ?g \ b * x - a * y = ?g" chaieb@26125: by blast chaieb@26125: hence "(a * x - b * y)*k = ?g*k \ (b * x - a * y)*k = ?g*k" by auto chaieb@26125: hence "a * x*k - b * y*k = ?g*k \ b * x * k - a * y*k = ?g*k" chaieb@26125: by (simp only: diff_mult_distrib) chaieb@26125: hence "a * (x*k) - b * (y*k) = d \ b * (x * k) - a * (y*k) = d" chaieb@26125: by (simp add: k[symmetric] mult_assoc) chaieb@26125: hence ?lhs by blast} chaieb@26125: moreover chaieb@26125: {fix x y assume H: "a * x - b * y = d \ b * x - a * y = d" chaieb@26125: have dv: "?g dvd a*x" "?g dvd b * y" "?g dvd b*x" "?g dvd a * y" chaieb@26125: using dvd_mult2[OF gcd_dvd1[of a b]] dvd_mult2[OF gcd_dvd2[of a b]] by simp_all chaieb@26125: from dvd_diff[OF dv(1,2)] dvd_diff[OF dv(3,4)] H chaieb@26125: have ?rhs by auto} chaieb@26125: ultimately show ?thesis by blast chaieb@26125: qed chaieb@26125: haftmann@27556: lemma gcd_bezout_sum: assumes H:"a * x + b * y = d" shows "gcd a b dvd d" chaieb@26125: proof- haftmann@27556: let ?g = "gcd a b" chaieb@26125: have dv: "?g dvd a*x" "?g dvd b * y" chaieb@26125: using dvd_mult2[OF gcd_dvd1[of a b]] dvd_mult2[OF gcd_dvd2[of a b]] by simp_all chaieb@26125: from dvd_add[OF dv] H chaieb@26125: show ?thesis by auto chaieb@26125: qed chaieb@26125: haftmann@27556: lemma gcd_mult': "gcd b (a * b) = b" chaieb@26125: by (simp add: gcd_mult mult_commute[of a b]) chaieb@26125: chaieb@27567: lemma gcd_add: "gcd(a + b) b = gcd a b" chaieb@27567: "gcd(b + a) b = gcd a b" "gcd a (a + b) = gcd a b" "gcd a (b + a) = gcd a b" chaieb@26125: apply (simp_all add: gcd_add1) chaieb@26125: by (simp add: gcd_commute gcd_add1) chaieb@26125: chaieb@27567: lemma gcd_sub: "b <= a ==> gcd(a - b) b = gcd a b" "a <= b ==> gcd a (b - a) = gcd a b" chaieb@26125: proof- chaieb@26125: {fix a b assume H: "b \ (a::nat)" chaieb@26125: hence th: "a - b + b = a" by arith chaieb@27567: from gcd_add(1)[of "a - b" b] th have "gcd(a - b) b = gcd a b" by simp} chaieb@26125: note th = this chaieb@26125: { chaieb@26125: assume ab: "b \ a" chaieb@27567: from th[OF ab] show "gcd (a - b) b = gcd a b" by blast chaieb@26125: next chaieb@26125: assume ab: "a \ b" haftmann@27556: from th[OF ab] show "gcd a (b - a) = gcd a b" chaieb@26125: by (simp add: gcd_commute)} chaieb@26125: qed chaieb@26125: wenzelm@26144: text {* Coprimality *} chaieb@26125: chaieb@26125: lemma coprime: "coprime a b \ (\d. d dvd a \ d dvd b \ d = 1)" chaieb@26125: using gcd_unique[of 1 a b, simplified] by (auto simp add: coprime_def) chaieb@26125: lemma coprime_commute: "coprime a b \ coprime b a" by (simp add: coprime_def gcd_commute) chaieb@26125: chaieb@26125: lemma coprime_bezout: "coprime a b \ (\x y. a * x - b * y = 1 \ b * x - a * y = 1)" chaieb@26125: using coprime_def gcd_bezout by auto chaieb@26125: chaieb@26125: lemma coprime_divprod: "d dvd a * b \ coprime d a \ d dvd b" chaieb@26125: using relprime_dvd_mult_iff[of d a b] by (auto simp add: coprime_def mult_commute) chaieb@26125: chaieb@26125: lemma coprime_1[simp]: "coprime a 1" by (simp add: coprime_def) chaieb@26125: lemma coprime_1'[simp]: "coprime 1 a" by (simp add: coprime_def) chaieb@26125: lemma coprime_Suc0[simp]: "coprime a (Suc 0)" by (simp add: coprime_def) chaieb@26125: lemma coprime_Suc0'[simp]: "coprime (Suc 0) a" by (simp add: coprime_def) chaieb@26125: chaieb@26125: lemma gcd_coprime: haftmann@27556: assumes z: "gcd a b \ 0" and a: "a = a' * gcd a b" and b: "b = b' * gcd a b" chaieb@26125: shows "coprime a' b'" chaieb@26125: proof- haftmann@27556: let ?g = "gcd a b" chaieb@26125: {assume bz: "a = 0" from b bz z a have ?thesis by (simp add: gcd_zero coprime_def)} chaieb@26125: moreover chaieb@26125: {assume az: "a\ 0" chaieb@26125: from z have z': "?g > 0" by simp chaieb@26125: from bezout_gcd_strong[OF az, of b] chaieb@26125: obtain x y where xy: "a*x = b*y + ?g" by blast chaieb@26125: from xy a b have "?g * a'*x = ?g * (b'*y + 1)" by (simp add: ring_simps) chaieb@26125: hence "?g * (a'*x) = ?g * (b'*y + 1)" by (simp add: mult_assoc) chaieb@26125: hence "a'*x = (b'*y + 1)" chaieb@26125: by (simp only: nat_mult_eq_cancel1[OF z']) chaieb@26125: hence "a'*x - b'*y = 1" by simp chaieb@26125: with coprime_bezout[of a' b'] have ?thesis by auto} chaieb@26125: ultimately show ?thesis by blast chaieb@26125: qed chaieb@26125: lemma coprime_0: "coprime d 0 \ d = 1" by (simp add: coprime_def) chaieb@26125: lemma coprime_mul: assumes da: "coprime d a" and db: "coprime d b" chaieb@26125: shows "coprime d (a * b)" chaieb@26125: proof- haftmann@27556: from da have th: "gcd a d = 1" by (simp add: coprime_def gcd_commute) chaieb@27567: from gcd_mult_cancel[of a d b, OF th] db[unfolded coprime_def] have "gcd d (a*b) = 1" chaieb@26125: by (simp add: gcd_commute) chaieb@26125: thus ?thesis unfolding coprime_def . chaieb@26125: qed chaieb@26125: lemma coprime_lmul2: assumes dab: "coprime d (a * b)" shows "coprime d b" chaieb@26125: using prems unfolding coprime_bezout chaieb@26125: apply clarsimp chaieb@26125: apply (case_tac "d * x - a * b * y = Suc 0 ", simp_all) chaieb@26125: apply (rule_tac x="x" in exI) chaieb@26125: apply (rule_tac x="a*y" in exI) chaieb@26125: apply (simp add: mult_ac) chaieb@26125: apply (rule_tac x="a*x" in exI) chaieb@26125: apply (rule_tac x="y" in exI) chaieb@26125: apply (simp add: mult_ac) chaieb@26125: done chaieb@26125: chaieb@26125: lemma coprime_rmul2: "coprime d (a * b) \ coprime d a" chaieb@26125: unfolding coprime_bezout chaieb@26125: apply clarsimp chaieb@26125: apply (case_tac "d * x - a * b * y = Suc 0 ", simp_all) chaieb@26125: apply (rule_tac x="x" in exI) chaieb@26125: apply (rule_tac x="b*y" in exI) chaieb@26125: apply (simp add: mult_ac) chaieb@26125: apply (rule_tac x="b*x" in exI) chaieb@26125: apply (rule_tac x="y" in exI) chaieb@26125: apply (simp add: mult_ac) chaieb@26125: done chaieb@26125: lemma coprime_mul_eq: "coprime d (a * b) \ coprime d a \ coprime d b" chaieb@26125: using coprime_rmul2[of d a b] coprime_lmul2[of d a b] coprime_mul[of d a b] chaieb@26125: by blast chaieb@26125: chaieb@26125: lemma gcd_coprime_exists: haftmann@27556: assumes nz: "gcd a b \ 0" haftmann@27556: shows "\a' b'. a = a' * gcd a b \ b = b' * gcd a b \ coprime a' b'" chaieb@26125: proof- haftmann@27556: let ?g = "gcd a b" chaieb@26125: from gcd_dvd1[of a b] gcd_dvd2[of a b] chaieb@26125: obtain a' b' where "a = ?g*a'" "b = ?g*b'" unfolding dvd_def by blast chaieb@26125: hence ab': "a = a'*?g" "b = b'*?g" by algebra+ chaieb@26125: from ab' gcd_coprime[OF nz ab'] show ?thesis by blast chaieb@26125: qed chaieb@26125: chaieb@26125: lemma coprime_exp: "coprime d a ==> coprime d (a^n)" chaieb@26125: by(induct n, simp_all add: coprime_mul) chaieb@26125: chaieb@26125: lemma coprime_exp_imp: "coprime a b ==> coprime (a ^n) (b ^n)" chaieb@26125: by (induct n, simp_all add: coprime_mul_eq coprime_commute coprime_exp) chaieb@26125: lemma coprime_refl[simp]: "coprime n n \ n = 1" by (simp add: coprime_def) chaieb@26125: lemma coprime_plus1[simp]: "coprime (n + 1) n" chaieb@26125: apply (simp add: coprime_bezout) chaieb@26125: apply (rule exI[where x=1]) chaieb@26125: apply (rule exI[where x=1]) chaieb@26125: apply simp chaieb@26125: done chaieb@26125: lemma coprime_minus1: "n \ 0 ==> coprime (n - 1) n" chaieb@26125: using coprime_plus1[of "n - 1"] coprime_commute[of "n - 1" n] by auto chaieb@26125: haftmann@27556: lemma bezout_gcd_pow: "\x y. a ^n * x - b ^ n * y = gcd a b ^ n \ b ^ n * x - a ^ n * y = gcd a b ^ n" chaieb@26125: proof- haftmann@27556: let ?g = "gcd a b" chaieb@26125: {assume z: "?g = 0" hence ?thesis chaieb@26125: apply (cases n, simp) chaieb@26125: apply arith chaieb@26125: apply (simp only: z power_0_Suc) chaieb@26125: apply (rule exI[where x=0]) chaieb@26125: apply (rule exI[where x=0]) chaieb@26125: by simp} chaieb@26125: moreover chaieb@26125: {assume z: "?g \ 0" chaieb@26125: from gcd_dvd1[of a b] gcd_dvd2[of a b] obtain a' b' where chaieb@26125: ab': "a = a'*?g" "b = b'*?g" unfolding dvd_def by (auto simp add: mult_ac) chaieb@26125: hence ab'': "?g*a' = a" "?g * b' = b" by algebra+ chaieb@26125: from coprime_exp_imp[OF gcd_coprime[OF z ab'], unfolded coprime_bezout, of n] chaieb@26125: obtain x y where "a'^n * x - b'^n * y = 1 \ b'^n * x - a'^n * y = 1" by blast chaieb@26125: hence "?g^n * (a'^n * x - b'^n * y) = ?g^n \ ?g^n*(b'^n * x - a'^n * y) = ?g^n" chaieb@26125: using z by auto chaieb@26125: then have "a^n * x - b^n * y = ?g^n \ b^n * x - a^n * y = ?g^n" chaieb@26125: using z ab'' by (simp only: power_mult_distrib[symmetric] chaieb@26125: diff_mult_distrib2 mult_assoc[symmetric]) chaieb@26125: hence ?thesis by blast } chaieb@26125: ultimately show ?thesis by blast chaieb@26125: qed chaieb@27567: chaieb@27567: lemma gcd_exp: "gcd (a^n) (b^n) = gcd a b^n" chaieb@26125: proof- haftmann@27556: let ?g = "gcd (a^n) (b^n)" chaieb@27567: let ?gn = "gcd a b^n" chaieb@26125: {fix e assume H: "e dvd a^n" "e dvd b^n" chaieb@26125: from bezout_gcd_pow[of a n b] obtain x y chaieb@26125: where xy: "a ^ n * x - b ^ n * y = ?gn \ b ^ n * x - a ^ n * y = ?gn" by blast chaieb@26125: from dvd_diff [OF dvd_mult2[OF H(1), of x] dvd_mult2[OF H(2), of y]] chaieb@26125: dvd_diff [OF dvd_mult2[OF H(2), of x] dvd_mult2[OF H(1), of y]] xy haftmann@27556: have "e dvd ?gn" by (cases "a ^ n * x - b ^ n * y = gcd a b ^ n", simp_all)} chaieb@26125: hence th: "\e. e dvd a^n \ e dvd b^n \ e dvd ?gn" by blast chaieb@26125: from divides_exp[OF gcd_dvd1[of a b], of n] divides_exp[OF gcd_dvd2[of a b], of n] th chaieb@26125: gcd_unique have "?gn = ?g" by blast thus ?thesis by simp chaieb@26125: qed chaieb@26125: chaieb@26125: lemma coprime_exp2: "coprime (a ^ Suc n) (b^ Suc n) \ coprime a b" chaieb@26125: by (simp only: coprime_def gcd_exp exp_eq_1) simp chaieb@26125: chaieb@26125: lemma division_decomp: assumes dc: "(a::nat) dvd b * c" chaieb@26125: shows "\b' c'. a = b' * c' \ b' dvd b \ c' dvd c" chaieb@26125: proof- haftmann@27556: let ?g = "gcd a b" chaieb@26125: {assume "?g = 0" with dc have ?thesis apply (simp add: gcd_zero) chaieb@26125: apply (rule exI[where x="0"]) chaieb@26125: by (rule exI[where x="c"], simp)} chaieb@26125: moreover chaieb@26125: {assume z: "?g \ 0" chaieb@26125: from gcd_coprime_exists[OF z] chaieb@26125: obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'" by blast chaieb@26125: from gcd_dvd2[of a b] have thb: "?g dvd b" . chaieb@26125: from ab'(1) have "a' dvd a" unfolding dvd_def by blast chaieb@26125: with dc have th0: "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp chaieb@26125: from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto chaieb@26125: hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult_assoc) chaieb@26125: with z have th_1: "a' dvd b'*c" by simp chaieb@26125: from coprime_divprod[OF th_1 ab'(3)] have thc: "a' dvd c" . chaieb@26125: from ab' have "a = ?g*a'" by algebra chaieb@26125: with thb thc have ?thesis by blast } chaieb@26125: ultimately show ?thesis by blast chaieb@26125: qed chaieb@26125: chaieb@26125: lemma nat_power_eq_0_iff: "(m::nat) ^ n = 0 \ n \ 0 \ m = 0" by (induct n, auto) chaieb@26125: chaieb@26125: lemma divides_rev: assumes ab: "(a::nat) ^ n dvd b ^n" and n:"n \ 0" shows "a dvd b" chaieb@26125: proof- haftmann@27556: let ?g = "gcd a b" chaieb@26125: from n obtain m where m: "n = Suc m" by (cases n, simp_all) chaieb@26125: {assume "?g = 0" with ab n have ?thesis by (simp add: gcd_zero)} chaieb@26125: moreover chaieb@26125: {assume z: "?g \ 0" chaieb@26125: hence zn: "?g ^ n \ 0" using n by (simp add: neq0_conv) chaieb@26125: from gcd_coprime_exists[OF z] chaieb@26125: obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'" by blast chaieb@26125: from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n" by (simp add: ab'(1,2)[symmetric]) chaieb@26125: hence "?g^n*a'^n dvd ?g^n *b'^n" by (simp only: power_mult_distrib mult_commute) chaieb@26125: with zn z n have th0:"a'^n dvd b'^n" by (auto simp add: nat_power_eq_0_iff) chaieb@26125: have "a' dvd a'^n" by (simp add: m) chaieb@26125: with th0 have "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by simp chaieb@26125: hence th1: "a' dvd b'^m * b'" by (simp add: m mult_commute) chaieb@26125: from coprime_divprod[OF th1 coprime_exp[OF ab'(3), of m]] chaieb@26125: have "a' dvd b'" . chaieb@26125: hence "a'*?g dvd b'*?g" by simp chaieb@26125: with ab'(1,2) have ?thesis by simp } chaieb@26125: ultimately show ?thesis by blast chaieb@26125: qed chaieb@26125: chaieb@26125: lemma divides_mul: assumes mr: "m dvd r" and nr: "n dvd r" and mn:"coprime m n" chaieb@26125: shows "m * n dvd r" chaieb@26125: proof- chaieb@26125: from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'" chaieb@26125: unfolding dvd_def by blast chaieb@26125: from mr n' have "m dvd n'*n" by (simp add: mult_commute) chaieb@26125: hence "m dvd n'" using relprime_dvd_mult_iff[OF mn[unfolded coprime_def]] by simp chaieb@26125: then obtain k where k: "n' = m*k" unfolding dvd_def by blast chaieb@26125: from n' k show ?thesis unfolding dvd_def by auto chaieb@26125: qed chaieb@26125: wenzelm@26144: wenzelm@26144: text {* A binary form of the Chinese Remainder Theorem. *} chaieb@26125: chaieb@26125: lemma chinese_remainder: assumes ab: "coprime a b" and a:"a \ 0" and b:"b \ 0" chaieb@26125: shows "\x q1 q2. x = u + q1 * a \ x = v + q2 * b" chaieb@26125: proof- chaieb@26125: from bezout_add_strong[OF a, of b] bezout_add_strong[OF b, of a] chaieb@26125: obtain d1 x1 y1 d2 x2 y2 where dxy1: "d1 dvd a" "d1 dvd b" "a * x1 = b * y1 + d1" chaieb@26125: and dxy2: "d2 dvd b" "d2 dvd a" "b * x2 = a * y2 + d2" by blast chaieb@26125: from gcd_unique[of 1 a b, simplified ab[unfolded coprime_def], simplified] chaieb@26125: dxy1(1,2) dxy2(1,2) have d12: "d1 = 1" "d2 =1" by auto chaieb@26125: let ?x = "v * a * x1 + u * b * x2" chaieb@26125: let ?q1 = "v * x1 + u * y2" chaieb@26125: let ?q2 = "v * y1 + u * x2" chaieb@26125: from dxy2(3)[simplified d12] dxy1(3)[simplified d12] chaieb@26125: have "?x = u + ?q1 * a" "?x = v + ?q2 * b" by algebra+ chaieb@26125: thus ?thesis by blast chaieb@26125: qed chaieb@26125: wenzelm@26144: text {* Primality *} wenzelm@26144: wenzelm@26144: text {* A few useful theorems about primes *} chaieb@26125: chaieb@26125: lemma prime_0[simp]: "~prime 0" by (simp add: prime_def) chaieb@26125: lemma prime_1[simp]: "~ prime 1" by (simp add: prime_def) chaieb@26125: lemma prime_Suc0[simp]: "~ prime (Suc 0)" by (simp add: prime_def) chaieb@26125: chaieb@26125: lemma prime_ge_2: "prime p ==> p \ 2" by (simp add: prime_def) chaieb@26125: lemma prime_factor: assumes n: "n \ 1" shows "\ p. prime p \ p dvd n" chaieb@26125: using n chaieb@26125: proof(induct n rule: nat_less_induct) chaieb@26125: fix n chaieb@26125: assume H: "\m 1 \ (\p. prime p \ p dvd m)" "n \ 1" chaieb@26125: let ?ths = "\p. prime p \ p dvd n" chaieb@26125: {assume "n=0" hence ?ths using two_is_prime by auto} chaieb@26125: moreover chaieb@26125: {assume nz: "n\0" chaieb@26125: {assume "prime n" hence ?ths by - (rule exI[where x="n"], simp)} chaieb@26125: moreover chaieb@26125: {assume n: "\ prime n" chaieb@26125: with nz H(2) chaieb@26125: obtain k where k:"k dvd n" "k \ 1" "k \ n" by (auto simp add: prime_def) chaieb@26125: from dvd_imp_le[OF k(1)] nz k(3) have kn: "k < n" by simp chaieb@26125: from H(1)[rule_format, OF kn k(2)] obtain p where p: "prime p" "p dvd k" by blast chaieb@26125: from dvd_trans[OF p(2) k(1)] p(1) have ?ths by blast} chaieb@26125: ultimately have ?ths by blast} chaieb@26125: ultimately show ?ths by blast chaieb@26125: qed chaieb@26125: chaieb@26125: lemma prime_factor_lt: assumes p: "prime p" and n: "n \ 0" and npm:"n = p * m" chaieb@26125: shows "m < n" chaieb@26125: proof- chaieb@26125: {assume "m=0" with n have ?thesis by simp} chaieb@26125: moreover chaieb@26125: {assume m: "m \ 0" chaieb@26125: from npm have mn: "m dvd n" unfolding dvd_def by auto chaieb@26125: from npm m have "n \ m" using p by auto chaieb@26125: with dvd_imp_le[OF mn] n have ?thesis by simp} chaieb@26125: ultimately show ?thesis by blast chaieb@26125: qed chaieb@26125: chaieb@26125: lemma euclid_bound: "\p. prime p \ n < p \ p <= Suc (fact n)" chaieb@26125: proof- chaieb@26125: have f1: "fact n + 1 \ 1" using fact_le[of n] by arith chaieb@26125: from prime_factor[OF f1] obtain p where p: "prime p" "p dvd fact n + 1" by blast chaieb@26125: from dvd_imp_le[OF p(2)] have pfn: "p \ fact n + 1" by simp chaieb@26125: {assume np: "p \ n" chaieb@26125: from p(1) have p1: "p \ 1" by (cases p, simp_all) chaieb@26125: from divides_fact[OF p1 np] have pfn': "p dvd fact n" . chaieb@26125: from divides_add_revr[OF pfn' p(2)] p(1) have False by simp} chaieb@26125: hence "n < p" by arith chaieb@26125: with p(1) pfn show ?thesis by auto chaieb@26125: qed chaieb@26125: chaieb@26125: lemma euclid: "\p. prime p \ p > n" using euclid_bound by auto chaieb@26125: lemma primes_infinite: "\ (finite {p. prime p})" chaieb@26125: proof (auto simp add: finite_conv_nat_seg_image) chaieb@26125: fix n f chaieb@26125: assume H: "Collect prime = f ` {i. i < (n::nat)}" chaieb@26125: let ?P = "Collect prime" chaieb@26125: let ?m = "Max ?P" chaieb@26125: have P0: "?P \ {}" using two_is_prime by auto chaieb@26125: from H have fP: "finite ?P" using finite_conv_nat_seg_image by blast haftmann@26757: from Max_in [OF fP P0] have "?m \ ?P" . haftmann@26757: from Max_ge [OF fP] have contr: "\ p. prime p \ p \ ?m" by blast haftmann@26757: from euclid [of ?m] obtain q where q: "prime q" "q > ?m" by blast chaieb@26125: with contr show False by auto chaieb@26125: qed chaieb@26125: chaieb@26125: lemma coprime_prime: assumes ab: "coprime a b" chaieb@26125: shows "~(prime p \ p dvd a \ p dvd b)" chaieb@26125: proof chaieb@26125: assume "prime p \ p dvd a \ p dvd b" chaieb@26125: thus False using ab gcd_greatest[of p a b] by (simp add: coprime_def) chaieb@26125: qed chaieb@26125: lemma coprime_prime_eq: "coprime a b \ (\p. ~(prime p \ p dvd a \ p dvd b))" chaieb@26125: (is "?lhs = ?rhs") chaieb@26125: proof- chaieb@26125: {assume "?lhs" with coprime_prime have ?rhs by blast} chaieb@26125: moreover chaieb@26125: {assume r: "?rhs" and c: "\ ?lhs" chaieb@26125: then obtain g where g: "g\1" "g dvd a" "g dvd b" unfolding coprime_def by blast chaieb@26125: from prime_factor[OF g(1)] obtain p where p: "prime p" "p dvd g" by blast chaieb@26125: from dvd_trans [OF p(2) g(2)] dvd_trans [OF p(2) g(3)] chaieb@26125: have "p dvd a" "p dvd b" . with p(1) r have False by blast} chaieb@26125: ultimately show ?thesis by blast chaieb@26125: qed chaieb@26125: chaieb@26125: lemma prime_coprime: assumes p: "prime p" chaieb@26125: shows "n = 1 \ p dvd n \ coprime p n" chaieb@26125: using p prime_imp_relprime[of p n] by (auto simp add: coprime_def) chaieb@26125: chaieb@26125: lemma prime_coprime_strong: "prime p \ p dvd n \ coprime p n" chaieb@26125: using prime_coprime[of p n] by auto chaieb@26125: chaieb@26125: declare coprime_0[simp] chaieb@26125: chaieb@26125: lemma coprime_0'[simp]: "coprime 0 d \ d = 1" by (simp add: coprime_commute[of 0 d]) chaieb@26125: lemma coprime_bezout_strong: assumes ab: "coprime a b" and b: "b \ 1" chaieb@26125: shows "\x y. a * x = b * y + 1" chaieb@26125: proof- chaieb@26125: from ab b have az: "a \ 0" by - (rule ccontr, auto) chaieb@26125: from bezout_gcd_strong[OF az, of b] ab[unfolded coprime_def] chaieb@26125: show ?thesis by auto chaieb@26125: qed chaieb@26125: chaieb@26125: lemma bezout_prime: assumes p: "prime p" and pa: "\ p dvd a" chaieb@26125: shows "\x y. a*x = p*y + 1" chaieb@26125: proof- chaieb@26125: from p have p1: "p \ 1" using prime_1 by blast chaieb@26125: from prime_coprime[OF p, of a] p1 pa have ap: "coprime a p" chaieb@26125: by (auto simp add: coprime_commute) chaieb@26125: from coprime_bezout_strong[OF ap p1] show ?thesis . chaieb@26125: qed chaieb@26125: lemma prime_divprod: assumes p: "prime p" and pab: "p dvd a*b" chaieb@26125: shows "p dvd a \ p dvd b" chaieb@26125: proof- chaieb@26125: {assume "a=1" hence ?thesis using pab by simp } chaieb@26125: moreover chaieb@26125: {assume "p dvd a" hence ?thesis by blast} chaieb@26125: moreover chaieb@26125: {assume pa: "coprime p a" from coprime_divprod[OF pab pa] have ?thesis .. } chaieb@26125: ultimately show ?thesis using prime_coprime[OF p, of a] by blast chaieb@26125: qed chaieb@26125: chaieb@26125: lemma prime_divprod_eq: assumes p: "prime p" chaieb@26125: shows "p dvd a*b \ p dvd a \ p dvd b" chaieb@26125: using p prime_divprod dvd_mult dvd_mult2 by auto chaieb@26125: chaieb@26125: lemma prime_divexp: assumes p:"prime p" and px: "p dvd x^n" chaieb@26125: shows "p dvd x" chaieb@26125: using px chaieb@26125: proof(induct n) chaieb@26125: case 0 thus ?case by simp chaieb@26125: next chaieb@26125: case (Suc n) chaieb@26125: hence th: "p dvd x*x^n" by simp chaieb@26125: {assume H: "p dvd x^n" chaieb@26125: from Suc.hyps[OF H] have ?case .} chaieb@26125: with prime_divprod[OF p th] show ?case by blast chaieb@26125: qed chaieb@26125: chaieb@26125: lemma prime_divexp_n: "prime p \ p dvd x^n \ p^n dvd x^n" chaieb@26125: using prime_divexp[of p x n] divides_exp[of p x n] by blast chaieb@26125: chaieb@26125: lemma coprime_prime_dvd_ex: assumes xy: "\coprime x y" chaieb@26125: shows "\p. prime p \ p dvd x \ p dvd y" chaieb@26125: proof- chaieb@26125: from xy[unfolded coprime_def] obtain g where g: "g \ 1" "g dvd x" "g dvd y" chaieb@26125: by blast chaieb@26125: from prime_factor[OF g(1)] obtain p where p: "prime p" "p dvd g" by blast chaieb@26125: from g(2,3) dvd_trans[OF p(2)] p(1) show ?thesis by auto chaieb@26125: qed chaieb@26125: lemma coprime_sos: assumes xy: "coprime x y" chaieb@26125: shows "coprime (x * y) (x^2 + y^2)" chaieb@26125: proof- chaieb@26125: {assume c: "\ coprime (x * y) (x^2 + y^2)" chaieb@26125: from coprime_prime_dvd_ex[OF c] obtain p chaieb@26125: where p: "prime p" "p dvd x*y" "p dvd x^2 + y^2" by blast chaieb@26125: {assume px: "p dvd x" haftmann@27651: from dvd_mult[OF px, of x] p(3) haftmann@27651: obtain r s where "x * x = p * r" and "x^2 + y^2 = p * s" haftmann@27651: by (auto elim!: dvdE) haftmann@27651: then have "y^2 = p * (s - r)" haftmann@27651: by (auto simp add: power2_eq_square diff_mult_distrib2) haftmann@27651: then have "p dvd y^2" .. chaieb@26125: with prime_divexp[OF p(1), of y 2] have py: "p dvd y" . chaieb@26125: from p(1) px py xy[unfolded coprime, rule_format, of p] prime_1 chaieb@26125: have False by simp } chaieb@26125: moreover chaieb@26125: {assume py: "p dvd y" haftmann@27651: from dvd_mult[OF py, of y] p(3) haftmann@27651: obtain r s where "y * y = p * r" and "x^2 + y^2 = p * s" haftmann@27651: by (auto elim!: dvdE) haftmann@27651: then have "x^2 = p * (s - r)" haftmann@27651: by (auto simp add: power2_eq_square diff_mult_distrib2) haftmann@27651: then have "p dvd x^2" .. chaieb@26125: with prime_divexp[OF p(1), of x 2] have px: "p dvd x" . chaieb@26125: from p(1) px py xy[unfolded coprime, rule_format, of p] prime_1 chaieb@26125: have False by simp } chaieb@26125: ultimately have False using prime_divprod[OF p(1,2)] by blast} chaieb@26125: thus ?thesis by blast chaieb@26125: qed chaieb@26125: chaieb@26125: lemma distinct_prime_coprime: "prime p \ prime q \ p \ q \ coprime p q" chaieb@26125: unfolding prime_def coprime_prime_eq by blast chaieb@26125: chaieb@26125: lemma prime_coprime_lt: assumes p: "prime p" and x: "0 < x" and xp: "x < p" chaieb@26125: shows "coprime x p" chaieb@26125: proof- chaieb@26125: {assume c: "\ coprime x p" chaieb@26125: then obtain g where g: "g \ 1" "g dvd x" "g dvd p" unfolding coprime_def by blast chaieb@26125: from dvd_imp_le[OF g(2)] x xp have gp: "g < p" by arith chaieb@26125: from g(2) x have "g \ 0" by - (rule ccontr, simp) chaieb@26125: with g gp p[unfolded prime_def] have False by blast} chaieb@26125: thus ?thesis by blast chaieb@26125: qed chaieb@26125: chaieb@26125: lemma even_dvd[simp]: "even (n::nat) \ 2 dvd n" by presburger chaieb@26125: lemma prime_odd: "prime p \ p = 2 \ odd p" unfolding prime_def by auto chaieb@26125: wenzelm@26144: wenzelm@26144: text {* One property of coprimality is easier to prove via prime factors. *} chaieb@26125: chaieb@26125: lemma prime_divprod_pow: chaieb@26125: assumes p: "prime p" and ab: "coprime a b" and pab: "p^n dvd a * b" chaieb@26125: shows "p^n dvd a \ p^n dvd b" chaieb@26125: proof- chaieb@26125: {assume "n = 0 \ a = 1 \ b = 1" with pab have ?thesis chaieb@26125: apply (cases "n=0", simp_all) chaieb@26125: apply (cases "a=1", simp_all) done} chaieb@26125: moreover chaieb@26125: {assume n: "n \ 0" and a: "a\1" and b: "b\1" chaieb@26125: then obtain m where m: "n = Suc m" by (cases n, auto) chaieb@26125: from divides_exp2[OF n pab] have pab': "p dvd a*b" . chaieb@26125: from prime_divprod[OF p pab'] chaieb@26125: have "p dvd a \ p dvd b" . chaieb@26125: moreover chaieb@26125: {assume pa: "p dvd a" chaieb@26125: have pnba: "p^n dvd b*a" using pab by (simp add: mult_commute) chaieb@26125: from coprime_prime[OF ab, of p] p pa have "\ p dvd b" by blast chaieb@26125: with prime_coprime[OF p, of b] b chaieb@26125: have cpb: "coprime b p" using coprime_commute by blast chaieb@26125: from coprime_exp[OF cpb] have pnb: "coprime (p^n) b" chaieb@26125: by (simp add: coprime_commute) chaieb@26125: from coprime_divprod[OF pnba pnb] have ?thesis by blast } chaieb@26125: moreover chaieb@26125: {assume pb: "p dvd b" chaieb@26125: have pnba: "p^n dvd b*a" using pab by (simp add: mult_commute) chaieb@26125: from coprime_prime[OF ab, of p] p pb have "\ p dvd a" by blast chaieb@26125: with prime_coprime[OF p, of a] a chaieb@26125: have cpb: "coprime a p" using coprime_commute by blast chaieb@26125: from coprime_exp[OF cpb] have pnb: "coprime (p^n) a" chaieb@26125: by (simp add: coprime_commute) chaieb@26125: from coprime_divprod[OF pab pnb] have ?thesis by blast } chaieb@26125: ultimately have ?thesis by blast} chaieb@26125: ultimately show ?thesis by blast chaieb@26125: qed chaieb@26125: chaieb@26125: lemma nat_mult_eq_one: "(n::nat) * m = 1 \ n = 1 \ m = 1" (is "?lhs \ ?rhs") chaieb@26125: proof chaieb@26125: assume H: "?lhs" chaieb@26125: hence "n dvd 1" "m dvd 1" unfolding dvd_def by (auto simp add: mult_commute) chaieb@26125: thus ?rhs by auto chaieb@26125: next chaieb@26125: assume ?rhs then show ?lhs by auto chaieb@26125: qed chaieb@26125: chaieb@26125: lemma power_Suc0[simp]: "Suc 0 ^ n = Suc 0" chaieb@26125: unfolding One_nat_def[symmetric] power_one .. chaieb@26125: lemma coprime_pow: assumes ab: "coprime a b" and abcn: "a * b = c ^n" chaieb@26125: shows "\r s. a = r^n \ b = s ^n" chaieb@26125: using ab abcn chaieb@26125: proof(induct c arbitrary: a b rule: nat_less_induct) chaieb@26125: fix c a b chaieb@26125: assume H: "\ma b. coprime a b \ a * b = m ^ n \ (\r s. a = r ^ n \ b = s ^ n)" "coprime a b" "a * b = c ^ n" chaieb@26125: let ?ths = "\r s. a = r^n \ b = s ^n" chaieb@26125: {assume n: "n = 0" chaieb@26125: with H(3) power_one have "a*b = 1" by simp chaieb@26125: hence "a = 1 \ b = 1" by simp chaieb@26125: hence ?ths chaieb@26125: apply - chaieb@26125: apply (rule exI[where x=1]) chaieb@26125: apply (rule exI[where x=1]) chaieb@26125: using power_one[of n] chaieb@26125: by simp} chaieb@26125: moreover chaieb@26125: {assume n: "n \ 0" then obtain m where m: "n = Suc m" by (cases n, auto) chaieb@26125: {assume c: "c = 0" chaieb@26125: with H(3) m H(2) have ?ths apply simp chaieb@26125: apply (cases "a=0", simp_all) chaieb@26125: apply (rule exI[where x="0"], simp) chaieb@26125: apply (rule exI[where x="0"], simp) chaieb@26125: done} chaieb@26125: moreover chaieb@26125: {assume "c=1" with H(3) power_one have "a*b = 1" by simp chaieb@26125: hence "a = 1 \ b = 1" by simp chaieb@26125: hence ?ths chaieb@26125: apply - chaieb@26125: apply (rule exI[where x=1]) chaieb@26125: apply (rule exI[where x=1]) chaieb@26125: using power_one[of n] chaieb@26125: by simp} chaieb@26125: moreover chaieb@26125: {assume c: "c\1" "c \ 0" chaieb@26125: from prime_factor[OF c(1)] obtain p where p: "prime p" "p dvd c" by blast chaieb@26125: from prime_divprod_pow[OF p(1) H(2), unfolded H(3), OF divides_exp[OF p(2), of n]] chaieb@26125: have pnab: "p ^ n dvd a \ p^n dvd b" . chaieb@26125: from p(2) obtain l where l: "c = p*l" unfolding dvd_def by blast chaieb@26125: have pn0: "p^n \ 0" using n prime_ge_2 [OF p(1)] by (simp add: neq0_conv) chaieb@26125: {assume pa: "p^n dvd a" chaieb@26125: then obtain k where k: "a = p^n * k" unfolding dvd_def by blast chaieb@26125: from l have "l dvd c" by auto chaieb@26125: with dvd_imp_le[of l c] c have "l \ c" by auto chaieb@26125: moreover {assume "l = c" with l c have "p = 1" by simp with p have False by simp} chaieb@26125: ultimately have lc: "l < c" by arith chaieb@26125: from coprime_lmul2 [OF H(2)[unfolded k coprime_commute[of "p^n*k" b]]] chaieb@26125: have kb: "coprime k b" by (simp add: coprime_commute) chaieb@26125: from H(3) l k pn0 have kbln: "k * b = l ^ n" chaieb@26125: by (auto simp add: power_mult_distrib) chaieb@26125: from H(1)[rule_format, OF lc kb kbln] chaieb@26125: obtain r s where rs: "k = r ^n" "b = s^n" by blast chaieb@26125: from k rs(1) have "a = (p*r)^n" by (simp add: power_mult_distrib) chaieb@26125: with rs(2) have ?ths by blast } chaieb@26125: moreover chaieb@26125: {assume pb: "p^n dvd b" chaieb@26125: then obtain k where k: "b = p^n * k" unfolding dvd_def by blast chaieb@26125: from l have "l dvd c" by auto chaieb@26125: with dvd_imp_le[of l c] c have "l \ c" by auto chaieb@26125: moreover {assume "l = c" with l c have "p = 1" by simp with p have False by simp} chaieb@26125: ultimately have lc: "l < c" by arith chaieb@26125: from coprime_lmul2 [OF H(2)[unfolded k coprime_commute[of "p^n*k" a]]] chaieb@26125: have kb: "coprime k a" by (simp add: coprime_commute) chaieb@26125: from H(3) l k pn0 n have kbln: "k * a = l ^ n" chaieb@26125: by (simp add: power_mult_distrib mult_commute) chaieb@26125: from H(1)[rule_format, OF lc kb kbln] chaieb@26125: obtain r s where rs: "k = r ^n" "a = s^n" by blast chaieb@26125: from k rs(1) have "b = (p*r)^n" by (simp add: power_mult_distrib) chaieb@26125: with rs(2) have ?ths by blast } chaieb@26125: ultimately have ?ths using pnab by blast} chaieb@26125: ultimately have ?ths by blast} chaieb@26125: ultimately show ?ths by blast chaieb@26125: qed chaieb@26125: wenzelm@26144: text {* More useful lemmas. *} chaieb@26125: lemma prime_product: haftmann@27651: assumes "prime (p * q)" haftmann@27651: shows "p = 1 \ q = 1" haftmann@27651: proof - haftmann@27651: from assms have haftmann@27651: "1 < p * q" and P: "\m. m dvd p * q \ m = 1 \ m = p * q" haftmann@27651: unfolding prime_def by auto haftmann@27651: from `1 < p * q` have "p \ 0" by (cases p) auto haftmann@27651: then have Q: "p = p * q \ q = 1" by auto haftmann@27651: have "p dvd p * q" by simp haftmann@27651: then have "p = 1 \ p = p * q" by (rule P) haftmann@27651: then show ?thesis by (simp add: Q) haftmann@27651: qed chaieb@26125: chaieb@26125: lemma prime_exp: "prime (p^n) \ prime p \ n = 1" chaieb@26125: proof(induct n) chaieb@26125: case 0 thus ?case by simp chaieb@26125: next chaieb@26125: case (Suc n) chaieb@26125: {assume "p = 0" hence ?case by simp} chaieb@26125: moreover chaieb@26125: {assume "p=1" hence ?case by simp} chaieb@26125: moreover chaieb@26125: {assume p: "p \ 0" "p\1" chaieb@26125: {assume pp: "prime (p^Suc n)" chaieb@26125: hence "p = 1 \ p^n = 1" using prime_product[of p "p^n"] by simp chaieb@26125: with p have n: "n = 0" chaieb@26125: by (simp only: exp_eq_1 ) simp chaieb@26125: with pp have "prime p \ Suc n = 1" by simp} chaieb@26125: moreover chaieb@26125: {assume n: "prime p \ Suc n = 1" hence "prime (p^Suc n)" by simp} chaieb@26125: ultimately have ?case by blast} chaieb@26125: ultimately show ?case by blast chaieb@26125: qed chaieb@26125: chaieb@26125: lemma prime_power_mult: chaieb@26125: assumes p: "prime p" and xy: "x * y = p ^ k" chaieb@26125: shows "\i j. x = p ^i \ y = p^ j" chaieb@26125: using xy chaieb@26125: proof(induct k arbitrary: x y) chaieb@26125: case 0 thus ?case apply simp by (rule exI[where x="0"], simp) chaieb@26125: next chaieb@26125: case (Suc k x y) chaieb@26125: from Suc.prems have pxy: "p dvd x*y" by auto chaieb@26125: from prime_divprod[OF p pxy] have pxyc: "p dvd x \ p dvd y" . chaieb@26125: from p have p0: "p \ 0" by - (rule ccontr, simp) chaieb@26125: {assume px: "p dvd x" chaieb@26125: then obtain d where d: "x = p*d" unfolding dvd_def by blast chaieb@26125: from Suc.prems d have "p*d*y = p^Suc k" by simp chaieb@26125: hence th: "d*y = p^k" using p0 by simp chaieb@26125: from Suc.hyps[OF th] obtain i j where ij: "d = p^i" "y = p^j" by blast chaieb@26125: with d have "x = p^Suc i" by simp chaieb@26125: with ij(2) have ?case by blast} chaieb@26125: moreover chaieb@26125: {assume px: "p dvd y" chaieb@26125: then obtain d where d: "y = p*d" unfolding dvd_def by blast chaieb@26125: from Suc.prems d have "p*d*x = p^Suc k" by (simp add: mult_commute) chaieb@26125: hence th: "d*x = p^k" using p0 by simp chaieb@26125: from Suc.hyps[OF th] obtain i j where ij: "d = p^i" "x = p^j" by blast chaieb@26125: with d have "y = p^Suc i" by simp chaieb@26125: with ij(2) have ?case by blast} chaieb@26125: ultimately show ?case using pxyc by blast chaieb@26125: qed chaieb@26125: chaieb@26125: lemma prime_power_exp: assumes p: "prime p" and n:"n \ 0" chaieb@26125: and xn: "x^n = p^k" shows "\i. x = p^i" chaieb@26125: using n xn chaieb@26125: proof(induct n arbitrary: k) chaieb@26125: case 0 thus ?case by simp chaieb@26125: next chaieb@26125: case (Suc n k) hence th: "x*x^n = p^k" by simp chaieb@26125: {assume "n = 0" with prems have ?case apply simp chaieb@26125: by (rule exI[where x="k"],simp)} chaieb@26125: moreover chaieb@26125: {assume n: "n \ 0" chaieb@26125: from prime_power_mult[OF p th] chaieb@26125: obtain i j where ij: "x = p^i" "x^n = p^j"by blast chaieb@26125: from Suc.hyps[OF n ij(2)] have ?case .} chaieb@26125: ultimately show ?case by blast chaieb@26125: qed chaieb@26125: chaieb@26125: lemma divides_primepow: assumes p: "prime p" chaieb@26125: shows "d dvd p^k \ (\ i. i \ k \ d = p ^i)" chaieb@26125: proof chaieb@26125: assume H: "d dvd p^k" then obtain e where e: "d*e = p^k" chaieb@26125: unfolding dvd_def apply (auto simp add: mult_commute) by blast chaieb@26125: from prime_power_mult[OF p e] obtain i j where ij: "d = p^i" "e=p^j" by blast chaieb@26125: from prime_ge_2[OF p] have p1: "p > 1" by arith chaieb@26125: from e ij have "p^(i + j) = p^k" by (simp add: power_add) chaieb@26125: hence "i + j = k" using power_inject_exp[of p "i+j" k, OF p1] by simp chaieb@26125: hence "i \ k" by arith chaieb@26125: with ij(1) show "\i\k. d = p ^ i" by blast chaieb@26125: next chaieb@26125: {fix i assume H: "i \ k" "d = p^i" chaieb@26125: hence "\j. k = i + j" by arith chaieb@26125: then obtain j where j: "k = i + j" by blast chaieb@26125: hence "p^k = p^j*d" using H(2) by (simp add: power_add) chaieb@26125: hence "d dvd p^k" unfolding dvd_def by auto} chaieb@26125: thus "\i\k. d = p ^ i \ d dvd p ^ k" by blast chaieb@26125: qed chaieb@26125: chaieb@26125: lemma coprime_divisors: "d dvd a \ e dvd b \ coprime a b \ coprime d e" chaieb@26125: by (auto simp add: dvd_def coprime) chaieb@26125: chaieb@26159: declare power_Suc0[simp del] chaieb@26159: declare even_dvd[simp del] haftmann@26757: paulson@11363: end