1(* Title: HOL/ex/Sqrt.thy 2 Author: Markus Wenzel, Tobias Nipkow, TU Muenchen 3*) 4 5section \<open>Square roots of primes are irrational\<close> 6 7theory Sqrt 8imports Complex_Main "HOL-Computational_Algebra.Primes" 9begin 10 11text \<open>The square root of any prime number (including 2) is irrational.\<close> 12 13theorem sqrt_prime_irrational: 14 assumes "prime (p::nat)" 15 shows "sqrt p \<notin> \<rat>" 16proof 17 from \<open>prime p\<close> have p: "1 < p" by (simp add: prime_nat_iff) 18 assume "sqrt p \<in> \<rat>" 19 then obtain m n :: nat where 20 n: "n \<noteq> 0" and sqrt_rat: "\<bar>sqrt p\<bar> = m / n" 21 and "coprime m n" by (rule Rats_abs_nat_div_natE) 22 have eq: "m\<^sup>2 = p * n\<^sup>2" 23 proof - 24 from n and sqrt_rat have "m = \<bar>sqrt p\<bar> * n" by simp 25 then have "m\<^sup>2 = (sqrt p)\<^sup>2 * n\<^sup>2" 26 by (auto simp add: power2_eq_square) 27 also have "(sqrt p)\<^sup>2 = p" by simp 28 also have "\<dots> * n\<^sup>2 = p * n\<^sup>2" by simp 29 finally show ?thesis using of_nat_eq_iff by blast 30 qed 31 have "p dvd m \<and> p dvd n" 32 proof 33 from eq have "p dvd m\<^sup>2" .. 34 with \<open>prime p\<close> show "p dvd m" by (rule prime_dvd_power_nat) 35 then obtain k where "m = p * k" .. 36 with eq have "p * n\<^sup>2 = p\<^sup>2 * k\<^sup>2" by (auto simp add: power2_eq_square ac_simps) 37 with p have "n\<^sup>2 = p * k\<^sup>2" by (simp add: power2_eq_square) 38 then have "p dvd n\<^sup>2" .. 39 with \<open>prime p\<close> show "p dvd n" by (rule prime_dvd_power_nat) 40 qed 41 then have "p dvd gcd m n" by simp 42 with \<open>coprime m n\<close> have "p = 1" by simp 43 with p show False by simp 44qed 45 46corollary sqrt_2_not_rat: "sqrt 2 \<notin> \<rat>" 47 using sqrt_prime_irrational[of 2] by simp 48 49 50subsection \<open>Variations\<close> 51 52text \<open> 53 Here is an alternative version of the main proof, using mostly 54 linear forward-reasoning. While this results in less top-down 55 structure, it is probably closer to proofs seen in mathematics. 56\<close> 57 58theorem 59 assumes "prime (p::nat)" 60 shows "sqrt p \<notin> \<rat>" 61proof 62 from \<open>prime p\<close> have p: "1 < p" by (simp add: prime_nat_iff) 63 assume "sqrt p \<in> \<rat>" 64 then obtain m n :: nat where 65 n: "n \<noteq> 0" and sqrt_rat: "\<bar>sqrt p\<bar> = m / n" 66 and "coprime m n" by (rule Rats_abs_nat_div_natE) 67 from n and sqrt_rat have "m = \<bar>sqrt p\<bar> * n" by simp 68 then have "m\<^sup>2 = (sqrt p)\<^sup>2 * n\<^sup>2" 69 by (auto simp add: power2_eq_square) 70 also have "(sqrt p)\<^sup>2 = p" by simp 71 also have "\<dots> * n\<^sup>2 = p * n\<^sup>2" by simp 72 finally have eq: "m\<^sup>2 = p * n\<^sup>2" using of_nat_eq_iff by blast 73 then have "p dvd m\<^sup>2" .. 74 with \<open>prime p\<close> have dvd_m: "p dvd m" by (rule prime_dvd_power_nat) 75 then obtain k where "m = p * k" .. 76 with eq have "p * n\<^sup>2 = p\<^sup>2 * k\<^sup>2" by (auto simp add: power2_eq_square ac_simps) 77 with p have "n\<^sup>2 = p * k\<^sup>2" by (simp add: power2_eq_square) 78 then have "p dvd n\<^sup>2" .. 79 with \<open>prime p\<close> have "p dvd n" by (rule prime_dvd_power_nat) 80 with dvd_m have "p dvd gcd m n" by (rule gcd_greatest) 81 with \<open>coprime m n\<close> have "p = 1" by simp 82 with p show False by simp 83qed 84 85 86text \<open>Another old chestnut, which is a consequence of the irrationality of 2.\<close> 87 88lemma "\<exists>a b::real. a \<notin> \<rat> \<and> b \<notin> \<rat> \<and> a powr b \<in> \<rat>" (is "\<exists>a b. ?P a b") 89proof cases 90 assume "sqrt 2 powr sqrt 2 \<in> \<rat>" 91 then have "?P (sqrt 2) (sqrt 2)" 92 by (metis sqrt_2_not_rat) 93 then show ?thesis by blast 94next 95 assume 1: "sqrt 2 powr sqrt 2 \<notin> \<rat>" 96 have "(sqrt 2 powr sqrt 2) powr sqrt 2 = 2" 97 using powr_realpow [of _ 2] 98 by (simp add: powr_powr power2_eq_square [symmetric]) 99 then have "?P (sqrt 2 powr sqrt 2) (sqrt 2)" 100 by (metis 1 Rats_number_of sqrt_2_not_rat) 101 then show ?thesis by blast 102qed 103 104end 105