(* Example proof by Markus Wenzel; see http://www.cs.kun.nl/~freek/comparison/ Taken from Isabelle2005 distribution. *) (* Title: HOL/Hyperreal/ex/Sqrt.thy ID: Root2_Isar.thy,v 9.0 2008/01/30 15:22:19 da Exp Author: Markus Wenzel, TU Muenchen *) header {* Square roots of primes are irrational *} theory Root2_Isar imports Primes Complex_Main begin subsection {* Preliminaries *} text {* The set of rational numbers, including the key representation theorem. *} constdefs rationals :: "real set" ("\<rat>") "\<rat> \<equiv> {x. \<exists>m n. n \<noteq> 0 \<and> \<bar>x\<bar> = real (m::nat) / real (n::nat)}" theorem rationals_rep: "x \<in> \<rat> \<Longrightarrow> \<exists>m n. n \<noteq> 0 \<and> \<bar>x\<bar> = real m / real n \<and> gcd (m, n) = 1" proof - assume "x \<in> \<rat>" then obtain m n :: nat where n: "n \<noteq> 0" and x_rat: "\<bar>x\<bar> = real m / real n" by (unfold rationals_def) blast let ?gcd = "gcd (m, n)" from n have gcd: "?gcd \<noteq> 0" by (simp add: gcd_zero) let ?k = "m div ?gcd" let ?l = "n div ?gcd" let ?gcd' = "gcd (?k, ?l)" have "?gcd dvd m" .. then have gcd_k: "?gcd * ?k = m" by (rule dvd_mult_div_cancel) have "?gcd dvd n" .. then have gcd_l: "?gcd * ?l = n" by (rule dvd_mult_div_cancel) from n and gcd_l have "?l \<noteq> 0" by (auto iff del: neq0_conv) moreover have "\<bar>x\<bar> = real ?k / real ?l" proof - from gcd have "real ?k / real ?l = real (?gcd * ?k) / real (?gcd * ?l)" by (simp add: mult_divide_cancel_left) also from gcd_k and gcd_l have "\<dots> = real m / real n" by simp also from x_rat have "\<dots> = \<bar>x\<bar>" .. finally show ?thesis .. qed moreover have "?gcd' = 1" proof - have "?gcd * ?gcd' = gcd (?gcd * ?k, ?gcd * ?l)" by (rule gcd_mult_distrib2) with gcd_k gcd_l have "?gcd * ?gcd' = ?gcd" by simp with gcd show ?thesis by simp qed ultimately show ?thesis by blast qed lemma [elim?]: "r \<in> \<rat> \<Longrightarrow> (\<And>m n. n \<noteq> 0 \<Longrightarrow> \<bar>r\<bar> = real m / real n \<Longrightarrow> gcd (m, n) = 1 \<Longrightarrow> C) \<Longrightarrow> C" using rationals_rep by blast subsection {* Main theorem *} text {* The square root of any prime number (including @{text 2}) is irrational. *} theorem sqrt_prime_irrational: "prime p \<Longrightarrow> sqrt (real p) \<notin> \<rat>" proof assume p_prime: "prime p" then have p: "1 < p" by (simp add: prime_def) assume "sqrt (real p) \<in> \<rat>" then obtain m n where n: "n \<noteq> 0" and sqrt_rat: "\<bar>sqrt (real p)\<bar> = real m / real n" and gcd: "gcd (m, n) = 1" .. have eq: "m\<twosuperior> = p * n\<twosuperior>" proof - from n and sqrt_rat have "real m = \<bar>sqrt (real p)\<bar> * real n" by simp then have "real (m\<twosuperior>) = (sqrt (real p))\<twosuperior> * real (n\<twosuperior>)" by (auto simp add: power2_eq_square) also have "(sqrt (real p))\<twosuperior> = real p" by simp also have "\<dots> * real (n\<twosuperior>) = real (p * n\<twosuperior>)" by simp finally show ?thesis .. qed have "p dvd m \<and> p dvd n" proof from eq have "p dvd m\<twosuperior>" .. with p_prime show "p dvd m" by (rule prime_dvd_power_two) then obtain k where "m = p * k" .. with eq have "p * n\<twosuperior> = p\<twosuperior> * k\<twosuperior>" by (auto simp add: power2_eq_square mult_ac) with p have "n\<twosuperior> = p * k\<twosuperior>" by (simp add: power2_eq_square) then have "p dvd n\<twosuperior>" .. with p_prime show "p dvd n" by (rule prime_dvd_power_two) qed then have "p dvd gcd (m, n)" .. with gcd have "p dvd 1" by simp then have "p \<le> 1" by (simp add: dvd_imp_le) with p show False by simp qed corollary "sqrt (real (2::nat)) \<notin> \<rat>" by (rule sqrt_prime_irrational) (rule two_is_prime) subsection {* Variations *} text {* Here is an alternative version of the main proof, using mostly linear forward-reasoning. While this results in less top-down structure, it is probably closer to proofs seen in mathematics. *} theorem "prime p \<Longrightarrow> sqrt (real p) \<notin> \<rat>" proof assume p_prime: "prime p" then have p: "1 < p" by (simp add: prime_def) assume "sqrt (real p) \<in> \<rat>" then obtain m n where n: "n \<noteq> 0" and sqrt_rat: "\<bar>sqrt (real p)\<bar> = real m / real n" and gcd: "gcd (m, n) = 1" .. from n and sqrt_rat have "real m = \<bar>sqrt (real p)\<bar> * real n" by simp then have "real (m\<twosuperior>) = (sqrt (real p))\<twosuperior> * real (n\<twosuperior>)" by (auto simp add: power2_eq_square) also have "(sqrt (real p))\<twosuperior> = real p" by simp also have "\<dots> * real (n\<twosuperior>) = real (p * n\<twosuperior>)" by simp finally have eq: "m\<twosuperior> = p * n\<twosuperior>" .. then have "p dvd m\<twosuperior>" .. with p_prime have dvd_m: "p dvd m" by (rule prime_dvd_power_two) then obtain k where "m = p * k" .. with eq have "p * n\<twosuperior> = p\<twosuperior> * k\<twosuperior>" by (auto simp add: power2_eq_square mult_ac) with p have "n\<twosuperior> = p * k\<twosuperior>" by (simp add: power2_eq_square) then have "p dvd n\<twosuperior>" .. with p_prime have "p dvd n" by (rule prime_dvd_power_two) with dvd_m have "p dvd gcd (m, n)" by (rule gcd_greatest) with gcd have "p dvd 1" by simp then have "p \<le> 1" by (simp add: dvd_imp_le) with p show False by simp qed end