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diff --git a/isar/Root2_Isar.thy b/isar/Root2_Isar.thy deleted file mode 100644 index 7a0123dd..00000000 --- a/isar/Root2_Isar.thy +++ /dev/null @@ -1,153 +0,0 @@ -(* Example proof by Markus Wenzel; see http://www.cs.kun.nl/~freek/comparison/ - Taken from Isabelle2005 distribution. *) - - -(* Title: HOL/Hyperreal/ex/Sqrt.thy - ID: $Id$ - 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 |