diff options
author | David Aspinall <da@inf.ed.ac.uk> | 2004-04-16 14:30:13 +0000 |
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committer | David Aspinall <da@inf.ed.ac.uk> | 2004-04-16 14:30:13 +0000 |
commit | 2b17630a04cc510ef6c504614ab59e99071c5cce (patch) | |
tree | 35646939b64142d48e323dd1686fb74fbd67f3dd | |
parent | 98b419b104a9c7b0a12ee2f00248ef983117a725 (diff) |
New files.
-rw-r--r-- | isar/Root2_Isar.thy | 151 | ||||
-rw-r--r-- | isar/Root2_Tactic.thy | 81 |
2 files changed, 232 insertions, 0 deletions
diff --git a/isar/Root2_Isar.thy b/isar/Root2_Isar.thy new file mode 100644 index 00000000..d09cc0d4 --- /dev/null +++ b/isar/Root2_Isar.thy @@ -0,0 +1,151 @@ +(* Example proof by Markus Wenzel; see http://www.cs.kun.nl/~freek/comparison/ + Taken from Isabelle2004 distribution. *) + + +(* Title: HOL/Hyperreal/ex/Sqrt.thy + ID: Id: Sqrt.thy,v 1.4 2004/01/12 15:51:47 paulson Exp + Author: Markus Wenzel, TU Muenchen + License: GPL (GNU GENERAL PUBLIC LICENSE) +*) + +header {* Square roots of primes are irrational *} + +theory Root2_Isar = Primes + Complex_Main: + +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: "p \<in> prime \<Longrightarrow> sqrt (real p) \<notin> \<rat>" +proof + assume p_prime: "p \<in> prime" + 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 "p \<in> prime \<Longrightarrow> sqrt (real p) \<notin> \<rat>" +proof + assume p_prime: "p \<in> prime" + 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 diff --git a/isar/Root2_Tactic.thy b/isar/Root2_Tactic.thy new file mode 100644 index 00000000..65f2a944 --- /dev/null +++ b/isar/Root2_Tactic.thy @@ -0,0 +1,81 @@ +(* Example proof by Larry Paulson; see http://www.cs.kun.nl/~freek/comparison/ + Taken from Isabelle2004 distribution. *) + + +(* Title: HOL/Hyperreal/ex/Sqrt_Script.thy + ID: Id: Sqrt_Script.thy,v 1.3 2003/12/10 14:59:35 paulson Exp + Author: Lawrence C Paulson, Cambridge University Computer Laboratory + Copyright 2001 University of Cambridge +*) + +header {* Square roots of primes are irrational (script version) *} + +theory Sqrt_Script = Primes + Complex_Main: + +text {* + \medskip Contrast this linear Isabelle/Isar script with Markus + Wenzel's more mathematical version. +*} + +subsection {* Preliminaries *} + +lemma prime_nonzero: "p \<in> prime \<Longrightarrow> p \<noteq> 0" + by (force simp add: prime_def) + +lemma prime_dvd_other_side: + "n * n = p * (k * k) \<Longrightarrow> p \<in> prime \<Longrightarrow> p dvd n" + apply (subgoal_tac "p dvd n * n", blast dest: prime_dvd_mult) + apply (rule_tac j = "k * k" in dvd_mult_left, simp) + done + +lemma reduction: "p \<in> prime \<Longrightarrow> + 0 < k \<Longrightarrow> k * k = p * (j * j) \<Longrightarrow> k < p * j \<and> 0 < j" + apply (rule ccontr) + apply (simp add: linorder_not_less) + apply (erule disjE) + apply (frule mult_le_mono, assumption) + apply auto + apply (force simp add: prime_def) + done + +lemma rearrange: "(j::nat) * (p * j) = k * k \<Longrightarrow> k * k = p * (j * j)" + by (simp add: mult_ac) + +lemma prime_not_square: + "p \<in> prime \<Longrightarrow> (\<And>k. 0 < k \<Longrightarrow> m * m \<noteq> p * (k * k))" + apply (induct m rule: nat_less_induct) + apply clarify + apply (frule prime_dvd_other_side, assumption) + apply (erule dvdE) + apply (simp add: nat_mult_eq_cancel_disj prime_nonzero) + apply (blast dest: rearrange reduction) + done + + +subsection {* The set of rational numbers *} + +constdefs + rationals :: "real set" ("\<rat>") + "\<rat> \<equiv> {x. \<exists>m n. n \<noteq> 0 \<and> \<bar>x\<bar> = real (m::nat) / real (n::nat)}" + + +subsection {* Main theorem *} + +text {* + The square root of any prime number (including @{text 2}) is + irrational. +*} + +theorem prime_sqrt_irrational: + "p \<in> prime \<Longrightarrow> x * x = real p \<Longrightarrow> 0 \<le> x \<Longrightarrow> x \<notin> \<rat>" + apply (simp add: rationals_def real_abs_def) + apply clarify + apply (erule_tac P = "real m / real n * ?x = ?y" in rev_mp) + apply (simp del: real_of_nat_mult + add: divide_eq_eq prime_not_square real_of_nat_mult [symmetric]) + done + +lemmas two_sqrt_irrational = + prime_sqrt_irrational [OF two_is_prime] + +end |