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author | wenzelm |

Sat, 05 Jun 2021 12:57:52 +0200 | |

changeset 73811 | f143d0a4cb6a |

parent 73810 | 1c5dcba6925f |

child 73812 | 90b64197bafd |

clarified examples;

src/HOL/Examples/Sqrt.thy | file | annotate | diff | comparison | revisions | |

src/HOL/ROOT | file | annotate | diff | comparison | revisions | |

src/HOL/ex/Sqrt.thy | file | annotate | diff | comparison | revisions | |

src/HOL/ex/Sqrt_Script.thy | file | annotate | diff | comparison | revisions |

--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/HOL/Examples/Sqrt.thy Sat Jun 05 12:57:52 2021 +0200 @@ -0,0 +1,104 @@ +(* Title: HOL/Examples/Sqrt.thy + Author: Makarius + Author: Tobias Nipkow, TU Muenchen +*) + +section \<open>Square roots of primes are irrational\<close> + +theory Sqrt + imports Complex_Main "HOL-Computational_Algebra.Primes" +begin + +text \<open> + The square root of any prime number (including 2) is irrational. +\<close> + +theorem sqrt_prime_irrational: + fixes p :: nat + assumes "prime p" + shows "sqrt p \<notin> \<rat>" +proof + from \<open>prime p\<close> have p: "p > 1" by (rule prime_gt_1_nat) + assume "sqrt p \<in> \<rat>" + then obtain m n :: nat + where n: "n \<noteq> 0" + and sqrt_rat: "\<bar>sqrt p\<bar> = m / n" + and "coprime m n" by (rule Rats_abs_nat_div_natE) + have eq: "m\<^sup>2 = p * n\<^sup>2" + proof - + from n and sqrt_rat have "m = \<bar>sqrt p\<bar> * n" by simp + then have "m\<^sup>2 = (sqrt p)\<^sup>2 * n\<^sup>2" by (simp add: power_mult_distrib) + also have "(sqrt p)\<^sup>2 = p" by simp + also have "\<dots> * n\<^sup>2 = p * n\<^sup>2" by simp + finally show ?thesis by linarith + qed + have "p dvd m \<and> p dvd n" + proof + from eq have "p dvd m\<^sup>2" .. + with \<open>prime p\<close> show "p dvd m" by (rule prime_dvd_power) + then obtain k where "m = p * k" .. + with eq have "p * n\<^sup>2 = p\<^sup>2 * k\<^sup>2" by algebra + with p have "n\<^sup>2 = p * k\<^sup>2" by (simp add: power2_eq_square) + then have "p dvd n\<^sup>2" .. + with \<open>prime p\<close> show "p dvd n" by (rule prime_dvd_power) + qed + then have "p dvd gcd m n" by simp + with \<open>coprime m n\<close> have "p = 1" by simp + with p show False by simp +qed + +corollary sqrt_2_not_rat: "sqrt 2 \<notin> \<rat>" + using sqrt_prime_irrational [of 2] by simp + +text \<open> + 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. +\<close> + +theorem + fixes p :: nat + assumes "prime p" + shows "sqrt p \<notin> \<rat>" +proof + from \<open>prime p\<close> have p: "p > 1" by (rule prime_gt_1_nat) + assume "sqrt p \<in> \<rat>" + then obtain m n :: nat + where n: "n \<noteq> 0" + and sqrt_rat: "\<bar>sqrt p\<bar> = m / n" + and "coprime m n" by (rule Rats_abs_nat_div_natE) + from n and sqrt_rat have "m = \<bar>sqrt p\<bar> * n" by simp + then have "m\<^sup>2 = (sqrt p)\<^sup>2 * n\<^sup>2" by (auto simp add: power2_eq_square) + also have "(sqrt p)\<^sup>2 = p" by simp + also have "\<dots> * n\<^sup>2 = p * n\<^sup>2" by simp + finally have eq: "m\<^sup>2 = p * n\<^sup>2" by linarith + then have "p dvd m\<^sup>2" .. + with \<open>prime p\<close> have dvd_m: "p dvd m" by (rule prime_dvd_power) + then obtain k where "m = p * k" .. + with eq have "p * n\<^sup>2 = p\<^sup>2 * k\<^sup>2" by algebra + with p have "n\<^sup>2 = p * k\<^sup>2" by (simp add: power2_eq_square) + then have "p dvd n\<^sup>2" .. + with \<open>prime p\<close> have "p dvd n" by (rule prime_dvd_power) + with dvd_m have "p dvd gcd m n" by (rule gcd_greatest) + with \<open>coprime m n\<close> have "p = 1" by simp + with p show False by simp +qed + + +text \<open> + Another old chestnut, which is a consequence of the irrationality of + \<^term>\<open>sqrt 2\<close>. +\<close> + +lemma "\<exists>a b::real. a \<notin> \<rat> \<and> b \<notin> \<rat> \<and> a powr b \<in> \<rat>" (is "\<exists>a b. ?P a b") +proof (cases "sqrt 2 powr sqrt 2 \<in> \<rat>") + case True + with sqrt_2_not_rat have "?P (sqrt 2) (sqrt 2)" by simp + then show ?thesis by blast +next + case False + with sqrt_2_not_rat powr_powr have "?P (sqrt 2 powr sqrt 2) (sqrt 2)" by simp + then show ?thesis by blast +qed + +end

--- a/src/HOL/ROOT Sat Jun 05 12:45:00 2021 +0200 +++ b/src/HOL/ROOT Sat Jun 05 12:57:52 2021 +0200 @@ -20,7 +20,7 @@ Notable Examples in Isabelle/HOL. " sessions - "HOL-Library" + "HOL-Computational_Algebra" theories Adhoc_Overloading_Examples Ackermann @@ -36,6 +36,7 @@ Peirce Records Seq + Sqrt document_files "root.bib" "root.tex" @@ -706,7 +707,6 @@ Sketch_and_Explore Sorting_Algorithms_Examples Specifications_with_bundle_mixins - Sqrt Sqrt_Script Sudoku Sum_of_Powers

--- a/src/HOL/ex/Sqrt.thy Sat Jun 05 12:45:00 2021 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,104 +0,0 @@ -(* Title: HOL/ex/Sqrt.thy - Author: Makarius - Author: Tobias Nipkow, TU Muenchen -*) - -section \<open>Square roots of primes are irrational\<close> - -theory Sqrt - imports Complex_Main "HOL-Computational_Algebra.Primes" -begin - -text \<open> - The square root of any prime number (including 2) is irrational. -\<close> - -theorem sqrt_prime_irrational: - fixes p :: nat - assumes "prime p" - shows "sqrt p \<notin> \<rat>" -proof - from \<open>prime p\<close> have p: "p > 1" by (rule prime_gt_1_nat) - assume "sqrt p \<in> \<rat>" - then obtain m n :: nat - where n: "n \<noteq> 0" - and sqrt_rat: "\<bar>sqrt p\<bar> = m / n" - and "coprime m n" by (rule Rats_abs_nat_div_natE) - have eq: "m\<^sup>2 = p * n\<^sup>2" - proof - - from n and sqrt_rat have "m = \<bar>sqrt p\<bar> * n" by simp - then have "m\<^sup>2 = (sqrt p)\<^sup>2 * n\<^sup>2" by (simp add: power_mult_distrib) - also have "(sqrt p)\<^sup>2 = p" by simp - also have "\<dots> * n\<^sup>2 = p * n\<^sup>2" by simp - finally show ?thesis by linarith - qed - have "p dvd m \<and> p dvd n" - proof - from eq have "p dvd m\<^sup>2" .. - with \<open>prime p\<close> show "p dvd m" by (rule prime_dvd_power) - then obtain k where "m = p * k" .. - with eq have "p * n\<^sup>2 = p\<^sup>2 * k\<^sup>2" by algebra - with p have "n\<^sup>2 = p * k\<^sup>2" by (simp add: power2_eq_square) - then have "p dvd n\<^sup>2" .. - with \<open>prime p\<close> show "p dvd n" by (rule prime_dvd_power) - qed - then have "p dvd gcd m n" by simp - with \<open>coprime m n\<close> have "p = 1" by simp - with p show False by simp -qed - -corollary sqrt_2_not_rat: "sqrt 2 \<notin> \<rat>" - using sqrt_prime_irrational [of 2] by simp - -text \<open> - 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. -\<close> - -theorem - fixes p :: nat - assumes "prime p" - shows "sqrt p \<notin> \<rat>" -proof - from \<open>prime p\<close> have p: "p > 1" by (rule prime_gt_1_nat) - assume "sqrt p \<in> \<rat>" - then obtain m n :: nat - where n: "n \<noteq> 0" - and sqrt_rat: "\<bar>sqrt p\<bar> = m / n" - and "coprime m n" by (rule Rats_abs_nat_div_natE) - from n and sqrt_rat have "m = \<bar>sqrt p\<bar> * n" by simp - then have "m\<^sup>2 = (sqrt p)\<^sup>2 * n\<^sup>2" by (auto simp add: power2_eq_square) - also have "(sqrt p)\<^sup>2 = p" by simp - also have "\<dots> * n\<^sup>2 = p * n\<^sup>2" by simp - finally have eq: "m\<^sup>2 = p * n\<^sup>2" by linarith - then have "p dvd m\<^sup>2" .. - with \<open>prime p\<close> have dvd_m: "p dvd m" by (rule prime_dvd_power) - then obtain k where "m = p * k" .. - with eq have "p * n\<^sup>2 = p\<^sup>2 * k\<^sup>2" by algebra - with p have "n\<^sup>2 = p * k\<^sup>2" by (simp add: power2_eq_square) - then have "p dvd n\<^sup>2" .. - with \<open>prime p\<close> have "p dvd n" by (rule prime_dvd_power) - with dvd_m have "p dvd gcd m n" by (rule gcd_greatest) - with \<open>coprime m n\<close> have "p = 1" by simp - with p show False by simp -qed - - -text \<open> - Another old chestnut, which is a consequence of the irrationality of - \<^term>\<open>sqrt 2\<close>. -\<close> - -lemma "\<exists>a b::real. a \<notin> \<rat> \<and> b \<notin> \<rat> \<and> a powr b \<in> \<rat>" (is "\<exists>a b. ?P a b") -proof (cases "sqrt 2 powr sqrt 2 \<in> \<rat>") - case True - with sqrt_2_not_rat have "?P (sqrt 2) (sqrt 2)" by simp - then show ?thesis by blast -next - case False - with sqrt_2_not_rat powr_powr have "?P (sqrt 2 powr sqrt 2) (sqrt 2)" by simp - then show ?thesis by blast -qed - -end

--- a/src/HOL/ex/Sqrt_Script.thy Sat Jun 05 12:45:00 2021 +0200 +++ b/src/HOL/ex/Sqrt_Script.thy Sat Jun 05 12:57:52 2021 +0200 @@ -5,14 +5,14 @@ section \<open>Square roots of primes are irrational (script version)\<close> -theory Sqrt_Script -imports Complex_Main "HOL-Computational_Algebra.Primes" -begin +text \<open> + Contrast this linear Isabelle/Isar script with the more mathematical version + in \<^file>\<open>~~/src/HOL/Examples/Sqrt.thy\<close> by Makarius Wenzel. +\<close> -text \<open> - \medskip Contrast this linear Isabelle/Isar script with Markus - Wenzel's more mathematical version. -\<close> +theory Sqrt_Script + imports Complex_Main "HOL-Computational_Algebra.Primes" +begin subsection \<open>Preliminaries\<close>