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src/HOL/ex/Sqrt.thy

author | wenzelm |

Tue, 10 Mar 2009 16:48:27 +0100 | |

changeset 30411 | 9c9b6511ad1b |

parent 28952 | 15a4b2cf8c34 |

child 31712 | 6f8aa9aea693 |

permissions | -rw-r--r-- |

tuned proofs;
tuned document;

(* Title: HOL/ex/Sqrt.thy Author: Markus Wenzel, TU Muenchen *) header {* Square roots of primes are irrational *} theory Sqrt imports Complex_Main Primes begin text {* The square root of any prime number (including @{text 2}) is irrational. *} theorem sqrt_prime_irrational: assumes "prime p" shows "sqrt (real p) \<notin> \<rat>" proof from `prime p` 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" by (rule Rats_abs_nat_div_natE) 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 `prime p` 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 `prime p` 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 assumes "prime p" shows "sqrt (real p) \<notin> \<rat>" proof from `prime p` 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" by (rule Rats_abs_nat_div_natE) 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 `prime p` 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 `prime p` 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