src/HOL/ex/Sqrt.thy
author haftmann
Fri Oct 10 19:55:32 2014 +0200 (2014-10-10)
changeset 58646 cd63a4b12a33
parent 57514 bdc2c6b40bf2
child 58889 5b7a9633cfa8
permissions -rw-r--r--
specialized specification: avoid trivial instances
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(*  Title:      HOL/ex/Sqrt.thy
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    Author:     Markus Wenzel, Tobias Nipkow, TU Muenchen
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*)
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header {*  Square roots of primes are irrational *}
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theory Sqrt
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imports Complex_Main "~~/src/HOL/Number_Theory/Primes"
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begin
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text {* The square root of any prime number (including 2) is irrational. *}
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theorem sqrt_prime_irrational:
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  assumes "prime (p::nat)"
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  shows "sqrt p \<notin> \<rat>"
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proof
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  from `prime p` have p: "1 < p" by (simp add: prime_nat_def)
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  assume "sqrt p \<in> \<rat>"
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  then obtain m n :: nat where
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      n: "n \<noteq> 0" and sqrt_rat: "\<bar>sqrt p\<bar> = m / n"
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    and gcd: "gcd m n = 1" by (rule Rats_abs_nat_div_natE)
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  have eq: "m\<^sup>2 = p * n\<^sup>2"
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  proof -
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    from n and sqrt_rat have "m = \<bar>sqrt p\<bar> * n" by simp
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    then have "m\<^sup>2 = (sqrt p)\<^sup>2 * n\<^sup>2"
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      by (auto simp add: power2_eq_square)
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    also have "(sqrt p)\<^sup>2 = p" by simp
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    also have "\<dots> * n\<^sup>2 = p * n\<^sup>2" by simp
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    finally show ?thesis ..
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  qed
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  have "p dvd m \<and> p dvd n"
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  proof
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    from eq have "p dvd m\<^sup>2" ..
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    with `prime p` show "p dvd m" by (rule prime_dvd_power_nat)
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    then obtain k where "m = p * k" ..
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    with eq have "p * n\<^sup>2 = p\<^sup>2 * k\<^sup>2" by (auto simp add: power2_eq_square ac_simps)
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    with p have "n\<^sup>2 = p * k\<^sup>2" by (simp add: power2_eq_square)
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    then have "p dvd n\<^sup>2" ..
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    with `prime p` show "p dvd n" by (rule prime_dvd_power_nat)
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  qed
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  then have "p dvd gcd m n" ..
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  with gcd have "p dvd 1" by simp
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  then have "p \<le> 1" by (simp add: dvd_imp_le)
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  with p show False by simp
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qed
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corollary sqrt_2_not_rat: "sqrt 2 \<notin> \<rat>"
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  using sqrt_prime_irrational[of 2] by simp
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subsection {* Variations *}
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text {*
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  Here is an alternative version of the main proof, using mostly
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  linear forward-reasoning.  While this results in less top-down
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  structure, it is probably closer to proofs seen in mathematics.
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*}
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theorem
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  assumes "prime (p::nat)"
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  shows "sqrt p \<notin> \<rat>"
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proof
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  from `prime p` have p: "1 < p" by (simp add: prime_nat_def)
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  assume "sqrt p \<in> \<rat>"
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  then obtain m n :: nat where
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      n: "n \<noteq> 0" and sqrt_rat: "\<bar>sqrt p\<bar> = m / n"
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    and gcd: "gcd m n = 1" by (rule Rats_abs_nat_div_natE)
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  from n and sqrt_rat have "m = \<bar>sqrt p\<bar> * n" by simp
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  then have "m\<^sup>2 = (sqrt p)\<^sup>2 * n\<^sup>2"
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    by (auto simp add: power2_eq_square)
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  also have "(sqrt p)\<^sup>2 = p" by simp
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  also have "\<dots> * n\<^sup>2 = p * n\<^sup>2" by simp
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  finally have eq: "m\<^sup>2 = p * n\<^sup>2" ..
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  then have "p dvd m\<^sup>2" ..
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  with `prime p` have dvd_m: "p dvd m" by (rule prime_dvd_power_nat)
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  then obtain k where "m = p * k" ..
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  with eq have "p * n\<^sup>2 = p\<^sup>2 * k\<^sup>2" by (auto simp add: power2_eq_square ac_simps)
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  with p have "n\<^sup>2 = p * k\<^sup>2" by (simp add: power2_eq_square)
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  then have "p dvd n\<^sup>2" ..
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  with `prime p` have "p dvd n" by (rule prime_dvd_power_nat)
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  with dvd_m have "p dvd gcd m n" by (rule gcd_greatest_nat)
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  with gcd have "p dvd 1" by simp
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  then have "p \<le> 1" by (simp add: dvd_imp_le)
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  with p show False by simp
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qed
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text {* Another old chestnut, which is a consequence of the irrationality of 2. *}
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lemma "\<exists>a b::real. a \<notin> \<rat> \<and> b \<notin> \<rat> \<and> a powr b \<in> \<rat>" (is "EX a b. ?P a b")
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proof cases
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  assume "sqrt 2 powr sqrt 2 \<in> \<rat>"
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  then have "?P (sqrt 2) (sqrt 2)"
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    by (metis sqrt_2_not_rat)
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  then show ?thesis by blast
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next
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  assume 1: "sqrt 2 powr sqrt 2 \<notin> \<rat>"
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  have "(sqrt 2 powr sqrt 2) powr sqrt 2 = 2"
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    using powr_realpow [of _ 2]
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    by (simp add: powr_powr power2_eq_square [symmetric])
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  then have "?P (sqrt 2 powr sqrt 2) (sqrt 2)"
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    by (metis 1 Rats_number_of sqrt_2_not_rat)
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  then show ?thesis by blast
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qed
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end