src/HOL/ex/Sqrt.thy
author wenzelm
Wed Jun 22 10:09:20 2016 +0200 (2016-06-22)
changeset 63343 fb5d8a50c641
parent 62348 9a5f43dac883
child 63534 523b488b15c9
permissions -rw-r--r--
bundle lifting_syntax;
     1 (*  Title:      HOL/ex/Sqrt.thy
     2     Author:     Markus Wenzel, Tobias Nipkow, TU Muenchen
     3 *)
     4 
     5 section \<open>Square roots of primes are irrational\<close>
     6 
     7 theory Sqrt
     8 imports Complex_Main "~~/src/HOL/Number_Theory/Primes"
     9 begin
    10 
    11 text \<open>The square root of any prime number (including 2) is irrational.\<close>
    12 
    13 theorem sqrt_prime_irrational:
    14   assumes "prime (p::nat)"
    15   shows "sqrt p \<notin> \<rat>"
    16 proof
    17   from \<open>prime p\<close> have p: "1 < p" by (simp add: prime_def)
    18   assume "sqrt p \<in> \<rat>"
    19   then obtain m n :: nat where
    20       n: "n \<noteq> 0" and sqrt_rat: "\<bar>sqrt p\<bar> = m / n"
    21     and "coprime m n" by (rule Rats_abs_nat_div_natE)
    22   have eq: "m\<^sup>2 = p * n\<^sup>2"
    23   proof -
    24     from n and sqrt_rat have "m = \<bar>sqrt p\<bar> * n" by simp
    25     then have "m\<^sup>2 = (sqrt p)\<^sup>2 * n\<^sup>2"
    26       by (auto simp add: power2_eq_square)
    27     also have "(sqrt p)\<^sup>2 = p" by simp
    28     also have "\<dots> * n\<^sup>2 = p * n\<^sup>2" by simp
    29     finally show ?thesis using of_nat_eq_iff by blast
    30   qed
    31   have "p dvd m \<and> p dvd n"
    32   proof
    33     from eq have "p dvd m\<^sup>2" ..
    34     with \<open>prime p\<close> show "p dvd m" by (rule prime_dvd_power_nat)
    35     then obtain k where "m = p * k" ..
    36     with eq have "p * n\<^sup>2 = p\<^sup>2 * k\<^sup>2" by (auto simp add: power2_eq_square ac_simps)
    37     with p have "n\<^sup>2 = p * k\<^sup>2" by (simp add: power2_eq_square)
    38     then have "p dvd n\<^sup>2" ..
    39     with \<open>prime p\<close> show "p dvd n" by (rule prime_dvd_power_nat)
    40   qed
    41   then have "p dvd gcd m n" by simp
    42   with \<open>coprime m n\<close> have "p = 1" by simp
    43   with p show False by simp
    44 qed
    45 
    46 corollary sqrt_2_not_rat: "sqrt 2 \<notin> \<rat>"
    47   using sqrt_prime_irrational[of 2] by simp
    48 
    49 
    50 subsection \<open>Variations\<close>
    51 
    52 text \<open>
    53   Here is an alternative version of the main proof, using mostly
    54   linear forward-reasoning.  While this results in less top-down
    55   structure, it is probably closer to proofs seen in mathematics.
    56 \<close>
    57 
    58 theorem
    59   assumes "prime (p::nat)"
    60   shows "sqrt p \<notin> \<rat>"
    61 proof
    62   from \<open>prime p\<close> have p: "1 < p" by (simp add: prime_def)
    63   assume "sqrt p \<in> \<rat>"
    64   then obtain m n :: nat where
    65       n: "n \<noteq> 0" and sqrt_rat: "\<bar>sqrt p\<bar> = m / n"
    66     and "coprime m n" by (rule Rats_abs_nat_div_natE)
    67   from n and sqrt_rat have "m = \<bar>sqrt p\<bar> * n" by simp
    68   then have "m\<^sup>2 = (sqrt p)\<^sup>2 * n\<^sup>2"
    69     by (auto simp add: power2_eq_square)
    70   also have "(sqrt p)\<^sup>2 = p" by simp
    71   also have "\<dots> * n\<^sup>2 = p * n\<^sup>2" by simp
    72   finally have eq: "m\<^sup>2 = p * n\<^sup>2" using of_nat_eq_iff by blast
    73   then have "p dvd m\<^sup>2" ..
    74   with \<open>prime p\<close> have dvd_m: "p dvd m" by (rule prime_dvd_power_nat)
    75   then obtain k where "m = p * k" ..
    76   with eq have "p * n\<^sup>2 = p\<^sup>2 * k\<^sup>2" by (auto simp add: power2_eq_square ac_simps)
    77   with p have "n\<^sup>2 = p * k\<^sup>2" by (simp add: power2_eq_square)
    78   then have "p dvd n\<^sup>2" ..
    79   with \<open>prime p\<close> have "p dvd n" by (rule prime_dvd_power_nat)
    80   with dvd_m have "p dvd gcd m n" by (rule gcd_greatest)
    81   with \<open>coprime m n\<close> have "p = 1" by simp
    82   with p show False by simp
    83 qed
    84 
    85 
    86 text \<open>Another old chestnut, which is a consequence of the irrationality of 2.\<close>
    87 
    88 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")
    89 proof cases
    90   assume "sqrt 2 powr sqrt 2 \<in> \<rat>"
    91   then have "?P (sqrt 2) (sqrt 2)"
    92     by (metis sqrt_2_not_rat)
    93   then show ?thesis by blast
    94 next
    95   assume 1: "sqrt 2 powr sqrt 2 \<notin> \<rat>"
    96   have "(sqrt 2 powr sqrt 2) powr sqrt 2 = 2"
    97     using powr_realpow [of _ 2]
    98     by (simp add: powr_powr power2_eq_square [symmetric])
    99   then have "?P (sqrt 2 powr sqrt 2) (sqrt 2)"
   100     by (metis 1 Rats_number_of sqrt_2_not_rat)
   101   then show ?thesis by blast
   102 qed
   103 
   104 end