src/HOL/ex/Sqrt.thy
author haftmann
Wed Feb 17 21:51:57 2016 +0100 (2016-02-17)
changeset 62348 9a5f43dac883
parent 61762 d50b993b4fb9
child 63534 523b488b15c9
permissions -rw-r--r--
dropped various legacy fact bindings
haftmann@28952
     1
(*  Title:      HOL/ex/Sqrt.thy
nipkow@45917
     2
    Author:     Markus Wenzel, Tobias Nipkow, TU Muenchen
paulson@13957
     3
*)
paulson@13957
     4
wenzelm@59031
     5
section \<open>Square roots of primes are irrational\<close>
paulson@13957
     6
nipkow@15149
     7
theory Sqrt
haftmann@32479
     8
imports Complex_Main "~~/src/HOL/Number_Theory/Primes"
nipkow@15149
     9
begin
paulson@13957
    10
wenzelm@59031
    11
text \<open>The square root of any prime number (including 2) is irrational.\<close>
paulson@13957
    12
wenzelm@19086
    13
theorem sqrt_prime_irrational:
huffman@31712
    14
  assumes "prime (p::nat)"
hoelzl@51708
    15
  shows "sqrt p \<notin> \<rat>"
paulson@13957
    16
proof
lp15@61762
    17
  from \<open>prime p\<close> have p: "1 < p" by (simp add: prime_def)
hoelzl@51708
    18
  assume "sqrt p \<in> \<rat>"
huffman@31712
    19
  then obtain m n :: nat where
hoelzl@51708
    20
      n: "n \<noteq> 0" and sqrt_rat: "\<bar>sqrt p\<bar> = m / n"
haftmann@60690
    21
    and "coprime m n" by (rule Rats_abs_nat_div_natE)
wenzelm@53015
    22
  have eq: "m\<^sup>2 = p * n\<^sup>2"
paulson@13957
    23
  proof -
hoelzl@51708
    24
    from n and sqrt_rat have "m = \<bar>sqrt p\<bar> * n" by simp
wenzelm@53015
    25
    then have "m\<^sup>2 = (sqrt p)\<^sup>2 * n\<^sup>2"
paulson@14353
    26
      by (auto simp add: power2_eq_square)
wenzelm@53015
    27
    also have "(sqrt p)\<^sup>2 = p" by simp
wenzelm@53015
    28
    also have "\<dots> * n\<^sup>2 = p * n\<^sup>2" by simp
lp15@61649
    29
    finally show ?thesis using of_nat_eq_iff by blast
paulson@13957
    30
  qed
paulson@13957
    31
  have "p dvd m \<and> p dvd n"
paulson@13957
    32
  proof
wenzelm@53015
    33
    from eq have "p dvd m\<^sup>2" ..
wenzelm@59031
    34
    with \<open>prime p\<close> show "p dvd m" by (rule prime_dvd_power_nat)
paulson@13957
    35
    then obtain k where "m = p * k" ..
haftmann@57514
    36
    with eq have "p * n\<^sup>2 = p\<^sup>2 * k\<^sup>2" by (auto simp add: power2_eq_square ac_simps)
wenzelm@53015
    37
    with p have "n\<^sup>2 = p * k\<^sup>2" by (simp add: power2_eq_square)
wenzelm@53015
    38
    then have "p dvd n\<^sup>2" ..
wenzelm@59031
    39
    with \<open>prime p\<close> show "p dvd n" by (rule prime_dvd_power_nat)
paulson@13957
    40
  qed
haftmann@60690
    41
  then have "p dvd gcd m n" by simp
haftmann@60690
    42
  with \<open>coprime m n\<close> have "p = 1" by simp
paulson@13957
    43
  with p show False by simp
paulson@13957
    44
qed
paulson@13957
    45
hoelzl@51708
    46
corollary sqrt_2_not_rat: "sqrt 2 \<notin> \<rat>"
hoelzl@51708
    47
  using sqrt_prime_irrational[of 2] by simp
paulson@13957
    48
paulson@13957
    49
wenzelm@59031
    50
subsection \<open>Variations\<close>
wenzelm@59031
    51
wenzelm@59031
    52
text \<open>
paulson@13957
    53
  Here is an alternative version of the main proof, using mostly
paulson@13957
    54
  linear forward-reasoning.  While this results in less top-down
paulson@13957
    55
  structure, it is probably closer to proofs seen in mathematics.
wenzelm@59031
    56
\<close>
paulson@13957
    57
wenzelm@19086
    58
theorem
huffman@31712
    59
  assumes "prime (p::nat)"
hoelzl@51708
    60
  shows "sqrt p \<notin> \<rat>"
paulson@13957
    61
proof
lp15@61762
    62
  from \<open>prime p\<close> have p: "1 < p" by (simp add: prime_def)
hoelzl@51708
    63
  assume "sqrt p \<in> \<rat>"
huffman@31712
    64
  then obtain m n :: nat where
hoelzl@51708
    65
      n: "n \<noteq> 0" and sqrt_rat: "\<bar>sqrt p\<bar> = m / n"
haftmann@60690
    66
    and "coprime m n" by (rule Rats_abs_nat_div_natE)
hoelzl@51708
    67
  from n and sqrt_rat have "m = \<bar>sqrt p\<bar> * n" by simp
wenzelm@53015
    68
  then have "m\<^sup>2 = (sqrt p)\<^sup>2 * n\<^sup>2"
paulson@14353
    69
    by (auto simp add: power2_eq_square)
wenzelm@53015
    70
  also have "(sqrt p)\<^sup>2 = p" by simp
wenzelm@53015
    71
  also have "\<dots> * n\<^sup>2 = p * n\<^sup>2" by simp
lp15@61649
    72
  finally have eq: "m\<^sup>2 = p * n\<^sup>2" using of_nat_eq_iff by blast
wenzelm@53015
    73
  then have "p dvd m\<^sup>2" ..
wenzelm@59031
    74
  with \<open>prime p\<close> have dvd_m: "p dvd m" by (rule prime_dvd_power_nat)
paulson@13957
    75
  then obtain k where "m = p * k" ..
haftmann@57514
    76
  with eq have "p * n\<^sup>2 = p\<^sup>2 * k\<^sup>2" by (auto simp add: power2_eq_square ac_simps)
wenzelm@53015
    77
  with p have "n\<^sup>2 = p * k\<^sup>2" by (simp add: power2_eq_square)
wenzelm@53015
    78
  then have "p dvd n\<^sup>2" ..
wenzelm@59031
    79
  with \<open>prime p\<close> have "p dvd n" by (rule prime_dvd_power_nat)
haftmann@62348
    80
  with dvd_m have "p dvd gcd m n" by (rule gcd_greatest)
haftmann@60690
    81
  with \<open>coprime m n\<close> have "p = 1" by simp
paulson@13957
    82
  with p show False by simp
paulson@13957
    83
qed
paulson@13957
    84
nipkow@45917
    85
wenzelm@59031
    86
text \<open>Another old chestnut, which is a consequence of the irrationality of 2.\<close>
nipkow@45917
    87
wenzelm@59031
    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")
nipkow@45917
    89
proof cases
nipkow@45917
    90
  assume "sqrt 2 powr sqrt 2 \<in> \<rat>"
wenzelm@46495
    91
  then have "?P (sqrt 2) (sqrt 2)"
hoelzl@51708
    92
    by (metis sqrt_2_not_rat)
wenzelm@46495
    93
  then show ?thesis by blast
nipkow@45917
    94
next
nipkow@45917
    95
  assume 1: "sqrt 2 powr sqrt 2 \<notin> \<rat>"
nipkow@45917
    96
  have "(sqrt 2 powr sqrt 2) powr sqrt 2 = 2"
wenzelm@46495
    97
    using powr_realpow [of _ 2]
wenzelm@46495
    98
    by (simp add: powr_powr power2_eq_square [symmetric])
wenzelm@46495
    99
  then have "?P (sqrt 2 powr sqrt 2) (sqrt 2)"
hoelzl@51708
   100
    by (metis 1 Rats_number_of sqrt_2_not_rat)
wenzelm@46495
   101
  then show ?thesis by blast
nipkow@45917
   102
qed
nipkow@45917
   103
paulson@13957
   104
end