Theory Sqrt

(*  Title:      HOL/Examples/Sqrt.thy
    Author:     Makarius
    Author:     Tobias Nipkow, TU Muenchen
*)

section ‹Square roots of primes are irrational›

theory Sqrt
  imports Complex_Main "HOL-Computational_Algebra.Primes"
begin

text ‹
  The square root of any prime number (including 2) is irrational.
›

theorem sqrt_prime_irrational:
  fixes p :: nat
  assumes "prime p"
  shows "sqrt p ∉ ℚ"
proof
  from ‹prime p› have p: "p > 1" by (rule prime_gt_1_nat)
  assume "sqrt p ∈ ℚ"
  then obtain m n :: nat
    where n: "n ≠ 0"
      and sqrt_rat: "¦sqrt p¦ = m / n"
      and "coprime m n" by (rule Rats_abs_nat_div_natE)
  have eq: "m⇧2 = p * n⇧2"
  proof -
    from n and sqrt_rat have "m = ¦sqrt p¦ * n" by simp
    then have "m⇧2 = (sqrt p)⇧2 * n⇧2" by (simp add: power_mult_distrib)
    also have "(sqrt p)⇧2 = p" by simp
    also have "… * n⇧2 = p * n⇧2" by simp
    finally show ?thesis by linarith
  qed
  have "p dvd m ∧ p dvd n"
  proof
    from eq have "p dvd m⇧2" ..
    with ‹prime p› show "p dvd m" by (rule prime_dvd_power)
    then obtain k where "m = p * k" ..
    with eq have "p * n⇧2 = p⇧2 * k⇧2" by algebra
    with p have "n⇧2 = p * k⇧2" by (simp add: power2_eq_square)
    then have "p dvd n⇧2" ..
    with ‹prime p› show "p dvd n" by (rule prime_dvd_power)
  qed
  then have "p dvd gcd m n" by simp
  with ‹coprime m n› have "p = 1" by simp
  with p show False by simp
qed

corollary sqrt_2_not_rat: "sqrt 2 ∉ ℚ"
  using sqrt_prime_irrational [of 2] by simp

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
  fixes p :: nat
  assumes "prime p"
  shows "sqrt p ∉ ℚ"
proof
  from ‹prime p› have p: "p > 1" by (rule prime_gt_1_nat)
  assume "sqrt p ∈ ℚ"
  then obtain m n :: nat
    where n: "n ≠ 0"
      and sqrt_rat: "¦sqrt p¦ = m / n"
      and "coprime m n" by (rule Rats_abs_nat_div_natE)
  from n and sqrt_rat have "m = ¦sqrt p¦ * n" by simp
  then have "m⇧2 = (sqrt p)⇧2 * n⇧2" by (auto simp add: power2_eq_square)
  also have "(sqrt p)⇧2 = p" by simp
  also have "… * n⇧2 = p * n⇧2" by simp
  finally have eq: "m⇧2 = p * n⇧2" by linarith
  then have "p dvd m⇧2" ..
  with ‹prime p› have dvd_m: "p dvd m" by (rule prime_dvd_power)
  then obtain k where "m = p * k" ..
  with eq have "p * n⇧2 = p⇧2 * k⇧2" by algebra
  with p have "n⇧2 = p * k⇧2" by (simp add: power2_eq_square)
  then have "p dvd n⇧2" ..
  with ‹prime p› have "p dvd n" by (rule prime_dvd_power)
  with dvd_m have "p dvd gcd m n" by (rule gcd_greatest)
  with ‹coprime m n› have "p = 1" by simp
  with p show False by simp
qed


text ‹
  Another old chestnut, which is a consequence of the irrationality of
  \<^term>‹sqrt 2›.
›

lemma "∃a b::real. a ∉ ℚ ∧ b ∉ ℚ ∧ a powr b ∈ ℚ" (is "∃a b. ?P a b")
proof (cases "sqrt 2 powr sqrt 2 ∈ ℚ")
  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