# Theory Sqrt

theory Sqrt
imports Complex_Main Primes
```(*  Title:      HOL/ex/Sqrt.thy
Author:     Markus Wenzel, 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:
assumes "prime (p::nat)"
shows "sqrt p ∉ ℚ"
proof
from ‹prime p› have p: "1 < p" by (simp add: prime_nat_iff)
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"
also have "(sqrt p)⇧2 = p" by simp
also have "… * n⇧2 = p * n⇧2" by simp
finally show ?thesis using of_nat_eq_iff by blast
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_nat)
then obtain k where "m = p * k" ..
with eq have "p * n⇧2 = p⇧2 * k⇧2" by (auto simp add: power2_eq_square ac_simps)
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_nat)
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

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::nat)"
shows "sqrt p ∉ ℚ"
proof
from ‹prime p› have p: "1 < p" by (simp add: prime_nat_iff)
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"
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" using of_nat_eq_iff by blast
then have "p dvd m⇧2" ..
with ‹prime p› have dvd_m: "p dvd m" by (rule prime_dvd_power_nat)
then obtain k where "m = p * k" ..
with eq have "p * n⇧2 = p⇧2 * k⇧2" by (auto simp add: power2_eq_square ac_simps)
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_nat)
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 2.›

lemma "∃a b::real. a ∉ ℚ ∧ b ∉ ℚ ∧ a powr b ∈ ℚ" (is "∃a b. ?P a b")
proof cases
assume "sqrt 2 powr sqrt 2 ∈ ℚ"
then have "?P (sqrt 2) (sqrt 2)"
by (metis sqrt_2_not_rat)
then show ?thesis by blast
next
assume 1: "sqrt 2 powr sqrt 2 ∉ ℚ"
have "(sqrt 2 powr sqrt 2) powr sqrt 2 = 2"
using powr_realpow [of _ 2]
by (simp add: powr_powr power2_eq_square [symmetric])
then have "?P (sqrt 2 powr sqrt 2) (sqrt 2)"
by (metis 1 Rats_number_of sqrt_2_not_rat)
then show ?thesis by blast
qed

end
```