# 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
```