(* Title: HOL/ex/Sqrt.thy
Author: Makarius
Author: Tobias Nipkow, TU Muenchen
*)
section \<open>Square roots of primes are irrational\<close>
theory Sqrt
imports Complex_Main "HOL-Computational_Algebra.Primes"
begin
text \<open>
The square root of any prime number (including 2) is irrational.
\<close>
theorem sqrt_prime_irrational:
fixes p :: nat
assumes "prime p"
shows "sqrt p \<notin> \<rat>"
proof
from \<open>prime p\<close> have p: "p > 1" by (rule prime_gt_1_nat)
assume "sqrt p \<in> \<rat>"
then obtain m n :: nat
where n: "n \<noteq> 0"
and sqrt_rat: "\<bar>sqrt p\<bar> = m / n"
and "coprime m n" by (rule Rats_abs_nat_div_natE)
have eq: "m\<^sup>2 = p * n\<^sup>2"
proof -
from n and sqrt_rat have "m = \<bar>sqrt p\<bar> * n" by simp
then have "m\<^sup>2 = (sqrt p)\<^sup>2 * n\<^sup>2" by (simp add: power_mult_distrib)
also have "(sqrt p)\<^sup>2 = p" by simp
also have "\<dots> * n\<^sup>2 = p * n\<^sup>2" by simp
finally show ?thesis by linarith
qed
have "p dvd m \<and> p dvd n"
proof
from eq have "p dvd m\<^sup>2" ..
with \<open>prime p\<close> show "p dvd m" by (rule prime_dvd_power)
then obtain k where "m = p * k" ..
with eq have "p * n\<^sup>2 = p\<^sup>2 * k\<^sup>2" by algebra
with p have "n\<^sup>2 = p * k\<^sup>2" by (simp add: power2_eq_square)
then have "p dvd n\<^sup>2" ..
with \<open>prime p\<close> show "p dvd n" by (rule prime_dvd_power)
qed
then have "p dvd gcd m n" by simp
with \<open>coprime m n\<close> have "p = 1" by simp
with p show False by simp
qed
corollary sqrt_2_not_rat: "sqrt 2 \<notin> \<rat>"
using sqrt_prime_irrational [of 2] by simp
text \<open>
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.
\<close>
theorem
fixes p :: nat
assumes "prime p"
shows "sqrt p \<notin> \<rat>"
proof
from \<open>prime p\<close> have p: "p > 1" by (rule prime_gt_1_nat)
assume "sqrt p \<in> \<rat>"
then obtain m n :: nat
where n: "n \<noteq> 0"
and sqrt_rat: "\<bar>sqrt p\<bar> = m / n"
and "coprime m n" by (rule Rats_abs_nat_div_natE)
from n and sqrt_rat have "m = \<bar>sqrt p\<bar> * n" by simp
then have "m\<^sup>2 = (sqrt p)\<^sup>2 * n\<^sup>2" by (auto simp add: power2_eq_square)
also have "(sqrt p)\<^sup>2 = p" by simp
also have "\<dots> * n\<^sup>2 = p * n\<^sup>2" by simp
finally have eq: "m\<^sup>2 = p * n\<^sup>2" by linarith
then have "p dvd m\<^sup>2" ..
with \<open>prime p\<close> have dvd_m: "p dvd m" by (rule prime_dvd_power)
then obtain k where "m = p * k" ..
with eq have "p * n\<^sup>2 = p\<^sup>2 * k\<^sup>2" by algebra
with p have "n\<^sup>2 = p * k\<^sup>2" by (simp add: power2_eq_square)
then have "p dvd n\<^sup>2" ..
with \<open>prime p\<close> have "p dvd n" by (rule prime_dvd_power)
with dvd_m have "p dvd gcd m n" by (rule gcd_greatest)
with \<open>coprime m n\<close> have "p = 1" by simp
with p show False by simp
qed
text \<open>
Another old chestnut, which is a consequence of the irrationality of
\<^term>\<open>sqrt 2\<close>.
\<close>
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")
proof (cases "sqrt 2 powr sqrt 2 \<in> \<rat>")
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