(* Title: HOL/ex/Sqrt.thy
Author: Markus Wenzel, Tobias Nipkow, TU Muenchen
*)
header {* Square roots of primes are irrational *}
theory Sqrt
imports Complex_Main "~~/src/HOL/Number_Theory/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 \<notin> \<rat>"
proof
from `prime p` have p: "1 < p" by (simp add: prime_nat_def)
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 gcd: "gcd m n = 1" 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 (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 show ?thesis ..
qed
have "p dvd m \<and> p dvd n"
proof
from eq have "p dvd m\<^sup>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\<^sup>2 = p\<^sup>2 * k\<^sup>2" by (auto simp add: power2_eq_square mult_ac)
with p have "n\<^sup>2 = p * k\<^sup>2" by (simp add: power2_eq_square)
then have "p dvd n\<^sup>2" ..
with `prime p` show "p dvd n" by (rule prime_dvd_power_nat)
qed
then have "p dvd gcd m n" ..
with gcd have "p dvd 1" by simp
then have "p \<le> 1" by (simp add: dvd_imp_le)
with p show False by simp
qed
corollary sqrt_2_not_rat: "sqrt 2 \<notin> \<rat>"
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 \<notin> \<rat>"
proof
from `prime p` have p: "1 < p" by (simp add: prime_nat_def)
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 gcd: "gcd m n = 1" 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" ..
then have "p dvd m\<^sup>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\<^sup>2 = p\<^sup>2 * k\<^sup>2" by (auto simp add: power2_eq_square mult_ac)
with p have "n\<^sup>2 = p * k\<^sup>2" by (simp add: power2_eq_square)
then have "p dvd n\<^sup>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_nat)
with gcd have "p dvd 1" by simp
then have "p \<le> 1" by (simp add: dvd_imp_le)
with p show False by simp
qed
text {* Another old chestnut, which is a consequence of the irrationality of 2. *}
lemma "\<exists>a b::real. a \<notin> \<rat> \<and> b \<notin> \<rat> \<and> a powr b \<in> \<rat>" (is "EX a b. ?P a b")
proof cases
assume "sqrt 2 powr sqrt 2 \<in> \<rat>"
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 \<notin> \<rat>"
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