(* Title: HOL/Hyperreal/ex/Sqrt.thy
ID: $Id$
Author: Markus Wenzel, TU Muenchen
*)
header {* Square roots of primes are irrational *}
theory Sqrt
imports Primes Complex_Main
begin
subsection {* Preliminaries *}
text {*
The set of rational numbers, including the key representation
theorem.
*}
definition
rationals ("\<rat>") where
"\<rat> = {x. \<exists>m n. n \<noteq> 0 \<and> \<bar>x\<bar> = real (m::nat) / real (n::nat)}"
theorem rationals_rep [elim?]:
assumes "x \<in> \<rat>"
obtains m n where "n \<noteq> 0" and "\<bar>x\<bar> = real m / real n" and "gcd m n = 1"
proof -
from `x \<in> \<rat>` obtain m n :: nat where
n: "n \<noteq> 0" and x_rat: "\<bar>x\<bar> = real m / real n"
unfolding rationals_def by blast
let ?gcd = "gcd m n"
from n have gcd: "?gcd \<noteq> 0" by (simp add: gcd_zero)
let ?k = "m div ?gcd"
let ?l = "n div ?gcd"
let ?gcd' = "gcd ?k ?l"
have "?gcd dvd m" .. then have gcd_k: "?gcd * ?k = m"
by (rule dvd_mult_div_cancel)
have "?gcd dvd n" .. then have gcd_l: "?gcd * ?l = n"
by (rule dvd_mult_div_cancel)
from n and gcd_l have "?l \<noteq> 0"
by (auto iff del: neq0_conv)
moreover
have "\<bar>x\<bar> = real ?k / real ?l"
proof -
from gcd have "real ?k / real ?l =
real (?gcd * ?k) / real (?gcd * ?l)" by simp
also from gcd_k and gcd_l have "\<dots> = real m / real n" by simp
also from x_rat have "\<dots> = \<bar>x\<bar>" ..
finally show ?thesis ..
qed
moreover
have "?gcd' = 1"
proof -
have "?gcd * ?gcd' = gcd (?gcd * ?k) (?gcd * ?l)"
by (rule gcd_mult_distrib2)
with gcd_k gcd_l have "?gcd * ?gcd' = ?gcd" by simp
with gcd show ?thesis by simp
qed
ultimately show ?thesis ..
qed
subsection {* Main theorem *}
text {*
The square root of any prime number (including @{text 2}) is
irrational.
*}
theorem sqrt_prime_irrational:
assumes "prime p"
shows "sqrt (real p) \<notin> \<rat>"
proof
from `prime p` have p: "1 < p" by (simp add: prime_def)
assume "sqrt (real p) \<in> \<rat>"
then obtain m n where
n: "n \<noteq> 0" and sqrt_rat: "\<bar>sqrt (real p)\<bar> = real m / real n"
and gcd: "gcd m n = 1" ..
have eq: "m\<twosuperior> = p * n\<twosuperior>"
proof -
from n and sqrt_rat have "real m = \<bar>sqrt (real p)\<bar> * real n" by simp
then have "real (m\<twosuperior>) = (sqrt (real p))\<twosuperior> * real (n\<twosuperior>)"
by (auto simp add: power2_eq_square)
also have "(sqrt (real p))\<twosuperior> = real p" by simp
also have "\<dots> * real (n\<twosuperior>) = real (p * n\<twosuperior>)" by simp
finally show ?thesis ..
qed
have "p dvd m \<and> p dvd n"
proof
from eq have "p dvd m\<twosuperior>" ..
with `prime p` show "p dvd m" by (rule prime_dvd_power_two)
then obtain k where "m = p * k" ..
with eq have "p * n\<twosuperior> = p\<twosuperior> * k\<twosuperior>" by (auto simp add: power2_eq_square mult_ac)
with p have "n\<twosuperior> = p * k\<twosuperior>" by (simp add: power2_eq_square)
then have "p dvd n\<twosuperior>" ..
with `prime p` show "p dvd n" by (rule prime_dvd_power_two)
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 (real (2::nat)) \<notin> \<rat>"
by (rule sqrt_prime_irrational) (rule two_is_prime)
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"
shows "sqrt (real p) \<notin> \<rat>"
proof
from `prime p` have p: "1 < p" by (simp add: prime_def)
assume "sqrt (real p) \<in> \<rat>"
then obtain m n where
n: "n \<noteq> 0" and sqrt_rat: "\<bar>sqrt (real p)\<bar> = real m / real n"
and gcd: "gcd m n = 1" ..
from n and sqrt_rat have "real m = \<bar>sqrt (real p)\<bar> * real n" by simp
then have "real (m\<twosuperior>) = (sqrt (real p))\<twosuperior> * real (n\<twosuperior>)"
by (auto simp add: power2_eq_square)
also have "(sqrt (real p))\<twosuperior> = real p" by simp
also have "\<dots> * real (n\<twosuperior>) = real (p * n\<twosuperior>)" by simp
finally have eq: "m\<twosuperior> = p * n\<twosuperior>" ..
then have "p dvd m\<twosuperior>" ..
with `prime p` have dvd_m: "p dvd m" by (rule prime_dvd_power_two)
then obtain k where "m = p * k" ..
with eq have "p * n\<twosuperior> = p\<twosuperior> * k\<twosuperior>" by (auto simp add: power2_eq_square mult_ac)
with p have "n\<twosuperior> = p * k\<twosuperior>" by (simp add: power2_eq_square)
then have "p dvd n\<twosuperior>" ..
with `prime p` have "p dvd n" by (rule prime_dvd_power_two)
with dvd_m have "p dvd gcd m n" by (rule gcd_greatest)
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
end