src/HOL/ex/Sqrt.thy
 author wenzelm Tue, 10 Mar 2009 16:48:27 +0100 changeset 30411 9c9b6511ad1b parent 28952 15a4b2cf8c34 child 31712 6f8aa9aea693 permissions -rw-r--r--
tuned proofs; tuned document;
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(*  Title:      HOL/ex/Sqrt.thy
Author:     Markus Wenzel, TU Muenchen
*)

header {*  Square roots of primes are irrational *}

theory Sqrt
imports Complex_Main Primes
begin

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" by (rule Rats_abs_nat_div_natE)
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>)"
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" by (rule Rats_abs_nat_div_natE)
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>)"
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
```