--- a/src/HOL/Complex/ex/Sqrt.thy Mon Jan 12 16:45:35 2004 +0100
+++ b/src/HOL/Complex/ex/Sqrt.thy Mon Jan 12 16:51:45 2004 +0100
@@ -84,7 +84,7 @@
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: power_two real_power_two)
+ 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 ..
@@ -94,8 +94,8 @@
from eq have "p dvd m\<twosuperior>" ..
with p_prime 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: power_two mult_ac)
- with p have "n\<twosuperior> = p * k\<twosuperior>" by (simp add: power_two)
+ 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 p_prime show "p dvd n" by (rule prime_dvd_power_two)
qed
@@ -127,15 +127,15 @@
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: power_two real_power_two)
+ 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 p_prime 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: power_two mult_ac)
- with p have "n\<twosuperior> = p * k\<twosuperior>" by (simp add: power_two)
+ 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 p_prime have "p dvd n" by (rule prime_dvd_power_two)
with dvd_m have "p dvd gcd (m, n)" by (rule gcd_greatest)