--- a/src/HOL/ex/Sqrt.thy Thu Jun 18 08:27:21 2009 -0700
+++ b/src/HOL/ex/Sqrt.thy Thu Jun 18 08:45:26 2009 -0700
@@ -5,7 +5,7 @@
header {* Square roots of primes are irrational *}
theory Sqrt
-imports Complex_Main Primes
+imports Complex_Main
begin
text {*
@@ -14,12 +14,12 @@
*}
theorem sqrt_prime_irrational:
- assumes "prime p"
+ assumes "prime (p::nat)"
shows "sqrt (real p) \<notin> \<rat>"
proof
- from `prime p` have p: "1 < p" by (simp add: prime_def)
+ from `prime p` have p: "1 < p" by (simp add: prime_nat_def)
assume "sqrt (real p) \<in> \<rat>"
- then obtain m n where
+ then obtain m n :: nat 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>"
@@ -34,12 +34,12 @@
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)
+ with `prime p` pos2 show "p dvd m" by (rule nat_prime_dvd_power)
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)
+ with `prime p` pos2 show "p dvd n" by (rule nat_prime_dvd_power)
qed
then have "p dvd gcd m n" ..
with gcd have "p dvd 1" by simp
@@ -48,7 +48,7 @@
qed
corollary "sqrt (real (2::nat)) \<notin> \<rat>"
- by (rule sqrt_prime_irrational) (rule two_is_prime)
+ by (rule sqrt_prime_irrational) (rule nat_two_is_prime)
subsection {* Variations *}
@@ -60,12 +60,12 @@
*}
theorem
- assumes "prime p"
+ assumes "prime (p::nat)"
shows "sqrt (real p) \<notin> \<rat>"
proof
- from `prime p` have p: "1 < p" by (simp add: prime_def)
+ from `prime p` have p: "1 < p" by (simp add: prime_nat_def)
assume "sqrt (real p) \<in> \<rat>"
- then obtain m n where
+ then obtain m n :: nat 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
@@ -75,13 +75,13 @@
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)
+ with `prime p` pos2 have dvd_m: "p dvd m" by (rule nat_prime_dvd_power)
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 `prime p` pos2 have "p dvd n" by (rule nat_prime_dvd_power)
+ with dvd_m have "p dvd gcd m n" by (rule nat_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