--- a/src/HOL/ex/Sqrt_Script.thy Tue Sep 01 14:13:34 2009 +0200
+++ b/src/HOL/ex/Sqrt_Script.thy Tue Sep 01 15:39:33 2009 +0200
@@ -6,7 +6,7 @@
header {* Square roots of primes are irrational (script version) *}
theory Sqrt_Script
-imports Complex_Main Primes
+imports Complex_Main "~~/src/HOL/Number_Theory/Primes"
begin
text {*
@@ -16,30 +16,30 @@
subsection {* Preliminaries *}
-lemma prime_nonzero: "prime p \<Longrightarrow> p \<noteq> 0"
- by (force simp add: prime_def)
+lemma prime_nonzero: "prime (p::nat) \<Longrightarrow> p \<noteq> 0"
+ by (force simp add: prime_nat_def)
lemma prime_dvd_other_side:
- "n * n = p * (k * k) \<Longrightarrow> prime p \<Longrightarrow> p dvd n"
- apply (subgoal_tac "p dvd n * n", blast dest: prime_dvd_mult)
+ "(n::nat) * n = p * (k * k) \<Longrightarrow> prime p \<Longrightarrow> p dvd n"
+ apply (subgoal_tac "p dvd n * n", blast dest: prime_dvd_mult_nat)
apply auto
done
-lemma reduction: "prime p \<Longrightarrow>
+lemma reduction: "prime (p::nat) \<Longrightarrow>
0 < k \<Longrightarrow> k * k = p * (j * j) \<Longrightarrow> k < p * j \<and> 0 < j"
apply (rule ccontr)
apply (simp add: linorder_not_less)
apply (erule disjE)
apply (frule mult_le_mono, assumption)
apply auto
- apply (force simp add: prime_def)
+ apply (force simp add: prime_nat_def)
done
lemma rearrange: "(j::nat) * (p * j) = k * k \<Longrightarrow> k * k = p * (j * j)"
by (simp add: mult_ac)
lemma prime_not_square:
- "prime p \<Longrightarrow> (\<And>k. 0 < k \<Longrightarrow> m * m \<noteq> p * (k * k))"
+ "prime (p::nat) \<Longrightarrow> (\<And>k. 0 < k \<Longrightarrow> m * m \<noteq> p * (k * k))"
apply (induct m rule: nat_less_induct)
apply clarify
apply (frule prime_dvd_other_side, assumption)
@@ -57,7 +57,7 @@
*}
theorem prime_sqrt_irrational:
- "prime p \<Longrightarrow> x * x = real p \<Longrightarrow> 0 \<le> x \<Longrightarrow> x \<notin> \<rat>"
+ "prime (p::nat) \<Longrightarrow> x * x = real p \<Longrightarrow> 0 \<le> x \<Longrightarrow> x \<notin> \<rat>"
apply (rule notI)
apply (erule Rats_abs_nat_div_natE)
apply (simp del: real_of_nat_mult
@@ -65,6 +65,6 @@
done
lemmas two_sqrt_irrational =
- prime_sqrt_irrational [OF two_is_prime]
+ prime_sqrt_irrational [OF two_is_prime_nat]
end