--- a/src/HOL/Complex/ex/Sqrt_Script.thy Fri Dec 05 11:26:07 2008 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,71 +0,0 @@
-(* Title: HOL/Hyperreal/ex/Sqrt_Script.thy
- ID: $Id$
- Author: Lawrence C Paulson, Cambridge University Computer Laboratory
- Copyright 2001 University of Cambridge
-*)
-
-header {* Square roots of primes are irrational (script version) *}
-
-theory Sqrt_Script
-imports Primes Complex_Main
-begin
-
-text {*
- \medskip Contrast this linear Isabelle/Isar script with Markus
- Wenzel's more mathematical version.
-*}
-
-subsection {* Preliminaries *}
-
-lemma prime_nonzero: "prime p \<Longrightarrow> p \<noteq> 0"
- by (force simp add: prime_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)
- apply auto
- done
-
-lemma reduction: "prime p \<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)
- 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))"
- apply (induct m rule: nat_less_induct)
- apply clarify
- apply (frule prime_dvd_other_side, assumption)
- apply (erule dvdE)
- apply (simp add: nat_mult_eq_cancel_disj prime_nonzero)
- apply (blast dest: rearrange reduction)
- done
-
-
-subsection {* Main theorem *}
-
-text {*
- The square root of any prime number (including @{text 2}) is
- irrational.
-*}
-
-theorem prime_sqrt_irrational:
- "prime p \<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
- add: real_abs_def divide_eq_eq prime_not_square real_of_nat_mult [symmetric])
- done
-
-lemmas two_sqrt_irrational =
- prime_sqrt_irrational [OF two_is_prime]
-
-end