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-(* Example proof by Larry Paulson; see http://www.cs.kun.nl/~freek/comparison/ 
-   Taken from Isabelle2005 distribution. *)
-
-
-(*  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 Root2_Tactic
-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 (rule_tac j = "k * k" in dvd_mult_left, simp)
-  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 {* The set of rational numbers *}
-
-constdefs
-  rationals :: "real set"    ("\<rat>")
-  "\<rat> \<equiv> {x. \<exists>m n. n \<noteq> 0 \<and> \<bar>x\<bar> = real (m::nat) / real (n::nat)}"
-
-
-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 (simp add: rationals_def real_abs_def)
-  apply clarify
-  apply (erule_tac P = "real m / real n * ?x = ?y" in rev_mp)
-  apply (simp del: real_of_nat_mult
-              add: divide_eq_eq prime_not_square real_of_nat_mult [symmetric])
-  done
-
-lemmas two_sqrt_irrational =
-  prime_sqrt_irrational [OF two_is_prime]
-
-end