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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
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