(* Title: HOL/ex/Sqrt_Script.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 2001 University of Cambridge
*)
section \<open>Square roots of primes are irrational (script version)\<close>
theory Sqrt_Script
imports Complex_Main "HOL-Computational_Algebra.Primes"
begin
text \<open>
\medskip Contrast this linear Isabelle/Isar script with Markus
Wenzel's more mathematical version.
\<close>
subsection \<open>Preliminaries\<close>
lemma prime_nonzero: "prime (p::nat) \ p \ 0"
by (force simp add: prime_nat_iff)
lemma prime_dvd_other_side:
"(n::nat) * n = p * (k * k) \ prime p \ p dvd n"
apply (subgoal_tac "p dvd n * n", blast dest: prime_dvd_mult_nat)
apply auto
done
lemma reduction: "prime (p::nat) \
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_nat_iff)
done
lemma rearrange: "(j::nat) * (p * j) = k * k \ k * k = p * (j * j)"
by (simp add: ac_simps)
lemma prime_not_square:
"prime (p::nat) \ (\k. 0 < k \ m * m \ 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 \<open>Main theorem\<close>
text \<open>
The square root of any prime number (including \<open>2\<close>) is
irrational.
\<close>
theorem prime_sqrt_irrational:
"prime (p::nat) \ x * x = real p \ 0 \ x \ x \ \"
apply (rule notI)
apply (erule Rats_abs_nat_div_natE)
apply (simp del: of_nat_mult
add: abs_if divide_eq_eq prime_not_square of_nat_mult [symmetric])
done
lemmas two_sqrt_irrational =
prime_sqrt_irrational [OF two_is_prime_nat]
end
¤ Dauer der Verarbeitung: 0.18 Sekunden
(vorverarbeitet)
¤
|
Haftungshinweis
Die Informationen auf dieser Webseite wurden
nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit,
noch Qualität der bereit gestellten Informationen zugesichert.
Bemerkung:
Die farbliche Syntaxdarstellung ist noch experimentell.
|
