(* Title: HOL/ex/Sqrt_Script.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 2001 University of Cambridge
*)
section ‹Square roots of primes are irrational (script version)
›
text ‹
Contrast this linear Isabelle/Isar script
with the more mathematical version
in 🍋‹~~/src/HOL/Examples/Sqrt.thy
› by Makarius Wenzel.
›
theory Sqrt_Script
imports Complex_Main
"HOL-Computational_Algebra.Primes"
begin
subsection ‹Preliminaries
›
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
==> k * k = p * (j * j)
==> k < p * j
∧ 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 ‹Main
theorem›
text ‹
The square root of any prime number (including
‹2
›)
is
irrational.
›
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
