header {* Square roots of primes are irrational (script version) *}
theory Sqrt_Script
imports Complex_Main Primes
begin
text {*
\medskip Contrast this linear Isabelle/Isar script with Markus
Wenzel's more mathematical version.
*}
subsection {* Preliminaries *}
lemma prime_nonzero: "prime p ==> p ≠ 0"
by (force simp add: prime_def)
lemma prime_dvd_other_side:
"n * n = p * (k * k) ==> prime p ==> p dvd n"
apply (subgoal_tac "p dvd n * n", blast dest: prime_dvd_mult)
apply auto
done
lemma reduction: "prime p ==>
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_def)
done
lemma rearrange: "(j::nat) * (p * j) = k * k ==> k * k = p * (j * j)"
by (simp add: mult_ac)
lemma prime_not_square:
"prime p ==> (!!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 @{text 2}) is
irrational.
*}
theorem prime_sqrt_irrational:
"prime p ==> x * x = real p ==> 0 ≤ x ==> x ∉ \<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