Theory Sqrt_Script

(*  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 ∉ ℚ"
  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