# Theory Sqrt_Script

Up to index of Isabelle/HOL/HOL-Complex/ex

theory Sqrt_Script
imports Primes Complex_Main
begin

    (*  Title:      HOL/Hyperreal/ex/Sqrt_Script.thy
        ID:         $Id: Sqrt_Script.thy,v 1.5 2005/07/01 15:41:15 nipkow Exp $
        Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
        Copyright   2001  University of Cambridge
    *)

    header {* Square roots of primes are irrational (script version) *}

    theory Sqrt_Script
    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 ==> 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 (rule_tac j = "k * k" in dvd_mult_left, simp)
      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 {* The set of rational numbers *}

    constdefs
      rationals :: "real set"    ("\<rat>")
      "\<rat> ≡ {x. ∃m n. n ≠ 0 ∧ ¦x¦ = 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 ==> x * x = real p ==> 0 ≤ x ==> x ∉ \<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

### Preliminaries

lemma prime_nonzero:

      prime p ==> p ≠ 0

lemma prime_dvd_other_side:

      [| n * n = p * (k * k); prime p |] ==> p dvd n

lemma reduction:

      [| prime p; 0 < k; k * k = p * (j * j) |] ==> k < p * j ∧ 0 < j

lemma rearrange:

      j * (p * j) = k * k ==> k * k = p * (j * j)

lemma prime_not_square:

      [| prime p; 0 < k |] ==> m * m ≠ p * (k * k)

### The set of rational numbers

### Main theorem

theorem prime_sqrt_irrational:

      [| prime p; x * x = real p; 0 ≤ x |] ==> x ∉ \<rat>

lemmas two_sqrt_irrational:

      [| x * x = real 2; 0 ≤ x |] ==> x ∉ \<rat>

lemmas two_sqrt_irrational:

      [| x * x = real 2; 0 ≤ x |] ==> x ∉ \<rat>