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 1.7 2006/11/17 01:20:24 wenzelm 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 *}

definition
  rationals :: "real set"    ("\<rat>") where
  "\<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 * j0 < 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>

lemma two_sqrt_irrational:

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