Up to index of Isabelle/HOL/HOL-Complex/ex
theory Sqrt_Script(* 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
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)
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>