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theory Sqrt(* Title: HOL/Hyperreal/ex/Sqrt.thy ID: $Id: Sqrt.thy,v 1.7 2005/07/01 15:41:15 nipkow Exp $ Author: Markus Wenzel, TU Muenchen *) header {* Square roots of primes are irrational *} theory Sqrt imports Primes Complex_Main begin subsection {* Preliminaries *} text {* The set of rational numbers, including the key representation theorem. *} constdefs rationals :: "real set" ("\<rat>") "\<rat> ≡ {x. ∃m n. n ≠ 0 ∧ ¦x¦ = real (m::nat) / real (n::nat)}" theorem rationals_rep: "x ∈ \<rat> ==> ∃m n. n ≠ 0 ∧ ¦x¦ = real m / real n ∧ gcd (m, n) = 1" proof - assume "x ∈ \<rat>" then obtain m n :: nat where n: "n ≠ 0" and x_rat: "¦x¦ = real m / real n" by (unfold rationals_def) blast let ?gcd = "gcd (m, n)" from n have gcd: "?gcd ≠ 0" by (simp add: gcd_zero) let ?k = "m div ?gcd" let ?l = "n div ?gcd" let ?gcd' = "gcd (?k, ?l)" have "?gcd dvd m" .. then have gcd_k: "?gcd * ?k = m" by (rule dvd_mult_div_cancel) have "?gcd dvd n" .. then have gcd_l: "?gcd * ?l = n" by (rule dvd_mult_div_cancel) from n and gcd_l have "?l ≠ 0" by (auto iff del: neq0_conv) moreover have "¦x¦ = real ?k / real ?l" proof - from gcd have "real ?k / real ?l = real (?gcd * ?k) / real (?gcd * ?l)" by (simp add: mult_divide_cancel_left) also from gcd_k and gcd_l have "… = real m / real n" by simp also from x_rat have "… = ¦x¦" .. finally show ?thesis .. qed moreover have "?gcd' = 1" proof - have "?gcd * ?gcd' = gcd (?gcd * ?k, ?gcd * ?l)" by (rule gcd_mult_distrib2) with gcd_k gcd_l have "?gcd * ?gcd' = ?gcd" by simp with gcd show ?thesis by simp qed ultimately show ?thesis by blast qed lemma [elim?]: "r ∈ \<rat> ==> (!!m n. n ≠ 0 ==> ¦r¦ = real m / real n ==> gcd (m, n) = 1 ==> C) ==> C" using rationals_rep by blast subsection {* Main theorem *} text {* The square root of any prime number (including @{text 2}) is irrational. *} theorem sqrt_prime_irrational: "prime p ==> sqrt (real p) ∉ \<rat>" proof assume p_prime: "prime p" then have p: "1 < p" by (simp add: prime_def) assume "sqrt (real p) ∈ \<rat>" then obtain m n where n: "n ≠ 0" and sqrt_rat: "¦sqrt (real p)¦ = real m / real n" and gcd: "gcd (m, n) = 1" .. have eq: "m² = p * n²" proof - from n and sqrt_rat have "real m = ¦sqrt (real p)¦ * real n" by simp then have "real (m²) = (sqrt (real p))² * real (n²)" by (auto simp add: power2_eq_square) also have "(sqrt (real p))² = real p" by simp also have "… * real (n²) = real (p * n²)" by simp finally show ?thesis .. qed have "p dvd m ∧ p dvd n" proof from eq have "p dvd m²" .. with p_prime show "p dvd m" by (rule prime_dvd_power_two) then obtain k where "m = p * k" .. with eq have "p * n² = p² * k²" by (auto simp add: power2_eq_square mult_ac) with p have "n² = p * k²" by (simp add: power2_eq_square) then have "p dvd n²" .. with p_prime show "p dvd n" by (rule prime_dvd_power_two) qed then have "p dvd gcd (m, n)" .. with gcd have "p dvd 1" by simp then have "p ≤ 1" by (simp add: dvd_imp_le) with p show False by simp qed corollary "sqrt (real (2::nat)) ∉ \<rat>" by (rule sqrt_prime_irrational) (rule two_is_prime) subsection {* Variations *} text {* Here is an alternative version of the main proof, using mostly linear forward-reasoning. While this results in less top-down structure, it is probably closer to proofs seen in mathematics. *} theorem "prime p ==> sqrt (real p) ∉ \<rat>" proof assume p_prime: "prime p" then have p: "1 < p" by (simp add: prime_def) assume "sqrt (real p) ∈ \<rat>" then obtain m n where n: "n ≠ 0" and sqrt_rat: "¦sqrt (real p)¦ = real m / real n" and gcd: "gcd (m, n) = 1" .. from n and sqrt_rat have "real m = ¦sqrt (real p)¦ * real n" by simp then have "real (m²) = (sqrt (real p))² * real (n²)" by (auto simp add: power2_eq_square) also have "(sqrt (real p))² = real p" by simp also have "… * real (n²) = real (p * n²)" by simp finally have eq: "m² = p * n²" .. then have "p dvd m²" .. with p_prime have dvd_m: "p dvd m" by (rule prime_dvd_power_two) then obtain k where "m = p * k" .. with eq have "p * n² = p² * k²" by (auto simp add: power2_eq_square mult_ac) with p have "n² = p * k²" by (simp add: power2_eq_square) then have "p dvd n²" .. with p_prime have "p dvd n" by (rule prime_dvd_power_two) with dvd_m have "p dvd gcd (m, n)" by (rule gcd_greatest) with gcd have "p dvd 1" by simp then have "p ≤ 1" by (simp add: dvd_imp_le) with p show False by simp qed end
theorem rationals_rep:
x ∈ \<rat> ==> ∃m n. n ≠ 0 ∧ ¦x¦ = real m / real n ∧ gcd (m, n) = 1
lemma
[| r ∈ \<rat>; !!m n. [| n ≠ 0; ¦r¦ = real m / real n; gcd (m, n) = 1 |] ==> C |] ==> C
theorem sqrt_prime_irrational:
prime p ==> sqrt (real p) ∉ \<rat>
corollary
sqrt (real 2) ∉ \<rat>
theorem
prime p ==> sqrt (real p) ∉ \<rat>