(* Title: HOL/Library/Primes.thy ID: $Id: Primes.thy,v 1.22 2007/07/18 09:43:06 paulson Exp $ Author: Christophe Tabacznyj and Lawrence C Paulson Copyright 1996 University of Cambridge *) header {* Primality on nat *} theory Primes imports GCD begin definition coprime :: "nat => nat => bool" where "coprime m n = (gcd (m, n) = 1)" definition prime :: "nat => bool" where "prime p = (1 < p ∧ (∀m. m dvd p --> m = 1 ∨ m = p))" lemma two_is_prime: "prime 2" apply (auto simp add: prime_def) apply (case_tac m) apply (auto dest!: dvd_imp_le) done lemma prime_imp_relprime: "prime p ==> ¬ p dvd n ==> gcd (p, n) = 1" apply (auto simp add: prime_def) apply (metis One_nat_def gcd_dvd1 gcd_dvd2) done text {* This theorem leads immediately to a proof of the uniqueness of factorization. If @{term p} divides a product of primes then it is one of those primes. *} lemma prime_dvd_mult: "prime p ==> p dvd m * n ==> p dvd m ∨ p dvd n" by (blast intro: relprime_dvd_mult prime_imp_relprime) lemma prime_dvd_square: "prime p ==> p dvd m^Suc (Suc 0) ==> p dvd m" by (auto dest: prime_dvd_mult) lemma prime_dvd_power_two: "prime p ==> p dvd m² ==> p dvd m" by (rule prime_dvd_square) (simp_all add: power2_eq_square) end
lemma two_is_prime:
prime 2
lemma prime_imp_relprime:
[| prime p; ¬ p dvd n |] ==> gcd (p, n) = 1
lemma prime_dvd_mult:
[| prime p; p dvd m * n |] ==> p dvd m ∨ p dvd n
lemma prime_dvd_square:
[| prime p; p dvd m ^ Suc (Suc 0) |] ==> p dvd m
lemma prime_dvd_power_two:
[| prime p; p dvd m ^ 2 |] ==> p dvd m