src/HOL/Library/Primes.thy
author paulson
Wed Jul 18 11:43:06 2007 +0200 (2007-07-18)
changeset 23839 d9fa0f457d9a
parent 22665 cf152ff55d16
child 25593 0b0df6c8646a
permissions -rw-r--r--
tidying using metis
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(*  Title:      HOL/Library/Primes.thy
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    ID:         $Id$
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    Author:     Christophe Tabacznyj and Lawrence C Paulson
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    Copyright   1996  University of Cambridge
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*)
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header {* Primality on nat *}
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theory Primes
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imports GCD
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begin
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definition
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  coprime :: "nat => nat => bool" where
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  "coprime m n = (gcd (m, n) = 1)"
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definition
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  prime :: "nat \<Rightarrow> bool" where
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  "prime p = (1 < p \<and> (\<forall>m. m dvd p --> m = 1 \<or> m = p))"
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lemma two_is_prime: "prime 2"
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  apply (auto simp add: prime_def)
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  apply (case_tac m)
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   apply (auto dest!: dvd_imp_le)
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  done
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lemma prime_imp_relprime: "prime p ==> \<not> p dvd n ==> gcd (p, n) = 1"
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  apply (auto simp add: prime_def)
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  apply (metis One_nat_def gcd_dvd1 gcd_dvd2)
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  done
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text {*
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  This theorem leads immediately to a proof of the uniqueness of
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  factorization.  If @{term p} divides a product of primes then it is
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  one of those primes.
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*}
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lemma prime_dvd_mult: "prime p ==> p dvd m * n ==> p dvd m \<or> p dvd n"
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  by (blast intro: relprime_dvd_mult prime_imp_relprime)
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lemma prime_dvd_square: "prime p ==> p dvd m^Suc (Suc 0) ==> p dvd m"
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  by (auto dest: prime_dvd_mult)
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lemma prime_dvd_power_two: "prime p ==> p dvd m\<twosuperior> ==> p dvd m"
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  by (rule prime_dvd_square) (simp_all add: power2_eq_square)
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end