src/HOL/Library/Primes.thy
author wenzelm
Thu Feb 16 21:12:00 2006 +0100 (2006-02-16)
changeset 19086 1b3780be6cc2
parent 16762 aafd23b47a5d
child 21256 47195501ecf7
permissions -rw-r--r--
new-style definitions/abbreviations;
wenzelm@11368
     1
(*  Title:      HOL/Library/Primes.thy
paulson@11363
     2
    ID:         $Id$
paulson@11363
     3
    Author:     Christophe Tabacznyj and Lawrence C Paulson
paulson@11363
     4
    Copyright   1996  University of Cambridge
paulson@11363
     5
*)
paulson@11363
     6
nipkow@16762
     7
header {* Primality on nat *}
paulson@11363
     8
nipkow@15131
     9
theory Primes
nipkow@15140
    10
imports Main
nipkow@15131
    11
begin
paulson@11363
    12
wenzelm@19086
    13
definition
paulson@11363
    14
  coprime :: "nat => nat => bool"
wenzelm@19086
    15
  "coprime m n = (gcd (m, n) = 1)"
paulson@11363
    16
nipkow@16663
    17
  prime :: "nat \<Rightarrow> bool"
wenzelm@19086
    18
  "prime p = (1 < p \<and> (\<forall>m. m dvd p --> m = 1 \<or> m = p))"
paulson@11363
    19
paulson@11363
    20
nipkow@16762
    21
lemma two_is_prime: "prime 2"
nipkow@16762
    22
  apply (auto simp add: prime_def)
nipkow@16762
    23
  apply (case_tac m)
nipkow@16762
    24
   apply (auto dest!: dvd_imp_le)
paulson@11363
    25
  done
paulson@11363
    26
nipkow@16663
    27
lemma prime_imp_relprime: "prime p ==> \<not> p dvd n ==> gcd (p, n) = 1"
paulson@11363
    28
  apply (auto simp add: prime_def)
paulson@11363
    29
  apply (drule_tac x = "gcd (p, n)" in spec)
paulson@11363
    30
  apply auto
paulson@11363
    31
  apply (insert gcd_dvd2 [of p n])
paulson@11363
    32
  apply simp
paulson@11363
    33
  done
paulson@11363
    34
paulson@11363
    35
text {*
paulson@11363
    36
  This theorem leads immediately to a proof of the uniqueness of
paulson@11363
    37
  factorization.  If @{term p} divides a product of primes then it is
paulson@11363
    38
  one of those primes.
paulson@11363
    39
*}
paulson@11363
    40
nipkow@16663
    41
lemma prime_dvd_mult: "prime p ==> p dvd m * n ==> p dvd m \<or> p dvd n"
wenzelm@12739
    42
  by (blast intro: relprime_dvd_mult prime_imp_relprime)
paulson@11363
    43
nipkow@16663
    44
lemma prime_dvd_square: "prime p ==> p dvd m^Suc (Suc 0) ==> p dvd m"
wenzelm@12739
    45
  by (auto dest: prime_dvd_mult)
wenzelm@12739
    46
nipkow@16663
    47
lemma prime_dvd_power_two: "prime p ==> p dvd m\<twosuperior> ==> p dvd m"
paulson@14353
    48
  by (rule prime_dvd_square) (simp_all add: power2_eq_square)
wenzelm@11368
    49
paulson@11363
    50
paulson@11363
    51
end