src/HOL/Number_Theory/Primes.thy
author wenzelm
Sat Sep 10 23:27:32 2011 +0200 (2011-09-10)
changeset 44872 a98ef45122f3
parent 44821 a92f65e174cf
child 45605 a89b4bc311a5
permissions -rw-r--r--
misc tuning;
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(*  Authors:    Christophe Tabacznyj, Lawrence C. Paulson, Amine Chaieb,
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                Thomas M. Rasmussen, Jeremy Avigad, Tobias Nipkow
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This file deals with properties of primes. Definitions and lemmas are
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proved uniformly for the natural numbers and integers.
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This file combines and revises a number of prior developments.
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The original theories "GCD" and "Primes" were by Christophe Tabacznyj
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and Lawrence C. Paulson, based on \cite{davenport92}. They introduced
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gcd, lcm, and prime for the natural numbers.
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The original theory "IntPrimes" was by Thomas M. Rasmussen, and
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extended gcd, lcm, primes to the integers. Amine Chaieb provided
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another extension of the notions to the integers, and added a number
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of results to "Primes" and "GCD". IntPrimes also defined and developed
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the congruence relations on the integers. The notion was extended to
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the natural numbers by Chaieb.
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Jeremy Avigad combined all of these, made everything uniform for the
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natural numbers and the integers, and added a number of new theorems.
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Tobias Nipkow cleaned up a lot.
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*)
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header {* Primes *}
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theory Primes
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imports "~~/src/HOL/GCD"
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begin
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class prime = one +
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  fixes prime :: "'a \<Rightarrow> bool"
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instantiation nat :: prime
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begin
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definition prime_nat :: "nat \<Rightarrow> bool"
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  where "prime_nat p = (1 < p \<and> (\<forall>m. m dvd p --> m = 1 \<or> m = p))"
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instance ..
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end
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instantiation int :: prime
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begin
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definition prime_int :: "int \<Rightarrow> bool"
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  where "prime_int p = prime (nat p)"
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instance ..
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end
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subsection {* Set up Transfer *}
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lemma transfer_nat_int_prime:
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  "(x::int) >= 0 \<Longrightarrow> prime (nat x) = prime x"
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  unfolding gcd_int_def lcm_int_def prime_int_def by auto
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declare transfer_morphism_nat_int[transfer add return:
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    transfer_nat_int_prime]
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lemma transfer_int_nat_prime: "prime (int x) = prime x"
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  unfolding gcd_int_def lcm_int_def prime_int_def by auto
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declare transfer_morphism_int_nat[transfer add return:
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    transfer_int_nat_prime]
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subsection {* Primes *}
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lemma prime_odd_nat: "prime (p::nat) \<Longrightarrow> p > 2 \<Longrightarrow> odd p"
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  unfolding prime_nat_def
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  by (metis gcd_lcm_complete_lattice_nat.bot_least nat_even_iff_2_dvd nat_neq_iff odd_1_nat)
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lemma prime_odd_int: "prime (p::int) \<Longrightarrow> p > 2 \<Longrightarrow> odd p"
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  unfolding prime_int_def
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  apply (frule prime_odd_nat)
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  apply (auto simp add: even_nat_def)
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  done
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(* FIXME Is there a better way to handle these, rather than making them elim rules? *)
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lemma prime_ge_0_nat [elim]: "prime (p::nat) \<Longrightarrow> p >= 0"
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  unfolding prime_nat_def by auto
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lemma prime_gt_0_nat [elim]: "prime (p::nat) \<Longrightarrow> p > 0"
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  unfolding prime_nat_def by auto
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lemma prime_ge_1_nat [elim]: "prime (p::nat) \<Longrightarrow> p >= 1"
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  unfolding prime_nat_def by auto
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lemma prime_gt_1_nat [elim]: "prime (p::nat) \<Longrightarrow> p > 1"
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  unfolding prime_nat_def by auto
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lemma prime_ge_Suc_0_nat [elim]: "prime (p::nat) \<Longrightarrow> p >= Suc 0"
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  unfolding prime_nat_def by auto
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lemma prime_gt_Suc_0_nat [elim]: "prime (p::nat) \<Longrightarrow> p > Suc 0"
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  unfolding prime_nat_def by auto
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lemma prime_ge_2_nat [elim]: "prime (p::nat) \<Longrightarrow> p >= 2"
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  unfolding prime_nat_def by auto
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lemma prime_ge_0_int [elim]: "prime (p::int) \<Longrightarrow> p >= 0"
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  unfolding prime_int_def prime_nat_def by auto
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lemma prime_gt_0_int [elim]: "prime (p::int) \<Longrightarrow> p > 0"
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  unfolding prime_int_def prime_nat_def by auto
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lemma prime_ge_1_int [elim]: "prime (p::int) \<Longrightarrow> p >= 1"
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  unfolding prime_int_def prime_nat_def by auto
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lemma prime_gt_1_int [elim]: "prime (p::int) \<Longrightarrow> p > 1"
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  unfolding prime_int_def prime_nat_def by auto
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lemma prime_ge_2_int [elim]: "prime (p::int) \<Longrightarrow> p >= 2"
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  unfolding prime_int_def prime_nat_def by auto
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lemma prime_int_altdef: "prime (p::int) = (1 < p \<and> (\<forall>m \<ge> 0. m dvd p \<longrightarrow>
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    m = 1 \<or> m = p))"
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  using prime_nat_def [transferred]
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  apply (cases "p >= 0")
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  apply blast
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  apply (auto simp add: prime_ge_0_int)
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  done
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lemma prime_imp_coprime_nat: "prime (p::nat) \<Longrightarrow> \<not> p dvd n \<Longrightarrow> coprime p n"
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  apply (unfold prime_nat_def)
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  apply (metis gcd_dvd1_nat gcd_dvd2_nat)
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  done
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lemma prime_imp_coprime_int: "prime (p::int) \<Longrightarrow> \<not> p dvd n \<Longrightarrow> coprime p n"
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  apply (unfold prime_int_altdef)
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  apply (metis gcd_dvd1_int gcd_dvd2_int gcd_ge_0_int)
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  done
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lemma prime_dvd_mult_nat: "prime (p::nat) \<Longrightarrow> p dvd m * n \<Longrightarrow> p dvd m \<or> p dvd n"
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  by (blast intro: coprime_dvd_mult_nat prime_imp_coprime_nat)
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lemma prime_dvd_mult_int: "prime (p::int) \<Longrightarrow> p dvd m * n \<Longrightarrow> p dvd m \<or> p dvd n"
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  by (blast intro: coprime_dvd_mult_int prime_imp_coprime_int)
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lemma prime_dvd_mult_eq_nat [simp]: "prime (p::nat) \<Longrightarrow>
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    p dvd m * n = (p dvd m \<or> p dvd n)"
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  by (rule iffI, rule prime_dvd_mult_nat, auto)
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lemma prime_dvd_mult_eq_int [simp]: "prime (p::int) \<Longrightarrow>
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    p dvd m * n = (p dvd m \<or> p dvd n)"
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  by (rule iffI, rule prime_dvd_mult_int, auto)
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lemma not_prime_eq_prod_nat: "(n::nat) > 1 \<Longrightarrow> ~ prime n \<Longrightarrow>
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    EX m k. n = m * k & 1 < m & m < n & 1 < k & k < n"
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  unfolding prime_nat_def dvd_def apply auto
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  by (metis mult_commute linorder_neq_iff linorder_not_le mult_1
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      n_less_n_mult_m one_le_mult_iff less_imp_le_nat)
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lemma not_prime_eq_prod_int: "(n::int) > 1 \<Longrightarrow> ~ prime n \<Longrightarrow>
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    EX m k. n = m * k & 1 < m & m < n & 1 < k & k < n"
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  unfolding prime_int_altdef dvd_def
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  apply auto
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  by (metis div_mult_self1_is_id div_mult_self2_is_id
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      int_div_less_self int_one_le_iff_zero_less zero_less_mult_pos less_le)
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lemma prime_dvd_power_nat [rule_format]: "prime (p::nat) -->
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    n > 0 --> (p dvd x^n --> p dvd x)"
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  by (induct n rule: nat_induct) auto
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lemma prime_dvd_power_int [rule_format]: "prime (p::int) -->
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    n > 0 --> (p dvd x^n --> p dvd x)"
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  apply (induct n rule: nat_induct)
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  apply auto
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  apply (frule prime_ge_0_int)
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  apply auto
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  done
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subsubsection {* Make prime naively executable *}
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lemma zero_not_prime_nat [simp]: "~prime (0::nat)"
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  by (simp add: prime_nat_def)
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lemma zero_not_prime_int [simp]: "~prime (0::int)"
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  by (simp add: prime_int_def)
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lemma one_not_prime_nat [simp]: "~prime (1::nat)"
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  by (simp add: prime_nat_def)
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lemma Suc_0_not_prime_nat [simp]: "~prime (Suc 0)"
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  by (simp add: prime_nat_def One_nat_def)
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lemma one_not_prime_int [simp]: "~prime (1::int)"
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  by (simp add: prime_int_def)
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lemma prime_nat_code [code]:
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    "prime (p::nat) \<longleftrightarrow> p > 1 \<and> (\<forall>n \<in> {1<..<p}. ~ n dvd p)"
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  apply (simp add: Ball_def)
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  apply (metis less_not_refl prime_nat_def dvd_triv_right not_prime_eq_prod_nat)
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  done
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lemma prime_nat_simp:
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    "prime (p::nat) \<longleftrightarrow> p > 1 \<and> (\<forall>n \<in> set [2..<p]. \<not> n dvd p)"
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  by (auto simp add: prime_nat_code)
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lemmas prime_nat_simp_number_of [simp] = prime_nat_simp [of "number_of m", standard]
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lemma prime_int_code [code]:
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  "prime (p::int) \<longleftrightarrow> p > 1 \<and> (\<forall>n \<in> {1<..<p}. ~ n dvd p)" (is "?L = ?R")
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proof
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  assume "?L"
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  then show "?R"
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    by (clarsimp simp: prime_gt_1_int) (metis int_one_le_iff_zero_less prime_int_altdef less_le)
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next
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  assume "?R"
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  then show "?L" by (clarsimp simp: Ball_def) (metis dvdI not_prime_eq_prod_int)
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qed
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lemma prime_int_simp: "prime (p::int) \<longleftrightarrow> p > 1 \<and> (\<forall>n \<in> set [2..p - 1]. ~ n dvd p)"
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  by (auto simp add: prime_int_code)
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lemmas prime_int_simp_number_of [simp] = prime_int_simp [of "number_of m", standard]
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lemma two_is_prime_nat [simp]: "prime (2::nat)"
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  by simp
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lemma two_is_prime_int [simp]: "prime (2::int)"
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  by simp
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text{* A bit of regression testing: *}
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lemma "prime(97::nat)" by simp
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lemma "prime(97::int)" by simp
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lemma "prime(997::nat)" by eval
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lemma "prime(997::int)" by eval
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lemma prime_imp_power_coprime_nat: "prime (p::nat) \<Longrightarrow> ~ p dvd a \<Longrightarrow> coprime a (p^m)"
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  apply (rule coprime_exp_nat)
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  apply (subst gcd_commute_nat)
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  apply (erule (1) prime_imp_coprime_nat)
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  done
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lemma prime_imp_power_coprime_int: "prime (p::int) \<Longrightarrow> ~ p dvd a \<Longrightarrow> coprime a (p^m)"
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  apply (rule coprime_exp_int)
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  apply (subst gcd_commute_int)
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  apply (erule (1) prime_imp_coprime_int)
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  done
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lemma primes_coprime_nat: "prime (p::nat) \<Longrightarrow> prime q \<Longrightarrow> p \<noteq> q \<Longrightarrow> coprime p q"
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  apply (rule prime_imp_coprime_nat, assumption)
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  apply (unfold prime_nat_def)
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  apply auto
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  done
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lemma primes_coprime_int: "prime (p::int) \<Longrightarrow> prime q \<Longrightarrow> p \<noteq> q \<Longrightarrow> coprime p q"
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  apply (rule prime_imp_coprime_int, assumption)
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  apply (unfold prime_int_altdef)
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  apply (metis int_one_le_iff_zero_less less_le)
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  done
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lemma primes_imp_powers_coprime_nat:
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    "prime (p::nat) \<Longrightarrow> prime q \<Longrightarrow> p ~= q \<Longrightarrow> coprime (p^m) (q^n)"
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  by (rule coprime_exp2_nat, rule primes_coprime_nat)
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lemma primes_imp_powers_coprime_int:
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    "prime (p::int) \<Longrightarrow> prime q \<Longrightarrow> p ~= q \<Longrightarrow> coprime (p^m) (q^n)"
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  by (rule coprime_exp2_int, rule primes_coprime_int)
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lemma prime_factor_nat: "n \<noteq> (1::nat) \<Longrightarrow> \<exists> p. prime p \<and> p dvd n"
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  apply (induct n rule: nat_less_induct)
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  apply (case_tac "n = 0")
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  using two_is_prime_nat
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  apply blast
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  apply (metis One_nat_def dvd.order_trans dvd_refl less_Suc0 linorder_neqE_nat
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    nat_dvd_not_less neq0_conv prime_nat_def)
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  done
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(* An Isar version:
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lemma prime_factor_b_nat:
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  fixes n :: nat
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  assumes "n \<noteq> 1"
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  shows "\<exists>p. prime p \<and> p dvd n"
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using `n ~= 1`
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proof (induct n rule: less_induct_nat)
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  fix n :: nat
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  assume "n ~= 1" and
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    ih: "\<forall>m<n. m \<noteq> 1 \<longrightarrow> (\<exists>p. prime p \<and> p dvd m)"
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  then show "\<exists>p. prime p \<and> p dvd n"
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  proof -
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  {
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    assume "n = 0"
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    moreover note two_is_prime_nat
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    ultimately have ?thesis
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      by (auto simp del: two_is_prime_nat)
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  }
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  moreover
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  {
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    assume "prime n"
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    then have ?thesis by auto
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  }
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  moreover
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  {
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    assume "n ~= 0" and "~ prime n"
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    with `n ~= 1` have "n > 1" by auto
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    with `~ prime n` and not_prime_eq_prod_nat obtain m k where
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      "n = m * k" and "1 < m" and "m < n" by blast
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    with ih obtain p where "prime p" and "p dvd m" by blast
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    with `n = m * k` have ?thesis by auto
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  }
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  ultimately show ?thesis by blast
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  qed
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qed
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*)
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text {* One property of coprimality is easier to prove via prime factors. *}
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lemma prime_divprod_pow_nat:
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  assumes p: "prime (p::nat)" and ab: "coprime a b" and pab: "p^n dvd a * b"
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  shows "p^n dvd a \<or> p^n dvd b"
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proof-
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  { assume "n = 0 \<or> a = 1 \<or> b = 1" with pab have ?thesis
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      apply (cases "n=0", simp_all)
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      apply (cases "a=1", simp_all)
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      done }
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  moreover
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  { assume n: "n \<noteq> 0" and a: "a\<noteq>1" and b: "b\<noteq>1"
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    then obtain m where m: "n = Suc m" by (cases n) auto
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    from n have "p dvd p^n" apply (intro dvd_power) apply auto done
huffman@31706
   336
    also note pab
huffman@31706
   337
    finally have pab': "p dvd a * b".
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   338
    from prime_dvd_mult_nat[OF p pab']
huffman@31706
   339
    have "p dvd a \<or> p dvd b" .
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   340
    moreover
nipkow@33946
   341
    { assume pa: "p dvd a"
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   342
      from coprime_common_divisor_nat [OF ab, OF pa] p have "\<not> p dvd b" by auto
huffman@31706
   343
      with p have "coprime b p"
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   344
        by (subst gcd_commute_nat, intro prime_imp_coprime_nat)
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   345
      then have pnb: "coprime (p^n) b"
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   346
        by (subst gcd_commute_nat, rule coprime_exp_nat)
nipkow@33946
   347
      from coprime_dvd_mult_nat[OF pnb pab] have ?thesis by blast }
huffman@31706
   348
    moreover
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   349
    { assume pb: "p dvd b"
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   350
      have pnba: "p^n dvd b*a" using pab by (simp add: mult_commute)
nipkow@31952
   351
      from coprime_common_divisor_nat [OF ab, of p] pb p have "\<not> p dvd a"
huffman@31706
   352
        by auto
huffman@31706
   353
      with p have "coprime a p"
nipkow@31952
   354
        by (subst gcd_commute_nat, intro prime_imp_coprime_nat)
wenzelm@44872
   355
      then have pna: "coprime (p^n) a"
nipkow@31952
   356
        by (subst gcd_commute_nat, rule coprime_exp_nat)
nipkow@33946
   357
      from coprime_dvd_mult_nat[OF pna pnba] have ?thesis by blast }
wenzelm@44872
   358
    ultimately have ?thesis by blast }
huffman@31706
   359
  ultimately show ?thesis by blast
nipkow@23983
   360
qed
nipkow@23983
   361
wenzelm@44872
   362
avigad@32036
   363
subsection {* Infinitely many primes *}
avigad@32036
   364
avigad@32036
   365
lemma next_prime_bound: "\<exists>(p::nat). prime p \<and> n < p \<and> p <= fact n + 1"
avigad@32036
   366
proof-
avigad@32036
   367
  have f1: "fact n + 1 \<noteq> 1" using fact_ge_one_nat [of n] by arith 
avigad@32036
   368
  from prime_factor_nat [OF f1]
wenzelm@44872
   369
  obtain p where "prime p" and "p dvd fact n + 1" by auto
wenzelm@44872
   370
  then have "p \<le> fact n + 1" apply (intro dvd_imp_le) apply auto done
wenzelm@44872
   371
  { assume "p \<le> n"
avigad@32036
   372
    from `prime p` have "p \<ge> 1" 
avigad@32036
   373
      by (cases p, simp_all)
avigad@32036
   374
    with `p <= n` have "p dvd fact n" 
avigad@32036
   375
      by (intro dvd_fact_nat)
avigad@32036
   376
    with `p dvd fact n + 1` have "p dvd fact n + 1 - fact n"
avigad@32036
   377
      by (rule dvd_diff_nat)
wenzelm@44872
   378
    then have "p dvd 1" by simp
wenzelm@44872
   379
    then have "p <= 1" by auto
avigad@32036
   380
    moreover from `prime p` have "p > 1" by auto
avigad@32036
   381
    ultimately have False by auto}
wenzelm@44872
   382
  then have "n < p" by presburger
avigad@32036
   383
  with `prime p` and `p <= fact n + 1` show ?thesis by auto
avigad@32036
   384
qed
avigad@32036
   385
avigad@32036
   386
lemma bigger_prime: "\<exists>p. prime p \<and> p > (n::nat)" 
wenzelm@44872
   387
  using next_prime_bound by auto
avigad@32036
   388
avigad@32036
   389
lemma primes_infinite: "\<not> (finite {(p::nat). prime p})"
avigad@32036
   390
proof
avigad@32036
   391
  assume "finite {(p::nat). prime p}"
avigad@32036
   392
  with Max_ge have "(EX b. (ALL x : {(p::nat). prime p}. x <= b))"
avigad@32036
   393
    by auto
avigad@32036
   394
  then obtain b where "ALL (x::nat). prime x \<longrightarrow> x <= b"
avigad@32036
   395
    by auto
wenzelm@44872
   396
  with bigger_prime [of b] show False
wenzelm@44872
   397
    by auto
avigad@32036
   398
qed
avigad@32036
   399
wenzelm@21256
   400
end