(* 011_the_infinitude_of_primes — isabelle *)

(*  Title:      HOL/Computational_Algebra/Primes.thy
    Author:     Christophe Tabacznyj
    Author:     Lawrence C. Paulson
    Author:     Amine Chaieb
    Author:     Thomas M. Rasmussen
    Author:     Jeremy Avigad
    Author:     Tobias Nipkow
    Author:     Manuel Eberl

This theory deals with properties of primes. Definitions and lemmas are
proved uniformly for the natural numbers and integers.

This file combines and revises a number of prior developments.

The original theories "GCD" and "Primes" were by Christophe Tabacznyj
and Lawrence C. Paulson, based on @{cite davenport92}. They introduced
gcd, lcm, and prime for the natural numbers.

The original theory "IntPrimes" was by Thomas M. Rasmussen, and
extended gcd, lcm, primes to the integers. Amine Chaieb provided
another extension of the notions to the integers, and added a number
of results to "Primes" and "GCD". IntPrimes also defined and developed
the congruence relations on the integers. The notion was extended to
the natural numbers by Chaieb.

Jeremy Avigad combined all of these, made everything uniform for the
natural numbers and the integers, and added a number of new theorems.

Tobias Nipkow cleaned up a lot.

Florian Haftmann and Manuel Eberl put primality and prime factorisation
onto an algebraic foundation and thus generalised these concepts to 
other rings, such as polynomials. (see also the Factorial_Ring theory).

There were also previous formalisations of unique factorisation by 
Thomas Marthedal Rasmussen, Jeremy Avigad, and David Gray.
*)

section \<open>Primes\<close>

theory Primes
imports Euclidean_Algorithm
begin

subsection \<open>Primes on \<^typ>\<open>nat\<close> and \<^typ>\<open>int\<close>\<close>

lemma Suc_0_not_prime_nat [simp]: "\<not> prime (Suc 0)"
  using not_prime_1 [where ?'a = nat] by simp

lemma prime_ge_2_nat:
  "p \<ge> 2" if "prime p" for p :: nat
proof -
  from that have "p \<noteq> 0" and "p \<noteq> 1"
    by (auto dest: prime_elem_not_zeroI prime_elem_not_unit)
  then show ?thesis
    by simp
qed

lemma prime_ge_2_int:
  "p \<ge> 2" if "prime p" for p :: int
proof -
  from that have "prime_elem p" and "\<bar>p\<bar> = p"
    by (auto dest: normalize_prime)
  then have "p \<noteq> 0" and "\<bar>p\<bar> \<noteq> 1" and "p \<ge> 0"
    by (auto dest: prime_elem_not_zeroI prime_elem_not_unit)
  then show ?thesis
    by simp
qed

lemma prime_ge_0_int: "prime p \<Longrightarrow> p \<ge> (0::int)"
  using prime_ge_2_int [of p] by simp

lemma prime_gt_0_nat: "prime p \<Longrightarrow> p > (0::nat)"
  using prime_ge_2_nat [of p] by simp

(* As a simp or intro rule,

     prime p \<Longrightarrow> p > 0

   wreaks havoc here. When the premise includes \<forall>x \<in># M. prime x, it
   leads to the backchaining

     x > 0
     prime x
     x \<in># M   which is, unfortunately,
     count M x > 0  FIXME no, this is obsolete
*)

lemma prime_gt_0_int: "prime p \<Longrightarrow> p > (0::int)"
  using prime_ge_2_int [of p] by simp

lemma prime_ge_1_nat: "prime p \<Longrightarrow> p \<ge> (1::nat)"
  using prime_ge_2_nat [of p] by simp

lemma prime_ge_Suc_0_nat: "prime p \<Longrightarrow> p \<ge> Suc 0"
  using prime_ge_1_nat [of p] by simp

lemma prime_ge_1_int: "prime p \<Longrightarrow> p \<ge> (1::int)"
  using prime_ge_2_int [of p] by simp

lemma prime_gt_1_nat: "prime p \<Longrightarrow> p > (1::nat)"
  using prime_ge_2_nat [of p] by simp

lemma prime_gt_Suc_0_nat: "prime p \<Longrightarrow> p > Suc 0"
  using prime_gt_1_nat [of p] by simp

lemma prime_gt_1_int: "prime p \<Longrightarrow> p > (1::int)"
  using prime_ge_2_int [of p] by simp

lemma prime_natI:
  "prime p" if "p \<ge> 2" and "\<And>m n. p dvd m * n \<Longrightarrow> p dvd m \<or> p dvd n" for p :: nat
  using that by (auto intro!: primeI prime_elemI)

lemma prime_intI:
  "prime p" if "p \<ge> 2" and "\<And>m n. p dvd m * n \<Longrightarrow> p dvd m \<or> p dvd n" for p :: int
  using that by (auto intro!: primeI prime_elemI)

lemma prime_elem_nat_iff [simp]:
  "prime_elem n \<longleftrightarrow> prime n" for n :: nat
  by (simp add: prime_def)

lemma prime_elem_iff_prime_abs [simp]:
  "prime_elem k \<longleftrightarrow> prime \<bar>k\<bar>" for k :: int
  by (auto intro: primeI)

lemma prime_nat_int_transfer [simp]:
  "prime (int n) \<longleftrightarrow> prime n" (is "?P \<longleftrightarrow> ?Q")
proof
  assume ?P
  then have "n \<ge> 2"
    by (auto dest: prime_ge_2_int)
  then show ?Q
  proof (rule prime_natI)
    fix r s
    assume "n dvd r * s"
    with of_nat_dvd_iff [of n "r * s"] have "int n dvd int r * int s"
      by simp
    with \<open>?P\<close> have "int n dvd int r \<or> int n dvd int s"
      using prime_dvd_mult_iff [of "int n" "int r" "int s"]
      by simp
    then show "n dvd r \<or> n dvd s"
      by simp
  qed
next
  assume ?Q
  then have "int n \<ge> 2"
    by (auto dest: prime_ge_2_nat)
  then show ?P
  proof (rule prime_intI)
    fix r s
    assume "int n dvd r * s"
    then have "n dvd nat \<bar>r * s\<bar>"
      by simp
    then have "n dvd nat \<bar>r\<bar> * nat \<bar>s\<bar>"
      by (simp add: nat_abs_mult_distrib)
    with \<open>?Q\<close> have "n dvd nat \<bar>r\<bar> \<or> n dvd nat \<bar>s\<bar>"
      using prime_dvd_mult_iff [of "n" "nat \<bar>r\<bar>" "nat \<bar>s\<bar>"]
      by simp
    then show "int n dvd r \<or> int n dvd s"
      by simp
  qed
qed

lemma prime_nat_iff_prime [simp]:
  "prime (nat k) \<longleftrightarrow> prime k"
proof (cases "k \<ge> 0")
  case True
  then show ?thesis
    using prime_nat_int_transfer [of "nat k"] by simp
next
  case False
  then show ?thesis
    by (auto dest: prime_ge_2_int)
qed

lemma prime_int_nat_transfer:
  "prime k \<longleftrightarrow> k \<ge> 0 \<and> prime (nat k)"
  by (auto dest: prime_ge_2_int)

lemma prime_nat_naiveI:
  "prime p" if "p \<ge> 2" and dvd: "\<And>n. n dvd p \<Longrightarrow> n = 1 \<or> n = p" for p :: nat
proof (rule primeI, rule prime_elemI)
  fix m n :: nat
  assume "p dvd m * n"
  then obtain r s where "p = r * s" "r dvd m" "s dvd n"
    by (blast dest: division_decomp)
  moreover have "r = 1 \<or> r = p"
    using \<open>r dvd m\<close> \<open>p = r * s\<close> dvd [of r] by simp
  ultimately show "p dvd m \<or> p dvd n"
    by auto
qed (use \<open>p \<ge> 2\<close> in simp_all)

lemma prime_int_naiveI:
  "prime p" if "p \<ge> 2" and dvd: "\<And>k. k dvd p \<Longrightarrow> \<bar>k\<bar> = 1 \<or> \<bar>k\<bar> = p" for p :: int
proof -
  from \<open>p \<ge> 2\<close> have "nat p \<ge> 2"
    by simp
  then have "prime (nat p)"
  proof (rule prime_nat_naiveI)
    fix n
    assume "n dvd nat p"
    with \<open>p \<ge> 2\<close> have "n dvd nat \<bar>p\<bar>"
      by simp
    then have "int n dvd p"
      by simp
    with dvd [of "int n"] show "n = 1 \<or> n = nat p"
      by auto
  qed
  then show ?thesis
    by simp
qed

lemma prime_nat_iff:
  "prime (n :: nat) \<longleftrightarrow> (1 < n \<and> (\<forall>m. m dvd n \<longrightarrow> m = 1 \<or> m = n))"
proof (safe intro!: prime_gt_1_nat)
  assume "prime n"
  then have *: "prime_elem n"
    by simp
  fix m assume m: "m dvd n" "m \<noteq> n"
  from * \<open>m dvd n\<close> have "n dvd m \<or> is_unit m"
    by (intro irreducibleD' prime_elem_imp_irreducible)
  with m show "m = 1" by (auto dest: dvd_antisym)
next
  assume "n > 1" "\<forall>m. m dvd n \<longrightarrow> m = 1 \<or> m = n"
  then show "prime n"
    using prime_nat_naiveI [of n] by auto
qed

lemma prime_nat_iff':
  "prime (p :: nat) \<longleftrightarrow> p > 1 \<and> (\<forall>n \<in> {2..<p}. \<not> n dvd p)"
proof safe
  assume "p > 1" and *: "\<forall>n\<in>{2..<p}. \<not>n dvd p"
  show "prime p" unfolding prime_nat_iff
  proof (intro conjI allI impI)
    fix m assume "m dvd p"
    with \<open>p > 1\<close> have "m \<noteq> 0" by (intro notI) auto
    hence "m \<ge> 1" by simp
    moreover from \<open>m dvd p\<close> and * have "m \<notin> {2..<p}" by blast
    with \<open>m dvd p\<close> and \<open>p > 1\<close> have "m \<le> 1 \<or> m = p" by (auto dest: dvd_imp_le)
    ultimately show "m = 1 \<or> m = p" by simp
  qed fact+
qed (auto simp: prime_nat_iff)

lemma prime_int_iff:
  "prime (n::int) \<longleftrightarrow> (1 < n \<and> (\<forall>m. m \<ge> 0 \<and> m dvd n \<longrightarrow> m = 1 \<or> m = n))"
proof (intro iffI conjI allI impI; (elim conjE)?)
  assume *: "prime n"
  hence irred: "irreducible n" by (auto intro: prime_elem_imp_irreducible)
  from * have "n \<ge> 0" "n \<noteq> 0" "n \<noteq> 1"
    by (auto simp add: prime_ge_0_int)
  thus "n > 1" by presburger
  fix m assume "m dvd n" \<open>m \<ge> 0\<close>
  with irred have "m dvd 1 \<or> n dvd m" by (auto simp: irreducible_altdef)
  with \<open>m dvd n\<close> \<open>m \<ge> 0\<close> \<open>n > 1\<close> show "m = 1 \<or> m = n"
    using associated_iff_dvd[of m n] by auto
next
  assume n: "1 < n" "\<forall>m. m \<ge> 0 \<and> m dvd n \<longrightarrow> m = 1 \<or> m = n"
  hence "nat n > 1" by simp
  moreover have "\<forall>m. m dvd nat n \<longrightarrow> m = 1 \<or> m = nat n"
  proof (intro allI impI)
    fix m assume "m dvd nat n"
    with \<open>n > 1\<close> have "m dvd nat \<bar>n\<bar>"
      by simp
    then have "int m dvd n"
      by simp
    with n(2) have "int m = 1 \<or> int m = n"
      using of_nat_0_le_iff by blast
    thus "m = 1 \<or> m = nat n" by auto
  qed
  ultimately show "prime n" 
    unfolding prime_int_nat_transfer prime_nat_iff by auto
qed

lemma prime_int_iff':
  "prime (p :: int) \<longleftrightarrow> p > 1 \<and> (\<forall>n \<in> {2..<p}. \<not> n dvd p)" (is "?P \<longleftrightarrow> ?Q")
proof (cases "p \<ge> 0")
  case True
  have "?P \<longleftrightarrow> prime (nat p)"
    by simp
  also have "\<dots> \<longleftrightarrow> p > 1 \<and> (\<forall>n\<in>{2..<nat p}. \<not> n dvd nat \<bar>p\<bar>)"
    using True by (simp add: prime_nat_iff')
  also have "{2..<nat p} = nat ` {2..<p}"
    using True int_eq_iff by fastforce
  finally show "?P \<longleftrightarrow> ?Q" by simp
next
  case False
  then show ?thesis
    by (auto simp add: prime_ge_0_int)
qed

lemma prime_nat_not_dvd:
  assumes "prime p" "p > n" "n \<noteq> (1::nat)"
  shows   "\<not>n dvd p"
proof
  assume "n dvd p"
  from assms(1) have "irreducible p" by (simp add: prime_elem_imp_irreducible)
  from irreducibleD'[OF this \<open>n dvd p\<close>] \<open>n dvd p\<close> \<open>p > n\<close> assms show False
    by (cases "n = 0") (auto dest!: dvd_imp_le)
qed

lemma prime_int_not_dvd:
  assumes "prime p" "p > n" "n > (1::int)"
  shows   "\<not>n dvd p"
proof
  assume "n dvd p"
  from assms(1) have "irreducible p" by (auto intro: prime_elem_imp_irreducible)
  from irreducibleD'[OF this \<open>n dvd p\<close>] \<open>n dvd p\<close> \<open>p > n\<close> assms show False
    by (auto dest!: zdvd_imp_le)
qed

lemma prime_odd_nat: "prime p \<Longrightarrow> p > (2::nat) \<Longrightarrow> odd p"
  by (intro prime_nat_not_dvd) auto

lemma prime_odd_int: "prime p \<Longrightarrow> p > (2::int) \<Longrightarrow> odd p"
  by (intro prime_int_not_dvd) auto

lemma prime_int_altdef:
  "prime p = (1 < p \<and> (\<forall>m::int. m \<ge> 0 \<longrightarrow> m dvd p \<longrightarrow>
    m = 1 \<or> m = p))"
  unfolding prime_int_iff by blast

lemma not_prime_eq_prod_nat:
  assumes "m > 1" "\<not> prime (m::nat)"
  shows   "\<exists>n k. n = m * k \<and> 1 < m \<and> m < n \<and> 1 < k \<and> k < n"
  using assms irreducible_altdef[of m]
  by (auto simp: prime_elem_iff_irreducible irreducible_altdef)


subsection \<open>Make prime naively executable\<close>

lemma prime_int_numeral_eq [simp]:
  "prime (numeral m :: int) \<longleftrightarrow> prime (numeral m :: nat)"
  by (simp add: prime_int_nat_transfer)

class check_prime_by_range = normalization_semidom + discrete_linordered_semidom +
  assumes prime_iff: \<open>prime a \<longleftrightarrow> 1 < a \<and> (\<forall>d\<in>{2..a div 2}. \<not> d dvd a)\<close>
begin

lemma two_is_prime [simp]:
  \<open>prime 2\<close>
  by (simp add: prime_iff)

end

lemma divisor_less_eq_half_nat:
  \<open>m \<le> n div 2\<close> if \<open>m dvd n\<close> \<open>m < n\<close> for m n :: nat
  using that by (auto simp add: less_eq_div_iff_mult_less_eq)

instance nat :: check_prime_by_range
  apply standard
  apply (auto simp add: prime_nat_iff)
  apply (rule ccontr)
  apply (auto simp add: neq_iff)
  apply (metis One_nat_def Suc_1 Suc_leI atLeastAtMost_iff divisor_less_eq_half_nat)
  done

lemma two_is_prime_nat [simp]:
  \<open>prime (2::nat)\<close>
  by (fact two_is_prime)

lemma divisor_less_eq_half_int:
  \<open>k \<le> l div 2\<close> if \<open>k dvd l\<close> \<open>k < l\<close> \<open>l \<ge> 0\<close> \<open>k \<ge> 0\<close> for k l :: int
proof -
  define m n where \<open>m = nat \<bar>k\<bar>\<close> \<open>n = nat \<bar>l\<bar>\<close>
  with \<open>l \<ge> 0\<close> \<open>k \<ge> 0\<close> have \<open>k = int m\<close> \<open>l = int n\<close>
    by simp_all
  with that show ?thesis
    using divisor_less_eq_half_nat [of m n] by simp
qed

instance int :: check_prime_by_range
  apply standard
  apply (auto simp add: prime_int_iff)
  apply (smt (verit) int_div_less_self)
  apply (rule ccontr)
  apply (auto simp add: neq_iff zdvd_not_zless)
  apply (metis div_by_0 dvd_div_eq_0_iff less_le_not_le one_dvd order_le_less
      zdvd_not_zless)
  apply (metis atLeastAtMost_iff divisor_less_eq_half_int dvd_div_eq_0_iff
      int_one_le_iff_zero_less nle_le one_add_one pos_imp_zdiv_nonneg_iff zdiv_eq_0_iff
      zless_imp_add1_zle)
  done

lemma prime_nat_numeral_eq [simp]: \<comment> \<open>TODO Sieve Of Erathosthenes might speed this up\<close>
  "prime (numeral m :: nat) \<longleftrightarrow>
    (1::nat) < numeral m \<and>
    (\<forall>n::nat \<in> set [2..<Suc (numeral m div 2)]. \<not> n dvd numeral m)"
  using prime_iff [of \<open>numeral m :: nat\<close>]
  by (simp only: set_upt atLeastLessThanSuc_atLeastAtMost)

context check_prime_by_range
begin

definition check_divisors :: \<open>'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool\<close>
  where \<open>check_divisors l u a \<longleftrightarrow> (\<forall>d\<in>{l..u}. \<not> d dvd a)\<close>

lemma check_divisors_rec [code]:
  \<open>check_divisors l u a \<longleftrightarrow> u < l \<or> (\<not> l dvd a \<and> check_divisors (l + 1) u a)\<close>
  apply (auto simp add: check_divisors_def not_less)
  apply (metis local.add_increasing2 local.atLeastAtMost_iff local.linear local.order_eq_iff
      local.zero_le_one)
  subgoal for d
    apply (cases \<open>l + 1 \<le> d\<close>)
     apply (auto simp add: not_le)
    apply (metis local.dual_order.antisym local.less_eq_iff_succ_less)
    done
  done

lemma prime_eq_check_divisors [code]:
  \<open>prime a \<longleftrightarrow> a > 1 \<and> check_divisors 2 (a div 2) a\<close>
  by (simp add: check_divisors_def prime_iff)

end


subsection \<open>Largest exponent of a prime factor\<close>

lemma prime_factor_nat:
  "n \<noteq> (1::nat) \<Longrightarrow> \<exists>p. prime p \<and> p dvd n"
  using prime_divisor_exists[of n]
  by (cases "n = 0") (auto intro: exI[of _ "2::nat"])

lemma prime_factor_int:
  fixes k :: int
  assumes "\<bar>k\<bar> \<noteq> 1"
  obtains p where "prime p" "p dvd k"
proof (cases "k = 0")
  case True
  then have "prime (2::int)" and "2 dvd k"
    by simp_all
  with that show thesis
    by blast
next
  case False
  with assms prime_divisor_exists [of k] obtain p where "prime p" "p dvd k"
    by auto
  with that show thesis
    by blast
qed

text\<open>Possibly duplicates other material, but avoid the complexities of multisets.\<close>
  
lemma prime_power_cancel_less:
  assumes "prime p" and eq: "m * (p ^ k) = m' * (p ^ k')" and less: "k < k'" and "\<not> p dvd m"
  shows False
proof -
  obtain l where l: "k' = k + l" and "l > 0"
    using less less_imp_add_positive by auto
  have "m = m * (p ^ k) div (p ^ k)"
    using \<open>prime p\<close> by simp
  also have "\<dots> = m' * (p ^ k') div (p ^ k)"
    using eq by simp
  also have "\<dots> = m' * (p ^ l) * (p ^ k) div (p ^ k)"
    by (simp add: l mult.commute mult.left_commute power_add)
  also have "... = m' * (p ^ l)"
    using \<open>prime p\<close> by simp
  finally have "p dvd m"
    using \<open>l > 0\<close> by simp
  with assms show False
    by simp
qed

lemma prime_power_cancel:
  assumes "prime p" and eq: "m * (p ^ k) = m' * (p ^ k')" and "\<not> p dvd m" "\<not> p dvd m'"
  shows "k = k'"
  using prime_power_cancel_less [OF \<open>prime p\<close>] assms
  by (metis linorder_neqE_nat)

lemma prime_power_cancel2:
  assumes "prime p" "m * (p ^ k) = m' * (p ^ k')" "\<not> p dvd m" "\<not> p dvd m'"
  obtains "m = m'" "k = k'"
  using prime_power_cancel [OF assms] assms by auto

lemma prime_power_canonical:
  fixes m :: nat
  assumes "prime p" "m > 0"
  shows "\<exists>k n. \<not> p dvd n \<and> m = n * p ^ k"
using \<open>m > 0\<close>
proof (induction m rule: less_induct)
  case (less m)
  show ?case
  proof (cases "p dvd m")
    case True
    then obtain m' where m': "m = p * m'"
      using dvdE by blast
    with \<open>prime p\<close> have "0 < m'" "m' < m"
      using less.prems prime_nat_iff by auto
    with m' less show ?thesis
      by (metis power_Suc mult.left_commute)
  next
    case False
    then show ?thesis
      by (metis mult.right_neutral power_0)
  qed
qed


subsection \<open>Infinitely many primes\<close>

lemma next_prime_bound: "\<exists>p::nat. prime p \<and> n < p \<and> p \<le> fact n + 1"
proof-
  have f1: "fact n + 1 \<noteq> (1::nat)" using fact_ge_1 [of n, where 'a=nat] by arith
  from prime_factor_nat [OF f1]
  obtain p :: nat where "prime p" and "p dvd fact n + 1" by auto
  then have "p \<le> fact n + 1" apply (intro dvd_imp_le) apply auto done
  { assume "p \<le> n"
    from \<open>prime p\<close> have "p \<ge> 1"
      by (cases p, simp_all)
    with \<open>p <= n\<close> have "p dvd fact n"
      by (intro dvd_fact)
    with \<open>p dvd fact n + 1\<close> have "p dvd fact n + 1 - fact n"
      by (rule dvd_diff_nat)
    then have "p dvd 1" by simp
    then have "p <= 1" by auto
    moreover from \<open>prime p\<close> have "p > 1"
      using prime_nat_iff by blast
    ultimately have False by auto}
  then have "n < p" by presburger
  with \<open>prime p\<close> and \<open>p <= fact n + 1\<close> show ?thesis by auto
qed

lemma bigger_prime: "\<exists>p. prime p \<and> p > (n::nat)"
  using next_prime_bound by auto

lemma primes_infinite: "\<not> (finite {(p::nat). prime p})"
proof
  assume "finite {(p::nat). prime p}"
  with Max_ge have "(\<exists>b. (\<forall>x \<in> {(p::nat). prime p}. x \<le> b))"
    by auto
  then obtain b where "\<forall>(x::nat). prime x \<longrightarrow> x \<le> b"
    by auto
  with bigger_prime [of b] show False
    by auto
qed

subsection \<open>Powers of Primes\<close>

text\<open>Versions for type nat only\<close>

lemma prime_product:
  fixes p::nat
  assumes "prime (p * q)"
  shows "p = 1 \<or> q = 1"
proof -
  from assms have
    "1 < p * q" and P: "\<And>m. m dvd p * q \<Longrightarrow> m = 1 \<or> m = p * q"
    unfolding prime_nat_iff by auto
  from \<open>1 < p * q\<close> have "p \<noteq> 0" by (cases p) auto
  then have Q: "p = p * q \<longleftrightarrow> q = 1" by auto
  have "p dvd p * q" by simp
  then have "p = 1 \<or> p = p * q" by (rule P)
  then show ?thesis by (simp add: Q)
qed

(* TODO: Generalise? *)
lemma prime_power_mult_nat:
  fixes p :: nat
  assumes p: "prime p" and xy: "x * y = p ^ k"
  shows "\<exists>i j. x = p ^ i \<and> y = p^ j"
using xy
proof(induct k arbitrary: x y)
  case 0 thus ?case apply simp by (rule exI[where x="0"], simp)
next
  case (Suc k x y)
  from Suc.prems have pxy: "p dvd x*y" by auto
  from prime_dvd_multD [OF p pxy] have pxyc: "p dvd x \<or> p dvd y" .
  from p have p0: "p \<noteq> 0" by - (rule ccontr, simp)
  {assume px: "p dvd x"
    then obtain d where d: "x = p*d" unfolding dvd_def by blast
    from Suc.prems d  have "p*d*y = p^Suc k" by simp
    hence th: "d*y = p^k" using p0 by simp
    from Suc.hyps[OF th] obtain i j where ij: "d = p^i" "y = p^j" by blast
    with d have "x = p^Suc i" by simp
    with ij(2) have ?case by blast}
  moreover
  {assume px: "p dvd y"
    then obtain d where d: "y = p*d" unfolding dvd_def by blast
    from Suc.prems d  have "p*d*x = p^Suc k" by (simp add: mult.commute)
    hence th: "d*x = p^k" using p0 by simp
    from Suc.hyps[OF th] obtain i j where ij: "d = p^i" "x = p^j" by blast
    with d have "y = p^Suc i" by simp
    with ij(2) have ?case by blast}
  ultimately show ?case  using pxyc by blast
qed

lemma prime_power_exp_nat:
  fixes p::nat
  assumes p: "prime p" and n: "n \<noteq> 0"
    and xn: "x^n = p^k" shows "\<exists>i. x = p^i"
  using n xn
proof(induct n arbitrary: k)
  case 0 thus ?case by simp
next
  case (Suc n k) hence th: "x*x^n = p^k" by simp
  {assume "n = 0" with Suc have ?case by simp (rule exI[where x="k"], simp)}
  moreover
  {assume n: "n \<noteq> 0"
    from prime_power_mult_nat[OF p th]
    obtain i j where ij: "x = p^i" "x^n = p^j"by blast
    from Suc.hyps[OF n ij(2)] have ?case .}
  ultimately show ?case by blast
qed

lemma divides_primepow_nat:
  fixes p :: nat
  assumes p: "prime p"
  shows "d dvd p ^ k \<longleftrightarrow> (\<exists>i\<le>k. d = p ^ i)"
  using assms divides_primepow [of p d k] by (auto intro: le_imp_power_dvd)

lemma gcd_prime_int:
  assumes "prime (p :: int)"
  shows   "gcd p k = (if p dvd k then p else 1)"
proof -
  have "p \<ge> 0"
    using assms prime_ge_0_int by auto
  show ?thesis
  proof (cases "p dvd k")
    case True
    thus ?thesis using assms \<open>p \<ge> 0\<close> by auto
  next
    case False
    hence "coprime p k"
      using assms by (simp add: prime_imp_coprime)
    with False show ?thesis
      by auto
  qed
qed


subsection \<open>Chinese Remainder Theorem Variants\<close>

lemma bezout_gcd_nat:
  fixes a::nat shows "\<exists>x y. a * x - b * y = gcd a b \<or> b * x - a * y = gcd a b"
  using bezout_nat[of a b]
by (metis bezout_nat diff_add_inverse gcd_add_mult gcd.commute
  gcd_nat.right_neutral mult_0)

lemma gcd_bezout_sum_nat:
  fixes a::nat
  assumes "a * x + b * y = d"
  shows "gcd a b dvd d"
proof-
  let ?g = "gcd a b"
    have dv: "?g dvd a*x" "?g dvd b * y"
      by simp_all
    from dvd_add[OF dv] assms
    show ?thesis by auto
qed


text \<open>A binary form of the Chinese Remainder Theorem.\<close>

(* TODO: Generalise? *)
lemma chinese_remainder:
  fixes a::nat  assumes ab: "coprime a b" and a: "a \<noteq> 0" and b: "b \<noteq> 0"
  shows "\<exists>x q1 q2. x = u + q1 * a \<and> x = v + q2 * b"
proof-
  from bezout_add_strong_nat[OF a, of b] bezout_add_strong_nat[OF b, of a]
  obtain d1 x1 y1 d2 x2 y2 where dxy1: "d1 dvd a" "d1 dvd b" "a * x1 = b * y1 + d1"
    and dxy2: "d2 dvd b" "d2 dvd a" "b * x2 = a * y2 + d2" by blast
  then have d12: "d1 = 1" "d2 = 1"
    using ab coprime_common_divisor_nat [of a b] by blast+
  let ?x = "v * a * x1 + u * b * x2"
  let ?q1 = "v * x1 + u * y2"
  let ?q2 = "v * y1 + u * x2"
  from dxy2(3)[simplified d12] dxy1(3)[simplified d12]
  have "?x = u + ?q1 * a" "?x = v + ?q2 * b"
    by algebra+
  thus ?thesis by blast
qed

text \<open>Primality\<close>

lemma coprime_bezout_strong:
  fixes a::nat assumes "coprime a b"  "b \<noteq> 1"
  shows "\<exists>x y. a * x = b * y + 1"
  by (metis add.commute add.right_neutral assms(1) assms(2) chinese_remainder coprime_1_left coprime_1_right coprime_crossproduct_nat mult.commute mult.right_neutral mult_cancel_left)

lemma bezout_prime:
  assumes p: "prime p" and pa: "\<not> p dvd a"
  shows "\<exists>x y. a*x = Suc (p*y)"
proof -
  have ap: "coprime a p"
    using coprime_commute p pa prime_imp_coprime by auto
  moreover from p have "p \<noteq> 1" by auto
  ultimately have "\<exists>x y. a * x = p * y + 1"
    by (rule coprime_bezout_strong)
  then show ?thesis by simp    
qed
(* END TODO *)



subsection \<open>Multiplicity and primality for natural numbers and integers\<close>

lemma prime_factors_gt_0_nat:
  "p \<in> prime_factors x \<Longrightarrow> p > (0::nat)"
  by (simp add: in_prime_factors_imp_prime prime_gt_0_nat)

lemma prime_factors_gt_0_int:
  "p \<in> prime_factors x \<Longrightarrow> p > (0::int)"
  by (simp add: in_prime_factors_imp_prime prime_gt_0_int)

lemma prime_factors_ge_0_int [elim]: (* FIXME !? *)
  fixes n :: int
  shows "p \<in> prime_factors n \<Longrightarrow> p \<ge> 0"
  by (drule prime_factors_gt_0_int) simp
  
lemma prod_mset_prime_factorization_int:
  fixes n :: int
  assumes "n > 0"
  shows   "prod_mset (prime_factorization n) = n"
  using assms by (simp add: prod_mset_prime_factorization)

lemma prime_factorization_exists_nat:
  "n > 0 \<Longrightarrow> (\<exists>M. (\<forall>p::nat \<in> set_mset M. prime p) \<and> n = (\<Prod>i \<in># M. i))"
  using prime_factorization_exists[of n] by auto

lemma prod_mset_prime_factorization_nat [simp]: 
  "(n::nat) > 0 \<Longrightarrow> prod_mset (prime_factorization n) = n"
  by (subst prod_mset_prime_factorization) simp_all

lemma prime_factorization_nat:
    "n > (0::nat) \<Longrightarrow> n = (\<Prod>p \<in> prime_factors n. p ^ multiplicity p n)"
  by (simp add: prod_prime_factors)

lemma prime_factorization_int:
    "n > (0::int) \<Longrightarrow> n = (\<Prod>p \<in> prime_factors n. p ^ multiplicity p n)"
  by (simp add: prod_prime_factors)

lemma prime_factorization_unique_nat:
  fixes f :: "nat \<Rightarrow> _"
  assumes S_eq: "S = {p. 0 < f p}"
    and "finite S"
    and S: "\<forall>p\<in>S. prime p" "n = (\<Prod>p\<in>S. p ^ f p)"
  shows "S = prime_factors n \<and> (\<forall>p. prime p \<longrightarrow> f p = multiplicity p n)"
  using assms by (intro prime_factorization_unique'') auto

lemma prime_factorization_unique_int:
  fixes f :: "int \<Rightarrow> _"
  assumes S_eq: "S = {p. 0 < f p}"
    and "finite S"
    and S: "\<forall>p\<in>S. prime p" "abs n = (\<Prod>p\<in>S. p ^ f p)"
  shows "S = prime_factors n \<and> (\<forall>p. prime p \<longrightarrow> f p = multiplicity p n)"
  using assms by (intro prime_factorization_unique'') auto

lemma prime_factors_characterization_nat:
  "S = {p. 0 < f (p::nat)} \<Longrightarrow>
    finite S \<Longrightarrow> \<forall>p\<in>S. prime p \<Longrightarrow> n = (\<Prod>p\<in>S. p ^ f p) \<Longrightarrow> prime_factors n = S"
  by (rule prime_factorization_unique_nat [THEN conjunct1, symmetric])

lemma prime_factors_characterization'_nat:
  "finite {p. 0 < f (p::nat)} \<Longrightarrow>
    (\<forall>p. 0 < f p \<longrightarrow> prime p) \<Longrightarrow>
      prime_factors (\<Prod>p | 0 < f p. p ^ f p) = {p. 0 < f p}"
  by (rule prime_factors_characterization_nat) auto

lemma prime_factors_characterization_int:
  "S = {p. 0 < f (p::int)} \<Longrightarrow> finite S \<Longrightarrow>
    \<forall>p\<in>S. prime p \<Longrightarrow> abs n = (\<Prod>p\<in>S. p ^ f p) \<Longrightarrow> prime_factors n = S"
  by (rule prime_factorization_unique_int [THEN conjunct1, symmetric])

(* TODO Move *)
lemma abs_prod: "abs (prod f A :: 'a :: linordered_idom) = prod (\<lambda>x. abs (f x)) A"
  by (cases "finite A", induction A rule: finite_induct) (simp_all add: abs_mult)

lemma primes_characterization'_int [rule_format]:
  "finite {p. p \<ge> 0 \<and> 0 < f (p::int)} \<Longrightarrow> \<forall>p. 0 < f p \<longrightarrow> prime p \<Longrightarrow>
      prime_factors (\<Prod>p | p \<ge> 0 \<and> 0 < f p. p ^ f p) = {p. p \<ge> 0 \<and> 0 < f p}"
  by (rule prime_factors_characterization_int) (auto simp: abs_prod prime_ge_0_int)

lemma multiplicity_characterization_nat:
  "S = {p. 0 < f (p::nat)} \<Longrightarrow> finite S \<Longrightarrow> \<forall>p\<in>S. prime p \<Longrightarrow> prime p \<Longrightarrow>
    n = (\<Prod>p\<in>S. p ^ f p) \<Longrightarrow> multiplicity p n = f p"
  by (frule prime_factorization_unique_nat [of S f n, THEN conjunct2, rule_format, symmetric]) auto

lemma multiplicity_characterization'_nat: "finite {p. 0 < f (p::nat)} \<longrightarrow>
    (\<forall>p. 0 < f p \<longrightarrow> prime p) \<longrightarrow> prime p \<longrightarrow>
      multiplicity p (\<Prod>p | 0 < f p. p ^ f p) = f p"
  by (intro impI, rule multiplicity_characterization_nat) auto

lemma multiplicity_characterization_int: "S = {p. 0 < f (p::int)} \<Longrightarrow>
    finite S \<Longrightarrow> \<forall>p\<in>S. prime p \<Longrightarrow> prime p \<Longrightarrow> n = (\<Prod>p\<in>S. p ^ f p) \<Longrightarrow> multiplicity p n = f p"
  by (frule prime_factorization_unique_int [of S f n, THEN conjunct2, rule_format, symmetric]) 
     (auto simp: abs_prod power_abs prime_ge_0_int intro!: prod.cong)

lemma multiplicity_characterization'_int [rule_format]:
  "finite {p. p \<ge> 0 \<and> 0 < f (p::int)} \<Longrightarrow>
    (\<forall>p. 0 < f p \<longrightarrow> prime p) \<Longrightarrow> prime p \<Longrightarrow>
      multiplicity p (\<Prod>p | p \<ge> 0 \<and> 0 < f p. p ^ f p) = f p"
  by (rule multiplicity_characterization_int) (auto simp: prime_ge_0_int)

lemma multiplicity_one_nat [simp]: "multiplicity p (Suc 0) = 0"
  unfolding One_nat_def [symmetric] by (rule multiplicity_one)

lemma multiplicity_eq_nat:
  fixes x and y::nat
  assumes "x > 0" "y > 0" "\<And>p. prime p \<Longrightarrow> multiplicity p x = multiplicity p y"
  shows "x = y"
  using multiplicity_eq_imp_eq[of x y] assms by simp

lemma multiplicity_eq_int:
  fixes x y :: int
  assumes "x > 0" "y > 0" "\<And>p. prime p \<Longrightarrow> multiplicity p x = multiplicity p y"
  shows "x = y"
  using multiplicity_eq_imp_eq[of x y] assms by simp

lemma multiplicity_prod_prime_powers:
  assumes "finite S" "\<And>x. x \<in> S \<Longrightarrow> prime x" "prime p"
  shows   "multiplicity p (\<Prod>p \<in> S. p ^ f p) = (if p \<in> S then f p else 0)"
proof -
  define g where "g = (\<lambda>x. if x \<in> S then f x else 0)"
  define A where "A = Abs_multiset g"
  have "{x. g x > 0} \<subseteq> S" by (auto simp: g_def)
  from finite_subset[OF this assms(1)] have [simp]: "finite {x. 0 < g x}"
    by simp
  from assms have count_A: "count A x = g x" for x unfolding A_def
    by simp
  have set_mset_A: "set_mset A = {x\<in>S. f x > 0}"
    unfolding set_mset_def count_A by (auto simp: g_def)
  with assms have prime: "prime x" if "x \<in># A" for x using that by auto
  from set_mset_A assms have "(\<Prod>p \<in> S. p ^ f p) = (\<Prod>p \<in> S. p ^ g p) "
    by (intro prod.cong) (auto simp: g_def)
  also from set_mset_A assms have "\<dots> = (\<Prod>p \<in> set_mset A. p ^ g p)"
    by (intro prod.mono_neutral_right) (auto simp: g_def set_mset_A)
  also have "\<dots> = prod_mset A"
    by (auto simp: prod_mset_multiplicity count_A set_mset_A intro!: prod.cong)
  also from assms have "multiplicity p \<dots> = sum_mset (image_mset (multiplicity p) A)"
    by (subst prime_elem_multiplicity_prod_mset_distrib) (auto dest: prime)
  also from assms have "image_mset (multiplicity p) A = image_mset (\<lambda>x. if x = p then 1 else 0) A"
    by (intro image_mset_cong) (auto simp: prime_multiplicity_other dest: prime)
  also have "sum_mset \<dots> = (if p \<in> S then f p else 0)" by (simp add: sum_mset_delta count_A g_def)
  finally show ?thesis .
qed

lemma prime_factorization_prod_mset:
  assumes "0 \<notin># A"
  shows "prime_factorization (prod_mset A) = \<Sum>\<^sub>#(image_mset prime_factorization A)"
  using assms by (induct A) (auto simp add: prime_factorization_mult)

lemma prime_factors_prod:
  assumes "finite A" and "0 \<notin> f ` A"
  shows "prime_factors (prod f A) = \<Union>((prime_factors \<circ> f) ` A)"
  using assms by (simp add: prod_unfold_prod_mset prime_factorization_prod_mset)

lemma prime_factors_fact:
  "prime_factors (fact n) = {p \<in> {2..n}. prime p}" (is "?M = ?N")
proof (rule set_eqI)
  fix p
  { fix m :: nat
    assume "p \<in> prime_factors m"
    then have "prime p" and "p dvd m" by auto
    moreover assume "m > 0" 
    ultimately have "2 \<le> p" and "p \<le> m"
      by (auto intro: prime_ge_2_nat dest: dvd_imp_le)
    moreover assume "m \<le> n"
    ultimately have "2 \<le> p" and "p \<le> n"
      by (auto intro: order_trans)
  } note * = this
  show "p \<in> ?M \<longleftrightarrow> p \<in> ?N"
    by (auto simp add: fact_prod prime_factors_prod Suc_le_eq dest!: prime_prime_factors intro: *)
qed

lemma prime_dvd_fact_iff:
  assumes "prime p"
  shows "p dvd fact n \<longleftrightarrow> p \<le> n"
  using assms
  by (auto simp add: prime_factorization_subset_iff_dvd [symmetric]
    prime_factorization_prime prime_factors_fact prime_ge_2_nat)

lemma dvd_choose_prime:
  assumes kn: "k < n" and k: "k \<noteq> 0" and n: "n \<noteq> 0" and prime_n: "prime n"
  shows "n dvd (n choose k)"
proof -
  have "n dvd (fact n)" by (simp add: fact_num_eq_if n)
  moreover have "\<not> n dvd (fact k * fact (n-k))"
    by (metis prime_dvd_fact_iff prime_dvd_mult_iff assms neq0_conv diff_less linorder_not_less)
  moreover have "(fact n::nat) = fact k * fact (n-k) * (n choose k)"
    using binomial_fact_lemma kn by auto
  ultimately show ?thesis using prime_n
    by (auto simp add: prime_dvd_mult_iff)
qed

lemma (in ring_1) minus_power_prime_CHAR:
  assumes "p = CHAR('a)" "prime p"
  shows "(-x :: 'a) ^ p = -(x ^ p)"
proof (cases "p = 2")
  case False
  have "prime p"
    using assms by blast
  hence "odd p"
    using prime_imp_coprime assms False coprime_right_2_iff_odd gcd_nat.strict_iff_not by blast
  thus ?thesis
    by simp
qed (use assms in \<open>auto simp: uminus_CHAR_2\<close>)


subsection \<open>Rings and fields with prime characteristic\<close>

text \<open>
  We introduce some type classes for rings and fields with prime characteristic.
\<close>

class semiring_prime_char = semiring_1 +
  assumes prime_char_aux: "\<exists>n. prime n \<and> of_nat n = (0 :: 'a)"
begin

lemma CHAR_pos [intro, simp]: "CHAR('a) > 0"
  using local.CHAR_pos_iff local.prime_char_aux prime_gt_0_nat by blast

lemma CHAR_nonzero [simp]: "CHAR('a) \<noteq> 0"
  using CHAR_pos by auto

lemma CHAR_prime [intro, simp]: "prime CHAR('a)"
  by (metis (mono_tags, lifting) gcd_nat.order_iff_strict local.of_nat_1 local.of_nat_eq_0_iff_char_dvd
        local.one_neq_zero local.prime_char_aux prime_nat_iff)

end

lemma semiring_prime_charI [intro?]:
  "prime CHAR('a :: semiring_1) \<Longrightarrow> OFCLASS('a, semiring_prime_char_class)"
  by standard auto

lemma idom_prime_charI [intro?]:
  assumes "CHAR('a :: idom) > 0"
  shows   "OFCLASS('a, semiring_prime_char_class)"
proof
  show "prime CHAR('a)"
    using assms prime_CHAR_semidom by blast
qed

class comm_semiring_prime_char = comm_semiring_1 + semiring_prime_char
class comm_ring_prime_char = comm_ring_1 + semiring_prime_char
begin
subclass comm_semiring_prime_char ..
end
class idom_prime_char = idom + semiring_prime_char
begin
subclass comm_ring_prime_char ..
end

class field_prime_char = field +
  assumes pos_char_exists: "\<exists>n>0. of_nat n = (0 :: 'a)"
begin
subclass idom_prime_char
  apply standard
  using pos_char_exists local.CHAR_pos_iff local.of_nat_CHAR local.prime_CHAR_semidom by blast
end

lemma field_prime_charI [intro?]:
  "n > 0 \<Longrightarrow> of_nat n = (0 :: 'a :: field) \<Longrightarrow> OFCLASS('a, field_prime_char_class)"
  by standard auto

lemma field_prime_charI' [intro?]:
  "CHAR('a :: field) > 0 \<Longrightarrow> OFCLASS('a, field_prime_char_class)"
  by standard auto


subsection \<open>Finite fields\<close>

class finite_field = field_prime_char + finite

lemma finite_fieldI [intro?]:
  assumes "finite (UNIV :: 'a :: field set)"
  shows   "OFCLASS('a, finite_field_class)"
proof standard
  show "\<exists>n>0. of_nat n = (0 :: 'a)"
    using assms prime_CHAR_semidom[where ?'a = 'a] finite_imp_CHAR_pos[OF assms]
    by (intro exI[of _ "CHAR('a)"]) auto
qed fact+

text \<open>
  On a finite field with \<open>n\<close> elements, taking the \<open>n\<close>-th power of an element
  is the identity. This is an obvious consequence of the fact that the multiplicative group of
  the field is a finite group of order \<open>n - 1\<close>, so \<open>x^n = 1\<close> for any non-zero \<open>x\<close>.

  Note that this result is sharp in the sense that the multiplicative group of a
  finite field is cyclic, i.e.\ it contains an element of order \<open>n - 1\<close>.
  (We don't prove this here.)
\<close>
lemma finite_field_power_card_eq_same:
  fixes x :: "'a :: finite_field"
  shows   "x ^ card (UNIV :: 'a set) = x"
proof (cases "x = 0")
  case False
  have "x * (\<Prod>y\<in>UNIV-{0}. x * y) = x * x ^ (card (UNIV :: 'a set) - 1) * \<Prod>(UNIV-{0})"
    by (simp add: prod.distrib mult_ac)
  also have "x * x ^ (card (UNIV :: 'a set) - 1) = x ^ Suc (card (UNIV :: 'a set) - 1)"
    by (subst power_Suc) auto
  also have "Suc (card (UNIV :: 'a set) - 1) = card (UNIV :: 'a set)"
    using finite_UNIV_card_ge_0[where ?'a = 'a] by simp
  also have "(\<Prod>y\<in>UNIV-{0}. x * y) = (\<Prod>y\<in>UNIV-{0}. y)"
    by (rule prod.reindex_bij_witness[of _ "\<lambda>y. y / x" "\<lambda>y. x * y"]) (use False in auto)
  finally show ?thesis
    by simp
qed (use finite_UNIV_card_ge_0[where ?'a = 'a] in auto)

lemma finite_field_power_card_power_eq_same:
  fixes x :: "'a :: finite_field"
  assumes "m = card (UNIV :: 'a set) ^ n"
  shows   "x ^ m = x"
  unfolding assms
  by (induction n) (simp_all add: finite_field_power_card_eq_same power_mult)

class enum_finite_field = finite_field +
  fixes enum_finite_field :: "nat \<Rightarrow> 'a"
  assumes enum_finite_field: "enum_finite_field ` {..<card (UNIV :: 'a set)} = UNIV"
begin

lemma inj_on_enum_finite_field: "inj_on enum_finite_field {..<card (UNIV :: 'a set)}"
  using enum_finite_field by (simp add: eq_card_imp_inj_on)

end

text \<open>
  To get rid of the pending sort hypotheses, we prove that the field with 2 elements is indeed
  a finite field.
\<close>
typedef gf2 = "{0, 1 :: nat}"
  by auto

setup_lifting type_definition_gf2

instantiation gf2 :: field
begin
lift_definition zero_gf2 :: "gf2" is "0" by auto
lift_definition one_gf2 :: "gf2" is "1" by auto
lift_definition uminus_gf2 :: "gf2 \<Rightarrow> gf2" is "\<lambda>x. x" .
lift_definition plus_gf2 :: "gf2 \<Rightarrow> gf2 \<Rightarrow> gf2" is "\<lambda>x y. if x = y then 0 else 1" by auto
lift_definition minus_gf2 :: "gf2 \<Rightarrow> gf2 \<Rightarrow> gf2" is "\<lambda>x y. if x = y then 0 else 1" by auto
lift_definition times_gf2 :: "gf2 \<Rightarrow> gf2 \<Rightarrow> gf2" is "\<lambda>x y. x * y" by auto
lift_definition inverse_gf2 :: "gf2 \<Rightarrow> gf2" is "\<lambda>x. x" .
lift_definition divide_gf2 :: "gf2 \<Rightarrow> gf2 \<Rightarrow> gf2" is "\<lambda>x y. x * y" by auto

instance
  by standard (transfer; fastforce)+

end

instance gf2 :: finite_field
proof
  interpret type_definition Rep_gf2 Abs_gf2 "{0, 1 :: nat}"
    by (rule type_definition_gf2)
  show "finite (UNIV :: gf2 set)"
    by (metis Abs_image finite.emptyI finite.insertI finite_imageI)
qed


subsection \<open>The Freshman's Dream in rings of prime characteristic\<close>

lemma (in comm_semiring_1) freshmans_dream:
  fixes x y :: 'a and n :: nat
  assumes "prime CHAR('a)"
  assumes n_def: "n = CHAR('a)"
  shows   "(x + y) ^ n = x ^ n + y ^ n"
proof -
  interpret comm_semiring_prime_char
    by standard (auto intro!: exI[of _ "CHAR('a)"] assms)
  have "n > 0"
    unfolding n_def by simp
  have "(x + y) ^ n = (\<Sum>k\<le>n. of_nat (n choose k) * x ^ k * y ^ (n - k))"
    by (rule binomial_ring)
  also have "\<dots> = (\<Sum>k\<in>{0,n}. of_nat (n choose k) * x ^ k * y ^ (n - k))"
  proof (intro sum.mono_neutral_right ballI)
    fix k assume "k \<in> {..n} - {0, n}"
    hence k: "k > 0" "k < n"
      by auto
    have "CHAR('a) dvd (n choose k)"
      unfolding n_def
      by (rule dvd_choose_prime) (use k in \<open>auto simp: n_def\<close>)
    hence "of_nat (n choose k) = (0 :: 'a)"
      using of_nat_eq_0_iff_char_dvd by blast
    thus "of_nat (n choose k) * x ^ k * y ^ (n - k) = 0"
      by simp
  qed auto
  finally show ?thesis
    using \<open>n > 0\<close> by (simp add: add_ac)
qed

lemma (in comm_semiring_1) freshmans_dream':
  assumes [simp]: "prime CHAR('a)" and "m = CHAR('a) ^ n"
  shows "(x + y :: 'a) ^ m = x ^ m + y ^ m"
  unfolding assms(2)
proof (induction n)
  case (Suc n)
  have "(x + y) ^ (CHAR('a) ^ n * CHAR('a)) = ((x + y) ^ (CHAR('a) ^ n)) ^ CHAR('a)"
    by (rule power_mult)
  thus ?case
    by (simp add: Suc.IH freshmans_dream Groups.mult_ac flip: power_mult)
qed auto

lemma (in comm_semiring_1) freshmans_dream_sum:
  fixes f :: "'b \<Rightarrow> 'a"
  assumes "prime CHAR('a)" and "n = CHAR('a)"
  shows "sum f A ^ n = sum (\<lambda>i. f i ^ n) A"
  using assms
  by (induct A rule: infinite_finite_induct)
     (auto simp add: power_0_left freshmans_dream)

lemma (in comm_semiring_1) freshmans_dream_sum':
  fixes f :: "'b \<Rightarrow> 'a"
  assumes "prime CHAR('a)" "m = CHAR('a) ^ n"
  shows   "sum f A ^ m = sum (\<lambda>i. f i ^ m) A"
  using assms
  by (induction A rule: infinite_finite_induct)
     (auto simp: freshmans_dream' power_0_left)



(* TODO Legacy names *)
lemmas prime_imp_coprime_nat = prime_imp_coprime[where ?'a = nat]
lemmas prime_imp_coprime_int = prime_imp_coprime[where ?'a = int]
lemmas prime_dvd_mult_nat = prime_dvd_mult_iff[where ?'a = nat]
lemmas prime_dvd_mult_int = prime_dvd_mult_iff[where ?'a = int]
lemmas prime_dvd_mult_eq_nat = prime_dvd_mult_iff[where ?'a = nat]
lemmas prime_dvd_mult_eq_int = prime_dvd_mult_iff[where ?'a = int]
lemmas prime_dvd_power_nat = prime_dvd_power[where ?'a = nat]
lemmas prime_dvd_power_int = prime_dvd_power[where ?'a = int]
lemmas prime_dvd_power_nat_iff = prime_dvd_power_iff[where ?'a = nat]
lemmas prime_dvd_power_int_iff = prime_dvd_power_iff[where ?'a = int]
lemmas prime_imp_power_coprime_nat = prime_imp_power_coprime[where ?'a = nat]
lemmas prime_imp_power_coprime_int = prime_imp_power_coprime[where ?'a = int]
lemmas primes_coprime_nat = primes_coprime[where ?'a = nat]
lemmas primes_coprime_int = primes_coprime[where ?'a = nat]
lemmas prime_divprod_pow_nat = prime_elem_divprod_pow[where ?'a = nat]
lemmas prime_exp = prime_elem_power_iff[where ?'a = nat]

end