(*
File: HOL/Number_Theory/Prime_Powers.thy
Author: Manuel Eberl <eberlm@in.tum.de>
Prime powers and the Mangoldt function
*)
section \<open>Prime powers\<close>
theory Prime_Powers
imports Complex_Main "~~/src/HOL/Computational_Algebra/Primes"
begin
definition aprimedivisor :: "'a :: normalization_semidom \<Rightarrow> 'a" where
"aprimedivisor q = (SOME p. prime p \<and> p dvd q)"
definition primepow :: "'a :: normalization_semidom \<Rightarrow> bool" where
"primepow n \<longleftrightarrow> (\<exists>p k. prime p \<and> k > 0 \<and> n = p ^ k)"
definition primepow_factors :: "'a :: normalization_semidom \<Rightarrow> 'a set" where
"primepow_factors n = {x. primepow x \<and> x dvd n}"
lemma primepow_gt_Suc_0: "primepow n \<Longrightarrow> n > Suc 0"
using one_less_power[of "p::nat" for p] by (auto simp: primepow_def prime_nat_iff)
lemma
assumes "prime p" "p dvd n"
shows prime_aprimedivisor: "prime (aprimedivisor n)"
and aprimedivisor_dvd: "aprimedivisor n dvd n"
proof -
from assms have "\<exists>p. prime p \<and> p dvd n" by auto
from someI_ex[OF this] show "prime (aprimedivisor n)" "aprimedivisor n dvd n"
unfolding aprimedivisor_def by (simp_all add: conj_commute)
qed
lemma
assumes "n \<noteq> 0" "\<not>is_unit (n :: 'a :: factorial_semiring)"
shows prime_aprimedivisor': "prime (aprimedivisor n)"
and aprimedivisor_dvd': "aprimedivisor n dvd n"
proof -
from someI_ex[OF prime_divisor_exists[OF assms]]
show "prime (aprimedivisor n)" "aprimedivisor n dvd n"
unfolding aprimedivisor_def by (simp_all add: conj_commute)
qed
lemma aprimedivisor_of_prime [simp]:
assumes "prime p"
shows "aprimedivisor p = p"
proof -
from assms have "\<exists>q. prime q \<and> q dvd p" by auto
from someI_ex[OF this, folded aprimedivisor_def] assms show ?thesis
by (auto intro: primes_dvd_imp_eq)
qed
lemma aprimedivisor_pos_nat: "(n::nat) > 1 \<Longrightarrow> aprimedivisor n > 0"
using aprimedivisor_dvd'[of n] by (auto elim: dvdE intro!: Nat.gr0I)
lemma aprimedivisor_primepow_power:
assumes "primepow n" "k > 0"
shows "aprimedivisor (n ^ k) = aprimedivisor n"
proof -
from assms obtain p l where l: "prime p" "l > 0" "n = p ^ l"
by (auto simp: primepow_def)
from l assms have *: "prime (aprimedivisor (n ^ k))" "aprimedivisor (n ^ k) dvd n ^ k"
by (intro prime_aprimedivisor[of p] aprimedivisor_dvd[of p] dvd_power;
simp add: power_mult [symmetric])+
from * l have "aprimedivisor (n ^ k) dvd p ^ (l * k)" by (simp add: power_mult)
with assms * l have "aprimedivisor (n ^ k) dvd p"
by (subst (asm) prime_dvd_power_iff) simp_all
with l assms have "aprimedivisor (n ^ k) = p"
by (intro primes_dvd_imp_eq prime_aprimedivisor l) (auto simp: power_mult [symmetric])
moreover from l have "aprimedivisor n dvd p ^ l"
by (auto intro: aprimedivisor_dvd simp: prime_gt_0_nat)
with assms l have "aprimedivisor n dvd p"
by (subst (asm) prime_dvd_power_iff) (auto intro!: prime_aprimedivisor simp: prime_gt_0_nat)
with l assms have "aprimedivisor n = p"
by (intro primes_dvd_imp_eq prime_aprimedivisor l) auto
ultimately show ?thesis by simp
qed
lemma aprimedivisor_prime_power:
assumes "prime p" "k > 0"
shows "aprimedivisor (p ^ k) = p"
proof -
from assms have *: "prime (aprimedivisor (p ^ k))" "aprimedivisor (p ^ k) dvd p ^ k"
by (intro prime_aprimedivisor[of p] aprimedivisor_dvd[of p]; simp add: prime_nat_iff)+
from assms * have "aprimedivisor (p ^ k) dvd p"
by (subst (asm) prime_dvd_power_iff) simp_all
with assms * show "aprimedivisor (p ^ k) = p" by (intro primes_dvd_imp_eq)
qed
lemma prime_factorization_primepow:
assumes "primepow n"
shows "prime_factorization n =
replicate_mset (multiplicity (aprimedivisor n) n) (aprimedivisor n)"
using assms
by (auto simp: primepow_def aprimedivisor_prime_power prime_factorization_prime_power)
lemma primepow_decompose:
assumes "primepow n"
shows "aprimedivisor n ^ multiplicity (aprimedivisor n) n = n"
proof -
from assms have "n \<noteq> 0" by (intro notI) (auto simp: primepow_def)
hence "n = unit_factor n * prod_mset (prime_factorization n)"
by (subst prod_mset_prime_factorization) simp_all
also from assms have "unit_factor n = 1" by (auto simp: primepow_def unit_factor_power)
also have "prime_factorization n =
replicate_mset (multiplicity (aprimedivisor n) n) (aprimedivisor n)"
by (intro prime_factorization_primepow assms)
also have "prod_mset \<dots> = aprimedivisor n ^ multiplicity (aprimedivisor n) n" by simp
finally show ?thesis by simp
qed
lemma prime_power_not_one:
assumes "prime p" "k > 0"
shows "p ^ k \<noteq> 1"
proof
assume "p ^ k = 1"
hence "is_unit (p ^ k)" by simp
thus False using assms by (simp add: is_unit_power_iff)
qed
lemma zero_not_primepow [simp]: "\<not>primepow 0"
by (auto simp: primepow_def)
lemma one_not_primepow [simp]: "\<not>primepow 1"
by (auto simp: primepow_def prime_power_not_one)
lemma primepow_not_unit [simp]: "primepow p \<Longrightarrow> \<not>is_unit p"
by (auto simp: primepow_def is_unit_power_iff)
lemma unit_factor_primepow: "primepow p \<Longrightarrow> unit_factor p = 1"
by (auto simp: primepow_def unit_factor_power)
lemma aprimedivisor_primepow:
assumes "prime p" "p dvd n" "primepow (n :: 'a :: factorial_semiring)"
shows "aprimedivisor (p * n) = p" "aprimedivisor n = p"
proof -
from assms have [simp]: "n \<noteq> 0" by auto
define q where "q = aprimedivisor n"
with assms have q: "prime q" by (auto simp: q_def intro!: prime_aprimedivisor)
from \<open>primepow n\<close> have n: "n = q ^ multiplicity q n"
by (simp add: primepow_decompose q_def)
have nz: "multiplicity q n \<noteq> 0"
proof
assume "multiplicity q n = 0"
with n have n': "n = unit_factor n" by simp
have "is_unit n" by (subst n', rule unit_factor_is_unit) (insert assms, auto)
with assms show False by auto
qed
with \<open>prime p\<close> \<open>p dvd n\<close> q have "p dvd q"
by (subst (asm) n) (auto intro: prime_dvd_power)
with \<open>prime p\<close> q have "p = q" by (intro primes_dvd_imp_eq)
thus "aprimedivisor n = p" by (simp add: q_def)
define r where "r = aprimedivisor (p * n)"
with assms have r: "r dvd (p * n)" "prime r" unfolding r_def
by (intro aprimedivisor_dvd[of p] prime_aprimedivisor[of p]; simp)+
hence "r dvd q ^ Suc (multiplicity q n)"
by (subst (asm) n) (auto simp: \<open>p = q\<close> dest: dvd_unit_imp_unit)
with r have "r dvd q"
by (auto intro: prime_dvd_power_nat simp: prime_dvd_mult_iff dest: prime_dvd_power)
with r q have "r = q" by (intro primes_dvd_imp_eq)
thus "aprimedivisor (p * n) = p" by (simp add: r_def \<open>p = q\<close>)
qed
lemma power_eq_prime_powerD:
fixes p :: "'a :: factorial_semiring"
assumes "prime p" "n > 0" "x ^ n = p ^ k"
shows "\<exists>i. normalize x = normalize (p ^ i)"
proof -
have "normalize x = normalize (p ^ multiplicity p x)"
proof (rule multiplicity_eq_imp_eq)
fix q :: 'a assume "prime q"
from assms have "multiplicity q (x ^ n) = multiplicity q (p ^ k)" by simp
with \<open>prime q\<close> and assms have "n * multiplicity q x = k * multiplicity q p"
by (subst (asm) (1 2) prime_elem_multiplicity_power_distrib) (auto simp: power_0_left)
with assms and \<open>prime q\<close> show "multiplicity q x = multiplicity q (p ^ multiplicity p x)"
by (cases "p = q") (auto simp: multiplicity_distinct_prime_power prime_multiplicity_other)
qed (insert assms, auto simp: power_0_left)
thus ?thesis by auto
qed
lemma primepow_power_iff:
assumes "unit_factor p = 1"
shows "primepow (p ^ n) \<longleftrightarrow> primepow (p :: 'a :: factorial_semiring) \<and> n > 0"
proof safe
assume "primepow (p ^ n)"
hence n: "n \<noteq> 0" by (auto intro!: Nat.gr0I)
thus "n > 0" by simp
from assms have [simp]: "normalize p = p"
using normalize_mult_unit_factor[of p] by (simp only: mult.right_neutral)
from \<open>primepow (p ^ n)\<close> obtain q k where *: "k > 0" "prime q" "p ^ n = q ^ k"
by (auto simp: primepow_def)
with power_eq_prime_powerD[of q n p k] n
obtain i where eq: "normalize p = normalize (q ^ i)" by auto
with primepow_not_unit[OF \<open>primepow (p ^ n)\<close>] have "i \<noteq> 0"
by (intro notI) (simp add: normalize_1_iff is_unit_power_iff del: primepow_not_unit)
with \<open>normalize p = normalize (q ^ i)\<close> \<open>prime q\<close> show "primepow p"
by (auto simp: normalize_power primepow_def intro!: exI[of _ q] exI[of _ i])
next
assume "primepow p" "n > 0"
then obtain q k where *: "k > 0" "prime q" "p = q ^ k" by (auto simp: primepow_def)
with \<open>n > 0\<close> show "primepow (p ^ n)"
by (auto simp: primepow_def power_mult intro!: exI[of _ q] exI[of _ "k * n"])
qed
lemma primepow_prime [simp]: "prime n \<Longrightarrow> primepow n"
by (auto simp: primepow_def intro!: exI[of _ n] exI[of _ "1::nat"])
lemma primepow_prime_power [simp]:
"prime (p :: 'a :: factorial_semiring) \<Longrightarrow> primepow (p ^ n) \<longleftrightarrow> n > 0"
by (subst primepow_power_iff) auto
(* TODO Generalise *)
lemma primepow_multD:
assumes "primepow (a * b :: nat)"
shows "a = 1 \<or> primepow a" "b = 1 \<or> primepow b"
proof -
from assms obtain p k where k: "k > 0" "a * b = p ^ k" "prime p"
unfolding primepow_def by auto
then obtain i j where "a = p ^ i" "b = p ^ j"
using prime_power_mult_nat[of p a b] by blast
with \<open>prime p\<close> show "a = 1 \<or> primepow a" "b = 1 \<or> primepow b" by auto
qed
lemma primepow_mult_aprimedivisorI:
assumes "primepow (n :: 'a :: factorial_semiring)"
shows "primepow (aprimedivisor n * n)"
by (subst (2) primepow_decompose[OF assms, symmetric], subst power_Suc [symmetric],
subst primepow_prime_power)
(insert assms, auto intro!: prime_aprimedivisor' dest: primepow_gt_Suc_0)
lemma aprimedivisor_vimage:
assumes "prime (p :: 'a :: factorial_semiring)"
shows "aprimedivisor -` {p} \<inter> primepow_factors n = {p ^ k |k. k > 0 \<and> p ^ k dvd n}"
proof safe
fix q assume q: "q \<in> primepow_factors n"
hence q': "q \<noteq> 0" "q \<noteq> 1" by (auto simp: primepow_def primepow_factors_def prime_power_not_one)
let ?n = "multiplicity (aprimedivisor q) q"
from q q' have "q = aprimedivisor q ^ ?n \<and> ?n > 0 \<and> aprimedivisor q ^ ?n dvd n"
using prime_aprimedivisor'[of q] aprimedivisor_dvd'[of q]
by (subst primepow_decompose [symmetric])
(auto simp: primepow_factors_def multiplicity_gt_zero_iff unit_factor_primepow
intro: dvd_trans[OF multiplicity_dvd])
thus "\<exists>k. q = aprimedivisor q ^ k \<and> k > 0 \<and> aprimedivisor q ^ k dvd n" ..
next
fix k :: nat assume k: "p ^ k dvd n" "k > 0"
with assms show "p ^ k \<in> aprimedivisor -` {p}"
by (auto simp: aprimedivisor_prime_power)
with assms k show "p ^ k \<in> primepow_factors n"
by (auto simp: primepow_factors_def primepow_def aprimedivisor_prime_power intro: Suc_leI)
qed
lemma primepow_factors_altdef:
fixes x :: "'a :: factorial_semiring"
assumes "x \<noteq> 0"
shows "primepow_factors x = {p ^ k |p k. p \<in> prime_factors x \<and> k \<in> {0<.. multiplicity p x}}"
proof (intro equalityI subsetI)
fix q assume "q \<in> primepow_factors x"
then obtain p k where pk: "prime p" "k > 0" "q = p ^ k" "q dvd x"
unfolding primepow_factors_def primepow_def by blast
moreover have "k \<le> multiplicity p x" using pk assms by (intro multiplicity_geI) auto
ultimately show "q \<in> {p ^ k |p k. p \<in> prime_factors x \<and> k \<in> {0<.. multiplicity p x}}"
by (auto simp: prime_factors_multiplicity intro!: exI[of _ p] exI[of _ k])
qed (auto simp: primepow_factors_def prime_factors_multiplicity multiplicity_dvd')
lemma finite_primepow_factors:
assumes "x \<noteq> (0 :: 'a :: factorial_semiring)"
shows "finite (primepow_factors x)"
proof -
have "finite (SIGMA p:prime_factors x. {0<..multiplicity p x})"
by (intro finite_SigmaI) simp_all
hence "finite ((\<lambda>(p,k). p ^ k) ` \<dots>)" (is "finite ?A") by (rule finite_imageI)
also have "?A = primepow_factors x"
using assms by (subst primepow_factors_altdef) fast+
finally show ?thesis .
qed
definition mangoldt :: "nat \<Rightarrow> 'a :: real_algebra_1" where
"mangoldt n = (if primepow n then of_real (ln (real (aprimedivisor n))) else 0)"
lemma of_real_mangoldt [simp]: "of_real (mangoldt n) = mangoldt n"
by (simp add: mangoldt_def)
lemma mangoldt_sum:
assumes "n \<noteq> 0"
shows "(\<Sum>d | d dvd n. mangoldt d :: 'a :: real_algebra_1) = of_real (ln (real n))"
proof -
have "(\<Sum>d | d dvd n. mangoldt d :: 'a) = of_real (\<Sum>d | d dvd n. mangoldt d)" by simp
also have "(\<Sum>d | d dvd n. mangoldt d) = (\<Sum>d \<in> primepow_factors n. ln (real (aprimedivisor d)))"
using assms by (intro sum.mono_neutral_cong_right) (auto simp: primepow_factors_def mangoldt_def)
also have "\<dots> = ln (real (\<Prod>d \<in> primepow_factors n. aprimedivisor d))"
using assms finite_primepow_factors[of n]
by (subst ln_prod [symmetric])
(auto simp: primepow_factors_def intro!: aprimedivisor_pos_nat
intro: Nat.gr0I primepow_gt_Suc_0)
also have "primepow_factors n =
(\<lambda>(p,k). p ^ k) ` (SIGMA p:prime_factors n. {0<..multiplicity p n})"
(is "_ = _ ` ?A") by (subst primepow_factors_altdef[OF assms]) fast+
also have "prod aprimedivisor \<dots> = (\<Prod>(p,k)\<in>?A. aprimedivisor (p ^ k))"
by (subst prod.reindex)
(auto simp: inj_on_def prime_power_inj'' prime_factors_multiplicity
prod.Sigma [symmetric] case_prod_unfold)
also have "\<dots> = (\<Prod>(p,k)\<in>?A. p)"
by (intro prod.cong refl) (auto simp: aprimedivisor_prime_power prime_factors_multiplicity)
also have "\<dots> = (\<Prod>x\<in>prime_factors n. \<Prod>k\<in>{0<..multiplicity x n}. x)"
by (rule prod.Sigma [symmetric]) auto
also have "\<dots> = (\<Prod>x\<in>prime_factors n. x ^ multiplicity x n)"
by (intro prod.cong refl) (simp add: prod_constant)
also have "\<dots> = n" using assms by (intro prime_factorization_nat [symmetric]) simp
finally show ?thesis .
qed
end