# Theory Prime_Powers

```(*
File:      HOL/Number_Theory/Prime_Powers.thy
Author:    Manuel Eberl <manuel@pruvisto.org>

Prime powers and the Mangoldt function
*)
section ‹Prime powers›
theory Prime_Powers
imports Complex_Main "HOL-Computational_Algebra.Primes" "HOL-Library.FuncSet"
begin

definition aprimedivisor :: "'a :: normalization_semidom ⇒ 'a" where
"aprimedivisor q = (SOME p. prime p ∧ p dvd q)"

definition primepow :: "'a :: normalization_semidom ⇒ bool" where
"primepow n ⟷ (∃p k. prime p ∧ k > 0 ∧ n = p ^ k)"

definition primepow_factors :: "'a :: normalization_semidom ⇒ 'a set" where
"primepow_factors n = {x. primepow x ∧ x dvd n}"

lemma primepow_gt_Suc_0: "primepow n ⟹ 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 "∃p. prime p ∧ 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 ≠ 0" "¬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 "∃q. prime q ∧ 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 ⟹ 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:
fixes n :: "'a :: factorial_semiring_multiplicative"
assumes "primepow n"
shows   "aprimedivisor n ^ multiplicity (aprimedivisor n) n = n"
proof -
from assms have "n ≠ 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 … = 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 ≠ 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]: "¬primepow 0"
by (auto simp: primepow_def)

lemma one_not_primepow [simp]: "¬primepow 1"
by (auto simp: primepow_def prime_power_not_one)

lemma primepow_not_unit [simp]: "primepow p ⟹ ¬is_unit p"
by (auto simp: primepow_def is_unit_power_iff)

lemma not_primepow_Suc_0_nat [simp]: "¬primepow (Suc 0)"
using primepow_gt_Suc_0[of "Suc 0"] by auto

lemma primepow_gt_0_nat: "primepow n ⟹ n > (0::nat)"
using primepow_gt_Suc_0[of n] by simp

lemma unit_factor_primepow:
fixes p :: "'a :: factorial_semiring_multiplicative"
shows "primepow p ⟹ 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_multiplicative)"
shows   "aprimedivisor (p * n) = p" "aprimedivisor n = p"
proof -
from assms have [simp]: "n ≠ 0" by auto
define q where "q = aprimedivisor n"
with assms have q: "prime q" by (auto simp: q_def intro!: prime_aprimedivisor)
from ‹primepow n› have n: "n = q ^ multiplicity q n"
by (simp add: primepow_decompose q_def)
have nz: "multiplicity q n ≠ 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 ‹prime p› ‹p dvd n› q have "p dvd q"
by (subst (asm) n) (auto intro: prime_dvd_power)
with ‹prime p› 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: ‹p = q› 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 ‹p = q›)
qed

lemma power_eq_prime_powerD:
fixes p :: "'a :: factorial_semiring"
assumes "prime p" "n > 0" "x ^ n = p ^ k"
shows   "∃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 ‹prime q› 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 ‹prime q› 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:
fixes p :: "'a :: factorial_semiring_multiplicative"
assumes "unit_factor p = 1"
shows   "primepow (p ^ n) ⟷ primepow p ∧ n > 0"
proof safe
assume "primepow (p ^ n)"
hence n: "n ≠ 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 ‹primepow (p ^ n)› 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 ‹primepow (p ^ n)›] have "i ≠ 0"
by (intro notI) (simp add: normalize_1_iff is_unit_power_iff del: primepow_not_unit)
with ‹normalize p = normalize (q ^ i)› ‹prime q› 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 ‹n > 0› show "primepow (p ^ n)"
by (auto simp: primepow_def power_mult intro!: exI[of _ q] exI[of _ "k * n"])
qed

lemma primepow_power_iff_nat:
"p > 0 ⟹ primepow (p ^ n) ⟷ primepow (p :: nat) ∧ n > 0"
by (rule primepow_power_iff) (simp_all add: unit_factor_nat_def)

lemma primepow_prime [simp]: "prime n ⟹ 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_multiplicative) ⟹ primepow (p ^ n) ⟷ n > 0"
by (subst primepow_power_iff) auto

lemma aprimedivisor_vimage:
assumes "prime (p :: 'a :: factorial_semiring_multiplicative)"
shows   "aprimedivisor -` {p} ∩ primepow_factors n = {p ^ k |k. k > 0 ∧ p ^ k dvd n}"
proof safe
fix q assume q: "q ∈ primepow_factors n"
hence q': "q ≠ 0" "q ≠ 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 ∧ ?n > 0 ∧ aprimedivisor q ^ ?n dvd n"
by (auto simp: primepow_decompose primepow_factors_def prime_multiplicity_gt_zero_iff
prime_aprimedivisor' prime_imp_prime_elem aprimedivisor_dvd')
thus "∃k. q = aprimedivisor q ^ k ∧ k > 0 ∧ aprimedivisor q ^ k dvd n" ..
next
fix k :: nat assume k: "p ^ k dvd n" "k > 0"
with assms show "p ^ k ∈ aprimedivisor -` {p}"
by (auto simp: aprimedivisor_prime_power)
with assms k show "p ^ k ∈ primepow_factors n"
by (auto simp: primepow_factors_def primepow_def aprimedivisor_prime_power intro: Suc_leI)
qed

lemma aprimedivisor_nat:
assumes "n ≠ (Suc 0::nat)"
shows   "prime (aprimedivisor n)" "aprimedivisor n dvd n"
proof -
from assms have "∃p. prime p ∧ p dvd n" by (intro prime_factor_nat) auto
from someI_ex[OF this, folded aprimedivisor_def]
show "prime (aprimedivisor n)" "aprimedivisor n dvd n" by blast+
qed

lemma aprimedivisor_gt_Suc_0:
assumes "n ≠ Suc 0"
shows   "aprimedivisor n > Suc 0"
proof -
from assms have "prime (aprimedivisor n)" by (rule aprimedivisor_nat)
thus "aprimedivisor n > Suc 0" by (simp add: prime_nat_iff)
qed

lemma aprimedivisor_le_nat:
assumes "n > Suc 0"
shows   "aprimedivisor n ≤ n"
proof -
from assms have "aprimedivisor n dvd n" by (intro aprimedivisor_nat) simp_all
with assms show "aprimedivisor n ≤ n"
by (intro dvd_imp_le) simp_all
qed

lemma bij_betw_primepows:
"bij_betw (λ(p,k). p ^ Suc k :: 'a :: factorial_semiring_multiplicative)
(Collect prime × UNIV) (Collect primepow)"
proof (rule bij_betwI [where ?g = "(λn. (aprimedivisor n, multiplicity (aprimedivisor n) n - 1))"],
goal_cases)
case 1
show "(λ(p, k). p ^ Suc k :: 'a) ∈ Collect prime × UNIV → Collect primepow"
by (auto intro!: primepow_prime_power simp del: power_Suc )
next
case 2
show ?case
by (auto simp: primepow_def prime_aprimedivisor)
next
case (3 n)
thus ?case
by (auto simp: aprimedivisor_prime_power simp del: power_Suc)
next
case (4 n)
hence *: "0 < multiplicity (aprimedivisor n) n"
by (subst prime_multiplicity_gt_zero_iff)
(auto intro!: prime_imp_prime_elem aprimedivisor_dvd simp: primepow_def prime_aprimedivisor)
have "aprimedivisor n * aprimedivisor n ^ (multiplicity (aprimedivisor n) n - Suc 0) =
aprimedivisor n ^ Suc (multiplicity (aprimedivisor n) n - Suc 0)" by simp
also from * have "Suc (multiplicity (aprimedivisor n) n - Suc 0) =
multiplicity (aprimedivisor n) n"
by (subst Suc_diff_Suc) (auto simp: prime_multiplicity_gt_zero_iff)
also have "aprimedivisor n ^ … = n"
using 4 by (subst primepow_decompose) auto
finally show ?case by auto
qed

(* TODO Generalise *)
lemma primepow_multD:
assumes "primepow (a * b :: nat)"
shows   "a = 1 ∨ primepow a" "b = 1 ∨ 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 ‹prime p› show "a = 1 ∨ primepow a" "b = 1 ∨ primepow b" by auto
qed

lemma primepow_mult_aprimedivisorI:
assumes "primepow (n :: 'a :: factorial_semiring_multiplicative)"
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 primepow_factors_altdef:
fixes x :: "'a :: factorial_semiring_multiplicative"
assumes "x ≠ 0"
shows "primepow_factors x = {p ^ k |p k. p ∈ prime_factors x ∧ k ∈ {0<.. multiplicity p x}}"
proof (intro equalityI subsetI)
fix q assume "q ∈ 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 ≤ multiplicity p x" using pk assms by (intro multiplicity_geI) auto
ultimately show "q ∈ {p ^ k |p k. p ∈ prime_factors x ∧ k ∈ {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 ≠ (0 :: 'a :: factorial_semiring_multiplicative)"
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 ((λ(p,k). p ^ k) ` …)" (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

lemma aprimedivisor_primepow_factors_conv_prime_factorization:
assumes [simp]: "n ≠ (0 :: 'a :: factorial_semiring_multiplicative)"
shows   "image_mset aprimedivisor (mset_set (primepow_factors n)) = prime_factorization n"
(is "?lhs = ?rhs")
proof (intro multiset_eqI)
fix p :: 'a
show "count ?lhs p = count ?rhs p"
proof (cases "prime p")
case False
have "p ∉# image_mset aprimedivisor (mset_set (primepow_factors n))"
proof
assume "p ∈# image_mset aprimedivisor (mset_set (primepow_factors n))"
then obtain q where "p = aprimedivisor q" "q ∈ primepow_factors n"
by (auto simp: finite_primepow_factors)
with False prime_aprimedivisor'[of q] have "q = 0 ∨ is_unit q" by auto
with ‹q ∈ primepow_factors n› show False by (auto simp: primepow_factors_def primepow_def)
qed
hence "count ?lhs p = 0" by (simp only: Multiset.not_in_iff)
with False show ?thesis by (simp add: count_prime_factorization)
next
case True
hence p: "p ≠ 0" "¬is_unit p" by auto
have "count ?lhs p = card (aprimedivisor -` {p} ∩ primepow_factors n)"
by (simp add: count_image_mset finite_primepow_factors)
also have "aprimedivisor -` {p} ∩ primepow_factors n = {p^k |k. k > 0 ∧ p ^ k dvd n}"
using True by (rule aprimedivisor_vimage)
also from True have "… = (λk. p ^ k) ` {0<..multiplicity p n}"
by (subst power_dvd_iff_le_multiplicity) auto
also from p True have "card … = multiplicity p n"
by (subst card_image) (auto intro!: inj_onI dest: prime_power_inj)
also from True have "… = count (prime_factorization n) p"
by (simp add: count_prime_factorization)
finally show ?thesis .
qed
qed

lemma prime_elem_aprimedivisor_nat: "d > Suc 0 ⟹ prime_elem (aprimedivisor d)"
using prime_aprimedivisor'[of d] by simp

lemma aprimedivisor_gt_0_nat [simp]: "d > Suc 0 ⟹ aprimedivisor d > 0"
using prime_aprimedivisor'[of d] by (simp add: prime_gt_0_nat)

lemma aprimedivisor_gt_Suc_0_nat [simp]: "d > Suc 0 ⟹ aprimedivisor d > Suc 0"
using prime_aprimedivisor'[of d] by (simp add: prime_gt_Suc_0_nat)

lemma aprimedivisor_not_Suc_0_nat [simp]: "d > Suc 0 ⟹ aprimedivisor d ≠ Suc 0"
using aprimedivisor_gt_Suc_0[of d] by (intro notI) auto

lemma multiplicity_aprimedivisor_gt_0_nat [simp]:
"d > Suc 0 ⟹ multiplicity (aprimedivisor d) d > 0"
by (subst multiplicity_gt_zero_iff) (auto intro: aprimedivisor_dvd')

lemma primepowI:
"prime p ⟹ k > 0 ⟹ p ^ k = n ⟹ primepow n ∧ aprimedivisor n = p"
unfolding primepow_def by (auto simp: aprimedivisor_prime_power)

lemma not_primepowI:
assumes "prime p" "prime q" "p ≠ q" "p dvd n" "q dvd n"
shows   "¬primepow n"
using assms by (auto simp: primepow_def dest!: prime_dvd_power[rotated] dest: primes_dvd_imp_eq)

lemma sum_prime_factorization_conv_sum_primepow_factors:
fixes n :: "'a :: factorial_semiring_multiplicative"
assumes "n ≠ 0"
shows "(∑q∈primepow_factors n. f (aprimedivisor q)) = (∑p∈#prime_factorization n. f p)"
proof -
from assms have "prime_factorization n = image_mset aprimedivisor (mset_set (primepow_factors n))"
by (rule aprimedivisor_primepow_factors_conv_prime_factorization [symmetric])
also have "(∑p∈#…. f p) = (∑q∈primepow_factors n. f (aprimedivisor q))"
by (simp add: image_mset.compositionality sum_unfold_sum_mset o_def)
finally show ?thesis ..
qed

lemma multiplicity_aprimedivisor_Suc_0_iff:
assumes "primepow (n :: 'a :: factorial_semiring_multiplicative)"
shows   "multiplicity (aprimedivisor n) n = Suc 0 ⟷ prime n"
by (subst (3) primepow_decompose [OF assms, symmetric])
(insert assms, auto simp add: prime_power_iff intro!: prime_aprimedivisor')

definition mangoldt :: "nat ⇒ 'a :: real_algebra_1" where
"mangoldt n = (if primepow n then of_real (ln (real (aprimedivisor n))) else 0)"

lemma mangoldt_0 [simp]: "mangoldt 0 = 0"
by (simp add: mangoldt_def)

lemma mangoldt_Suc_0 [simp]: "mangoldt (Suc 0) = 0"
by (simp add: mangoldt_def)

lemma of_real_mangoldt [simp]: "of_real (mangoldt n) = mangoldt n"
by (simp add: mangoldt_def)

lemma mangoldt_sum:
assumes "n ≠ 0"
shows   "(∑d | d dvd n. mangoldt d :: 'a :: real_algebra_1) = of_real (ln (real n))"
proof -
have "(∑d | d dvd n. mangoldt d :: 'a) = of_real (∑d | d dvd n. mangoldt d)" by simp
also have "(∑d | d dvd n. mangoldt d) = (∑d ∈ 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 "… = ln (real (∏d ∈ 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 =
(λ(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 … = (∏(p,k)∈?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 "… = (∏(p,k)∈?A. p)"
by (intro prod.cong refl) (auto simp: aprimedivisor_prime_power prime_factors_multiplicity)
also have "… = (∏x∈prime_factors n. ∏k∈{0<..multiplicity x n}. x)"
by (rule prod.Sigma [symmetric]) auto
also have "… = (∏x∈prime_factors n. x ^ multiplicity x n)"
by (intro prod.cong refl) (simp add: prod_constant)
also have "… = n" using assms by (intro prime_factorization_nat [symmetric]) simp
finally show ?thesis .
qed

lemma mangoldt_primepow:
"prime p ⟹ mangoldt (p ^ k) = (if k > 0 then of_real (ln (real p)) else 0)"
by (simp add: mangoldt_def aprimedivisor_prime_power)

lemma mangoldt_primepow' [simp]: "prime p ⟹ k > 0 ⟹ mangoldt (p ^ k) = of_real (ln (real p))"
by (subst mangoldt_primepow) auto

lemma mangoldt_prime [simp]: "prime p ⟹ mangoldt p = of_real (ln (real p))"
using mangoldt_primepow[of p 1] by simp

lemma mangoldt_nonneg: "0 ≤ (mangoldt d :: real)"
using aprimedivisor_gt_Suc_0_nat[of d]
by (auto simp: mangoldt_def of_nat_le_iff[of 1 x for x, unfolded of_nat_1] Suc_le_eq
intro!: ln_ge_zero dest: primepow_gt_Suc_0)

lemma norm_mangoldt [simp]:
"norm (mangoldt n :: 'a :: real_normed_algebra_1) = mangoldt n"
proof (cases "primepow n")
case True
hence "prime (aprimedivisor n)"
by (intro prime_aprimedivisor')
(auto simp: primepow_def prime_gt_0_nat)
hence "aprimedivisor n > 1" by (simp add: prime_gt_Suc_0_nat)
with True show ?thesis by (auto simp: mangoldt_def abs_if)
qed (auto simp: mangoldt_def)

lemma Re_mangoldt [simp]: "Re (mangoldt n) = mangoldt n"
and Im_mangoldt [simp]: "Im (mangoldt n) = 0"
by (simp_all add: mangoldt_def)

lemma abs_mangoldt [simp]: "abs (mangoldt n :: real) = mangoldt n"
using norm_mangoldt[of n, where ?'a = real, unfolded real_norm_def] .

lemma mangoldt_le:
assumes "n > 0"
shows   "mangoldt n ≤ ln n"
proof (cases "primepow n")
case True
from True have "prime (aprimedivisor n)"
by (intro prime_aprimedivisor')
(auto simp: primepow_def prime_gt_0_nat)
hence gt_1: "aprimedivisor n > 1" by (simp add: prime_gt_Suc_0_nat)
from True have "mangoldt n = ln (aprimedivisor n)"
by (simp add: mangoldt_def)
also have "… ≤ ln n" using True gt_1
by (subst ln_le_cancel_iff) (auto intro!: Nat.gr0I dvd_imp_le aprimedivisor_dvd')
finally show ?thesis .
qed (insert assms, auto simp: mangoldt_def)

end
```