(* File: Squarefree_Nat.thy Author: Manuel Eberl <eberlm@in.tum.de> The unique decomposition of a natural number into a squarefree part and a square. *) section ‹Squarefree decomposition of natural numbers› theory Squarefree_Nat imports Main "HOL-Number_Theory.Number_Theory" Prime_Harmonic_Misc begin text ‹ The squarefree part of a natural number is the set of all prime factors that appear with odd multiplicity. The square part, correspondingly, is what remains after dividing by the squarefree part. › definition squarefree_part :: "nat ⇒ nat set" where "squarefree_part n = {p∈prime_factors n. odd (multiplicity p n)}" definition square_part :: "nat ⇒ nat" where "square_part n = (if n = 0 then 0 else (∏p∈prime_factors n. p ^ (multiplicity p n div 2)))" lemma squarefree_part_0 [simp]: "squarefree_part 0 = {}" by (simp add: squarefree_part_def) lemma square_part_0 [simp]: "square_part 0 = 0" by (simp add: square_part_def) lemma squarefree_decompose: "∏(squarefree_part n) * square_part n ^ 2 = n" proof (cases "n = 0") case False define A s where "A = squarefree_part n" and "s = square_part n" have "(∏A) = (∏p∈A. p ^ (multiplicity p n mod 2))" by (intro prod.cong) (auto simp: A_def squarefree_part_def elim!: oddE) also have "… = (∏p∈prime_factors n. p ^ (multiplicity p n mod 2))" by (intro prod.mono_neutral_left) (auto simp: A_def squarefree_part_def) also from False have "… * s^2 = n" by (simp add: s_def square_part_def prod.distrib [symmetric] power_add [symmetric] power_mult [symmetric] prime_factorization_nat [symmetric] algebra_simps prod_power_distrib) finally show "∏A * s^2 = n" . qed simp lemma squarefree_part_pos [simp]: "∏(squarefree_part n) > 0" using prime_gt_0_nat unfolding squarefree_part_def by auto lemma squarefree_part_ge_Suc_0 [simp]: "∏(squarefree_part n) ≥ Suc 0" using squarefree_part_pos[of n] by presburger lemma squarefree_part_subset [intro]: "squarefree_part n ⊆ prime_factors n" unfolding squarefree_part_def by auto lemma squarefree_part_finite [simp]: "finite (squarefree_part n)" by (rule finite_subset[OF squarefree_part_subset]) simp lemma squarefree_part_dvd [simp]: "∏(squarefree_part n) dvd n" by (subst (2) squarefree_decompose [of n, symmetric]) simp lemma squarefree_part_dvd' [simp]: "p ∈ squarefree_part n ⟹ p dvd n" by (rule dvd_prodD[OF _ squarefree_part_dvd]) simp_all lemma square_part_dvd [simp]: "square_part n ^ 2 dvd n" by (subst (2) squarefree_decompose [of n, symmetric]) simp lemma square_part_dvd' [simp]: "square_part n dvd n" by (subst (2) squarefree_decompose [of n, symmetric]) simp lemma squarefree_part_le: "p ∈ squarefree_part n ⟹ p ≤ n" by (cases "n = 0") (simp_all add: dvd_imp_le) lemma square_part_le: "square_part n ≤ n" by (cases "n = 0") (simp_all add: dvd_imp_le) lemma square_part_le_sqrt: "square_part n ≤ nat ⌊sqrt (real n)⌋" proof - have "1 * square_part n ^ 2 ≤ ∏(squarefree_part n) * square_part n ^ 2" by (intro mult_right_mono) simp_all also have "… = n" by (rule squarefree_decompose) finally have "real (square_part n ^ 2) ≤ real n" by (subst of_nat_le_iff) simp hence "sqrt (real (square_part n ^ 2)) ≤ sqrt (real n)" by (subst real_sqrt_le_iff) simp_all also have "sqrt (real (square_part n ^ 2)) = real (square_part n)" by simp finally show ?thesis by linarith qed lemma square_part_pos [simp]: "n > 0 ⟹ square_part n > 0" unfolding square_part_def using prime_gt_0_nat by auto lemma square_part_ge_Suc_0 [simp]: "n > 0 ⟹ square_part n ≥ Suc 0" using square_part_pos[of n] by presburger lemma zero_not_in_squarefree_part [simp]: "0 ∉ squarefree_part n" unfolding squarefree_part_def by auto lemma multiplicity_squarefree_part: "prime p ⟹ multiplicity p (∏(squarefree_part n)) = (if p ∈ squarefree_part n then 1 else 0)" using squarefree_part_subset[of n] by (intro multiplicity_prod_prime_powers_nat') auto text ‹ The squarefree part really is square, its only square divisor is 1. › lemma square_dvd_squarefree_part_iff: "x^2 dvd ∏(squarefree_part n) ⟷ x = 1" proof (rule iffI, rule multiplicity_eq_nat) assume dvd: "x^2 dvd ∏(squarefree_part n)" hence "x ≠ 0" using squarefree_part_subset[of n] prime_gt_0_nat by (intro notI) auto thus x: "x > 0" by simp fix p :: nat assume p: "prime p" from p x have "2 * multiplicity p x = multiplicity p (x^2)" by (simp add: multiplicity_power_nat) also from dvd have "… ≤ multiplicity p (∏(squarefree_part n))" by (intro dvd_imp_multiplicity_le) simp_all also have "… ≤ 1" using multiplicity_squarefree_part[of p n] p by simp finally show "multiplicity p x = multiplicity p 1" by simp qed simp_all lemma squarefree_decomposition_unique1: assumes "squarefree_part m = squarefree_part n" assumes "square_part m = square_part n" shows "m = n" by (subst (1 2) squarefree_decompose [symmetric]) (simp add: assms) lemma squarefree_decomposition_unique2: assumes n: "n > 0" assumes decomp: "n = (∏A2 * s2^2)" assumes prime: "⋀x. x ∈ A2 ⟹ prime x" assumes fin: "finite A2" assumes s2_nonneg: "s2 ≥ 0" shows "A2 = squarefree_part n" and "s2 = square_part n" proof - define A1 s1 where "A1 = squarefree_part n" and "s1 = square_part n" have "finite A1" unfolding A1_def by simp note fin = ‹finite A1› ‹finite A2› have "A1 ⊆ prime_factors n" unfolding A1_def using squarefree_part_subset . note subset = this prime have "∏A1 > 0" "∏A2 > 0" using subset(1) prime_gt_0_nat by (auto intro!: prod_pos dest: prime) from n have "s1 > 0" unfolding s1_def by simp from assms have "s2 ≠ 0" by (intro notI) simp hence "s2 > 0" by simp note pos = ‹∏A1 > 0› ‹∏A2 > 0› ‹s1 > 0› ‹s2 > 0› have eq': "multiplicity p s1 = multiplicity p s2" "multiplicity p (∏A1) = multiplicity p (∏A2)" if p: "prime p" for p proof - define m where "m = multiplicity p" from decomp have "m (∏A1 * s1^2) = m (∏A2 * s2^2)" unfolding A1_def s1_def by (simp add: A1_def s1_def squarefree_decompose) with p pos have eq: "m (∏A1) + 2 * m s1 = m (∏A2) + 2 * m s2" by (simp add: m_def prime_elem_multiplicity_mult_distrib multiplicity_power_nat) moreover from fin subset p have "m (∏A1) ≤ 1" "m (∏A2) ≤ 1" unfolding m_def by ((subst multiplicity_prod_prime_powers_nat', auto)[])+ ultimately show "m s1 = m s2" by linarith with eq show "m (∏A1) = m (∏A2)" by simp qed show "s2 = square_part n" by (rule multiplicity_eq_nat) (insert pos eq'(1), auto simp: s1_def) have "∏A2 = ∏(squarefree_part n)" by (rule multiplicity_eq_nat) (insert pos eq'(2), auto simp: A1_def) with fin subset show "A2 = squarefree_part n" unfolding A1_def by (intro prod_prime_eq) auto qed lemma squarefree_decomposition_unique2': assumes decomp: "(∏A1 * s1^2) = (∏A2 * s2^2 :: nat)" assumes fin: "finite A1" "finite A2" assumes subset: "⋀x. x ∈ A1 ⟹ prime x" "⋀x. x ∈ A2 ⟹ prime x" assumes pos: "s1 > 0" "s2 > 0" defines "n ≡ ∏A1 * s1^2" shows "A1 = A2" "s1 = s2" proof - from pos have n: "n > 0" using prime_gt_0_nat by (auto simp: n_def intro!: prod_pos dest: subset) have "A1 = squarefree_part n" "s1 = square_part n" by ((rule squarefree_decomposition_unique2[of n A1 s1], insert assms n, simp_all)[])+ moreover have "A2 = squarefree_part n" "s2 = square_part n" by ((rule squarefree_decomposition_unique2[of n A2 s2], insert assms n, simp_all)[])+ ultimately show "A1 = A2" "s1 = s2" by simp_all qed text ‹ The following is a nice and simple lower bound on the number of prime numbers less than a given number due to Erd\H{o}s. In particular, it implies that there are infinitely many primes. › lemma primes_lower_bound: fixes n :: nat assumes "n > 0" defines "π ≡ λn. card {p. prime p ∧ p ≤ n}" shows "real (π n) ≥ ln (real n) / ln 4" proof - have "real n = real (card {1..n})" by simp also have "{1..n} = (λ(A, b). ∏A * b^2) ` (λn. (squarefree_part n, square_part n)) ` {1..n}" unfolding image_comp o_def squarefree_decompose case_prod_unfold fst_conv snd_conv by simp also have "card … ≤ card ((λn. (squarefree_part n, square_part n)) ` {1..n})" by (rule card_image_le) simp_all also have "… ≤ card (squarefree_part ` {1..n} × square_part ` {1..n})" by (rule card_mono) auto also have "real … = real (card (squarefree_part ` {1..n})) * real (card (square_part ` {1..n}))" by simp also have "… ≤ 2 ^ π n * sqrt (real n)" proof (rule mult_mono) have "card (squarefree_part ` {1..n}) ≤ card (Pow {p. prime p ∧ p ≤ n})" using squarefree_part_subset squarefree_part_le by (intro card_mono) force+ also have "real … = 2 ^ π n" by (simp add: π_def card_Pow) finally show "real (card (squarefree_part ` {1..n})) ≤ 2 ^ π n" by - simp_all next have "square_part k ≤ nat ⌊sqrt n⌋" if "k ≤ n" for k by (rule order.trans[OF square_part_le_sqrt]) (insert that, auto intro!: nat_mono floor_mono) hence "card (square_part ` {1..n}) ≤ card {1..nat ⌊sqrt n⌋}" by (intro card_mono) (auto intro: order.trans[OF square_part_le_sqrt]) also have "… = nat ⌊sqrt n⌋" by simp also have "real … ≤ sqrt n" by simp finally show "real (card (square_part ` {1..n})) ≤ sqrt (real n)" by - simp_all qed simp_all finally have "real n ≤ 2 ^ π n * sqrt (real n)" by - simp_all with ‹n > 0› have "ln (real n) ≤ ln (2 ^ π n * sqrt (real n))" by (subst ln_le_cancel_iff) simp_all moreover have "ln (4 :: real) = real 2 * ln 2" by (subst ln_realpow [symmetric]) simp_all ultimately show ?thesis using ‹n > 0› by (simp add: ln_mult ln_realpow[of _ "π n"] ln_sqrt field_simps) qed end