(* Title: HOL/Number_Theory/Totient.thy Author: Jeremy Avigad Author: Florian Haftmann Author: Manuel Eberl *) section ‹Fundamental facts about Euler's totient function› theory Totient imports Complex_Main "HOL-Computational_Algebra.Primes" "~~/src/HOL/Number_Theory/Cong" begin definition totatives :: "nat ⇒ nat set" where "totatives n = {k ∈ {0<..n}. coprime k n}" lemma in_totatives_iff: "k ∈ totatives n ⟷ k > 0 ∧ k ≤ n ∧ coprime k n" by (simp add: totatives_def) lemma totatives_code [code]: "totatives n = Set.filter (λk. coprime k n) {0<..n}" by (simp add: totatives_def Set.filter_def) lemma finite_totatives [simp]: "finite (totatives n)" by (simp add: totatives_def) lemma totatives_subset: "totatives n ⊆ {0<..n}" by (auto simp: totatives_def) lemma zero_not_in_totatives [simp]: "0 ∉ totatives n" by (auto simp: totatives_def) lemma totatives_le: "x ∈ totatives n ⟹ x ≤ n" by (auto simp: totatives_def) lemma totatives_less: assumes "x ∈ totatives n" "n > 1" shows "x < n" proof - from assms have "x ≠ n" by (auto simp: totatives_def) with totatives_le[OF assms(1)] show ?thesis by simp qed lemma totatives_0 [simp]: "totatives 0 = {}" by (auto simp: totatives_def) lemma totatives_1 [simp]: "totatives 1 = {Suc 0}" by (auto simp: totatives_def) lemma totatives_Suc_0 [simp]: "totatives (Suc 0) = {Suc 0}" by (auto simp: totatives_def) lemma one_in_totatives [simp]: "n > 0 ⟹ Suc 0 ∈ totatives n" by (auto simp: totatives_def) lemma totatives_eq_empty_iff [simp]: "totatives n = {} ⟷ n = 0" using one_in_totatives[of n] by (auto simp del: one_in_totatives) lemma minus_one_in_totatives: assumes "n ≥ 2" shows "n - 1 ∈ totatives n" using assms coprime_minus_one_nat [of n] by (simp add: in_totatives_iff) lemma totatives_prime_power_Suc: assumes "prime p" shows "totatives (p ^ Suc n) = {0<..p^Suc n} - (λm. p * m) ` {0<..p^n}" proof safe fix m assume m: "p * m ∈ totatives (p ^ Suc n)" and m: "m ∈ {0<..p^n}" thus False using assms by (auto simp: totatives_def gcd_mult_left) next fix k assume k: "k ∈ {0<..p^Suc n}" "k ∉ (λm. p * m) ` {0<..p^n}" from k have "¬(p dvd k)" by (auto elim!: dvdE) hence "coprime k (p ^ Suc n)" using prime_imp_coprime[OF assms, of k] by (intro coprime_exp) (simp_all add: gcd.commute) with k show "k ∈ totatives (p ^ Suc n)" by (simp add: totatives_def) qed (auto simp: totatives_def) lemma totatives_prime: "prime p ⟹ totatives p = {0<..<p}" using totatives_prime_power_Suc[of p 0] by fastforce lemma bij_betw_totatives: assumes "m1 > 1" "m2 > 1" "coprime m1 m2" shows "bij_betw (λx. (x mod m1, x mod m2)) (totatives (m1 * m2)) (totatives m1 × totatives m2)" unfolding bij_betw_def proof show "inj_on (λx. (x mod m1, x mod m2)) (totatives (m1 * m2))" proof (intro inj_onI, clarify) fix x y assume xy: "x ∈ totatives (m1 * m2)" "y ∈ totatives (m1 * m2)" "x mod m1 = y mod m1" "x mod m2 = y mod m2" have ex: "∃!z. z < m1 * m2 ∧ [z = x] (mod m1) ∧ [z = x] (mod m2)" by (rule binary_chinese_remainder_unique_nat) (insert assms, simp_all) have "x < m1 * m2 ∧ [x = x] (mod m1) ∧ [x = x] (mod m2)" "y < m1 * m2 ∧ [y = x] (mod m1) ∧ [y = x] (mod m2)" using xy assms by (simp_all add: totatives_less one_less_mult cong_nat_def) from this[THEN the1_equality[OF ex]] show "x = y" by simp qed next show "(λx. (x mod m1, x mod m2)) ` totatives (m1 * m2) = totatives m1 × totatives m2" proof safe fix x assume "x ∈ totatives (m1 * m2)" with assms show "x mod m1 ∈ totatives m1" "x mod m2 ∈ totatives m2" by (auto simp: totatives_def coprime_mul_eq not_le simp del: One_nat_def intro!: Nat.gr0I) next fix a b assume ab: "a ∈ totatives m1" "b ∈ totatives m2" with assms have ab': "a < m1" "b < m2" by (auto simp: totatives_less) with binary_chinese_remainder_unique_nat[OF assms(3), of a b] obtain x where x: "x < m1 * m2" "x mod m1 = a" "x mod m2 = b" by (auto simp: cong_nat_def) from x ab assms(3) have "x ∈ totatives (m1 * m2)" by (auto simp: totatives_def coprime_mul_eq simp del: One_nat_def intro!: Nat.gr0I) with x show "(a, b) ∈ (λx. (x mod m1, x mod m2)) ` totatives (m1*m2)" by blast qed qed lemma bij_betw_totatives_gcd_eq: fixes n d :: nat assumes "d dvd n" "n > 0" shows "bij_betw (λk. k * d) (totatives (n div d)) {k∈{0<..n}. gcd k n = d}" unfolding bij_betw_def proof show "inj_on (λk. k * d) (totatives (n div d))" by (auto simp: inj_on_def) next show "(λk. k * d) ` totatives (n div d) = {k∈{0<..n}. gcd k n = d}" proof (intro equalityI subsetI, goal_cases) case (1 k) thus ?case using assms by (auto elim!: dvdE simp: inj_on_def totatives_def mult.commute[of d] gcd_mult_right gcd.commute) next case (2 k) hence "d dvd k" by auto then obtain l where k: "k = l * d" by (elim dvdE) auto from 2 and assms show ?case unfolding k by (intro imageI) (auto simp: totatives_def gcd.commute mult.commute[of d] gcd_mult_right elim!: dvdE) qed qed definition totient :: "nat ⇒ nat" where "totient n = card (totatives n)" primrec totient_naive :: "nat ⇒ nat ⇒ nat ⇒ nat" where "totient_naive 0 acc n = acc" | "totient_naive (Suc k) acc n = (if coprime (Suc k) n then totient_naive k (acc + 1) n else totient_naive k acc n)" lemma totient_naive: "totient_naive k acc n = card {x ∈ {0<..k}. coprime x n} + acc" proof (induction k arbitrary: acc) case (Suc k acc) have "totient_naive (Suc k) acc n = (if coprime (Suc k) n then 1 else 0) + card {x ∈ {0<..k}. coprime x n} + acc" using Suc by simp also have "(if coprime (Suc k) n then 1 else 0) = card (if coprime (Suc k) n then {Suc k} else {})" by auto also have "… + card {x ∈ {0<..k}. coprime x n} = card ((if coprime (Suc k) n then {Suc k} else {}) ∪ {x ∈ {0<..k}. coprime x n})" by (intro card_Un_disjoint [symmetric]) auto also have "((if coprime (Suc k) n then {Suc k} else {}) ∪ {x ∈ {0<..k}. coprime x n}) = {x ∈ {0<..Suc k}. coprime x n}" by (auto elim: le_SucE) finally show ?case . qed simp_all lemma totient_code_naive [code]: "totient n = totient_naive n 0 n" by (subst totient_naive) (simp add: totient_def totatives_def) lemma totient_le: "totient n ≤ n" proof - have "card (totatives n) ≤ card {0<..n}" by (intro card_mono) (auto simp: totatives_def) thus ?thesis by (simp add: totient_def) qed lemma totient_less: assumes "n > 1" shows "totient n < n" proof - from assms have "card (totatives n) ≤ card {0<..<n}" using totatives_less[of _ n] totatives_subset[of n] by (intro card_mono) auto with assms show ?thesis by (simp add: totient_def) qed lemma totient_0 [simp]: "totient 0 = 0" by (simp add: totient_def) lemma totient_Suc_0 [simp]: "totient (Suc 0) = Suc 0" by (simp add: totient_def) lemma totient_1 [simp]: "totient 1 = Suc 0" by simp lemma totient_0_iff [simp]: "totient n = 0 ⟷ n = 0" by (auto simp: totient_def) lemma totient_gt_0_iff [simp]: "totient n > 0 ⟷ n > 0" by (auto intro: Nat.gr0I) lemma card_gcd_eq_totient: "n > 0 ⟹ d dvd n ⟹ card {k∈{0<..n}. gcd k n = d} = totient (n div d)" unfolding totient_def by (rule sym, rule bij_betw_same_card[OF bij_betw_totatives_gcd_eq]) lemma totient_divisor_sum: "(∑d | d dvd n. totient d) = n" proof (cases "n = 0") case False hence "n > 0" by simp define A where "A = (λd. {k∈{0<..n}. gcd k n = d})" have *: "card (A d) = totient (n div d)" if d: "d dvd n" for d using ‹n > 0› and d unfolding A_def by (rule card_gcd_eq_totient) have "n = card {1..n}" by simp also have "{1..n} = (⋃d∈{d. d dvd n}. A d)" by safe (auto simp: A_def) also have "card … = (∑d | d dvd n. card (A d))" using ‹n > 0› by (intro card_UN_disjoint) (auto simp: A_def) also have "… = (∑d | d dvd n. totient (n div d))" by (intro sum.cong refl *) auto also have "… = (∑d | d dvd n. totient d)" using ‹n > 0› by (intro sum.reindex_bij_witness[of _ "op div n" "op div n"]) (auto elim: dvdE) finally show ?thesis .. qed auto lemma totient_mult_coprime: assumes "coprime m n" shows "totient (m * n) = totient m * totient n" proof (cases "m > 1 ∧ n > 1") case True hence mn: "m > 1" "n > 1" by simp_all have "totient (m * n) = card (totatives (m * n))" by (simp add: totient_def) also have "… = card (totatives m × totatives n)" using bij_betw_totatives [OF mn ‹coprime m n›] by (rule bij_betw_same_card) also have "… = totient m * totient n" by (simp add: totient_def) finally show ?thesis . next case False with assms show ?thesis by (cases m; cases n) auto qed lemma totient_prime_power_Suc: assumes "prime p" shows "totient (p ^ Suc n) = p ^ n * (p - 1)" proof - from assms have "totient (p ^ Suc n) = card ({0<..p ^ Suc n} - op * p ` {0<..p ^ n})" unfolding totient_def by (subst totatives_prime_power_Suc) simp_all also from assms have "… = p ^ Suc n - card (op * p ` {0<..p^n})" by (subst card_Diff_subset) (auto intro: prime_gt_0_nat) also from assms have "card (op * p ` {0<..p^n}) = p ^ n" by (subst card_image) (auto simp: inj_on_def) also have "p ^ Suc n - p ^ n = p ^ n * (p - 1)" by (simp add: algebra_simps) finally show ?thesis . qed lemma totient_prime_power: assumes "prime p" "n > 0" shows "totient (p ^ n) = p ^ (n - 1) * (p - 1)" using totient_prime_power_Suc[of p "n - 1"] assms by simp lemma totient_imp_prime: assumes "totient p = p - 1" "p > 0" shows "prime p" proof (cases "p = 1") case True with assms show ?thesis by auto next case False with assms have p: "p > 1" by simp have "x ∈ {0<..<p}" if "x ∈ totatives p" for x using that and p by (cases "x = p") (auto simp: totatives_def) with assms have *: "totatives p = {0<..<p}" by (intro card_subset_eq) (auto simp: totient_def) have **: False if "x ≠ 1" "x ≠ p" "x dvd p" for x proof - from that have nz: "x ≠ 0" by (auto intro!: Nat.gr0I) from that and p have le: "x ≤ p" by (intro dvd_imp_le) auto from that and nz have "¬coprime x p" by auto hence "x ∉ totatives p" by (simp add: totatives_def) also note * finally show False using that and le by auto qed hence "(∀m. m dvd p ⟶ m = 1 ∨ m = p)" by blast with p show ?thesis by (subst prime_nat_iff) (auto dest: **) qed lemma totient_prime: assumes "prime p" shows "totient p = p - 1" using totient_prime_power_Suc[of p 0] assms by simp lemma totient_2 [simp]: "totient 2 = 1" and totient_3 [simp]: "totient 3 = 2" and totient_5 [simp]: "totient 5 = 4" and totient_7 [simp]: "totient 7 = 6" by (subst totient_prime; simp)+ lemma totient_4 [simp]: "totient 4 = 2" and totient_8 [simp]: "totient 8 = 4" and totient_9 [simp]: "totient 9 = 6" using totient_prime_power[of 2 2] totient_prime_power[of 2 3] totient_prime_power[of 3 2] by simp_all lemma totient_6 [simp]: "totient 6 = 2" using totient_mult_coprime[of 2 3] by (simp add: gcd_non_0_nat) lemma totient_even: assumes "n > 2" shows "even (totient n)" proof (cases "∃p. prime p ∧ p ≠ 2 ∧ p dvd n") case True then obtain p where p: "prime p" "p ≠ 2" "p dvd n" by auto from ‹p ≠ 2› have "p = 0 ∨ p = 1 ∨ p > 2" by auto with p(1) have "odd p" using prime_odd_nat[of p] by auto define k where "k = multiplicity p n" from p assms have k_pos: "k > 0" unfolding k_def by (subst multiplicity_gt_zero_iff) auto have "p ^ k dvd n" unfolding k_def by (simp add: multiplicity_dvd) then obtain m where m: "n = p ^ k * m" by (elim dvdE) with assms have m_pos: "m > 0" by (auto intro!: Nat.gr0I) from k_def m_pos p have "¬p dvd m" by (subst (asm) m) (auto intro!: Nat.gr0I simp: prime_elem_multiplicity_mult_distrib prime_elem_multiplicity_eq_zero_iff) hence "coprime (p ^ k) m" by (intro coprime_exp_left prime_imp_coprime[OF p(1)]) thus ?thesis using p k_pos ‹odd p› by (auto simp add: m totient_mult_coprime totient_prime_power) next case False from assms have "n = (∏p∈prime_factors n. p ^ multiplicity p n)" by (intro Primes.prime_factorization_nat) auto also from False have "… = (∏p∈prime_factors n. if p = 2 then 2 ^ multiplicity 2 n else 1)" by (intro prod.cong refl) auto also have "… = 2 ^ multiplicity 2 n" by (subst prod.delta[OF finite_set_mset]) (auto simp: prime_factors_multiplicity) finally have n: "n = 2 ^ multiplicity 2 n" . have "multiplicity 2 n = 0 ∨ multiplicity 2 n = 1 ∨ multiplicity 2 n > 1" by force with n assms have "multiplicity 2 n > 1" by auto thus ?thesis by (subst n) (simp add: totient_prime_power) qed lemma totient_prod_coprime: assumes "pairwise_coprime (f ` A)" "inj_on f A" shows "totient (prod f A) = prod (λx. totient (f x)) A" using assms proof (induction A rule: infinite_finite_induct) case (insert x A) from insert.prems and insert.hyps have *: "coprime (prod f A) (f x)" by (intro prod_coprime[OF pairwise_coprimeD[OF insert.prems(1)]]) (auto simp: inj_on_def) from insert.hyps have "prod f (insert x A) = prod f A * f x" by simp also have "totient … = totient (prod f A) * totient (f x)" using insert.hyps insert.prems by (intro totient_mult_coprime *) also have "totient (prod f A) = (∏x∈A. totient (f x))" using insert.prems by (intro insert.IH) (auto dest: pairwise_coprime_subset) also from insert.hyps have "… * totient (f x) = (∏x∈insert x A. totient (f x))" by simp finally show ?case . qed simp_all (* TODO Move *) lemma prime_power_eq_imp_eq: fixes p q :: "'a :: factorial_semiring" assumes "prime p" "prime q" "m > 0" assumes "p ^ m = q ^ n" shows "p = q" proof (rule ccontr) assume pq: "p ≠ q" from assms have "m = multiplicity p (p ^ m)" by (subst multiplicity_prime_power) auto also note ‹p ^ m = q ^ n› also from assms pq have "multiplicity p (q ^ n) = 0" by (subst multiplicity_distinct_prime_power) auto finally show False using ‹m > 0› by simp qed lemma totient_formula1: assumes "n > 0" shows "totient n = (∏p∈prime_factors n. p ^ (multiplicity p n - 1) * (p - 1))" proof - from assms have "n = (∏p∈prime_factors n. p ^ multiplicity p n)" by (rule prime_factorization_nat) also have "totient … = (∏x∈prime_factors n. totient (x ^ multiplicity x n))" proof (rule totient_prod_coprime) show "pairwise_coprime ((λp. p ^ multiplicity p n) ` prime_factors n)" proof (standard, clarify, goal_cases) fix p q assume "p ∈# prime_factorization n" "q ∈# prime_factorization n" "p ^ multiplicity p n ≠ q ^ multiplicity q n" thus "coprime (p ^ multiplicity p n) (q ^ multiplicity q n)" by (intro coprime_exp2 primes_coprime[of p q]) auto qed next show "inj_on (λp. p ^ multiplicity p n) (prime_factors n)" proof fix p q assume pq: "p ∈# prime_factorization n" "q ∈# prime_factorization n" "p ^ multiplicity p n = q ^ multiplicity q n" from assms and pq have "prime p" "prime q" "multiplicity p n > 0" by (simp_all add: prime_factors_multiplicity) from prime_power_eq_imp_eq[OF this pq(3)] show "p = q" . qed qed also have "… = (∏p∈prime_factors n. p ^ (multiplicity p n - 1) * (p - 1))" by (intro prod.cong refl totient_prime_power) (auto simp: prime_factors_multiplicity) finally show ?thesis . qed lemma totient_dvd: assumes "m dvd n" shows "totient m dvd totient n" proof (cases "m = 0 ∨ n = 0") case False let ?M = "λp m :: nat. multiplicity p m - 1" have "(∏p∈prime_factors m. p ^ ?M p m * (p - 1)) dvd (∏p∈prime_factors n. p ^ ?M p n * (p - 1))" using assms False by (intro prod_dvd_prod_subset2 mult_dvd_mono dvd_refl le_imp_power_dvd diff_le_mono dvd_prime_factors dvd_imp_multiplicity_le) auto with False show ?thesis by (simp add: totient_formula1) qed (insert assms, auto) lemma totient_dvd_mono: assumes "m dvd n" "n > 0" shows "totient m ≤ totient n" by (cases "m = 0") (insert assms, auto intro: dvd_imp_le totient_dvd) (* TODO Move *) lemma prime_factors_power: "n > 0 ⟹ prime_factors (x ^ n) = prime_factors x" by (cases "x = 0"; cases "n = 0") (auto simp: prime_factors_multiplicity prime_elem_multiplicity_power_distrib zero_power) lemma totient_formula2: "real (totient n) = real n * (∏p∈prime_factors n. 1 - 1 / real p)" proof (cases "n = 0") case False have "real (totient n) = (∏p∈prime_factors n. real (p ^ (multiplicity p n - 1) * (p - 1)))" using False by (subst totient_formula1) simp_all also have "… = (∏p∈prime_factors n. real (p ^ multiplicity p n) * (1 - 1 / real p))" by (intro prod.cong refl) (auto simp add: field_simps prime_factors_multiplicity prime_ge_Suc_0_nat of_nat_diff power_Suc [symmetric] simp del: power_Suc) also have "… = real (∏p∈prime_factors n. p ^ multiplicity p n) * (∏p∈prime_factors n. 1 - 1 / real p)" by (subst prod.distrib) auto also have "(∏p∈prime_factors n. p ^ multiplicity p n) = n" using False by (intro Primes.prime_factorization_nat [symmetric]) auto finally show ?thesis . qed auto lemma totient_gcd: "totient (a * b) * totient (gcd a b) = totient a * totient b * gcd a b" proof (cases "a = 0 ∨ b = 0") case False let ?P = "prime_factors :: nat ⇒ nat set" have "real (totient a * totient b * gcd a b) = real (a * b * gcd a b) * ((∏p∈?P a. 1 - 1 / real p) * (∏p∈?P b. 1 - 1 / real p))" by (simp add: totient_formula2) also have "?P a = (?P a - ?P b) ∪ (?P a ∩ ?P b)" by auto also have "(∏p∈…. 1 - 1 / real p) = (∏p∈?P a - ?P b. 1 - 1 / real p) * (∏p∈?P a ∩ ?P b. 1 - 1 / real p)" by (rule prod.union_disjoint) blast+ also have "… * (∏p∈?P b. 1 - 1 / real p) = (∏p∈?P a - ?P b. 1 - 1 / real p) * (∏p∈?P b. 1 - 1 / real p) * (∏p∈?P a ∩ ?P b. 1 - 1 / real p)" (is "_ = ?A * _") by (simp only: mult_ac) also have "?A = (∏p∈?P a - ?P b ∪ ?P b. 1 - 1 / real p)" by (rule prod.union_disjoint [symmetric]) blast+ also have "?P a - ?P b ∪ ?P b = ?P a ∪ ?P b" by blast also have "real (a * b * gcd a b) * ((∏p∈…. 1 - 1 / real p) * (∏p∈?P a ∩ ?P b. 1 - 1 / real p)) = real (totient (a * b) * totient (gcd a b))" using False by (simp add: totient_formula2 prime_factors_product prime_factorization_gcd) finally show ?thesis by (simp only: of_nat_eq_iff) qed auto lemma totient_mult: "totient (a * b) = totient a * totient b * gcd a b div totient (gcd a b)" by (subst totient_gcd [symmetric]) simp lemma of_nat_eq_1_iff: "of_nat x = (1 :: 'a :: {semiring_1, semiring_char_0}) ⟷ x = 1" using of_nat_eq_iff[of x 1] by (simp del: of_nat_eq_iff) (* TODO Move *) lemma gcd_2_odd: assumes "odd (n::nat)" shows "gcd n 2 = 1" proof - from assms obtain k where n: "n = Suc (2 * k)" by (auto elim!: oddE) have "coprime (Suc (2 * k)) (2 * k)" by (rule coprime_Suc_nat) thus ?thesis using n by (subst (asm) coprime_mul_eq) simp_all qed lemma totient_double: "totient (2 * n) = (if even n then 2 * totient n else totient n)" by (subst totient_mult) (auto simp: gcd.commute[of 2] gcd_2_odd) lemma totient_power_Suc: "totient (n ^ Suc m) = n ^ m * totient n" proof (induction m arbitrary: n) case (Suc m n) have "totient (n ^ Suc (Suc m)) = totient (n * n ^ Suc m)" by simp also have "… = n ^ Suc m * totient n" using Suc.IH by (subst totient_mult) simp finally show ?case . qed simp_all lemma totient_power: "m > 0 ⟹ totient (n ^ m) = n ^ (m - 1) * totient n" using totient_power_Suc[of n "m - 1"] by (cases m) simp_all lemma totient_gcd_lcm: "totient (gcd a b) * totient (lcm a b) = totient a * totient b" proof (cases "a = 0 ∨ b = 0") case False let ?P = "prime_factors :: nat ⇒ nat set" and ?f = "λp::nat. 1 - 1 / real p" have "real (totient (gcd a b) * totient (lcm a b)) = real (gcd a b * lcm a b) * (prod ?f (?P a ∩ ?P b) * prod ?f (?P a ∪ ?P b))" using False unfolding of_nat_mult by (simp add: totient_formula2 prime_factorization_gcd prime_factorization_lcm) also have "gcd a b * lcm a b = a * b" by simp also have "?P a ∪ ?P b = (?P a - ?P a ∩ ?P b) ∪ ?P b" by blast also have "prod ?f … = prod ?f (?P a - ?P a ∩ ?P b) * prod ?f (?P b)" by (rule prod.union_disjoint) blast+ also have "prod ?f (?P a ∩ ?P b) * … = prod ?f (?P a ∩ ?P b ∪ (?P a - ?P a ∩ ?P b)) * prod ?f (?P b)" by (subst prod.union_disjoint) auto also have "?P a ∩ ?P b ∪ (?P a - ?P a ∩ ?P b) = ?P a" by blast also have "real (a * b) * (prod ?f (?P a) * prod ?f (?P b)) = real (totient a * totient b)" using False by (simp add: totient_formula2) finally show ?thesis by (simp only: of_nat_eq_iff) qed auto end