author eberlm Thu, 04 May 2017 16:49:29 +0200 changeset 65726 f5d64d094efe parent 65720 c5b19f997214 child 65727 33368a2296aa
More material on totient function
 src/HOL/Number_Theory/Pocklington.thy file | annotate | diff | comparison | revisions src/HOL/Number_Theory/Residues.thy file | annotate | diff | comparison | revisions src/HOL/Number_Theory/Totient.thy file | annotate | diff | comparison | revisions
```--- a/src/HOL/Number_Theory/Pocklington.thy	Thu May 04 15:15:07 2017 +0100
+++ b/src/HOL/Number_Theory/Pocklington.thy	Thu May 04 16:49:29 2017 +0200
@@ -144,7 +144,7 @@
assumes n: "n \<ge> 2" and an: "[a ^ (n - 1) = 1] (mod n)"
and nm: "\<forall>m. 0 < m \<and> m < n - 1 \<longrightarrow> \<not> [a ^ m = 1] (mod n)"
shows "prime n"
-using \<open>n \<ge> 2\<close> proof (rule totient_prime)
+proof (rule totient_imp_prime)
show "totient n = n - 1"
proof (rule ccontr)
have "[a ^ totient n = 1] (mod n)"
@@ -152,11 +152,11 @@
(use n an in auto)
moreover assume "totient n \<noteq> n - 1"
then have "totient n > 0 \<and> totient n < n - 1"
-      using \<open>n \<ge> 2\<close> by (simp add: order_less_le)
+      using \<open>n \<ge> 2\<close> and totient_less[of n] by simp
ultimately show False
using nm by auto
qed
-qed
+qed (insert n, auto)

lemma nat_exists_least_iff: "(\<exists>(n::nat). P n) \<longleftrightarrow> (\<exists>n. P n \<and> (\<forall>m < n. \<not> P m))"
by (metis ex_least_nat_le not_less0)
@@ -527,7 +527,7 @@
by auto}
hence d1: "d = 1" by blast
hence o: "ord p (a^r) = q" using d by simp
-  from pp prime_totient [of p]
+  from pp totient_prime [of p]
have totient_eq: "totient p = p - 1" by simp
{fix d assume d: "d dvd p" "d dvd a" "d \<noteq> 1"
from pp[unfolded prime_nat_iff] d have dp: "d = p" by blast```
```--- a/src/HOL/Number_Theory/Residues.thy	Thu May 04 15:15:07 2017 +0100
+++ b/src/HOL/Number_Theory/Residues.thy	Thu May 04 16:49:29 2017 +0200
@@ -31,7 +31,7 @@
where
"residue_ring m =
\<lparr>carrier = {0..m - 1},
-     mult = \<lambda>x y. (x * y) mod m,
+     monoid.mult = \<lambda>x y. (x * y) mod m,
one = 1,
zero = 0,
add = \<lambda>x y. (x + y) mod m\<rparr>"
@@ -247,18 +247,22 @@

lemma (in residues) totient_eq:
"totient (nat m) = card (Units R)"
+  thm R_def
proof -
have *: "inj_on nat (Units R)"
by (rule inj_onI) (auto simp add: res_units_eq)
-  have "totatives (nat m) = nat ` Units R"
-    by (auto simp add: res_units_eq totatives_def transfer_nat_int_gcd(1))
-      (smt One_nat_def image_def mem_Collect_eq nat_0 nat_eq_iff nat_less_iff of_nat_1 transfer_int_nat_gcd(1))
+  define m' where "m' = nat m"
+  from m_gt_one have m: "m = int m'" "m' > 1" by (simp_all add: m'_def)
+  from m have "x \<in> Units R \<longleftrightarrow> x \<in> int ` totatives m'" for x
+    unfolding res_units_eq
+    by (cases x; cases "x = m") (auto simp: totatives_def transfer_int_nat_gcd)
+  hence "Units R = int ` totatives m'" by blast
+  hence "totatives m' = nat ` Units R" by (simp add: image_image)
then have "card (totatives (nat m)) = card (nat ` Units R)"
-    by simp
+    by (simp add: m'_def)
also have "\<dots> = card (Units R)"
using * card_image [of nat "Units R"] by auto
-  finally show ?thesis
-    by simp
+  finally show ?thesis by (simp add: totient_def)
qed

lemma (in residues_prime) totient_eq: "totient p = p - 1"
@@ -298,7 +302,7 @@
then have "[a ^ totient p = 1] (mod p)"
by (rule euler_theorem)
also have "totient p = p - 1"
-    by (rule prime_totient) (rule assms)
+    by (rule totient_prime) (rule assms)
finally show ?thesis .
qed
```
```--- a/src/HOL/Number_Theory/Totient.thy	Thu May 04 15:15:07 2017 +0100
+++ b/src/HOL/Number_Theory/Totient.thy	Thu May 04 16:49:29 2017 +0200
@@ -1,164 +1,515 @@
(*  Title:      HOL/Number_Theory/Totient.thy
Author:     Jeremy Avigad
Author:     Florian Haftmann
+    Author:     Manuel Eberl
*)

section \<open>Fundamental facts about Euler's totient function\<close>

theory Totient
imports
+  Complex_Main
"~~/src/HOL/Computational_Algebra/Primes"
+  "~~/src/HOL/Number_Theory/Cong"
begin
-
-definition totatives :: "nat \<Rightarrow> nat set"
-  where "totatives n = {m. 0 < m \<and> m < n \<and> coprime m n}"
-
-lemma in_totatives_iff [simp]:
-  "m \<in> totatives n \<longleftrightarrow> 0 < m \<and> m < n \<and> coprime m n"
+
+definition totatives :: "nat \<Rightarrow> nat set" where
+  "totatives n = {k \<in> {0<..n}. coprime k n}"
+
+lemma in_totatives_iff: "k \<in> totatives n \<longleftrightarrow> k > 0 \<and> k \<le> n \<and> coprime k n"
+  by (simp add: totatives_def)
+
+lemma totatives_code [code]: "totatives n = Set.filter (\<lambda>k. 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 finite_totatives [simp]:
-  "finite (totatives n)"
-  by (simp add: totatives_def)
-
-lemma totatives_subset:
-  "totatives n \<subseteq> {1..<n}"
-  by auto
+
+lemma totatives_subset: "totatives n \<subseteq> {0<..n}"
+  by (auto simp: totatives_def)
+
+lemma zero_not_in_totatives [simp]: "0 \<notin> totatives n"
+  by (auto simp: totatives_def)
+
+lemma totatives_le: "x \<in> totatives n \<Longrightarrow> x \<le> n"
+  by (auto simp: totatives_def)
+
+lemma totatives_less:
+  assumes "x \<in> totatives n" "n > 1"
+  shows   "x < n"
+proof -
+  from assms have "x \<noteq> n" by (auto simp: totatives_def)
+  with totatives_le[OF assms(1)] show ?thesis by simp
+qed

-lemma totatives_subset_Suc_0 [simp]:
-  "totatives n \<subseteq> {Suc 0..<n}"
-  using totatives_subset [of n] by simp
+lemma totatives_0 [simp]: "totatives 0 = {}"
+  by (auto simp: totatives_def)

-lemma totatives_0 [simp]:
-  "totatives 0 = {}"
-  using totatives_subset [of 0] by simp
+lemma totatives_1 [simp]: "totatives 1 = {Suc 0}"
+  by (auto simp: totatives_def)

-lemma totatives_1 [simp]:
-  "totatives 1 = {}"
-  using totatives_subset [of 1] by simp
+lemma totatives_Suc_0 [simp]: "totatives (Suc 0) = {Suc 0}"
+  by (auto simp: totatives_def)

-lemma totatives_Suc_0 [simp]:
-  "totatives (Suc 0) = {}"
-  using totatives_1 by simp
+lemma one_in_totatives [simp]: "n > 0 \<Longrightarrow> Suc 0 \<in> totatives n"
+  by (auto simp: totatives_def)

-lemma one_in_totatives:
-  assumes "n \<ge> 2"
-  shows "1 \<in> totatives n"
-  using assms by simp
-
+lemma totatives_eq_empty_iff [simp]: "totatives n = {} \<longleftrightarrow> n = 0"
+  using one_in_totatives[of n] by (auto simp del: one_in_totatives)
+
lemma minus_one_in_totatives:
assumes "n \<ge> 2"
shows "n - 1 \<in> totatives n"
-  using assms coprime_minus_one_nat [of n] by simp
+  using assms coprime_minus_one_nat [of n] by (simp add: in_totatives_iff)

-lemma totatives_eq_empty_iff [simp]:
-  "totatives n = {} \<longleftrightarrow> n \<in> {0, 1}"
+lemma totatives_prime_power_Suc:
+  assumes "prime p"
+  shows   "totatives (p ^ Suc n) = {0<..p^Suc n} - (\<lambda>m. p * m) ` {0<..p^n}"
+proof safe
+  fix m assume m: "p * m \<in> totatives (p ^ Suc n)" and m: "m \<in> {0<..p^n}"
+  thus False using assms by (auto simp: totatives_def gcd_mult_left)
+next
+  fix k assume k: "k \<in> {0<..p^Suc n}" "k \<notin> (\<lambda>m. p * m) ` {0<..p^n}"
+  from k have "\<not>(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 \<in> totatives (p ^ Suc n)" by (simp add: totatives_def)
+qed (auto simp: totatives_def)
+
+lemma totatives_prime: "prime p \<Longrightarrow> 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 (\<lambda>x. (x mod m1, x mod m2)) (totatives (m1 * m2))
+             (totatives m1 \<times> totatives m2)"
+  unfolding bij_betw_def
proof
-  assume "totatives n = {}"
-  show "n \<in> {0, 1}"
-  proof (rule ccontr)
-    assume "n \<notin> {0, 1}"
-    then have "n \<ge> 2"
-      by simp
-    then have "1 \<in> totatives n"
-      by (rule one_in_totatives)
-    with \<open>totatives n = {}\<close> show False
-      by simp
+  show "inj_on (\<lambda>x. (x mod m1, x mod m2)) (totatives (m1 * m2))"
+  proof (intro inj_onI, clarify)
+    fix x y assume xy: "x \<in> totatives (m1 * m2)" "y \<in> totatives (m1 * m2)"
+                       "x mod m1 = y mod m1" "x mod m2 = y mod m2"
+    have ex: "\<exists>!z. z < m1 * m2 \<and> [z = x] (mod m1) \<and> [z = x] (mod m2)"
+      by (rule binary_chinese_remainder_unique_nat) (insert assms, simp_all)
+    have "x < m1 * m2 \<and> [x = x] (mod m1) \<and> [x = x] (mod m2)"
+         "y < m1 * m2 \<and> [y = x] (mod m1) \<and> [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 "(\<lambda>x. (x mod m1, x mod m2)) ` totatives (m1 * m2) = totatives m1 \<times> totatives m2"
+  proof safe
+    fix x assume "x \<in> totatives (m1 * m2)"
+    with assms show "x mod m1 \<in> totatives m1" "x mod m2 \<in> 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 \<in> totatives m1" "b \<in> 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 \<in> totatives (m1 * m2)"
+      by (auto simp: totatives_def coprime_mul_eq simp del: One_nat_def intro!: Nat.gr0I)
+    with x show "(a, b) \<in> (\<lambda>x. (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 (\<lambda>k. k * d) (totatives (n div d)) {k\<in>{0<..n}. gcd k n = d}"
+  unfolding bij_betw_def
+proof
+  show "inj_on (\<lambda>k. k * d) (totatives (n div d))"
+    by (auto simp: inj_on_def)
+next
+  show "(\<lambda>k. k * d) ` totatives (n div d) = {k\<in>{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 auto
+qed
+
+
+
+definition totient :: "nat \<Rightarrow> nat" where
+  "totient n = card (totatives n)"
+
+primrec totient_naive :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 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 \<in> {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 \<in> {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 "\<dots> + card {x \<in> {0<..k}. coprime x n} =
+               card ((if coprime (Suc k) n then {Suc k} else {}) \<union> {x \<in> {0<..k}. coprime x n})"
+    by (intro card_Un_disjoint [symmetric]) auto
+  also have "((if coprime (Suc k) n then {Suc k} else {}) \<union> {x \<in> {0<..k}. coprime x n}) =
+               {x \<in> {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 prime_totatives:
-  assumes "prime p"
-  shows "totatives p = {1..<p}"
-proof
-  show "{1..<p} \<subseteq> totatives p"
-  proof
-    fix n
-    assume "n \<in> {1..<p}"
-    with nat_dvd_not_less have "\<not> p dvd n"
-      by auto
-    with assms prime_imp_coprime [of p n] have "coprime p n"
-      by simp
-    with \<open>n \<in> {1..<p}\<close> show "n \<in> totatives p"
-      by (auto simp add: totatives_def ac_simps)
-  qed
+lemma totient_le: "totient n \<le> n"
+proof -
+  have "card (totatives n) \<le> 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) \<le> 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 \<longleftrightarrow> n = 0"
+  by (auto simp: totient_def)
+
+lemma totient_gt_0_iff [simp]: "totient n > 0 \<longleftrightarrow> n > 0"
+  by (auto intro: Nat.gr0I)
+
+lemma card_gcd_eq_totient:
+  "n > 0 \<Longrightarrow> d dvd n \<Longrightarrow> card {k\<in>{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: "(\<Sum>d | d dvd n. totient d) = n"
+proof (cases "n = 0")
+  case False
+  hence "n > 0" by simp
+  define A where "A = (\<lambda>d. {k\<in>{0<..n}. gcd k n = d})"
+  have *: "card (A d) = totient (n div d)" if d: "d dvd n" for d
+    using \<open>n > 0\<close> and d unfolding A_def by (rule card_gcd_eq_totient)
+  have "n = card {1..n}" by simp
+  also have "{1..n} = (\<Union>d\<in>{d. d dvd n}. A d)" by safe (auto simp: A_def)
+  also have "card \<dots> = (\<Sum>d | d dvd n. card (A d))"
+    using \<open>n > 0\<close> by (intro card_UN_disjoint) (auto simp: A_def)
+  also have "\<dots> = (\<Sum>d | d dvd n. totient (n div d))" by (intro sum.cong refl *) auto
+  also have "\<dots> = (\<Sum>d | d dvd n. totient d)" using \<open>n > 0\<close>
+    by (intro sum.reindex_bij_witness[of _ "op div n" "op div n"]) (auto elim: dvdE)
+  finally show ?thesis ..
qed auto

-lemma totatives_prime:
-  assumes "p \<ge> 2" and "totatives p = {1..<p}"
-  shows "prime p"
-proof (rule ccontr)
-  from \<open>2 \<le> p\<close> have "1 < p"
-    by simp
-  moreover assume "\<not> prime p"
-  ultimately obtain n where "1 < n" "n < p" "n dvd p"
-    by (auto simp add: prime_nat_iff)
-      (metis Suc_lessD Suc_lessI dvd_imp_le dvd_pos_nat le_neq_implies_less)
-  then have "n \<in> {1..<p}"
-    by simp
-  with assms have "n \<in> totatives p"
-    by simp
-  then have "coprime n p"
-    by simp
-  with \<open>1 < n\<close> \<open>n dvd p\<close> show False
-    by simp
+lemma totient_mult_coprime:
+  assumes "coprime m n"
+  shows   "totient (m * n) = totient m * totient n"
+proof (cases "m > 1 \<and> 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 "\<dots> = card (totatives m \<times> totatives n)"
+    using bij_betw_totatives [OF mn \<open>coprime m n\<close>] by (rule bij_betw_same_card)
+  also have "\<dots> = 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

-definition totient :: "nat \<Rightarrow> nat"
-  where "totient = card \<circ> totatives"
-
-lemma card_totatives [simp]:
-  "card (totatives n) = totient n"
-  by (simp add: totient_def)
-
-lemma totient_0 [simp]:
-  "totient 0 = 0"
-  by (simp add: totient_def)
-
-lemma totient_1 [simp]:
-  "totient 1 = 0"
-  by (simp add: totient_def)
-
-lemma totient_Suc_0 [simp]:
-  "totient (Suc 0) = 0"
-  using totient_1 by simp
-
-lemma prime_totient:
+lemma totient_prime_power_Suc:
assumes "prime p"
-  shows "totient p = p - 1"
-  using assms prime_totatives
-  by (simp add: totient_def)
-
-lemma totient_eq_0_iff [simp]:
-  "totient n = 0 \<longleftrightarrow> n \<in> {0, 1}"
-  by (simp only: totient_def comp_def card_eq_0_iff) auto
-
-lemma totient_noneq_0_iff [simp]:
-  "totient n > 0 \<longleftrightarrow> 2 \<le> n"
+  shows   "totient (p ^ Suc n) = p ^ n * (p - 1)"
proof -
-  have "totient n > 0 \<longleftrightarrow> totient n \<noteq> 0"
-    by blast
-  also have "\<dots> \<longleftrightarrow> 2 \<le> n"
-    by auto
+  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 "\<dots> = 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_less_eq:
-  "totient n \<le> n - 1"
-  using card_mono [of "{1..<n}" "totatives n"] by auto
+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_less_eq_Suc_0 [simp]:
-  "totient n \<le> n - Suc 0"
-  using totient_less_eq [of n] 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 \<in> {0<..<p}" if "x \<in> 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 \<noteq> 1" "x \<noteq> p" "x dvd p" for x
+  proof -
+    from that have nz: "x \<noteq> 0" by (auto intro!: Nat.gr0I)
+    from that and p have le: "x \<le> p" by (intro dvd_imp_le) auto
+    from that and nz have "\<not>coprime x p" by auto
+    hence "x \<notin> totatives p" by (simp add: totatives_def)
+    also note *
+    finally show False using that and le by auto
+  qed
+  hence "(\<forall>m. m dvd p \<longrightarrow> m = 1 \<or> 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_prime:
-  assumes "2 \<le> p" "totient p = p - 1"
-  shows "prime p"
-proof -
-  have "totatives p = {1..<p}"
-    by (rule card_subset_eq) (simp_all add: assms)
-  with assms show ?thesis
-    by (auto intro: totatives_prime)
+lemma totient_even:
+  assumes "n > 2"
+  shows   "even (totient n)"
+proof (cases "\<exists>p. prime p \<and> p \<noteq> 2 \<and> p dvd n")
+  case True
+  then obtain p where p: "prime p" "p \<noteq> 2" "p dvd n" by auto
+  from \<open>p \<noteq> 2\<close> have "p = 0 \<or> p = 1 \<or> 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 "\<not>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 \<open>odd p\<close>
+    by (auto simp add: m totient_mult_coprime totient_prime_power)
+next
+  case False
+  from assms have "n = (\<Prod>p\<in>prime_factors n. p ^ multiplicity p n)"
+    by (intro Primes.prime_factorization_nat) auto
+  also from False have "\<dots> = (\<Prod>p\<in>prime_factors n. if p = 2 then 2 ^ multiplicity 2 n else 1)"
+    by (intro prod.cong refl) auto
+  also have "\<dots> = 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 \<or> multiplicity 2 n = 1 \<or> 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 (\<lambda>x. 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 \<dots> = totient (prod f A) * totient (f x)"
+    using insert.hyps insert.prems by (intro totient_mult_coprime *)
+  also have "totient (prod f A) = (\<Prod>x\<in>A. totient (f x))"
+    using insert.prems by (intro insert.IH) (auto dest: pairwise_coprime_subset)
+  also from insert.hyps have "\<dots> * totient (f x) = (\<Prod>x\<in>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 \<noteq> q"
+  from assms have "m = multiplicity p (p ^ m)"
+    by (subst multiplicity_prime_power) auto
+  also note \<open>p ^ m = q ^ n\<close>
+  also from assms pq have "multiplicity p (q ^ n) = 0"
+    by (subst multiplicity_distinct_prime_power) auto
+  finally show False using \<open>m > 0\<close> by simp
qed

+lemma totient_formula1:
+  assumes "n > 0"
+  shows   "totient n = (\<Prod>p\<in>prime_factors n. p ^ (multiplicity p n - 1) * (p - 1))"
+proof -
+  from assms have "n = (\<Prod>p\<in>prime_factors n. p ^ multiplicity p n)"
+    by (rule prime_factorization_nat)
+  also have "totient \<dots> = (\<Prod>x\<in>prime_factors n. totient (x ^ multiplicity x n))"
+  proof (rule totient_prod_coprime)
+    show "pairwise_coprime ((\<lambda>p. p ^ multiplicity p n) ` prime_factors n)"
+    proof (standard, clarify, goal_cases)
+      fix p q assume "p \<in># prime_factorization n" "q \<in># prime_factorization n"
+                     "p ^ multiplicity p n \<noteq> 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 (\<lambda>p. p ^ multiplicity p n) (prime_factors n)"
+    proof
+      fix p q assume pq: "p \<in># prime_factorization n" "q \<in># 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 "\<dots> = (\<Prod>p\<in>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 \<or> n = 0")
+  case False
+  let ?M = "\<lambda>p m :: nat. multiplicity p m - 1"
+  have "(\<Prod>p\<in>prime_factors m. p ^ ?M p m * (p - 1)) dvd
+          (\<Prod>p\<in>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 \<le> totient n"
+  by (cases "m = 0") (insert assms, auto intro: dvd_imp_le totient_dvd)
+
+(* TODO Move *)
+lemma prime_factors_power: "n > 0 \<Longrightarrow> 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 * (\<Prod>p\<in>prime_factors n. 1 - 1 / real p)"
+proof (cases "n = 0")
+  case False
+  have "real (totient n) = (\<Prod>p\<in>prime_factors n. real
+          (p ^ (multiplicity p n - 1) * (p - 1)))"
+    using False by (subst totient_formula1) simp_all
+  also have "\<dots> = (\<Prod>p\<in>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 "\<dots> = real (\<Prod>p\<in>prime_factors n. p ^ multiplicity p n) *
+                    (\<Prod>p\<in>prime_factors n. 1 - 1 / real p)" by (subst prod.distrib) auto
+  also have "(\<Prod>p\<in>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 \<or> b = 0")
+  case False
+  let ?P = "prime_factors :: nat \<Rightarrow> nat set"
+  have "real (totient a * totient b * gcd a b) = real (a * b * gcd a b) *
+          ((\<Prod>p\<in>?P a. 1 - 1 / real p) * (\<Prod>p\<in>?P b. 1 - 1 / real p))"
+    by (simp add: totient_formula2)
+  also have "?P a = (?P a - ?P b) \<union> (?P a \<inter> ?P b)" by auto
+  also have "(\<Prod>p\<in>\<dots>. 1 - 1 / real p) =
+                 (\<Prod>p\<in>?P a - ?P b. 1 - 1 / real p) * (\<Prod>p\<in>?P a \<inter> ?P b. 1 - 1 / real p)"
+    by (rule prod.union_disjoint) blast+
+  also have "\<dots> * (\<Prod>p\<in>?P b. 1 - 1 / real p) = (\<Prod>p\<in>?P a - ?P b. 1 - 1 / real p) *
+               (\<Prod>p\<in>?P b. 1 - 1 / real p) * (\<Prod>p\<in>?P a \<inter> ?P b. 1 - 1 / real p)" (is "_ = ?A * _")
+    by (simp only: mult_ac)
+  also have "?A = (\<Prod>p\<in>?P a - ?P b \<union> ?P b. 1 - 1 / real p)"
+    by (rule prod.union_disjoint [symmetric]) blast+
+  also have "?P a - ?P b \<union> ?P b = ?P a \<union> ?P b" by blast
+  also have "real (a * b * gcd a b) * ((\<Prod>p\<in>\<dots>. 1 - 1 / real p) *
+                 (\<Prod>p\<in>?P a \<inter> ?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}) \<longleftrightarrow> 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 "\<dots> = n ^ Suc m * totient n"
+    using Suc.IH by (subst totient_mult) simp
+  finally show ?case .
+qed simp_all
+
+lemma totient_power: "m > 0 \<Longrightarrow> 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 \<or> b = 0")
+  case False
+  let ?P = "prime_factors :: nat \<Rightarrow> nat set" and ?f = "\<lambda>p::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 \<inter> ?P b) * prod ?f (?P a \<union> ?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 \<union> ?P b = (?P a - ?P a \<inter> ?P b) \<union> ?P b" by blast
+  also have "prod ?f \<dots> = prod ?f (?P a - ?P a \<inter> ?P b) * prod ?f (?P b)"
+    by (rule prod.union_disjoint) blast+
+  also have "prod ?f (?P a \<inter> ?P b) * \<dots> =
+               prod ?f (?P a \<inter> ?P b \<union> (?P a - ?P a \<inter> ?P b)) * prod ?f (?P b)"
+    by (subst prod.union_disjoint) auto
+  also have "?P a \<inter> ?P b \<union> (?P a - ?P a \<inter> ?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```