src/HOL/Algebra/Multiplicative_Group.thy

some new results in group theory

(*  Title:      HOL/Algebra/Multiplicative_Group.thy
    Author:     Simon Wimmer
    Author:     Lars Noschinski
*)

theory Multiplicative_Group
imports
  Complex_Main
  Group
  Coset
  UnivPoly
  Generated_Groups
begin

section \<open>Simplification Rules for Polynomials\<close>
text_raw \<open>\label{sec:simp-rules}\<close>

lemma (in ring_hom_cring) hom_sub[simp]:
  assumes "x \<in> carrier R" "y \<in> carrier R"
  shows "h (x \<ominus> y) = h x \<ominus>\<^bsub>S\<^esub> h y"
  using assms by (simp add: R.minus_eq S.minus_eq)

context UP_ring begin

lemma deg_nzero_nzero:
  assumes deg_p_nzero: "deg R p \<noteq> 0"
  shows "p \<noteq> \<zero>\<^bsub>P\<^esub>"
  using deg_zero deg_p_nzero by auto

lemma deg_add_eq:
  assumes c: "p \<in> carrier P" "q \<in> carrier P"
  assumes "deg R q \<noteq> deg R p"
  shows "deg R (p \<oplus>\<^bsub>P\<^esub> q) = max (deg R p) (deg R q)"
proof -
  let ?m = "max (deg R p) (deg R q)"
  from assms have "coeff P p ?m = \<zero> \<longleftrightarrow> coeff P q ?m \<noteq> \<zero>"
    by (metis deg_belowI lcoeff_nonzero[OF deg_nzero_nzero] linear max.absorb_iff2 max.absorb1)
  then have "coeff P (p \<oplus>\<^bsub>P\<^esub> q) ?m \<noteq> \<zero>"
    using assms by auto
  then have "deg R (p \<oplus>\<^bsub>P\<^esub> q) \<ge> ?m"
    using assms by (blast intro: deg_belowI)
  with deg_add[OF c] show ?thesis by arith
qed

lemma deg_minus_eq:
  assumes "p \<in> carrier P" "q \<in> carrier P" "deg R q \<noteq> deg R p"
  shows "deg R (p \<ominus>\<^bsub>P\<^esub> q) = max (deg R p) (deg R q)"
  using assms by (simp add: deg_add_eq a_minus_def)

end

context UP_cring begin

lemma evalRR_add:
  assumes "p \<in> carrier P" "q \<in> carrier P"
  assumes x:"x \<in> carrier R"
  shows "eval R R id x (p \<oplus>\<^bsub>P\<^esub> q) = eval R R id x p \<oplus> eval R R id x q"
proof -
  interpret UP_pre_univ_prop R R id by unfold_locales simp
  interpret ring_hom_cring P R "eval R R id x" by unfold_locales (rule eval_ring_hom[OF x])
  show ?thesis using assms by simp
qed

lemma evalRR_sub:
  assumes "p \<in> carrier P" "q \<in> carrier P"
  assumes x:"x \<in> carrier R"
  shows "eval R R id x (p \<ominus>\<^bsub>P\<^esub> q) = eval R R id x p \<ominus> eval R R id x q"
proof -
  interpret UP_pre_univ_prop R R id by unfold_locales simp
  interpret ring_hom_cring P R "eval R R id x" by unfold_locales (rule eval_ring_hom[OF x])
  show ?thesis using assms by simp
qed

lemma evalRR_mult:
  assumes "p \<in> carrier P" "q \<in> carrier P"
  assumes x:"x \<in> carrier R"
  shows "eval R R id x (p \<otimes>\<^bsub>P\<^esub> q) = eval R R id x p \<otimes> eval R R id x q"
proof -
  interpret UP_pre_univ_prop R R id by unfold_locales simp
  interpret ring_hom_cring P R "eval R R id x" by unfold_locales (rule eval_ring_hom[OF x])
  show ?thesis using assms by simp
qed

lemma evalRR_monom:
  assumes a: "a \<in> carrier R" and x: "x \<in> carrier R"
  shows "eval R R id x (monom P a d) = a \<otimes> x [^] d"
proof -
  interpret UP_pre_univ_prop R R id by unfold_locales simp
  show ?thesis using assms by (simp add: eval_monom)
qed

lemma evalRR_one:
  assumes x: "x \<in> carrier R"
  shows "eval R R id x \<one>\<^bsub>P\<^esub> = \<one>"
proof -
  interpret UP_pre_univ_prop R R id by unfold_locales simp
  interpret ring_hom_cring P R "eval R R id x" by unfold_locales (rule eval_ring_hom[OF x])
  show ?thesis using assms by simp
qed

lemma carrier_evalRR:
  assumes x: "x \<in> carrier R" and "p \<in> carrier P"
  shows "eval R R id x p \<in> carrier R"
proof -
  interpret UP_pre_univ_prop R R id by unfold_locales simp
  interpret ring_hom_cring P R "eval R R id x" by unfold_locales (rule eval_ring_hom[OF x])
  show ?thesis using assms by simp
qed

lemmas evalRR_simps = evalRR_add evalRR_sub evalRR_mult evalRR_monom evalRR_one carrier_evalRR

end



section \<open>Properties of the Euler \<open>\<phi>\<close>-function\<close>
text_raw \<open>\label{sec:euler-phi}\<close>

text\<open>
  In this section we prove that for every positive natural number the equation
  $\sum_{d | n}^n \varphi(d) = n$ holds.
\<close>

lemma dvd_div_ge_1:
  fixes a b :: nat
  assumes "a \<ge> 1" "b dvd a"
  shows "a div b \<ge> 1"
proof -
  from \<open>b dvd a\<close> obtain c where "a = b * c" ..
  with \<open>a \<ge> 1\<close> show ?thesis by simp
qed

lemma dvd_nat_bounds:
 fixes n p :: nat
 assumes "p > 0" "n dvd p"
 shows "n > 0 \<and> n \<le> p"
 using assms by (simp add: dvd_pos_nat dvd_imp_le)

(* Deviates from the definition given in the library in number theory *)
definition phi' :: "nat => nat"
  where "phi' m = card {x. 1 \<le> x \<and> x \<le> m \<and> coprime x m}"

notation (latex output)
  phi' ("\<phi> _")

lemma phi'_nonzero:
  assumes "m > 0"
  shows "phi' m > 0"
proof -
  have "1 \<in> {x. 1 \<le> x \<and> x \<le> m \<and> coprime x m}" using assms by simp
  hence "card {x. 1 \<le> x \<and> x \<le> m \<and> coprime x m} > 0" by (auto simp: card_gt_0_iff)
  thus ?thesis unfolding phi'_def by simp
qed

lemma dvd_div_eq_1:
  fixes a b c :: nat
  assumes "c dvd a" "c dvd b" "a div c = b div c"
  shows "a = b" using assms dvd_mult_div_cancel[OF \<open>c dvd a\<close>] dvd_mult_div_cancel[OF \<open>c dvd b\<close>]
                by presburger

lemma dvd_div_eq_2:
  fixes a b c :: nat
  assumes "c>0" "a dvd c" "b dvd c" "c div a = c div b"
  shows "a = b"
  proof -
  have "a > 0" "a \<le> c" using dvd_nat_bounds[OF assms(1-2)] by auto
  have "a*(c div a) = c" using assms dvd_mult_div_cancel by fastforce
  also have "\<dots> = b*(c div a)" using assms dvd_mult_div_cancel by fastforce
  finally show "a = b" using \<open>c>0\<close> dvd_div_ge_1[OF _ \<open>a dvd c\<close>] by fastforce
qed

lemma div_mult_mono:
  fixes a b c :: nat
  assumes "a > 0" "a\<le>d"
  shows "a * b div d \<le> b"
proof -
  have "a*b div d \<le> b*a div a" using assms div_le_mono2 mult.commute[of a b] by presburger
  thus ?thesis using assms by force
qed

text\<open>
  We arrive at the main result of this section:
  For every positive natural number the equation $\sum_{d | n}^n \varphi(d) = n$ holds.

  The outline of the proof for this lemma is as follows:
  We count the $n$ fractions $1/n$, $\ldots$, $(n-1)/n$, $n/n$.
  We analyze the reduced form $a/d = m/n$ for any of those fractions.
  We want to know how many fractions $m/n$ have the reduced form denominator $d$.
  The condition $1 \leq m \leq n$ is equivalent to the condition $1 \leq a \leq d$.
  Therefore we want to know how many $a$ with $1 \leq a \leq d$ exist, s.t. \<^term>\<open>gcd a d = 1\<close>.
  This number is exactly \<^term>\<open>phi' d\<close>.

  Finally, by counting the fractions $m/n$ according to their reduced form denominator,
  we get: @{term [display] "(\<Sum>d | d dvd n . phi' d) = n"}.
  To formalize this proof in Isabelle, we analyze for an arbitrary divisor $d$ of $n$
  \begin{itemize}
    \item the set of reduced form numerators \<^term>\<open>{a. (1::nat) \<le> a \<and> a \<le> d \<and> coprime a d}\<close>
    \item the set of numerators $m$, for which $m/n$ has the reduced form denominator $d$,
      i.e. the set \<^term>\<open>{m \<in> {1::nat .. n}. n div gcd m n = d}\<close>
  \end{itemize}
  We show that \<^term>\<open>\<lambda>a. a*n div d\<close> with the inverse \<^term>\<open>\<lambda>a. a div gcd a n\<close> is
  a bijection between theses sets, thus yielding the equality
  @{term [display] "phi' d = card {m \<in> {1 .. n}. n div gcd m n = d}"}
  This gives us
  @{term [display] "(\<Sum>d | d dvd n . phi' d)
          = card (\<Union>d \<in> {d. d dvd n}. {m \<in> {1 .. n}. n div gcd m n = d})"}
  and by showing
  \<^term>\<open>(\<Union>d \<in> {d. d dvd n}. {m \<in> {1::nat .. n}. n div gcd m n = d}) \<supseteq> {1 .. n}\<close>
  (this is our counting argument) the thesis follows.
\<close>
lemma sum_phi'_factors:
 fixes n :: nat
 assumes "n > 0"
 shows "(\<Sum>d | d dvd n. phi' d) = n"
proof -
  { fix d assume "d dvd n" then obtain q where q: "n = d * q" ..
    have "card {a. 1 \<le> a \<and> a \<le> d \<and> coprime a d} = card {m \<in> {1 .. n}.  n div gcd m n = d}"
         (is "card ?RF = card ?F")
    proof (rule card_bij_eq)
      { fix a b assume "a * n div d = b * n div d"
        hence "a * (n div d) = b * (n div d)"
          using dvd_div_mult[OF \<open>d dvd n\<close>] by (fastforce simp add: mult.commute)
        hence "a = b" using dvd_div_ge_1[OF _ \<open>d dvd n\<close>] \<open>n>0\<close>
          by (simp add: mult.commute nat_mult_eq_cancel1)
      } thus "inj_on (\<lambda>a. a*n div d) ?RF" unfolding inj_on_def by blast
      { fix a assume a:"a\<in>?RF"
        hence "a * (n div d) \<ge> 1" using \<open>n>0\<close> dvd_div_ge_1[OF _ \<open>d dvd n\<close>] by simp
        hence ge_1:"a * n div d \<ge> 1" by (simp add: \<open>d dvd n\<close> div_mult_swap)
        have le_n:"a * n div d \<le> n" using div_mult_mono a by simp
        have "gcd (a * n div d) n = n div d * gcd a d"
          by (simp add: gcd_mult_distrib_nat q ac_simps)
        hence "n div gcd (a * n div d) n = d*n div (d*(n div d))" using a by simp
        hence "a * n div d \<in> ?F"
          using ge_1 le_n by (fastforce simp add: \<open>d dvd n\<close>)
      } thus "(\<lambda>a. a*n div d) ` ?RF \<subseteq> ?F" by blast
      { fix m l assume A: "m \<in> ?F" "l \<in> ?F" "m div gcd m n = l div gcd l n"
        hence "gcd m n = gcd l n" using dvd_div_eq_2[OF assms] by fastforce
        hence "m = l" using dvd_div_eq_1[of "gcd m n" m l] A(3) by fastforce
      } thus "inj_on (\<lambda>a. a div gcd a n) ?F" unfolding inj_on_def by blast
      { fix m assume "m \<in> ?F"
        hence "m div gcd m n \<in> ?RF" using dvd_div_ge_1
          by (fastforce simp add: div_le_mono div_gcd_coprime)
      } thus "(\<lambda>a. a div gcd a n) ` ?F \<subseteq> ?RF" by blast
    qed force+
  } hence phi'_eq:"\<And>d. d dvd n \<Longrightarrow> phi' d = card {m \<in> {1 .. n}. n div gcd m n = d}"
      unfolding phi'_def by presburger
  have fin:"finite {d. d dvd n}" using dvd_nat_bounds[OF \<open>n>0\<close>] by force
  have "(\<Sum>d | d dvd n. phi' d)
                 = card (\<Union>d \<in> {d. d dvd n}. {m \<in> {1 .. n}. n div gcd m n = d})"
    using card_UN_disjoint[OF fin, of "(\<lambda>d. {m \<in> {1 .. n}. n div gcd m n = d})"] phi'_eq
    by fastforce
  also have "(\<Union>d \<in> {d. d dvd n}. {m \<in> {1 .. n}. n div gcd m n = d}) = {1 .. n}" (is "?L = ?R")
  proof
    show "?L \<supseteq> ?R"
    proof
      fix m assume m: "m \<in> ?R"
      thus "m \<in> ?L" using dvd_triv_right[of "n div gcd m n" "gcd m n"]
        by simp
    qed
  qed fastforce
  finally show ?thesis by force
qed

section \<open>Order of an Element of a Group\<close>
text_raw \<open>\label{sec:order-elem}\<close>


context group begin

lemma pow_eq_div2:
  fixes m n :: nat
  assumes x_car: "x \<in> carrier G"
  assumes pow_eq: "x [^] m = x [^] n"
  shows "x [^] (m - n) = \<one>"
proof (cases "m < n")
  case False
  have "\<one> \<otimes> x [^] m = x [^] m" by (simp add: x_car)
  also have "\<dots> = x [^] (m - n) \<otimes> x [^] n"
    using False by (simp add: nat_pow_mult x_car)
  also have "\<dots> = x [^] (m - n) \<otimes> x [^] m"
    by (simp add: pow_eq)
  finally show ?thesis by (simp add: x_car)
qed simp

definition ord where "ord a = Min {d \<in> {1 .. order G} . a [^] d = \<one>}"

lemma
  assumes finite:"finite (carrier G)"
  assumes a:"a \<in> carrier G"
  shows ord_ge_1: "1 \<le> ord a" and ord_le_group_order: "ord a \<le> order G"
    and pow_ord_eq_1: "a [^] ord a = \<one>"
proof -
  have "\<not>inj_on (\<lambda>x. a [^] x) {0 .. order G}"
  proof (rule notI)
    assume A: "inj_on (\<lambda>x. a [^] x) {0 .. order G}"
    have "order G + 1 = card {0 .. order G}" by simp
    also have "\<dots> = card ((\<lambda>x. a [^] x) ` {0 .. order G})" (is "_ = card ?S")
      using A by (simp add: card_image)
    also have "?S = {a [^] x | x. x \<in> {0 .. order G}}" by blast
    also have "\<dots> \<subseteq> carrier G" (is "?S \<subseteq> _") using a by blast
    then have "card ?S \<le> order G" unfolding order_def
      by (rule card_mono[OF finite])
    finally show False by arith
  qed

  then obtain x y where x_y:"x \<noteq> y" "x \<in> {0 .. order G}" "y \<in> {0 .. order G}"
                        "a [^] x = a [^] y" unfolding inj_on_def by blast
  obtain d where "1 \<le> d" "a [^] d = \<one>" "d \<le> order G"
  proof cases
    assume "y < x" with x_y show ?thesis
      by (intro that[where d="x - y"]) (auto simp add: pow_eq_div2[OF a])
  next
    assume "\<not>y < x" with x_y show ?thesis
      by (intro that[where d="y - x"]) (auto simp add: pow_eq_div2[OF a])
  qed
  hence "ord a \<in> {d \<in> {1 .. order G} . a [^] d = \<one>}"
    unfolding ord_def using Min_in[of "{d \<in> {1 .. order G} . a [^] d = \<one>}"]
    by fastforce
  then show "1 \<le> ord a" and "ord a \<le> order G" and "a [^] ord a = \<one>"
    by (auto simp: order_def)
qed

lemma finite_group_elem_finite_ord:
  assumes "finite (carrier G)" "x \<in> carrier G"
  shows "\<exists> d::nat. d \<ge> 1 \<and> x [^] d = \<one>"
  using assms ord_ge_1 pow_ord_eq_1 by auto

lemma ord_min:
  assumes  "finite (carrier G)" "1 \<le> d" "a \<in> carrier G" "a [^] d = \<one>" shows "ord a \<le> d"
proof -
  define Ord where "Ord = {d \<in> {1..order G}. a [^] d = \<one>}"
  have fin: "finite Ord" by (auto simp: Ord_def)
  have in_ord: "ord a \<in> Ord"
    using assms pow_ord_eq_1 ord_ge_1 ord_le_group_order by (auto simp: Ord_def)
  then have "Ord \<noteq> {}" by auto

  show ?thesis
  proof (cases "d \<le> order G")
    case True
    then have "d \<in> Ord" using assms by (auto simp: Ord_def)
    with fin in_ord show ?thesis
      unfolding ord_def Ord_def[symmetric] by simp
  next
    case False
    then show ?thesis using in_ord by (simp add: Ord_def)
  qed
qed

lemma ord_inj:
  assumes finite: "finite (carrier G)"
  assumes a: "a \<in> carrier G"
  shows "inj_on (\<lambda> x . a [^] x) {0 .. ord a - 1}"
proof (rule inj_onI, rule ccontr)
  fix x y assume A: "x \<in> {0 .. ord a - 1}" "y \<in> {0 .. ord a - 1}" "a [^] x= a [^] y" "x \<noteq> y"

  have "finite {d \<in> {1..order G}. a [^] d = \<one>}" by auto

  { fix x y assume A: "x < y" "x \<in> {0 .. ord a - 1}" "y \<in> {0 .. ord a - 1}"
        "a [^] x = a [^] y"
    hence "y - x < ord a" by auto
    also have "\<dots> \<le> order G" using assms by (simp add: ord_le_group_order)
    finally have y_x_range:"y - x \<in> {1 .. order G}" using A by force
    have "a [^] (y-x) = \<one>" using a A by (simp add: pow_eq_div2)

    hence y_x:"y - x \<in> {d \<in> {1.. order G}. a [^] d = \<one>}" using y_x_range by blast
    have "min (y - x) (ord a) = ord a"
      using Min.in_idem[OF \<open>finite {d \<in> {1 .. order G} . a [^] d = \<one>}\<close> y_x] ord_def by auto
    with \<open>y - x < ord a\<close> have False by linarith
  }
  note X = this

  { assume "x < y" with A X have False by blast }
  moreover
  { assume "x > y" with A X  have False by metis }
  moreover
  { assume "x = y" then have False using A by auto}
  ultimately
  show False by fastforce
qed

lemma ord_inj':
  assumes finite: "finite (carrier G)"
  assumes a: "a \<in> carrier G"
  shows "inj_on (\<lambda> x . a [^] x) {1 .. ord a}"
proof (rule inj_onI, rule ccontr)
  fix x y :: nat
  assume A:"x \<in> {1 .. ord a}" "y \<in> {1 .. ord a}" "a [^] x = a [^] y" "x\<noteq>y"
  { assume "x < ord a" "y < ord a"
    hence False using ord_inj[OF assms] A unfolding inj_on_def by fastforce
  }
  moreover
  { assume "x = ord a" "y < ord a"
    hence "a [^] y = a [^] (0::nat)" using pow_ord_eq_1[OF assms] A by auto
    hence "y=0" using ord_inj[OF assms] \<open>y < ord a\<close> unfolding inj_on_def by force
    hence False using A by fastforce
  }
  moreover
  { assume "y = ord a" "x < ord a"
    hence "a [^] x = a [^] (0::nat)" using pow_ord_eq_1[OF assms] A by auto
    hence "x=0" using ord_inj[OF assms] \<open>x < ord a\<close> unfolding inj_on_def by force
    hence False using A by fastforce
  }
  ultimately show False using A  by force
qed

lemma ord_elems:
  assumes "finite (carrier G)" "a \<in> carrier G"
  shows "{a[^]x | x. x \<in> (UNIV :: nat set)} = {a[^]x | x. x \<in> {0 .. ord a - 1}}" (is "?L = ?R")
proof
  show "?R \<subseteq> ?L" by blast
  { fix y assume "y \<in> ?L"
    then obtain x::nat where x:"y = a[^]x" by auto
    define r q where "r = x mod ord a" and "q = x div ord a"
    then have "x = q * ord a + r"
      by (simp add: div_mult_mod_eq)
    hence "y = (a[^]ord a)[^]q \<otimes> a[^]r"
      using x assms by (simp add: mult.commute nat_pow_mult nat_pow_pow)
    hence "y = a[^]r" using assms by (simp add: pow_ord_eq_1)
    have "r < ord a" using ord_ge_1[OF assms] by (simp add: r_def)
    hence "r \<in> {0 .. ord a - 1}" by (force simp: r_def)
    hence "y \<in> {a[^]x | x. x \<in> {0 .. ord a - 1}}" using \<open>y=a[^]r\<close> by blast
  }
  thus "?L \<subseteq> ?R" by auto
qed

(* Next three lemmas contributed by Paulo Emílio de Vilhena*)
lemma generate_pow_on_finite_carrier:
  assumes "finite (carrier G)" and "a \<in> carrier G"
  shows "generate G { a } = { a [^] k | k. k \<in> (UNIV :: nat set) }"
proof
  show "{ a [^] k | k. k \<in> (UNIV :: nat set) } \<subseteq> generate G { a }"
  proof
    fix b assume "b \<in> { a [^] k | k. k \<in> (UNIV :: nat set) }"
    then obtain k :: nat where "b = a [^] k" by blast
    hence "b = a [^] (int k)"
      by (simp add: int_pow_int)
    thus "b \<in> generate G { a }"
      unfolding generate_pow[OF assms(2)] by blast
  qed
next
  show "generate G { a } \<subseteq> { a [^] k | k. k \<in> (UNIV :: nat set) }"
  proof
    fix b assume "b \<in> generate G { a }"
    then obtain k :: int where k: "b = a [^] k"
      unfolding generate_pow[OF assms(2)] by blast
    show "b \<in> { a [^] k | k. k \<in> (UNIV :: nat set) }"
    proof (cases "k < 0")
      assume "\<not> k < 0"
      hence "b = a [^] (nat k)"
        by (simp add: int_pow_def2 k)
      thus ?thesis by blast
    next
      assume "k < 0"
      hence b: "b = inv (a [^] (nat (- k)))"
        using k int_pow_def2[of G] by auto

      obtain m where m: "ord a * m \<ge> nat (- k)"
        by (metis assms mult.left_neutral mult_le_mono1 ord_ge_1)
      hence "a [^] (ord a * m) = \<one>"
        by (metis assms nat_pow_one nat_pow_pow pow_ord_eq_1)
      then obtain k' :: nat where "(a [^] (nat (- k))) \<otimes> (a [^] k') = \<one>"
        using m assms(2) nat_le_iff_add nat_pow_mult by auto
      hence "b = a [^] k'"
        using b assms(2) by (metis inv_unique' nat_pow_closed nat_pow_comm)
      thus "b \<in> { a [^] k | k. k \<in> (UNIV :: nat set) }" by blast
    qed
  qed
qed

lemma generate_pow_card:
  assumes "finite (carrier G)" and "a \<in> carrier G"
  shows "ord a = card (generate G { a })"
proof -
  have "generate G { a } = (([^]) a) ` {0..ord a - 1}"
    using generate_pow_on_finite_carrier[OF assms] unfolding ord_elems[OF assms] by auto
  thus ?thesis
    using ord_inj[OF assms] ord_ge_1[OF assms] by (simp add: card_image)
qed

(* This lemma was a suggestion of generalization given by Jeremy Avigad
   at the end of the theory FiniteProduct. *)
corollary power_order_eq_one_group_version:
  assumes "finite (carrier G)" and "a \<in> carrier G"
  shows "a [^] (order G) = \<one>"
proof -
  have "(ord a) dvd (order G)"
    using lagrange[OF generate_is_subgroup[of " { a }"]] assms(2)
    unfolding generate_pow_card[OF assms]
    by (metis dvd_triv_right empty_subsetI insert_subset)
  then obtain k :: nat where "order G = ord a * k" by blast
  thus ?thesis
    using assms(2) pow_ord_eq_1[OF assms] by (metis nat_pow_one nat_pow_pow)
qed

lemma ord_dvd_pow_eq_1 :
  assumes "finite (carrier G)" "a \<in> carrier G" "a [^] k = \<one>"
  shows "ord a dvd k"
proof -
  define r where "r = k mod ord a"

  define r q where "r = k mod ord a" and "q = k div ord a"
  then have q: "k = q * ord a + r"
    by (simp add: div_mult_mod_eq)
  hence "a[^]k = (a[^]ord a)[^]q \<otimes> a[^]r"
      using assms by (simp add: mult.commute nat_pow_mult nat_pow_pow)
  hence "a[^]k = a[^]r" using assms by (simp add: pow_ord_eq_1)
  hence "a[^]r = \<one>" using assms(3) by simp
  have "r < ord a" using ord_ge_1[OF assms(1-2)] by (simp add: r_def)
  hence "r = 0" using \<open>a[^]r = \<one>\<close> ord_def[of a] ord_min[of r a] assms(1-2) by linarith
  thus ?thesis using q by simp
qed

lemma dvd_gcd :
  fixes a b :: nat
  obtains q where "a * (b div gcd a b) = b*q"
proof
  have "a * (b div gcd a b) = (a div gcd a b) * b" by (simp add:  div_mult_swap dvd_div_mult)
  also have "\<dots> = b * (a div gcd a b)" by simp
  finally show "a * (b div gcd a b) = b * (a div gcd a b) " .
qed

lemma ord_pow_dvd_ord_elem :
  assumes finite[simp]: "finite (carrier G)"
  assumes a[simp]:"a \<in> carrier G"
  shows "ord (a[^]n) = ord a div gcd n (ord a)"
proof -
  have "(a[^]n) [^] ord a = (a [^] ord a) [^] n"
    by (simp add: mult.commute nat_pow_pow)
  hence "(a[^]n) [^] ord a = \<one>" by (simp add: pow_ord_eq_1)
  obtain q where "n * (ord a div gcd n (ord a)) = ord a * q" by (rule dvd_gcd)
  hence "(a[^]n) [^] (ord a div gcd n (ord a)) = (a [^] ord a)[^]q"  by (simp add : nat_pow_pow)
  hence pow_eq_1: "(a[^]n) [^] (ord a div gcd n (ord a)) = \<one>"
     by (auto simp add : pow_ord_eq_1[of a])
  have "ord a \<ge> 1" using ord_ge_1 by simp
  have ge_1:"ord a div gcd n (ord a) \<ge> 1"
  proof -
    have "gcd n (ord a) dvd ord a" by blast
    thus ?thesis by (rule dvd_div_ge_1[OF \<open>ord a \<ge> 1\<close>])
  qed
  have "ord a \<le> order G" by (simp add: ord_le_group_order)
  have "ord a div gcd n (ord a) \<le> order G"
  proof -
    have "ord a div gcd n (ord a) \<le> ord a" by simp
    thus ?thesis using \<open>ord a \<le> order G\<close> by linarith
  qed
  hence ord_gcd_elem:"ord a div gcd n (ord a) \<in> {d \<in> {1..order G}. (a[^]n) [^] d = \<one>}"
    using ge_1 pow_eq_1 by force
  { fix d :: nat
    assume d_elem:"d \<in> {d \<in> {1..order G}. (a[^]n) [^] d = \<one>}"
    assume d_lt:"d < ord a div gcd n (ord a)"
    hence pow_nd:"a[^](n*d)  = \<one>" using d_elem
      by (simp add : nat_pow_pow)
    hence "ord a dvd n*d" using assms by (auto simp add : ord_dvd_pow_eq_1)
    then obtain q where "ord a * q = n*d" by (metis dvd_mult_div_cancel)
    hence prod_eq:"(ord a div gcd n (ord a)) * q = (n div gcd n (ord a)) * d"
      by (simp add: dvd_div_mult)
    have cp:"coprime (ord a div gcd n (ord a)) (n div gcd n (ord a))"
    proof -
      have "coprime (n div gcd n (ord a)) (ord a div gcd n (ord a))"
        using div_gcd_coprime[of n "ord a"] ge_1 by fastforce
      thus ?thesis by (simp add: ac_simps)
    qed
    have dvd_d:"(ord a div gcd n (ord a)) dvd d"
    proof -
      have "ord a div gcd n (ord a) dvd (n div gcd n (ord a)) * d" using prod_eq
        by (metis dvd_triv_right mult.commute)
      hence "ord a div gcd n (ord a) dvd d * (n div gcd n (ord a))"
        by (simp add: mult.commute)
      then show ?thesis
        using cp by (simp add: coprime_dvd_mult_left_iff)
    qed
    have "d > 0" using d_elem by simp
    hence "ord a div gcd n (ord a) \<le> d" using dvd_d by (simp add : Nat.dvd_imp_le)
    hence False using d_lt by simp
  } hence ord_gcd_min: "\<And> d . d \<in> {d \<in> {1..order G}. (a[^]n) [^] d = \<one>}
                        \<Longrightarrow> d\<ge>ord a div gcd n (ord a)" by fastforce
  have fin:"finite {d \<in> {1..order G}. (a[^]n) [^] d = \<one>}" by auto
  thus ?thesis using Min_eqI[OF fin ord_gcd_min ord_gcd_elem]
    unfolding ord_def by simp
qed

lemma ord_1_eq_1 :
  assumes "finite (carrier G)"
  shows "ord \<one> = 1"
 using assms ord_ge_1 ord_min[of 1 \<one>] by force

lemma element_generates_subgroup:
  assumes finite[simp]: "finite (carrier G)"
  assumes a[simp]: "a \<in> carrier G"
  shows "subgroup {a [^] i | i. i \<in> {0 .. ord a - 1}} G"
  using generate_is_subgroup[of "{ a }"] assms(2)
        generate_pow_on_finite_carrier[OF assms]
  unfolding ord_elems[OF assms] by auto

lemma ord_dvd_group_order: (* <- DELETE *)
  assumes "finite (carrier G)" and "a \<in> carrier G"
  shows "(ord a) dvd (order G)"
  using lagrange[OF generate_is_subgroup[of " { a }"]] assms(2)
  unfolding generate_pow_card[OF assms]
  by (metis dvd_triv_right empty_subsetI insert_subset)

end


section \<open>Number of Roots of a Polynomial\<close>
text_raw \<open>\label{sec:number-roots}\<close>


definition mult_of :: "('a, 'b) ring_scheme \<Rightarrow> 'a monoid" where
  "mult_of R \<equiv> \<lparr> carrier = carrier R - {\<zero>\<^bsub>R\<^esub>}, mult = mult R, one = \<one>\<^bsub>R\<^esub>\<rparr>"

lemma carrier_mult_of [simp]: "carrier (mult_of R) = carrier R - {\<zero>\<^bsub>R\<^esub>}"
  by (simp add: mult_of_def)

lemma mult_mult_of [simp]: "mult (mult_of R) = mult R"
 by (simp add: mult_of_def)

lemma nat_pow_mult_of: "([^]\<^bsub>mult_of R\<^esub>) = (([^]\<^bsub>R\<^esub>) :: _ \<Rightarrow> nat \<Rightarrow> _)"
  by (simp add: mult_of_def fun_eq_iff nat_pow_def)

lemma one_mult_of [simp]: "\<one>\<^bsub>mult_of R\<^esub> = \<one>\<^bsub>R\<^esub>"
  by (simp add: mult_of_def)

lemmas mult_of_simps = carrier_mult_of mult_mult_of nat_pow_mult_of one_mult_of

context field
begin

lemma mult_of_is_Units: "mult_of R = units_of R"
  unfolding mult_of_def units_of_def using field_Units by auto

lemma m_inv_mult_of :
"\<And>x. x \<in> carrier (mult_of R) \<Longrightarrow> m_inv (mult_of R) x = m_inv R x"
  using mult_of_is_Units units_of_inv unfolding units_of_def
  by simp

lemma field_mult_group :
  shows "group (mult_of R)"
  apply (rule groupI)
  apply (auto simp: mult_of_simps m_assoc dest: integral)
  by (metis Diff_iff Units_inv_Units Units_l_inv field_Units singletonE)

lemma finite_mult_of: "finite (carrier R) \<Longrightarrow> finite (carrier (mult_of R))"
  by (auto simp: mult_of_simps)

lemma order_mult_of: "finite (carrier R) \<Longrightarrow> order (mult_of R) = order R - 1"
  unfolding order_def carrier_mult_of by (simp add: card.remove)

end



lemma (in monoid) Units_pow_closed :
  fixes d :: nat
  assumes "x \<in> Units G"
  shows "x [^] d \<in> Units G"
    by (metis assms group.is_monoid monoid.nat_pow_closed units_group units_of_carrier units_of_pow)

lemma (in comm_monoid) is_monoid:
  shows "monoid G" by unfold_locales

declare comm_monoid.is_monoid[intro?]

lemma (in ring) r_right_minus_eq[simp]:
  assumes "a \<in> carrier R" "b \<in> carrier R"
  shows "a \<ominus> b = \<zero> \<longleftrightarrow> a = b"
  using assms by (metis a_minus_def add.inv_closed minus_equality r_neg)

context UP_cring begin

lemma is_UP_cring:"UP_cring R" by (unfold_locales)
lemma is_UP_ring :
  shows "UP_ring R" by (unfold_locales)

end

context UP_domain begin


lemma roots_bound:
  assumes f [simp]: "f \<in> carrier P"
  assumes f_not_zero: "f \<noteq> \<zero>\<^bsub>P\<^esub>"
  assumes finite: "finite (carrier R)"
  shows "finite {a \<in> carrier R . eval R R id a f = \<zero>} \<and>
         card {a \<in> carrier R . eval R R id a f = \<zero>} \<le> deg R f" using f f_not_zero
proof (induction "deg R f" arbitrary: f)
  case 0
  have "\<And>x. eval R R id x f \<noteq> \<zero>"
  proof -
    fix x
    have "(\<Oplus>i\<in>{..deg R f}. id (coeff P f i) \<otimes> x [^] i) \<noteq> \<zero>"
      using 0 lcoeff_nonzero_nonzero[where p = f] by simp
    thus "eval R R id x f \<noteq> \<zero>" using 0 unfolding eval_def P_def by simp
  qed
  then have *: "{a \<in> carrier R. eval R R (\<lambda>a. a) a f = \<zero>} = {}"
    by (auto simp: id_def)
  show ?case by (simp add: *)
next
  case (Suc x)
  show ?case
  proof (cases "\<exists> a \<in> carrier R . eval R R id a f = \<zero>")
    case True
    then obtain a where a_carrier[simp]: "a \<in> carrier R" and a_root:"eval R R id a f = \<zero>" by blast
    have R_not_triv: "carrier R \<noteq> {\<zero>}"
      by (metis R.one_zeroI R.zero_not_one)
    obtain q  where q:"(q \<in> carrier P)" and
      f:"f = (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub> P\<^esub> monom P a 0) \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> monom P (eval R R id a f) 0"
     using remainder_theorem[OF Suc.prems(1) a_carrier R_not_triv] by auto
    hence lin_fac: "f = (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub> P\<^esub> monom P a 0) \<otimes>\<^bsub>P\<^esub> q" using q by (simp add: a_root)
    have deg:"deg R (monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub> P\<^esub> monom P a 0) = 1"
      using a_carrier by (simp add: deg_minus_eq)
    hence mon_not_zero:"(monom P \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub> P\<^esub> monom P a 0) \<noteq> \<zero>\<^bsub>P\<^esub>"
      by (fastforce simp del: r_right_minus_eq)
    have q_not_zero:"q \<noteq> \<zero>\<^bsub>P\<^esub>" using Suc by (auto simp add : lin_fac)
    hence "deg R q = x" using Suc deg deg_mult[OF mon_not_zero q_not_zero _ q]
      by (simp add : lin_fac)
    hence q_IH:"finite {a \<in> carrier R . eval R R id a q = \<zero>}
                \<and> card {a \<in> carrier R . eval R R id a q = \<zero>} \<le> x" using Suc q q_not_zero by blast
    have subs:"{a \<in> carrier R . eval R R id a f = \<zero>}
                \<subseteq> {a \<in> carrier R . eval R R id a q = \<zero>} \<union> {a}" (is "?L \<subseteq> ?R \<union> {a}")
      using a_carrier \<open>q \<in> _\<close>
      by (auto simp: evalRR_simps lin_fac R.integral_iff)
    have "{a \<in> carrier R . eval R R id a f = \<zero>} \<subseteq> insert a {a \<in> carrier R . eval R R id a q = \<zero>}"
     using subs by auto
    hence "card {a \<in> carrier R . eval R R id a f = \<zero>} \<le>
           card (insert a {a \<in> carrier R . eval R R id a q = \<zero>})" using q_IH by (blast intro: card_mono)
    also have "\<dots> \<le> deg R f" using q_IH \<open>Suc x = _\<close>
      by (simp add: card_insert_if)
    finally show ?thesis using q_IH \<open>Suc x = _\<close> using finite by force
  next
    case False
    hence "card {a \<in> carrier R. eval R R id a f = \<zero>} = 0" using finite by auto
    also have "\<dots> \<le>  deg R f" by simp
    finally show ?thesis using finite by auto
  qed
qed

end

lemma (in domain) num_roots_le_deg :
  fixes p d :: nat
  assumes finite:"finite (carrier R)"
  assumes d_neq_zero : "d \<noteq> 0"
  shows "card {x \<in> carrier R. x [^] d = \<one>} \<le> d"
proof -
  let ?f = "monom (UP R) \<one>\<^bsub>R\<^esub> d \<ominus>\<^bsub> (UP R)\<^esub> monom (UP R) \<one>\<^bsub>R\<^esub> 0"
  have one_in_carrier:"\<one> \<in> carrier R" by simp
  interpret R: UP_domain R "UP R" by (unfold_locales)
  have "deg R ?f = d"
    using d_neq_zero by (simp add: R.deg_minus_eq)
  hence f_not_zero:"?f \<noteq> \<zero>\<^bsub>UP R\<^esub>" using  d_neq_zero by (auto simp add : R.deg_nzero_nzero)
  have roots_bound:"finite {a \<in> carrier R . eval R R id a ?f = \<zero>} \<and>
                    card {a \<in> carrier R . eval R R id a ?f = \<zero>} \<le> deg R ?f"
                    using finite by (intro R.roots_bound[OF _ f_not_zero]) simp
  have subs:"{x \<in> carrier R. x [^] d = \<one>} \<subseteq> {a \<in> carrier R . eval R R id a ?f = \<zero>}"
    by (auto simp: R.evalRR_simps)
  then have "card {x \<in> carrier R. x [^] d = \<one>} \<le>
        card {a \<in> carrier R. eval R R id a ?f = \<zero>}" using finite by (simp add : card_mono)
  thus ?thesis using \<open>deg R ?f = d\<close> roots_bound by linarith
qed



section \<open>The Multiplicative Group of a Field\<close>
text_raw \<open>\label{sec:mult-group}\<close>


text \<open>
  In this section we show that the multiplicative group of a finite field
  is generated by a single element, i.e. it is cyclic. The proof is inspired
  by the first proof given in the survey~@{cite "conrad-cyclicity"}.
\<close>

lemma (in group) pow_order_eq_1:
  assumes "finite (carrier G)" "x \<in> carrier G" shows "x [^] order G = \<one>"
  using assms by (metis nat_pow_pow ord_dvd_group_order pow_ord_eq_1 dvdE nat_pow_one)

(* XXX remove in AFP devel, replaced by div_eq_dividend_iff *)
lemma nat_div_eq: "a \<noteq> 0 \<Longrightarrow> (a :: nat) div b = a \<longleftrightarrow> b = 1"
  apply rule
  apply (cases "b = 0")
  apply simp_all
  apply (metis (full_types) One_nat_def Suc_lessI div_less_dividend less_not_refl3)
  done

lemma (in group)
  assumes finite': "finite (carrier G)"
  assumes "a \<in> carrier G"
  shows pow_ord_eq_ord_iff: "group.ord G (a [^] k) = ord a \<longleftrightarrow> coprime k (ord a)" (is "?L \<longleftrightarrow> ?R")
proof
  assume A: ?L then show ?R
    using assms ord_ge_1 [OF assms]
    by (auto simp: nat_div_eq ord_pow_dvd_ord_elem coprime_iff_gcd_eq_1)
next
  assume ?R then show ?L
    using ord_pow_dvd_ord_elem[OF assms, of k] by auto
qed

context field begin

lemma num_elems_of_ord_eq_phi':
  assumes finite: "finite (carrier R)" and dvd: "d dvd order (mult_of R)"
      and exists: "\<exists>a\<in>carrier (mult_of R). group.ord (mult_of R) a = d"
  shows "card {a \<in> carrier (mult_of R). group.ord (mult_of R) a = d} = phi' d"
proof -
  note mult_of_simps[simp]
  have finite': "finite (carrier (mult_of R))" using finite by (rule finite_mult_of)

  interpret G:group "mult_of R" rewrites "([^]\<^bsub>mult_of R\<^esub>) = (([^]) :: _ \<Rightarrow> nat \<Rightarrow> _)" and "\<one>\<^bsub>mult_of R\<^esub> = \<one>"
    by (rule field_mult_group) simp_all

  from exists
  obtain a where a:"a \<in> carrier (mult_of R)" and ord_a: "group.ord (mult_of R) a = d"
    by (auto simp add: card_gt_0_iff)

  have set_eq1:"{a[^]n| n. n \<in> {1 .. d}} = {x \<in> carrier (mult_of R). x [^] d = \<one>}"
  proof (rule card_seteq)
    show "finite {x \<in> carrier (mult_of R). x [^] d = \<one>}" using finite by auto

    show "{a[^]n| n. n \<in> {1 ..d}} \<subseteq> {x \<in> carrier (mult_of R). x[^]d = \<one>}"
    proof
      fix x assume "x \<in> {a[^]n | n. n \<in> {1 .. d}}"
      then obtain n where n:"x = a[^]n \<and> n \<in> {1 .. d}" by auto
      have "x[^]d =(a[^]d)[^]n" using n a ord_a by (simp add:nat_pow_pow mult.commute)
      hence "x[^]d = \<one>" using ord_a G.pow_ord_eq_1[OF finite' a] by fastforce
      thus "x \<in> {x \<in> carrier (mult_of R). x[^]d = \<one>}" using G.nat_pow_closed[OF a] n by blast
    qed

    show "card {x \<in> carrier (mult_of R). x [^] d = \<one>} \<le> card {a[^]n | n. n \<in> {1 .. d}}"
    proof -
      have *:"{a[^]n | n. n \<in> {1 .. d }} = ((\<lambda> n. a[^]n) ` {1 .. d})" by auto
      have "0 < order (mult_of R)" unfolding order_mult_of[OF finite]
        using card_mono[OF finite, of "{\<zero>, \<one>}"] by (simp add: order_def)
      have "card {x \<in> carrier (mult_of R). x [^] d = \<one>} \<le> card {x \<in> carrier R. x [^] d = \<one>}"
        using finite by (auto intro: card_mono)
      also have "\<dots> \<le> d" using \<open>0 < order (mult_of R)\<close> num_roots_le_deg[OF finite, of d]
        by (simp add : dvd_pos_nat[OF _ \<open>d dvd order (mult_of R)\<close>])
      finally show ?thesis using G.ord_inj'[OF finite' a] ord_a * by (simp add: card_image)
    qed
  qed

  have set_eq2:"{x \<in> carrier (mult_of R) . group.ord (mult_of R) x = d}
                = (\<lambda> n . a[^]n) ` {n \<in> {1 .. d}. group.ord (mult_of R) (a[^]n) = d}" (is "?L = ?R")
  proof
    { fix x assume x:"x \<in> (carrier (mult_of R)) \<and> group.ord (mult_of R) x = d"
      hence "x \<in> {x \<in> carrier (mult_of R). x [^] d = \<one>}"
        by (simp add: G.pow_ord_eq_1[OF finite', of x, symmetric])
      then obtain n where n:"x = a[^]n \<and> n \<in> {1 .. d}" using set_eq1 by blast
      hence "x \<in> ?R" using x by fast
    } thus "?L \<subseteq> ?R" by blast
    show "?R \<subseteq> ?L" using a by (auto simp add: carrier_mult_of[symmetric] simp del: carrier_mult_of)
  qed
  have "inj_on (\<lambda> n . a[^]n) {n \<in> {1 .. d}. group.ord (mult_of R) (a[^]n) = d}"
    using G.ord_inj'[OF finite' a, unfolded ord_a] unfolding inj_on_def by fast
  hence "card ((\<lambda>n. a[^]n) ` {n \<in> {1 .. d}. group.ord (mult_of R) (a[^]n) = d})
         = card {k \<in> {1 .. d}. group.ord (mult_of R) (a[^]k) = d}"
         using card_image by blast
  thus ?thesis using set_eq2 G.pow_ord_eq_ord_iff[OF finite' \<open>a \<in> _\<close>, unfolded ord_a]
    by (simp add: phi'_def)
qed

end


theorem (in field) finite_field_mult_group_has_gen :
  assumes finite:"finite (carrier R)"
  shows "\<exists> a \<in> carrier (mult_of R) . carrier (mult_of R) = {a[^]i | i::nat . i \<in> UNIV}"
proof -
  note mult_of_simps[simp]
  have finite': "finite (carrier (mult_of R))" using finite by (rule finite_mult_of)

  interpret G: group "mult_of R" rewrites
      "([^]\<^bsub>mult_of R\<^esub>) = (([^]) :: _ \<Rightarrow> nat \<Rightarrow> _)" and "\<one>\<^bsub>mult_of R\<^esub> = \<one>"
    by (rule field_mult_group) (simp_all add: fun_eq_iff nat_pow_def)

  let ?N = "\<lambda> x . card {a \<in> carrier (mult_of R). group.ord (mult_of R) a  = x}"
  have "0 < order R - 1" unfolding order_def using card_mono[OF finite, of "{\<zero>, \<one>}"] by simp
  then have *: "0 < order (mult_of R)" using assms by (simp add: order_mult_of)
  have fin: "finite {d. d dvd order (mult_of R) }" using dvd_nat_bounds[OF *] by force

  have "(\<Sum>d | d dvd order (mult_of R). ?N d)
      = card (UN d:{d . d dvd order (mult_of R) }. {a \<in> carrier (mult_of R). group.ord (mult_of R) a  = d})"
      (is "_ = card ?U")
    using fin finite by (subst card_UN_disjoint) auto
  also have "?U = carrier (mult_of R)"
  proof
    { fix x assume x:"x \<in> carrier (mult_of R)"
      hence x':"x\<in>carrier (mult_of R)" by simp
      then have "group.ord (mult_of R) x dvd order (mult_of R)"
          using finite' G.ord_dvd_group_order[OF _ x'] by (simp add: order_mult_of)
      hence "x \<in> ?U" using dvd_nat_bounds[of "order (mult_of R)" "group.ord (mult_of R) x"] x by blast
    } thus "carrier (mult_of R) \<subseteq> ?U" by blast
  qed auto
  also have "card ... = order (mult_of R)"
    using order_mult_of finite' by (simp add: order_def)
  finally have sum_Ns_eq: "(\<Sum>d | d dvd order (mult_of R). ?N d) = order (mult_of R)" .

  { fix d assume d:"d dvd order (mult_of R)"
    have "card {a \<in> carrier (mult_of R). group.ord (mult_of R) a = d} \<le> phi' d"
    proof cases
      assume "card {a \<in> carrier (mult_of R). group.ord (mult_of R) a = d} = 0" thus ?thesis by presburger
      next
      assume "card {a \<in> carrier (mult_of R). group.ord (mult_of R) a = d} \<noteq> 0"
      hence "\<exists>a \<in> carrier (mult_of R). group.ord (mult_of R) a = d" by (auto simp: card_eq_0_iff)
      thus ?thesis using num_elems_of_ord_eq_phi'[OF finite d] by auto
    qed
  }
  hence all_le:"\<And>i. i \<in> {d. d dvd order (mult_of R) }
        \<Longrightarrow> (\<lambda>i. card {a \<in> carrier (mult_of R). group.ord (mult_of R) a = i}) i \<le> (\<lambda>i. phi' i) i" by fast
  hence le:"(\<Sum>i | i dvd order (mult_of R). ?N i)
            \<le> (\<Sum>i | i dvd order (mult_of R). phi' i)"
            using sum_mono[of "{d .  d dvd order (mult_of R)}"
                  "\<lambda>i. card {a \<in> carrier (mult_of R). group.ord (mult_of R) a = i}"] by presburger
  have "order (mult_of R) = (\<Sum>d | d dvd order (mult_of R). phi' d)" using *
    by (simp add: sum_phi'_factors)
  hence eq:"(\<Sum>i | i dvd order (mult_of R). ?N i)
          = (\<Sum>i | i dvd order (mult_of R). phi' i)" using le sum_Ns_eq by presburger
  have "\<And>i. i \<in> {d. d dvd order (mult_of R) } \<Longrightarrow> ?N i = (\<lambda>i. phi' i) i"
  proof (rule ccontr)
    fix i
    assume i1:"i \<in> {d. d dvd order (mult_of R)}" and "?N i \<noteq> phi' i"
    hence "?N i = 0"
      using num_elems_of_ord_eq_phi'[OF finite, of i] by (auto simp: card_eq_0_iff)
    moreover  have "0 < i" using * i1 by (simp add: dvd_nat_bounds[of "order (mult_of R)" i])
    ultimately have "?N i < phi' i" using phi'_nonzero by presburger
    hence "(\<Sum>i | i dvd order (mult_of R). ?N i)
         < (\<Sum>i | i dvd order (mult_of R). phi' i)"
      using sum_strict_mono_ex1[OF fin, of "?N" "\<lambda> i . phi' i"]
            i1 all_le by auto
    thus False using eq by force
  qed
  hence "?N (order (mult_of R)) > 0" using * by (simp add: phi'_nonzero)
  then obtain a where a:"a \<in> carrier (mult_of R)" and a_ord:"group.ord (mult_of R) a = order (mult_of R)"
    by (auto simp add: card_gt_0_iff)
  hence set_eq:"{a[^]i | i::nat. i \<in> UNIV} = (\<lambda>x. a[^]x) ` {0 .. group.ord (mult_of R) a - 1}"
    using G.ord_elems[OF finite'] by auto
  have card_eq:"card ((\<lambda>x. a[^]x) ` {0 .. group.ord (mult_of R) a - 1}) = card {0 .. group.ord (mult_of R) a - 1}"
    by (intro card_image G.ord_inj finite' a)
  hence "card ((\<lambda> x . a[^]x) ` {0 .. group.ord (mult_of R) a - 1}) = card {0 ..order (mult_of R) - 1}"
    using assms by (simp add: card_eq a_ord)
  hence card_R_minus_1:"card {a[^]i | i::nat. i \<in> UNIV} =  order (mult_of R)"
    using * by (subst set_eq) auto
  have **:"{a[^]i | i::nat. i \<in> UNIV} \<subseteq> carrier (mult_of R)"
    using G.nat_pow_closed[OF a] by auto
  with _ have "carrier (mult_of R) = {a[^]i|i::nat. i \<in> UNIV}"
    by (rule card_seteq[symmetric]) (simp_all add: card_R_minus_1 finite order_def del: UNIV_I)
  thus ?thesis using a by blast
qed

end