(* 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)
(* TODO FIXME: This is the "totient" function from HOL-Number_Theory, but since part of
HOL-Number_Theory depends on HOL-Algebra.Multiplicative_Group, there would be a cyclic
dependency. *)
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
lemma generate_pow_on_finite_carrier: \<^marker>\<open>contributor \<open>Paulo Emílio de Vilhena\<close>\<close>
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: \<^marker>\<open>contributor \<open>Paulo Emílio de Vilhena\<close>\<close>
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: \<^marker>\<open>contributor \<open>Paulo Emílio de Vilhena\<close>\<close>
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