Theory Multiplicative_Group
theory Multiplicative_Group
imports
Complex_Main
Group
Coset
UnivPoly
Generated_Groups
Elementary_Groups
begin
section ‹Simplification Rules for Polynomials›
text_raw ‹\label{sec:simp-rules}›
lemma (in ring_hom_cring) hom_sub[simp]:
assumes "x ∈ carrier R" "y ∈ carrier R"
shows "h (x ⊖ y) = h x ⊖⇘S⇙ 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 ≠ 0"
shows "p ≠ 𝟬⇘P⇙"
using deg_zero deg_p_nzero by auto
lemma deg_add_eq:
assumes c: "p ∈ carrier P" "q ∈ carrier P"
assumes "deg R q ≠ deg R p"
shows "deg R (p ⊕⇘P⇙ 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 = 𝟬 ⟷ coeff P q ?m ≠ 𝟬"
by (metis deg_belowI lcoeff_nonzero[OF deg_nzero_nzero] linear max.absorb_iff2 max.absorb1)
then have "coeff P (p ⊕⇘P⇙ q) ?m ≠ 𝟬"
using assms by auto
then have "deg R (p ⊕⇘P⇙ q) ≥ ?m"
using assms by (blast intro: deg_belowI)
with deg_add[OF c] show ?thesis by arith
qed
lemma deg_minus_eq:
assumes "p ∈ carrier P" "q ∈ carrier P" "deg R q ≠ deg R p"
shows "deg R (p ⊖⇘P⇙ 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 ∈ carrier P" "q ∈ carrier P"
assumes x: "x ∈ carrier R"
shows "eval R R id x (p ⊕⇘P⇙ q) = eval R R id x p ⊕ 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 ∈ carrier P" "q ∈ carrier P"
assumes x: "x ∈ carrier R"
shows "eval R R id x (p ⊖⇘P⇙ q) = eval R R id x p ⊖ 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 ∈ carrier P" "q ∈ carrier P"
assumes x: "x ∈ carrier R"
shows "eval R R id x (p ⊗⇘P⇙ q) = eval R R id x p ⊗ 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 ∈ carrier R" and x: "x ∈ carrier R"
shows "eval R R id x (monom P a d) = a ⊗ 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 ∈ carrier R"
shows "eval R R id x 𝟭⇘P⇙ = 𝟭"
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 ∈ carrier R" and "p ∈ carrier P"
shows "eval R R id x p ∈ 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 ‹Properties of the Euler ‹φ›-function›
text_raw ‹\label{sec:euler-phi}›
text‹
In this section we prove that for every positive natural number the equation
$\sum_{d | n}^n \varphi(d) = n$ holds.
›
lemma dvd_div_ge_1:
fixes a b :: nat
assumes "a ≥ 1" "b dvd a"
shows "a div b ≥ 1"
proof -
from ‹b dvd a› obtain c where "a = b * c" ..
with ‹a ≥ 1› show ?thesis by simp
qed
lemma dvd_nat_bounds:
fixes n p :: nat
assumes "p > 0" "n dvd p"
shows "n > 0 ∧ n ≤ p"
using assms by (simp add: dvd_pos_nat dvd_imp_le)
definition phi' :: "nat => nat"
where "phi' m = card {x. 1 ≤ x ∧ x ≤ m ∧ coprime x m}"
notation (latex output)
phi' (‹φ _›)
lemma phi'_nonzero:
assumes "m > 0"
shows "phi' m > 0"
proof -
have "1 ∈ {x. 1 ≤ x ∧ x ≤ m ∧ coprime x m}" using assms by simp
hence "card {x. 1 ≤ x ∧ x ≤ m ∧ 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 ‹c dvd a›] dvd_mult_div_cancel[OF ‹c dvd b›]
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 ≤ 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 "… = b*(c div a)" using assms dvd_mult_div_cancel by fastforce
finally show "a = b" using ‹c>0› dvd_div_ge_1[OF _ ‹a dvd c›] by fastforce
qed
lemma div_mult_mono:
fixes a b c :: nat
assumes "a > 0" "a≤d"
shows "a * b div d ≤ b"
proof -
have "a*b div d ≤ b*a div a" using assms div_le_mono2 mult.commute[of a b] by presburger
thus ?thesis using assms by force
qed
text‹
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>‹gcd a d = 1›.
This number is exactly \<^term>‹phi' d›.
Finally, by counting the fractions $m/n$ according to their reduced form denominator,
we get: @{term [display] "(∑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>‹{a. (1::nat) ≤ a ∧ a ≤ d ∧ coprime a d}›
\item the set of numerators $m$, for which $m/n$ has the reduced form denominator $d$,
i.e. the set \<^term>‹{m ∈ {1::nat .. n}. n div gcd m n = d}›
\end{itemize}
We show that \<^term>‹λa. a*n div d› with the inverse \<^term>‹λa. a div gcd a n› is
a bijection between theses sets, thus yielding the equality
@{term [display] "phi' d = card {m ∈ {1 .. n}. n div gcd m n = d}"}
This gives us
@{term [display] "(∑d | d dvd n . phi' d)
= card (⋃d ∈ {d. d dvd n}. {m ∈ {1 .. n}. n div gcd m n = d})"}
and by showing
\<^term>‹(⋃d ∈ {d. d dvd n}. {m ∈ {1::nat .. n}. n div gcd m n = d}) ⊇ {1 .. n}›
(this is our counting argument) the thesis follows.
›
lemma sum_phi'_factors:
fixes n :: nat
assumes "n > 0"
shows "(∑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 ≤ a ∧ a ≤ d ∧ coprime a d} = card {m ∈ {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 ‹d dvd n›] by (fastforce simp add: mult.commute)
hence "a = b" using dvd_div_ge_1[OF _ ‹d dvd n›] ‹n>0›
by (simp add: mult.commute nat_mult_eq_cancel1)
} thus "inj_on (λa. a*n div d) ?RF" unfolding inj_on_def by blast
{ fix a assume a: "a∈?RF"
hence "a * (n div d) ≥ 1" using ‹n>0› dvd_div_ge_1[OF _ ‹d dvd n›] by simp
hence ge_1: "a * n div d ≥ 1" by (simp add: ‹d dvd n› div_mult_swap)
have le_n: "a * n div d ≤ 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 ∈ ?F"
using ge_1 le_n by (fastforce simp add: ‹d dvd n›)
} thus "(λa. a*n div d) ` ?RF ⊆ ?F" by blast
{ fix m l assume A: "m ∈ ?F" "l ∈ ?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 (λa. a div gcd a n) ?F" unfolding inj_on_def by blast
{ fix m assume "m ∈ ?F"
hence "m div gcd m n ∈ ?RF" using dvd_div_ge_1
by (fastforce simp add: div_le_mono div_gcd_coprime)
} thus "(λa. a div gcd a n) ` ?F ⊆ ?RF" by blast
qed force+
} hence phi'_eq: "⋀d. d dvd n ⟹ phi' d = card {m ∈ {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 ‹n>0›] by force
have "(∑d | d dvd n. phi' d)
= card (⋃d ∈ {d. d dvd n}. {m ∈ {1 .. n}. n div gcd m n = d})"
using card_UN_disjoint[OF fin, of "(λd. {m ∈ {1 .. n}. n div gcd m n = d})"] phi'_eq
by fastforce
also have "(⋃d ∈ {d. d dvd n}. {m ∈ {1 .. n}. n div gcd m n = d}) = {1 .. n}" (is "?L = ?R")
proof
show "?L ⊇ ?R"
proof
fix m assume m: "m ∈ ?R"
thus "m ∈ ?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 ‹Order of an Element of a Group›
text_raw ‹\label{sec:order-elem}›
context group begin
definition (in group) ord :: "'a ⇒ nat" where
"ord x ≡ (@d. ∀n::nat. x [^] n = 𝟭 ⟷ d dvd n)"
lemma (in group) pow_eq_id:
assumes "x ∈ carrier G"
shows "x [^] n = 𝟭 ⟷ (ord x) dvd n"
proof (cases "∀n::nat. pow G x n = one G ⟶ n = 0")
case True
show ?thesis
unfolding ord_def
by (rule someI2 [where a=0]) (auto simp: True)
next
case False
define N where "N ≡ LEAST n::nat. x [^] n = 𝟭 ∧ n > 0"
have N: "x [^] N = 𝟭 ∧ N > 0"
using False
apply (simp add: N_def)
by (metis (mono_tags, lifting) LeastI)
have eq0: "n = 0" if "x [^] n = 𝟭" "n < N" for n
using N_def not_less_Least that by fastforce
show ?thesis
unfolding ord_def
proof (rule someI2 [where a = N], rule allI)
fix n :: "nat"
show "(x [^] n = 𝟭) ⟷ (N dvd n)"
proof (cases "n = 0")
case False
show ?thesis
unfolding dvd_def
proof safe
assume 1: "x [^] n = 𝟭"
have "x [^] n = x [^] (n mod N + N * (n div N))"
by simp
also have "… = x [^] (n mod N) ⊗ x [^] (N * (n div N))"
by (simp add: assms nat_pow_mult)
also have "… = x [^] (n mod N)"
by (metis N assms l_cancel_one nat_pow_closed nat_pow_one nat_pow_pow)
finally have "x [^] (n mod N) = 𝟭"
by (simp add: "1")
then have "n mod N = 0"
using N eq0 mod_less_divisor by blast
then show "∃k. n = N * k"
by blast
next
fix k :: "nat"
assume "n = N * k"
with N show "x [^] (N * k) = 𝟭"
by (metis assms nat_pow_one nat_pow_pow)
qed
qed simp
qed blast
qed
lemma (in group) pow_ord_eq_1 [simp]:
"x ∈ carrier G ⟹ x [^] ord x = 𝟭"
by (simp add: pow_eq_id)
lemma (in group) int_pow_eq_id:
assumes "x ∈ carrier G"
shows "(pow G x i = one G ⟷ int (ord x) dvd i)"
proof (cases i rule: int_cases2)
case (nonneg n)
then show ?thesis
by (simp add: int_pow_int pow_eq_id assms)
next
case (nonpos n)
then have "x [^] i = inv (x [^] n)"
by (simp add: assms int_pow_int int_pow_neg)
then show ?thesis
by (simp add: assms pow_eq_id nonpos)
qed
lemma (in group) int_pow_eq:
"x ∈ carrier G ⟹ (x [^] m = x [^] n) ⟷ int (ord x) dvd (n - m)"
apply (simp flip: int_pow_eq_id)
by (metis int_pow_closed int_pow_diff inv_closed r_inv right_cancel)
lemma (in group) ord_eq_0:
"x ∈ carrier G ⟹ (ord x = 0 ⟷ (∀n::nat. n ≠ 0 ⟶ x [^] n ≠ 𝟭))"
by (auto simp: pow_eq_id)
lemma (in group) ord_unique:
"x ∈ carrier G ⟹ ord x = d ⟷ (∀n. pow G x n = one G ⟷ d dvd n)"
by (meson dvd_antisym dvd_refl pow_eq_id)
lemma (in group) ord_eq_1:
"x ∈ carrier G ⟹ (ord x = 1 ⟷ x = 𝟭)"
by (metis pow_eq_id nat_dvd_1_iff_1 nat_pow_eone)
lemma (in group) ord_id [simp]:
"ord (one G) = 1"
using ord_eq_1 by blast
lemma (in group) ord_inv [simp]:
"x ∈ carrier G
⟹ ord (m_inv G x) = ord x"
by (simp add: ord_unique pow_eq_id nat_pow_inv)
lemma (in group) ord_pow:
assumes "x ∈ carrier G" "k dvd ord x" "k ≠ 0"
shows "ord (pow G x k) = ord x div k"
proof -
have "(x [^] k) [^] (ord x div k) = 𝟭"
using assms by (simp add: nat_pow_pow)
moreover have "ord x dvd k * ord (x [^] k)"
by (metis assms(1) pow_ord_eq_1 pow_eq_id nat_pow_closed nat_pow_pow)
ultimately show ?thesis
by (metis assms div_dvd_div dvd_antisym dvd_triv_left pow_eq_id nat_pow_closed nonzero_mult_div_cancel_left)
qed
lemma (in group) ord_mul_divides:
assumes eq: "x ⊗ y = y ⊗ x" and xy: "x ∈ carrier G" "y ∈ carrier G"
shows "ord (x ⊗ y) dvd (ord x * ord y)"
apply (simp add: xy flip: pow_eq_id eq)
by (metis dvd_triv_left dvd_triv_right eq pow_eq_id one_closed pow_mult_distrib r_one xy)
lemma (in comm_group) abelian_ord_mul_divides:
"⟦x ∈ carrier G; y ∈ carrier G⟧
⟹ ord (x ⊗ y) dvd (ord x * ord y)"
by (simp add: ord_mul_divides m_comm)
lemma ord_inj:
assumes a: "a ∈ carrier G"
shows "inj_on (λ x . a [^] x) {0 .. ord a - 1}"
proof -
let ?M = "Max (ord ` carrier G)"
have "finite {d ∈ {..?M}. a [^] d = 𝟭}" by auto
have *: False if A: "x < y" "x ∈ {0 .. ord a - 1}" "y ∈ {0 .. ord a - 1}"
"a [^] x = a [^] y" for x y
proof -
have "y - x < ord a" using that by auto
moreover have "a [^] (y-x) = 𝟭" using a A by (simp add: pow_eq_div2)
ultimately have "min (y - x) (ord a) = ord a"
using A(1) a pow_eq_id by auto
with ‹y - x < ord a› show False by linarith
qed
show ?thesis
unfolding inj_on_def by (metis nat_neq_iff *)
qed
lemma ord_inj':
assumes a: "a ∈ carrier G"
shows "inj_on (λ x . a [^] x) {1 .. ord a}"
proof (rule inj_onI, rule ccontr)
fix x y :: nat
assume A: "x ∈ {1 .. ord a}" "y ∈ {1 .. ord a}" "a [^] x = a [^] y" "x≠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 A by (simp add: a)
hence "y=0" using ord_inj[OF assms] ‹y < ord a› 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 A by (simp add: a)
hence "x=0" using ord_inj[OF assms] ‹x < ord a› unfolding inj_on_def by force
hence False using A by fastforce
}
ultimately show False using A by force
qed
lemma (in group) ord_ge_1:
assumes finite: "finite (carrier G)" and a: "a ∈ carrier G"
shows "ord a ≥ 1"
proof -
have "((λn::nat. a [^] n) ` {0<..}) ⊆ carrier G"
using a by blast
then have "finite ((λn::nat. a [^] n) ` {0<..})"
using finite_subset finite by auto
then have "¬ inj_on (λn::nat. a [^] n) {0<..}"
using finite_imageD infinite_Ioi by blast
then obtain i j::nat where "i ≠ j" "a [^] i = a [^] j"
by (auto simp: inj_on_def)
then have "∃n::nat. n>0 ∧ a [^] n = 𝟭"
by (metis a diffs0_imp_equal pow_eq_div2 neq0_conv)
then have "ord a ≠ 0"
by (simp add: ord_eq_0 [OF a])
then show ?thesis
by simp
qed
lemma ord_elems:
assumes "finite (carrier G)" "a ∈ carrier G"
shows "{a[^]x | x. x ∈ (UNIV :: nat set)} = {a[^]x | x. x ∈ {0 .. ord a - 1}}" (is "?L = ?R")
proof
show "?R ⊆ ?L" by blast
{ fix y assume "y ∈ ?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 ⊗ a[^]r"
using x assms by (metis 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 ∈ {0 .. ord a - 1}" by (force simp: r_def)
hence "y ∈ {a[^]x | x. x ∈ {0 .. ord a - 1}}" using ‹y=a[^]r› by blast
}
thus "?L ⊆ ?R" by auto
qed
lemma (in group)
assumes "x ∈ carrier G"
shows finite_cyclic_subgroup:
"finite(carrier(subgroup_generated G {x})) ⟷ (∃n::nat. n ≠ 0 ∧ x [^] n = 𝟭)" (is "?fin ⟷ ?nat1")
and infinite_cyclic_subgroup:
"infinite(carrier(subgroup_generated G {x})) ⟷ (∀m n::nat. x [^] m = x [^] n ⟶ m = n)" (is "¬ ?fin ⟷ ?nateq")
and finite_cyclic_subgroup_int:
"finite(carrier(subgroup_generated G {x})) ⟷ (∃i::int. i ≠ 0 ∧ x [^] i = 𝟭)" (is "?fin ⟷ ?int1")
and infinite_cyclic_subgroup_int:
"infinite(carrier(subgroup_generated G {x})) ⟷ (∀i j::int. x [^] i = x [^] j ⟶ i = j)" (is "¬ ?fin ⟷ ?inteq")
proof -
have 1: "¬ ?fin" if ?nateq
proof -
have "infinite (range (λn::nat. x [^] n))"
using that range_inj_infinite [of "(λn::nat. x [^] n)"] by (auto simp: inj_on_def)
moreover have "range (λn::nat. x [^] n) ⊆ range (λi::int. x [^] i)"
apply clarify
by (metis assms group.int_pow_neg int_pow_closed int_pow_neg_int is_group local.inv_equality nat_pow_closed r_inv rangeI)
ultimately show ?thesis
using carrier_subgroup_generated_by_singleton [OF assms] finite_subset by auto
qed
have 2: "m = n" if mn: "x [^] m = x [^] n" and eq [rule_format]: "?inteq" for m n::nat
using eq [of "int m" "int n"]
by (simp add: int_pow_int mn)
have 3: ?nat1 if non: "¬ ?inteq"
proof -
obtain i j::int where eq: "x [^] i = x [^] j" and "i ≠ j"
using non by auto
show ?thesis
proof (cases i j rule: linorder_cases)
case less
then have [simp]: "x [^] (j - i) = 𝟭"
by (simp add: eq assms int_pow_diff)
show ?thesis
using less by (rule_tac x="nat (j-i)" in exI) auto
next
case greater
then have [simp]: "x [^] (i - j) = 𝟭"
by (simp add: eq assms int_pow_diff)
then show ?thesis
using greater by (rule_tac x="nat (i-j)" in exI) auto
qed (use ‹i ≠ j› in auto)
qed
have 4: "∃i::int. (i ≠ 0) ∧ x [^] i = 𝟭" if "n ≠ 0" "x [^] n = 𝟭" for n::nat
apply (rule_tac x="int n" in exI)
by (simp add: int_pow_int that)
have 5: "finite (carrier (subgroup_generated G {x}))" if "i ≠ 0" and 1: "x [^] i = 𝟭" for i::int
proof -
obtain n::nat where n: "n > 0" "x [^] n = 𝟭"
using "1" "3" ‹i ≠ 0› by fastforce
have "x [^] a ∈ ([^]) x ` {0..<n}" for a::int
proof
show "x [^] a = x [^] nat (a mod int n)"
using n
by simp (metis (no_types, lifting) assms dvd_minus_mod dvd_trans int_pow_eq int_pow_eq_id int_pow_int)
show "nat (a mod int n) ∈ {0..<n}"
using n by (simp add: nat_less_iff)
qed
then have "carrier (subgroup_generated G {x}) ⊆ ([^]) x ` {0..<n}"
using carrier_subgroup_generated_by_singleton [OF assms] by auto
then show ?thesis
using finite_surj by blast
qed
show "?fin ⟷ ?nat1" "¬ ?fin ⟷ ?nateq" "?fin ⟷ ?int1" "¬ ?fin ⟷ ?inteq"
using 1 2 3 4 5 by meson+
qed
lemma (in group) finite_cyclic_subgroup_order:
"x ∈ carrier G ⟹ finite(carrier(subgroup_generated G {x})) ⟷ ord x ≠ 0"
by (simp add: finite_cyclic_subgroup ord_eq_0)
lemma (in group) infinite_cyclic_subgroup_order:
"x ∈ carrier G ⟹ infinite (carrier(subgroup_generated G {x})) ⟷ ord x = 0"
by (simp add: finite_cyclic_subgroup_order)
lemma generate_pow_on_finite_carrier:
assumes "finite (carrier G)" and a: "a ∈ carrier G"
shows "generate G { a } = { a [^] k | k. k ∈ (UNIV :: nat set) }"
proof
show "{ a [^] k | k. k ∈ (UNIV :: nat set) } ⊆ generate G { a }"
proof
fix b assume "b ∈ { a [^] k | k. k ∈ (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 ∈ generate G { a }"
unfolding generate_pow[OF a] by blast
qed
next
show "generate G { a } ⊆ { a [^] k | k. k ∈ (UNIV :: nat set) }"
proof
fix b assume "b ∈ generate G { a }"
then obtain k :: int where k: "b = a [^] k"
unfolding generate_pow[OF a] by blast
show "b ∈ { a [^] k | k. k ∈ (UNIV :: nat set) }"
proof (cases "k < 0")
assume "¬ k < 0"
hence "b = a [^] (nat k)"
by (simp add: k)
thus ?thesis by blast
next
assume "k < 0"
hence b: "b = inv (a [^] (nat (- k)))"
using k a by (auto simp: int_pow_neg)
obtain m where m: "ord a * m ≥ nat (- k)"
by (metis assms mult.left_neutral mult_le_mono1 ord_ge_1)
hence "a [^] (ord a * m) = 𝟭"
by (metis a nat_pow_one nat_pow_pow pow_ord_eq_1)
then obtain k' :: nat where "(a [^] (nat (- k))) ⊗ (a [^] k') = 𝟭"
using m a nat_le_iff_add nat_pow_mult by auto
hence "b = a [^] k'"
using b a by (metis inv_unique' nat_pow_closed nat_pow_comm)
thus "b ∈ { a [^] k | k. k ∈ (UNIV :: nat set) }" by blast
qed
qed
qed
lemma ord_elems_inf_carrier:
assumes "a ∈ carrier G" "ord a ≠ 0"
shows "{a[^]x | x. x ∈ (UNIV :: nat set)} = {a[^]x | x. x ∈ {0 .. ord a - 1}}" (is "?L = ?R")
proof
show "?R ⊆ ?L" by blast
{ fix y assume "y ∈ ?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 ⊗ a[^]r"
using x assms by (metis mult.commute nat_pow_mult nat_pow_pow)
hence "y = a[^]r" using assms by simp
have "r < ord a" using assms by (simp add: r_def)
hence "r ∈ {0 .. ord a - 1}" by (force simp: r_def)
hence "y ∈ {a[^]x | x. x ∈ {0 .. ord a - 1}}" using ‹y=a[^]r› by blast
}
thus "?L ⊆ ?R" by auto
qed
lemma generate_pow_nat:
assumes a: "a ∈ carrier G" and "ord a ≠ 0"
shows "generate G { a } = { a [^] k | k. k ∈ (UNIV :: nat set) }"
proof
show "{ a [^] k | k. k ∈ (UNIV :: nat set) } ⊆ generate G { a }"
proof
fix b assume "b ∈ { a [^] k | k. k ∈ (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 ∈ generate G { a }"
unfolding generate_pow[OF a] by blast
qed
next
show "generate G { a } ⊆ { a [^] k | k. k ∈ (UNIV :: nat set) }"
proof
fix b assume "b ∈ generate G { a }"
then obtain k :: int where k: "b = a [^] k"
unfolding generate_pow[OF a] by blast
show "b ∈ { a [^] k | k. k ∈ (UNIV :: nat set) }"
proof (cases "k < 0")
assume "¬ k < 0"
hence "b = a [^] (nat k)"
by (simp add: k)
thus ?thesis by blast
next
assume "k < 0"
hence b: "b = inv (a [^] (nat (- k)))"
using k a by (auto simp: int_pow_neg)
obtain m where m: "ord a * m ≥ nat (- k)"
by (metis assms(2) dvd_imp_le dvd_triv_right le_zero_eq mult_eq_0_iff not_gr_zero)
hence "a [^] (ord a * m) = 𝟭"
by (metis a nat_pow_one nat_pow_pow pow_ord_eq_1)
then obtain k' :: nat where "(a [^] (nat (- k))) ⊗ (a [^] k') = 𝟭"
using m a nat_le_iff_add nat_pow_mult by auto
hence "b = a [^] k'"
using b a by (metis inv_unique' nat_pow_closed nat_pow_comm)
thus "b ∈ { a [^] k | k. k ∈ (UNIV :: nat set) }" by blast
qed
qed
qed
lemma generate_pow_card:
assumes a: "a ∈ carrier G"
shows "ord a = card (generate G { a })"
proof (cases "ord a = 0")
case True
then have "infinite (carrier (subgroup_generated G {a}))"
using infinite_cyclic_subgroup_order[OF a] by auto
then have "infinite (generate G {a})"
unfolding subgroup_generated_def
using a by simp
then show ?thesis
using ‹ord a = 0› by auto
next
case False
note finite_subgroup = this
then have "generate G { a } = (([^]) a) ` {0..ord a - 1}"
using generate_pow_nat ord_elems_inf_carrier a by auto
hence "card (generate G {a}) = card {0..ord a - 1}"
using ord_inj[OF a] card_image by metis
also have "... = ord a" using finite_subgroup by auto
finally show ?thesis..
qed
lemma (in group) cyclic_order_is_ord:
assumes "g ∈ carrier G"
shows "ord g = order (subgroup_generated G {g})"
unfolding order_def subgroup_generated_def
using assms generate_pow_card by simp
lemma ord_dvd_group_order:
assumes "a ∈ carrier G" shows "(ord a) dvd (order G)"
using lagrange[OF generate_is_subgroup[of "{a}"]] assms
unfolding generate_pow_card[OF assms]
by (metis dvd_triv_right empty_subsetI insert_subset)
lemma (in group) pow_order_eq_1:
assumes "a ∈ carrier G" shows "a [^] order G = 𝟭"
using assms by (metis nat_pow_pow ord_dvd_group_order pow_ord_eq_1 dvdE nat_pow_one)
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 "… = b * (a div gcd a b)" by simp
finally show "a * (b div gcd a b) = b * (a div gcd a b) " .
qed
lemma (in group) ord_le_group_order:
assumes finite: "finite (carrier G)" and a: "a ∈ carrier G"
shows "ord a ≤ order G"
by (simp add: a dvd_imp_le local.finite ord_dvd_group_order order_gt_0_iff_finite)
lemma (in group) ord_pow_gen:
assumes "x ∈ carrier G"
shows "ord (pow G x k) = (if k = 0 then 1 else ord x div gcd (ord x) k)"
proof -
have "ord (x [^] k) = ord x div gcd (ord x) k"
if "0 < k"
proof -
have "(d dvd k * n) = (d div gcd (d) k dvd n)" for d n
using that by (simp add: div_dvd_iff_mult gcd_mult_distrib_nat mult.commute)
then show ?thesis
using that by (auto simp add: assms ord_unique nat_pow_pow pow_eq_id)
qed
then show ?thesis by auto
qed
lemma (in group)
assumes finite': "finite (carrier G)" "a ∈ carrier G"
shows pow_ord_eq_ord_iff: "group.ord G (a [^] k) = ord a ⟷ coprime k (ord a)" (is "?L ⟷ ?R")
using assms ord_ge_1 [OF assms]
by (auto simp: div_eq_dividend_iff ord_pow_gen coprime_iff_gcd_eq_1 gcd.commute split: if_split_asm)
lemma element_generates_subgroup:
assumes finite[simp]: "finite (carrier G)"
assumes a[simp]: "a ∈ carrier G"
shows "subgroup {a [^] i | i. i ∈ {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
end
section ‹Number of Roots of a Polynomial›
text_raw ‹\label{sec:number-roots}›
definition mult_of :: "('a, 'b) ring_scheme ⇒ 'a monoid" where
"mult_of R ≡ ⦇ carrier = carrier R - {𝟬⇘R⇙}, mult = mult R, one = 𝟭⇘R⇙⦈"
lemma carrier_mult_of [simp]: "carrier (mult_of R) = carrier R - {𝟬⇘R⇙}"
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: "([^]⇘mult_of R⇙) = (([^]⇘R⇙) :: _ ⇒ nat ⇒ _)"
by (simp add: mult_of_def fun_eq_iff nat_pow_def)
lemma one_mult_of [simp]: "𝟭⇘mult_of R⇙ = 𝟭⇘R⇙"
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:
"⋀x. x ∈ carrier (mult_of R) ⟹ 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 (in field) field_mult_group: "group (mult_of R)"
proof (rule groupI)
show "∃y∈carrier (mult_of R). y ⊗⇘mult_of R⇙ x = 𝟭⇘mult_of R⇙"
if "x ∈ carrier (mult_of R)" for x
using group.l_inv_ex mult_of_is_Units that units_group by fastforce
qed (auto simp: m_assoc dest: integral)
lemma finite_mult_of: "finite (carrier R) ⟹ finite (carrier (mult_of R))"
by simp
lemma order_mult_of: "finite (carrier R) ⟹ 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 ∈ Units G"
shows "x [^] d ∈ Units G"
by (metis assms group.is_monoid monoid.nat_pow_closed units_group units_of_carrier units_of_pow)
lemma (in ring) r_right_minus_eq[simp]:
assumes "a ∈ carrier R" "b ∈ carrier R"
shows "a ⊖ b = 𝟬 ⟷ 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 ∈ carrier P"
assumes f_not_zero: "f ≠ 𝟬⇘P⇙"
assumes finite: "finite (carrier R)"
shows "finite {a ∈ carrier R . eval R R id a f = 𝟬} ∧
card {a ∈ carrier R . eval R R id a f = 𝟬} ≤ deg R f" using f f_not_zero
proof (induction "deg R f" arbitrary: f)
case 0
have "⋀x. eval R R id x f ≠ 𝟬"
proof -
fix x
have "(⨁i∈{..deg R f}. id (coeff P f i) ⊗ x [^] i) ≠ 𝟬"
using 0 lcoeff_nonzero_nonzero[where p = f] by simp
thus "eval R R id x f ≠ 𝟬" using 0 unfolding eval_def P_def by simp
qed
then have *: "{a ∈ carrier R. eval R R (λa. a) a f = 𝟬} = {}"
by (auto simp: id_def)
show ?case by (simp add: *)
next
case (Suc x)
show ?case
proof (cases "∃ a ∈ carrier R . eval R R id a f = 𝟬")
case True
then obtain a where a_carrier[simp]: "a ∈ carrier R" and a_root: "eval R R id a f = 𝟬" by blast
have R_not_triv: "carrier R ≠ {𝟬}"
by (metis R.one_zeroI R.zero_not_one)
obtain q where q: "(q ∈ carrier P)" and
f: "f = (monom P 𝟭⇘R⇙ 1 ⊖⇘ P⇙ monom P a 0) ⊗⇘P⇙ q ⊕⇘P⇙ 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 𝟭⇘R⇙ 1 ⊖⇘ P⇙ monom P a 0) ⊗⇘P⇙ q" using q by (simp add: a_root)
have deg: "deg R (monom P 𝟭⇘R⇙ 1 ⊖⇘ P⇙ monom P a 0) = 1"
using a_carrier by (simp add: deg_minus_eq)
hence mon_not_zero: "(monom P 𝟭⇘R⇙ 1 ⊖⇘ P⇙ monom P a 0) ≠ 𝟬⇘P⇙"
by (fastforce simp del: r_right_minus_eq)
have q_not_zero: "q ≠ 𝟬⇘P⇙" 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 ∈ carrier R . eval R R id a q = 𝟬}
∧ card {a ∈ carrier R . eval R R id a q = 𝟬} ≤ x" using Suc q q_not_zero by blast
have subs: "{a ∈ carrier R . eval R R id a f = 𝟬}
⊆ {a ∈ carrier R . eval R R id a q = 𝟬} ∪ {a}" (is "?L ⊆ ?R ∪ {a}")
using a_carrier ‹q ∈ _›
by (auto simp: evalRR_simps lin_fac R.integral_iff)
have "{a ∈ carrier R . eval R R id a f = 𝟬} ⊆ insert a {a ∈ carrier R . eval R R id a q = 𝟬}"
using subs by auto
hence "card {a ∈ carrier R . eval R R id a f = 𝟬} ≤
card (insert a {a ∈ carrier R . eval R R id a q = 𝟬})" using q_IH by (blast intro: card_mono)
also have "… ≤ deg R f" using q_IH ‹Suc x = _›
by (simp add: card_insert_if)
finally show ?thesis using q_IH ‹Suc x = _› using finite by force
next
case False
hence "card {a ∈ carrier R. eval R R id a f = 𝟬} = 0" using finite by auto
also have "… ≤ 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 ≠ 0"
shows "card {x ∈ carrier R. x [^] d = 𝟭} ≤ d"
proof -
let ?f = "monom (UP R) 𝟭⇘R⇙ d ⊖⇘ (UP R)⇙ monom (UP R) 𝟭⇘R⇙ 0"
have one_in_carrier: "𝟭 ∈ 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 ≠ 𝟬⇘UP R⇙" using d_neq_zero by (auto simp add : R.deg_nzero_nzero)
have roots_bound: "finite {a ∈ carrier R . eval R R id a ?f = 𝟬} ∧
card {a ∈ carrier R . eval R R id a ?f = 𝟬} ≤ deg R ?f"
using finite by (intro R.roots_bound[OF _ f_not_zero]) simp
have subs: "{x ∈ carrier R. x [^] d = 𝟭} ⊆ {a ∈ carrier R . eval R R id a ?f = 𝟬}"
by (auto simp: R.evalRR_simps)
then have "card {x ∈ carrier R. x [^] d = 𝟭} ≤
card {a ∈ carrier R. eval R R id a ?f = 𝟬}" using finite by (simp add : card_mono)
thus ?thesis using ‹deg R ?f = d› roots_bound by linarith
qed
section ‹The Multiplicative Group of a Field›
text_raw ‹\label{sec:mult-group}›
text ‹
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"›.
›
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: "∃a∈carrier (mult_of R). group.ord (mult_of R) a = d"
shows "card {a ∈ 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 "([^]⇘mult_of R⇙) = (([^]) :: _ ⇒ nat ⇒ _)" and "𝟭⇘mult_of R⇙ = 𝟭"
by (rule field_mult_group) simp_all
from exists
obtain a where a: "a ∈ 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 ∈ {1 .. d}} = {x ∈ carrier (mult_of R). x [^] d = 𝟭}"
proof (rule card_seteq)
show "finite {x ∈ carrier (mult_of R). x [^] d = 𝟭}" using finite by auto
show "{a[^]n| n. n ∈ {1 ..d}} ⊆ {x ∈ carrier (mult_of R). x[^]d = 𝟭}"
proof
fix x assume "x ∈ {a[^]n | n. n ∈ {1 .. d}}"
then obtain n where n: "x = a[^]n ∧ n ∈ {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 = 𝟭" using ord_a G.pow_ord_eq_1[OF a] by fastforce
thus "x ∈ {x ∈ carrier (mult_of R). x[^]d = 𝟭}" using G.nat_pow_closed[OF a] n by blast
qed
show "card {x ∈ carrier (mult_of R). x [^] d = 𝟭} ≤ card {a[^]n | n. n ∈ {1 .. d}}"
proof -
have *: "{a[^]n | n. n ∈ {1 .. d }} = ((λ 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 "{𝟬, 𝟭}"] by (simp add: order_def)
have "card {x ∈ carrier (mult_of R). x [^] d = 𝟭} ≤ card {x ∈ carrier R. x [^] d = 𝟭}"
using finite by (auto intro: card_mono)
also have "… ≤ d" using ‹0 < order (mult_of R)› num_roots_le_deg[OF finite, of d]
by (simp add : dvd_pos_nat[OF _ ‹d dvd order (mult_of R)›])
finally show ?thesis using G.ord_inj'[OF a] ord_a * by (simp add: card_image)
qed
qed
have set_eq2: "{x ∈ carrier (mult_of R) . group.ord (mult_of R) x = d}
= (λ n . a[^]n) ` {n ∈ {1 .. d}. group.ord (mult_of R) (a[^]n) = d}" (is "?L = ?R")
proof
{ fix x assume x: "x ∈ (carrier (mult_of R)) ∧ group.ord (mult_of R) x = d"
hence "x ∈ {x ∈ carrier (mult_of R). x [^] d = 𝟭}"
by (simp add: G.pow_ord_eq_1[of x, symmetric])
then obtain n where n: "x = a[^]n ∧ n ∈ {1 .. d}" using set_eq1 by blast
hence "x ∈ ?R" using x by fast
} thus "?L ⊆ ?R" by blast
show "?R ⊆ ?L" using a by (auto simp add: carrier_mult_of[symmetric] simp del: carrier_mult_of)
qed
have "inj_on (λ n . a[^]n) {n ∈ {1 .. d}. group.ord (mult_of R) (a[^]n) = d}"
using G.ord_inj'[OF a, unfolded ord_a] unfolding inj_on_def by fast
hence "card ((λn. a[^]n) ` {n ∈ {1 .. d}. group.ord (mult_of R) (a[^]n) = d})
= card {k ∈ {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' ‹a ∈ _›, 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 "∃ a ∈ carrier (mult_of R) . carrier (mult_of R) = {a[^]i | i::nat . i ∈ 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
"([^]⇘mult_of R⇙) = (([^]) :: _ ⇒ nat ⇒ _)" and "𝟭⇘mult_of R⇙ = 𝟭"
by (rule field_mult_group) (simp_all add: fun_eq_iff nat_pow_def)
let ?N = "λ x . card {a ∈ 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 "{𝟬, 𝟭}"] 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 "(∑d | d dvd order (mult_of R). ?N d)
= card (UN d:{d . d dvd order (mult_of R) }. {a ∈ 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 ∈ carrier (mult_of R)"
hence x': "x∈carrier (mult_of R)" by simp
then have "group.ord (mult_of R) x dvd order (mult_of R)"
using G.ord_dvd_group_order by blast
hence "x ∈ ?U" using dvd_nat_bounds[of "order (mult_of R)" "group.ord (mult_of R) x"] x by blast
} thus "carrier (mult_of R) ⊆ ?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: "(∑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 ∈ carrier (mult_of R). group.ord (mult_of R) a = d} ≤ phi' d"
proof cases
assume "card {a ∈ carrier (mult_of R). group.ord (mult_of R) a = d} = 0" thus ?thesis by presburger
next
assume "card {a ∈ carrier (mult_of R). group.ord (mult_of R) a = d} ≠ 0"
hence "∃a ∈ 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: "⋀i. i ∈ {d. d dvd order (mult_of R) }
⟹ (λi. card {a ∈ carrier (mult_of R). group.ord (mult_of R) a = i}) i ≤ (λi. phi' i) i" by fast
hence le: "(∑i | i dvd order (mult_of R). ?N i)
≤ (∑i | i dvd order (mult_of R). phi' i)"
using sum_mono[of "{d . d dvd order (mult_of R)}"
"λi. card {a ∈ carrier (mult_of R). group.ord (mult_of R) a = i}"] by presburger
have "order (mult_of R) = (∑d | d dvd order (mult_of R). phi' d)" using *
by (simp add: sum_phi'_factors)
hence eq: "(∑i | i dvd order (mult_of R). ?N i)
= (∑i | i dvd order (mult_of R). phi' i)" using le sum_Ns_eq by presburger
have "⋀i. i ∈ {d. d dvd order (mult_of R) } ⟹ ?N i = (λi. phi' i) i"
proof (rule ccontr)
fix i
assume i1: "i ∈ {d. d dvd order (mult_of R)}" and "?N i ≠ 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 "(∑i | i dvd order (mult_of R). ?N i)
< (∑i | i dvd order (mult_of R). phi' i)"
using sum_strict_mono_ex1[OF fin, of "?N" "λ 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 ∈ 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 ∈ UNIV} = (λx. a[^]x) ` {0 .. group.ord (mult_of R) a - 1}"
using G.ord_elems[OF finite'] by auto
have card_eq: "card ((λx. 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 ((λ 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 ∈ UNIV} = order (mult_of R)"
using * by (subst set_eq) auto
have **: "{a[^]i | i::nat. i ∈ UNIV} ⊆ carrier (mult_of R)"
using G.nat_pow_closed[OF a] by auto
with _ have "carrier (mult_of R) = {a[^]i|i::nat. i ∈ 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