(* Title: HOL/Number_Theory/Residues.thy
Author: Jeremy Avigad
An algebraic treatment of residue rings, and resulting proofs of
Euler's theorem and Wilson's theorem.
*)
section \<open>Residue rings\<close>
theory Residues
imports UniqueFactorization MiscAlgebra
begin
subsection \<open>A locale for residue rings\<close>
definition residue_ring :: "int \<Rightarrow> int ring"
where
"residue_ring m =
\<lparr>carrier = {0..m - 1},
mult = \<lambda>x y. (x * y) mod m,
one = 1,
zero = 0,
add = \<lambda>x y. (x + y) mod m\<rparr>"
locale residues =
fixes m :: int and R (structure)
assumes m_gt_one: "m > 1"
defines "R \<equiv> residue_ring m"
begin
lemma abelian_group: "abelian_group R"
apply (insert m_gt_one)
apply (rule abelian_groupI)
apply (unfold R_def residue_ring_def)
apply (auto simp add: mod_add_right_eq [symmetric] ac_simps)
apply (case_tac "x = 0")
apply force
apply (subgoal_tac "(x + (m - x)) mod m = 0")
apply (erule bexI)
apply auto
done
lemma comm_monoid: "comm_monoid R"
apply (insert m_gt_one)
apply (unfold R_def residue_ring_def)
apply (rule comm_monoidI)
apply auto
apply (subgoal_tac "x * y mod m * z mod m = z * (x * y mod m) mod m")
apply (erule ssubst)
apply (subst mod_mult_right_eq [symmetric])+
apply (simp_all only: ac_simps)
done
lemma cring: "cring R"
apply (rule cringI)
apply (rule abelian_group)
apply (rule comm_monoid)
apply (unfold R_def residue_ring_def, auto)
apply (subst mod_add_eq [symmetric])
apply (subst mult.commute)
apply (subst mod_mult_right_eq [symmetric])
apply (simp add: field_simps)
done
end
sublocale residues < cring
by (rule cring)
context residues
begin
text \<open>
These lemmas translate back and forth between internal and
external concepts.
\<close>
lemma res_carrier_eq: "carrier R = {0..m - 1}"
unfolding R_def residue_ring_def by auto
lemma res_add_eq: "x \<oplus> y = (x + y) mod m"
unfolding R_def residue_ring_def by auto
lemma res_mult_eq: "x \<otimes> y = (x * y) mod m"
unfolding R_def residue_ring_def by auto
lemma res_zero_eq: "\<zero> = 0"
unfolding R_def residue_ring_def by auto
lemma res_one_eq: "\<one> = 1"
unfolding R_def residue_ring_def units_of_def by auto
lemma res_units_eq: "Units R = {x. 0 < x \<and> x < m \<and> coprime x m}"
apply (insert m_gt_one)
apply (unfold Units_def R_def residue_ring_def)
apply auto
apply (subgoal_tac "x \<noteq> 0")
apply auto
apply (metis invertible_coprime_int)
apply (subst (asm) coprime_iff_invertible'_int)
apply (auto simp add: cong_int_def mult.commute)
done
lemma res_neg_eq: "\<ominus> x = (- x) mod m"
apply (insert m_gt_one)
apply (unfold R_def a_inv_def m_inv_def residue_ring_def)
apply auto
apply (rule the_equality)
apply auto
apply (subst mod_add_right_eq [symmetric])
apply auto
apply (subst mod_add_left_eq [symmetric])
apply auto
apply (subgoal_tac "y mod m = - x mod m")
apply simp
apply (metis minus_add_cancel mod_mult_self1 mult.commute)
done
lemma finite [iff]: "finite (carrier R)"
by (subst res_carrier_eq) auto
lemma finite_Units [iff]: "finite (Units R)"
by (subst res_units_eq) auto
text \<open>
The function \<open>a \<mapsto> a mod m\<close> maps the integers to the
residue classes. The following lemmas show that this mapping
respects addition and multiplication on the integers.
\<close>
lemma mod_in_carrier [iff]: "a mod m \<in> carrier R"
unfolding res_carrier_eq
using insert m_gt_one by auto
lemma add_cong: "(x mod m) \<oplus> (y mod m) = (x + y) mod m"
unfolding R_def residue_ring_def
apply auto
apply presburger
done
lemma mult_cong: "(x mod m) \<otimes> (y mod m) = (x * y) mod m"
unfolding R_def residue_ring_def
by auto (metis mod_mult_eq)
lemma zero_cong: "\<zero> = 0"
unfolding R_def residue_ring_def by auto
lemma one_cong: "\<one> = 1 mod m"
using m_gt_one unfolding R_def residue_ring_def by auto
(* FIXME revise algebra library to use 1? *)
lemma pow_cong: "(x mod m) (^) n = x^n mod m"
apply (insert m_gt_one)
apply (induct n)
apply (auto simp add: nat_pow_def one_cong)
apply (metis mult.commute mult_cong)
done
lemma neg_cong: "\<ominus> (x mod m) = (- x) mod m"
by (metis mod_minus_eq res_neg_eq)
lemma (in residues) prod_cong: "finite A \<Longrightarrow> (\<Otimes>i\<in>A. (f i) mod m) = (\<Prod>i\<in>A. f i) mod m"
by (induct set: finite) (auto simp: one_cong mult_cong)
lemma (in residues) sum_cong: "finite A \<Longrightarrow> (\<Oplus>i\<in>A. (f i) mod m) = (\<Sum>i\<in>A. f i) mod m"
by (induct set: finite) (auto simp: zero_cong add_cong)
lemma mod_in_res_units [simp]:
assumes "1 < m" and "coprime a m"
shows "a mod m \<in> Units R"
proof (cases "a mod m = 0")
case True with assms show ?thesis
by (auto simp add: res_units_eq gcd_red_int [symmetric])
next
case False
from assms have "0 < m" by simp
with pos_mod_sign [of m a] have "0 \<le> a mod m" .
with False have "0 < a mod m" by simp
with assms show ?thesis
by (auto simp add: res_units_eq gcd_red_int [symmetric] ac_simps)
qed
lemma res_eq_to_cong: "(a mod m) = (b mod m) \<longleftrightarrow> [a = b] (mod m)"
unfolding cong_int_def by auto
text \<open>Simplifying with these will translate a ring equation in R to a congruence.\<close>
lemmas res_to_cong_simps = add_cong mult_cong pow_cong one_cong
prod_cong sum_cong neg_cong res_eq_to_cong
text \<open>Other useful facts about the residue ring.\<close>
lemma one_eq_neg_one: "\<one> = \<ominus> \<one> \<Longrightarrow> m = 2"
apply (simp add: res_one_eq res_neg_eq)
apply (metis add.commute add_diff_cancel mod_mod_trivial one_add_one uminus_add_conv_diff
zero_neq_one zmod_zminus1_eq_if)
done
end
subsection \<open>Prime residues\<close>
locale residues_prime =
fixes p :: nat and R (structure)
assumes p_prime [intro]: "prime p"
defines "R \<equiv> residue_ring (int p)"
sublocale residues_prime < residues p
apply (unfold R_def residues_def)
using p_prime apply auto
apply (metis (full_types) of_nat_1 of_nat_less_iff prime_gt_1_nat)
done
context residues_prime
begin
lemma is_field: "field R"
apply (rule cring.field_intro2)
apply (rule cring)
apply (auto simp add: res_carrier_eq res_one_eq res_zero_eq res_units_eq)
apply (rule classical)
apply (erule notE)
apply (subst gcd.commute)
apply (rule prime_imp_coprime_int)
apply (simp add: p_prime)
apply (rule notI)
apply (frule zdvd_imp_le)
apply auto
done
lemma res_prime_units_eq: "Units R = {1..p - 1}"
apply (subst res_units_eq)
apply auto
apply (subst gcd.commute)
apply (auto simp add: p_prime prime_imp_coprime_int zdvd_not_zless)
done
end
sublocale residues_prime < field
by (rule is_field)
section \<open>Test cases: Euler's theorem and Wilson's theorem\<close>
subsection \<open>Euler's theorem\<close>
text \<open>The definition of the phi function.\<close>
definition phi :: "int \<Rightarrow> nat"
where "phi m = card {x. 0 < x \<and> x < m \<and> gcd x m = 1}"
lemma phi_def_nat: "phi m = card {x. 0 < x \<and> x < nat m \<and> gcd x (nat m) = 1}"
apply (simp add: phi_def)
apply (rule bij_betw_same_card [of nat])
apply (auto simp add: inj_on_def bij_betw_def image_def)
apply (metis dual_order.irrefl dual_order.strict_trans leI nat_1 transfer_nat_int_gcd(1))
apply (metis One_nat_def of_nat_0 of_nat_1 of_nat_less_0_iff int_nat_eq nat_int
transfer_int_nat_gcd(1) of_nat_less_iff)
done
lemma prime_phi:
assumes "2 \<le> p" "phi p = p - 1"
shows "prime p"
proof -
have *: "{x. 0 < x \<and> x < p \<and> coprime x p} = {1..p - 1}"
using assms unfolding phi_def_nat
by (intro card_seteq) fastforce+
have False if **: "1 < x" "x < p" and "x dvd p" for x :: nat
proof -
from * have cop: "x \<in> {1..p - 1} \<Longrightarrow> coprime x p"
by blast
have "coprime x p"
apply (rule cop)
using ** apply auto
done
with \<open>x dvd p\<close> \<open>1 < x\<close> show ?thesis
by auto
qed
then show ?thesis
using \<open>2 \<le> p\<close>
by (simp add: is_prime_nat_iff)
(metis One_nat_def dvd_pos_nat nat_dvd_not_less nat_neq_iff not_gr0
not_numeral_le_zero one_dvd)
qed
lemma phi_zero [simp]: "phi 0 = 0"
unfolding phi_def
(* Auto hangs here. Once again, where is the simplification rule
1 \<equiv> Suc 0 coming from? *)
apply (auto simp add: card_eq_0_iff)
(* Add card_eq_0_iff as a simp rule? delete card_empty_imp? *)
done
lemma phi_one [simp]: "phi 1 = 0"
by (auto simp add: phi_def card_eq_0_iff)
lemma (in residues) phi_eq: "phi m = card (Units R)"
by (simp add: phi_def res_units_eq)
lemma (in residues) euler_theorem1:
assumes a: "gcd a m = 1"
shows "[a^phi m = 1] (mod m)"
proof -
from a m_gt_one have [simp]: "a mod m \<in> Units R"
by (intro mod_in_res_units)
from phi_eq have "(a mod m) (^) (phi m) = (a mod m) (^) (card (Units R))"
by simp
also have "\<dots> = \<one>"
by (intro units_power_order_eq_one) auto
finally show ?thesis
by (simp add: res_to_cong_simps)
qed
(* In fact, there is a two line proof!
lemma (in residues) euler_theorem1:
assumes a: "gcd a m = 1"
shows "[a^phi m = 1] (mod m)"
proof -
have "(a mod m) (^) (phi m) = \<one>"
by (simp add: phi_eq units_power_order_eq_one a m_gt_one)
then show ?thesis
by (simp add: res_to_cong_simps)
qed
*)
text \<open>Outside the locale, we can relax the restriction \<open>m > 1\<close>.\<close>
lemma euler_theorem:
assumes "m \<ge> 0"
and "gcd a m = 1"
shows "[a^phi m = 1] (mod m)"
proof (cases "m = 0 | m = 1")
case True
then show ?thesis by auto
next
case False
with assms show ?thesis
by (intro residues.euler_theorem1, unfold residues_def, auto)
qed
lemma (in residues_prime) phi_prime: "phi p = nat p - 1"
apply (subst phi_eq)
apply (subst res_prime_units_eq)
apply auto
done
lemma phi_prime: "prime (int p) \<Longrightarrow> phi p = nat p - 1"
apply (rule residues_prime.phi_prime)
apply simp
apply (erule residues_prime.intro)
done
lemma fermat_theorem:
fixes a :: int
assumes "prime (int p)"
and "\<not> p dvd a"
shows "[a^(p - 1) = 1] (mod p)"
proof -
from assms have "[a ^ phi p = 1] (mod p)"
by (auto intro!: euler_theorem intro!: prime_imp_coprime_int[of p]
simp: gcd.commute[of _ "int p"])
also have "phi p = nat p - 1"
by (rule phi_prime) (rule assms)
finally show ?thesis
by (metis nat_int)
qed
lemma fermat_theorem_nat:
assumes "prime (int p)" and "\<not> p dvd a"
shows "[a ^ (p - 1) = 1] (mod p)"
using fermat_theorem [of p a] assms
by (metis of_nat_1 of_nat_power transfer_int_nat_cong zdvd_int)
subsection \<open>Wilson's theorem\<close>
lemma (in field) inv_pair_lemma: "x \<in> Units R \<Longrightarrow> y \<in> Units R \<Longrightarrow>
{x, inv x} \<noteq> {y, inv y} \<Longrightarrow> {x, inv x} \<inter> {y, inv y} = {}"
apply auto
apply (metis Units_inv_inv)+
done
lemma (in residues_prime) wilson_theorem1:
assumes a: "p > 2"
shows "[fact (p - 1) = (-1::int)] (mod p)"
proof -
let ?Inverse_Pairs = "{{x, inv x}| x. x \<in> Units R - {\<one>, \<ominus> \<one>}}"
have UR: "Units R = {\<one>, \<ominus> \<one>} \<union> \<Union>?Inverse_Pairs"
by auto
have "(\<Otimes>i\<in>Units R. i) = (\<Otimes>i\<in>{\<one>, \<ominus> \<one>}. i) \<otimes> (\<Otimes>i\<in>\<Union>?Inverse_Pairs. i)"
apply (subst UR)
apply (subst finprod_Un_disjoint)
apply (auto intro: funcsetI)
using inv_one apply auto[1]
using inv_eq_neg_one_eq apply auto
done
also have "(\<Otimes>i\<in>{\<one>, \<ominus> \<one>}. i) = \<ominus> \<one>"
apply (subst finprod_insert)
apply auto
apply (frule one_eq_neg_one)
using a apply force
done
also have "(\<Otimes>i\<in>(\<Union>?Inverse_Pairs). i) = (\<Otimes>A\<in>?Inverse_Pairs. (\<Otimes>y\<in>A. y))"
apply (subst finprod_Union_disjoint)
apply auto
apply (metis Units_inv_inv)+
done
also have "\<dots> = \<one>"
apply (rule finprod_one)
apply auto
apply (subst finprod_insert)
apply auto
apply (metis inv_eq_self)
done
finally have "(\<Otimes>i\<in>Units R. i) = \<ominus> \<one>"
by simp
also have "(\<Otimes>i\<in>Units R. i) = (\<Otimes>i\<in>Units R. i mod p)"
apply (rule finprod_cong')
apply auto
apply (subst (asm) res_prime_units_eq)
apply auto
done
also have "\<dots> = (\<Prod>i\<in>Units R. i) mod p"
apply (rule prod_cong)
apply auto
done
also have "\<dots> = fact (p - 1) mod p"
apply (simp add: fact_setprod)
apply (insert assms)
apply (subst res_prime_units_eq)
apply (simp add: int_setprod zmod_int setprod_int_eq)
done
finally have "fact (p - 1) mod p = \<ominus> \<one>" .
then show ?thesis
by (metis of_nat_fact Divides.transfer_int_nat_functions(2)
cong_int_def res_neg_eq res_one_eq)
qed
lemma wilson_theorem:
assumes "prime p"
shows "[fact (p - 1) = - 1] (mod p)"
proof (cases "p = 2")
case True
then show ?thesis
by (simp add: cong_int_def fact_setprod)
next
case False
then show ?thesis
using assms prime_ge_2_nat
by (metis residues_prime.wilson_theorem1 residues_prime.intro le_eq_less_or_eq)
qed
end