Theory Residues

(*  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 ‹Residue rings›

theory Residues
imports UniqueFactorization MiscAlgebra
begin

subsection ‹A locale for residue rings›

definition residue_ring :: "int ⇒ int ring"
where
  "residue_ring m =
    ⦇carrier = {0..m - 1},
     mult = λx y. (x * y) mod m,
     one = 1,
     zero = 0,
     add = λx y. (x + y) mod m⦈"

locale residues =
  fixes m :: int and R (structure)
  assumes m_gt_one: "m > 1"
  defines "R ≡ 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 ‹
  These lemmas translate back and forth between internal and
  external concepts.
›

lemma res_carrier_eq: "carrier R = {0..m - 1}"
  unfolding R_def residue_ring_def by auto

lemma res_add_eq: "x ⊕ y = (x + y) mod m"
  unfolding R_def residue_ring_def by auto

lemma res_mult_eq: "x ⊗ y = (x * y) mod m"
  unfolding R_def residue_ring_def by auto

lemma res_zero_eq: "𝟬 = 0"
  unfolding R_def residue_ring_def by auto

lemma res_one_eq: "𝟭 = 1"
  unfolding R_def residue_ring_def units_of_def by auto

lemma res_units_eq: "Units R = {x. 0 < x ∧ x < m ∧ coprime x m}"
  apply (insert m_gt_one)
  apply (unfold Units_def R_def residue_ring_def)
  apply auto
  apply (subgoal_tac "x ≠ 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: "⊖ 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 ‹
  The function @{text "a ↦ a mod m"} maps the integers to the
  residue classes. The following lemmas show that this mapping
  respects addition and multiplication on the integers.
›

lemma mod_in_carrier [iff]: "a mod m ∈ carrier R"
  unfolding res_carrier_eq
  using insert m_gt_one by auto

lemma add_cong: "(x mod m) ⊕ (y mod m) = (x + y) mod m"
  unfolding R_def residue_ring_def
  apply auto
  apply presburger
  done

lemma mult_cong: "(x mod m) ⊗ (y mod m) = (x * y) mod m"
  unfolding R_def residue_ring_def
  by auto (metis mod_mult_eq)

lemma zero_cong: "𝟬 = 0"
  unfolding R_def residue_ring_def by auto

lemma one_cong: "𝟭 = 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: "⊖ (x mod m) = (- x) mod m"
  by (metis mod_minus_eq res_neg_eq)

lemma (in residues) prod_cong: "finite A ⟹ (⨂i∈A. (f i) mod m) = (∏i∈A. f i) mod m"
  by (induct set: finite) (auto simp: one_cong mult_cong)

lemma (in residues) sum_cong: "finite A ⟹ (⨁i∈A. (f i) mod m) = (∑i∈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 ∈ 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 ≤ 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) ⟷ [a = b] (mod m)"
  unfolding cong_int_def by auto


text ‹Simplifying with these will translate a ring equation in R to a congruence.›
lemmas res_to_cong_simps = add_cong mult_cong pow_cong one_cong
    prod_cong sum_cong neg_cong res_eq_to_cong

text ‹Other useful facts about the residue ring.›
lemma one_eq_neg_one: "𝟭 = ⊖ 𝟭 ⟹ 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 ‹Prime residues›

locale residues_prime =
  fixes p and R (structure)
  assumes p_prime [intro]: "prime p"
  defines "R ≡ residue_ring 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 (rule 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 ‹Test cases: Euler's theorem and Wilson's theorem›

subsection ‹Euler's theorem›

text ‹The definition of the phi function.›

definition phi :: "int ⇒ nat"
  where "phi m = card {x. 0 < x ∧ x < m ∧ gcd x m = 1}"

lemma phi_def_nat: "phi m = card {x. 0 < x ∧ x < nat m ∧ 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 ≤ p" "phi p = p - 1"
  shows "prime p"
proof -
  have *: "{x. 0 < x ∧ x < p ∧ 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 ∈ {1..p - 1} ⟹ coprime x p"
      by blast
    have "coprime x p"
      apply (rule cop)
      using ** apply auto
      done
    with ‹x dvd p› ‹1 < x› show ?thesis
      by auto
  qed
  then show ?thesis
    using ‹2 ≤ p›
    by (simp add: prime_def)
       (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 ≡ 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 ∈ 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 "… = 𝟭"
    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) = 𝟭"
    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 ‹Outside the locale, we can relax the restriction @{text "m > 1"}.›
lemma euler_theorem:
  assumes "m ≥ 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 p ⟹ phi p = nat p - 1"
  apply (rule residues_prime.phi_prime)
  apply (erule residues_prime.intro)
  done

lemma fermat_theorem:
  fixes a :: int
  assumes "prime p"
    and "¬ 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 dest!: prime_imp_coprime_int simp add: ac_simps)
  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 p" and "¬ 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 ‹Wilson's theorem›

lemma (in field) inv_pair_lemma: "x ∈ Units R ⟹ y ∈ Units R ⟹
    {x, inv x} ≠ {y, inv y} ⟹ {x, inv x} ∩ {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 ∈ Units R - {𝟭, ⊖ 𝟭}}"
  have UR: "Units R = {𝟭, ⊖ 𝟭} ∪ ⋃?Inverse_Pairs"
    by auto
  have "(⨂i∈Units R. i) = (⨂i∈{𝟭, ⊖ 𝟭}. i) ⊗ (⨂i∈⋃?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 "(⨂i∈{𝟭, ⊖ 𝟭}. i) = ⊖ 𝟭"
    apply (subst finprod_insert)
    apply auto
    apply (frule one_eq_neg_one)
    using a apply force
    done
  also have "(⨂i∈(⋃?Inverse_Pairs). i) = (⨂A∈?Inverse_Pairs. (⨂y∈A. y))"
    apply (subst finprod_Union_disjoint)
    apply auto
    apply (metis Units_inv_inv)+
    done
  also have "… = 𝟭"
    apply (rule finprod_one)
    apply auto
    apply (subst finprod_insert)
    apply auto
    apply (metis inv_eq_self)
    done
  finally have "(⨂i∈Units R. i) = ⊖ 𝟭"
    by simp
  also have "(⨂i∈Units R. i) = (⨂i∈Units R. i mod p)"
    apply (rule finprod_cong')
    apply auto
    apply (subst (asm) res_prime_units_eq)
    apply auto
    done
  also have "… = (∏i∈Units R. i) mod p"
    apply (rule prod_cong)
    apply auto
    done
  also have "… = fact (p - 1) mod p"
    apply (subst fact_altdef_nat)
    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 = ⊖ 𝟭" .
  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_altdef_nat)
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