src/HOL/Number_Theory/Residues.thy
author paulson <lp15@cam.ac.uk>
Thu Jun 14 15:20:10 2018 +0100 (10 months ago)
changeset 68447 0beb927eed89
parent 67341 df79ef3b3a41
child 68458 023b353911c5
permissions -rw-r--r--
Adjusting Number_Theory for new Algebra
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(*  Title:      HOL/Number_Theory/Residues.thy
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    Author:     Jeremy Avigad
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An algebraic treatment of residue rings, and resulting proofs of
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Euler's theorem and Wilson's theorem.
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*)
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section \<open>Residue rings\<close>
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theory Residues
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imports
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  Cong
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  "HOL-Algebra.Multiplicative_Group"
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  Totient
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begin
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definition QuadRes :: "int \<Rightarrow> int \<Rightarrow> bool"
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  where "QuadRes p a = (\<exists>y. ([y^2 = a] (mod p)))"
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definition Legendre :: "int \<Rightarrow> int \<Rightarrow> int"
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  where "Legendre a p =
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    (if ([a = 0] (mod p)) then 0
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     else if QuadRes p a then 1
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     else -1)"
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subsection \<open>A locale for residue rings\<close>
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definition residue_ring :: "int \<Rightarrow> int ring"
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  where
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    "residue_ring m =
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      \<lparr>carrier = {0..m - 1},
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       monoid.mult = \<lambda>x y. (x * y) mod m,
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       one = 1,
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       zero = 0,
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       add = \<lambda>x y. (x + y) mod m\<rparr>"
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locale residues =
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  fixes m :: int and R (structure)
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  assumes m_gt_one: "m > 1"
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  defines "R \<equiv> residue_ring m"
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begin
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lemma abelian_group: "abelian_group R"
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proof -
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  have "\<exists>y\<in>{0..m - 1}. (x + y) mod m = 0" if "0 \<le> x" "x < m" for x
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  proof (cases "x = 0")
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    case True
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    with m_gt_one show ?thesis by simp
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  next
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    case False
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    then have "(x + (m - x)) mod m = 0"
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      by simp
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    with m_gt_one that show ?thesis
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      by (metis False atLeastAtMost_iff diff_ge_0_iff_ge diff_left_mono int_one_le_iff_zero_less less_le)
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  qed
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  with m_gt_one show ?thesis
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    by (fastforce simp add: R_def residue_ring_def mod_add_right_eq ac_simps  intro!: abelian_groupI)
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qed
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lemma comm_monoid: "comm_monoid R"
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  unfolding R_def residue_ring_def
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  apply (rule comm_monoidI)
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    using m_gt_one  apply auto
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  apply (metis mod_mult_right_eq mult.assoc mult.commute)
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  apply (metis mult.commute)
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  done
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lemma cring: "cring R"
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  apply (intro cringI abelian_group comm_monoid)
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  unfolding R_def residue_ring_def
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  apply (auto simp add: comm_semiring_class.distrib mod_add_eq mod_mult_left_eq)
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  done
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end
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sublocale residues < cring
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  by (rule cring)
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context residues
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begin
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text \<open>
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  These lemmas translate back and forth between internal and
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  external concepts.
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\<close>
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lemma res_carrier_eq: "carrier R = {0..m - 1}"
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  by (auto simp: R_def residue_ring_def)
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lemma res_add_eq: "x \<oplus> y = (x + y) mod m"
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  by (auto simp: R_def residue_ring_def)
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lemma res_mult_eq: "x \<otimes> y = (x * y) mod m"
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  by (auto simp: R_def residue_ring_def)
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lemma res_zero_eq: "\<zero> = 0"
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  by (auto simp: R_def residue_ring_def)
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lemma res_one_eq: "\<one> = 1"
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  by (auto simp: R_def residue_ring_def units_of_def)
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lemma res_units_eq: "Units R = {x. 0 < x \<and> x < m \<and> coprime x m}"
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  using m_gt_one
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  apply (auto simp add: Units_def R_def residue_ring_def ac_simps invertible_coprime intro: ccontr)
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  apply (subst (asm) coprime_iff_invertible'_int)
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   apply (auto simp add: cong_def)
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  done
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lemma res_neg_eq: "\<ominus> x = (- x) mod m"
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  using m_gt_one unfolding R_def a_inv_def m_inv_def residue_ring_def
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  apply simp
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  apply (rule the_equality)
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   apply (simp add: mod_add_right_eq)
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   apply (simp add: add.commute mod_add_right_eq)
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  apply (metis add.right_neutral minus_add_cancel mod_add_right_eq mod_pos_pos_trivial)
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  done
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lemma finite [iff]: "finite (carrier R)"
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  by (simp add: res_carrier_eq)
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lemma finite_Units [iff]: "finite (Units R)"
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  by (simp add: finite_ring_finite_units)
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text \<open>
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  The function \<open>a \<mapsto> a mod m\<close> maps the integers to the
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  residue classes. The following lemmas show that this mapping
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  respects addition and multiplication on the integers.
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\<close>
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lemma mod_in_carrier [iff]: "a mod m \<in> carrier R"
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  unfolding res_carrier_eq
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  using insert m_gt_one by auto
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lemma add_cong: "(x mod m) \<oplus> (y mod m) = (x + y) mod m"
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  by (auto simp: R_def residue_ring_def mod_simps)
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lemma mult_cong: "(x mod m) \<otimes> (y mod m) = (x * y) mod m"
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  by (auto simp: R_def residue_ring_def mod_simps)
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lemma zero_cong: "\<zero> = 0"
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  by (auto simp: R_def residue_ring_def)
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lemma one_cong: "\<one> = 1 mod m"
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  using m_gt_one by (auto simp: R_def residue_ring_def)
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(* FIXME revise algebra library to use 1? *)
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lemma pow_cong: "(x mod m) [^] n = x^n mod m"
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  using m_gt_one
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  apply (induct n)
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  apply (auto simp add: nat_pow_def one_cong)
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  apply (metis mult.commute mult_cong)
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  done
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lemma neg_cong: "\<ominus> (x mod m) = (- x) mod m"
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  by (metis mod_minus_eq res_neg_eq)
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lemma (in residues) prod_cong: "finite A \<Longrightarrow> (\<Otimes>i\<in>A. (f i) mod m) = (\<Prod>i\<in>A. f i) mod m"
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  by (induct set: finite) (auto simp: one_cong mult_cong)
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lemma (in residues) sum_cong: "finite A \<Longrightarrow> (\<Oplus>i\<in>A. (f i) mod m) = (\<Sum>i\<in>A. f i) mod m"
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  by (induct set: finite) (auto simp: zero_cong add_cong)
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lemma mod_in_res_units [simp]:
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  assumes "1 < m" and "coprime a m"
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  shows "a mod m \<in> Units R"
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proof (cases "a mod m = 0")
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  case True
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  with assms show ?thesis
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    by (auto simp add: res_units_eq gcd_red_int [symmetric])
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next
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  case False
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  from assms have "0 < m" by simp
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  then have "0 \<le> a mod m" by (rule pos_mod_sign [of m a])
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  with False have "0 < a mod m" by simp
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  with assms show ?thesis
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    by (auto simp add: res_units_eq gcd_red_int [symmetric] ac_simps)
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qed
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lemma res_eq_to_cong: "(a mod m) = (b mod m) \<longleftrightarrow> [a = b] (mod m)"
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  by (auto simp: cong_def)
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text \<open>Simplifying with these will translate a ring equation in R to a congruence.\<close>
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lemmas res_to_cong_simps =
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  add_cong mult_cong pow_cong one_cong
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  prod_cong sum_cong neg_cong res_eq_to_cong
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text \<open>Other useful facts about the residue ring.\<close>
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lemma one_eq_neg_one: "\<one> = \<ominus> \<one> \<Longrightarrow> m = 2"
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  apply (simp add: res_one_eq res_neg_eq)
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  apply (metis add.commute add_diff_cancel mod_mod_trivial one_add_one uminus_add_conv_diff
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    zero_neq_one zmod_zminus1_eq_if)
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  done
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end
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subsection \<open>Prime residues\<close>
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locale residues_prime =
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  fixes p :: nat and R (structure)
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  assumes p_prime [intro]: "prime p"
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  defines "R \<equiv> residue_ring (int p)"
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sublocale residues_prime < residues p
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  unfolding R_def residues_def
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  using p_prime apply auto
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  apply (metis (full_types) of_nat_1 of_nat_less_iff prime_gt_1_nat)
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  done
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context residues_prime
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begin
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lemma p_coprime_left:
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  "coprime p a \<longleftrightarrow> \<not> p dvd a"
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  using p_prime by (auto intro: prime_imp_coprime dest: coprime_common_divisor)
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lemma p_coprime_right:
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  "coprime a p  \<longleftrightarrow> \<not> p dvd a"
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  using p_coprime_left [of a] by (simp add: ac_simps)
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lemma p_coprime_left_int:
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  "coprime (int p) a \<longleftrightarrow> \<not> int p dvd a"
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  using p_prime by (auto intro: prime_imp_coprime dest: coprime_common_divisor)
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lemma p_coprime_right_int:
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  "coprime a (int p) \<longleftrightarrow> \<not> int p dvd a"
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  using p_coprime_left_int [of a] by (simp add: ac_simps)
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lemma is_field: "field R"
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proof -
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  have "0 < x \<Longrightarrow> x < int p \<Longrightarrow> coprime (int p) x" for x
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    by (rule prime_imp_coprime) (auto simp add: zdvd_not_zless)
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  then show ?thesis
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    by (intro cring.field_intro2 cring)
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      (auto simp add: res_carrier_eq res_one_eq res_zero_eq res_units_eq ac_simps)
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qed
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lemma res_prime_units_eq: "Units R = {1..p - 1}"
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  apply (subst res_units_eq)
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  apply (auto simp add: p_coprime_right_int zdvd_not_zless)
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  done
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end
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sublocale residues_prime < field
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  by (rule is_field)
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section \<open>Test cases: Euler's theorem and Wilson's theorem\<close>
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subsection \<open>Euler's theorem\<close>
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lemma (in residues) totatives_eq:
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  "totatives (nat m) = nat ` Units R"
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proof -
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  from m_gt_one have "\<bar>m\<bar> > 1"
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    by simp
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  then have "totatives (nat \<bar>m\<bar>) = nat ` abs ` Units R"
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    by (auto simp add: totatives_def res_units_eq image_iff le_less)
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      (use m_gt_one zless_nat_eq_int_zless in force)
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  moreover have "\<bar>m\<bar> = m" "abs ` Units R = Units R"
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    using m_gt_one by (auto simp add: res_units_eq image_iff)
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  ultimately show ?thesis
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    by simp
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qed
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lemma (in residues) totient_eq:
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  "totient (nat m) = card (Units R)"
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proof  -
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  have *: "inj_on nat (Units R)"
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    by (rule inj_onI) (auto simp add: res_units_eq)
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  then show ?thesis
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    by (simp add: totient_def totatives_eq card_image)
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qed
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lemma (in residues_prime) totient_eq: "totient p = p - 1"
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  using totient_eq by (simp add: res_prime_units_eq)
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lemma (in residues) euler_theorem:
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  assumes "coprime a m"
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  shows "[a ^ totient (nat m) = 1] (mod m)"
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proof -
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  have "a ^ totient (nat m) mod m = 1 mod m"
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    by (metis assms finite_Units m_gt_one mod_in_res_units one_cong totient_eq pow_cong units_power_order_eq_one)
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  then show ?thesis
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    using res_eq_to_cong by blast
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qed
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lemma euler_theorem:
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  fixes a m :: nat
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  assumes "coprime a m"
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  shows "[a ^ totient m = 1] (mod m)"
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proof (cases "m = 0 \<or> m = 1")
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  case True
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  then show ?thesis by auto
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next
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  case False
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  with assms show ?thesis
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    using residues.euler_theorem [of "int m" "int a"] cong_int_iff
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    by (auto simp add: residues_def gcd_int_def) fastforce
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qed
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lemma fermat_theorem:
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  fixes p a :: nat
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  assumes "prime p" and "\<not> p dvd a"
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  shows "[a ^ (p - 1) = 1] (mod p)"
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proof -
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  from assms prime_imp_coprime [of p a] have "coprime a p"
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    by (auto simp add: ac_simps)
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  then have "[a ^ totient p = 1] (mod p)"
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     by (rule euler_theorem)
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  also have "totient p = p - 1"
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    by (rule totient_prime) (rule assms)
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  finally show ?thesis .
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qed
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subsection \<open>Wilson's theorem\<close>
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lemma (in field) inv_pair_lemma: "x \<in> Units R \<Longrightarrow> y \<in> Units R \<Longrightarrow>
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    {x, inv x} \<noteq> {y, inv y} \<Longrightarrow> {x, inv x} \<inter> {y, inv y} = {}"
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  apply auto
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  apply (metis Units_inv_inv)+
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  done
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lemma (in residues_prime) wilson_theorem1:
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  assumes a: "p > 2"
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  shows "[fact (p - 1) = (-1::int)] (mod p)"
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proof -
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  let ?Inverse_Pairs = "{{x, inv x}| x. x \<in> Units R - {\<one>, \<ominus> \<one>}}"
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  have UR: "Units R = {\<one>, \<ominus> \<one>} \<union> \<Union>?Inverse_Pairs"
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    by auto
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  have "(\<Otimes>i\<in>Units R. i) = (\<Otimes>i\<in>{\<one>, \<ominus> \<one>}. i) \<otimes> (\<Otimes>i\<in>\<Union>?Inverse_Pairs. i)"
nipkow@31732
   337
    apply (subst UR)
nipkow@31719
   338
    apply (subst finprod_Un_disjoint)
wenzelm@66305
   339
         apply (auto intro: funcsetI)
wenzelm@60527
   340
    using inv_one apply auto[1]
wenzelm@60527
   341
    using inv_eq_neg_one_eq apply auto
nipkow@31719
   342
    done
wenzelm@60527
   343
  also have "(\<Otimes>i\<in>{\<one>, \<ominus> \<one>}. i) = \<ominus> \<one>"
nipkow@31719
   344
    apply (subst finprod_insert)
wenzelm@66305
   345
        apply auto
nipkow@31719
   346
    apply (frule one_eq_neg_one)
wenzelm@60527
   347
    using a apply force
nipkow@31719
   348
    done
wenzelm@60527
   349
  also have "(\<Otimes>i\<in>(\<Union>?Inverse_Pairs). i) = (\<Otimes>A\<in>?Inverse_Pairs. (\<Otimes>y\<in>A. y))"
wenzelm@60527
   350
    apply (subst finprod_Union_disjoint)
wenzelm@66305
   351
       apply auto
wenzelm@66305
   352
     apply (metis Units_inv_inv)+
nipkow@31719
   353
    done
nipkow@31719
   354
  also have "\<dots> = \<one>"
lp15@68447
   355
    apply (rule finprod_one_eqI)
wenzelm@66305
   356
     apply auto
wenzelm@60527
   357
    apply (subst finprod_insert)
wenzelm@66305
   358
        apply auto
lp15@55352
   359
    apply (metis inv_eq_self)
nipkow@31719
   360
    done
wenzelm@60527
   361
  finally have "(\<Otimes>i\<in>Units R. i) = \<ominus> \<one>"
nipkow@31719
   362
    by simp
wenzelm@60527
   363
  also have "(\<Otimes>i\<in>Units R. i) = (\<Otimes>i\<in>Units R. i mod p)"
lp15@65066
   364
    by (rule finprod_cong') (auto simp: res_units_eq)
wenzelm@60527
   365
  also have "\<dots> = (\<Prod>i\<in>Units R. i) mod p"
lp15@65066
   366
    by (rule prod_cong) auto
nipkow@31719
   367
  also have "\<dots> = fact (p - 1) mod p"
nipkow@64272
   368
    apply (simp add: fact_prod)
lp15@65066
   369
    using assms
lp15@55242
   370
    apply (subst res_prime_units_eq)
nipkow@64272
   371
    apply (simp add: int_prod zmod_int prod_int_eq)
nipkow@31719
   372
    done
wenzelm@60527
   373
  finally have "fact (p - 1) mod p = \<ominus> \<one>" .
wenzelm@60527
   374
  then show ?thesis
haftmann@66888
   375
    by (simp add: cong_def res_neg_eq res_one_eq zmod_int)
nipkow@31719
   376
qed
nipkow@31719
   377
lp15@55352
   378
lemma wilson_theorem:
wenzelm@60527
   379
  assumes "prime p"
wenzelm@60527
   380
  shows "[fact (p - 1) = - 1] (mod p)"
lp15@55352
   381
proof (cases "p = 2")
lp15@59667
   382
  case True
lp15@55352
   383
  then show ?thesis
haftmann@66888
   384
    by (simp add: cong_def fact_prod)
lp15@55352
   385
next
lp15@55352
   386
  case False
lp15@55352
   387
  then show ?thesis
lp15@55352
   388
    using assms prime_ge_2_nat
lp15@55352
   389
    by (metis residues_prime.wilson_theorem1 residues_prime.intro le_eq_less_or_eq)
lp15@55352
   390
qed
nipkow@31719
   391
wenzelm@66304
   392
text \<open>
haftmann@65416
   393
  This result can be transferred to the multiplicative group of
wenzelm@66305
   394
  \<open>\<int>/p\<int>\<close> for \<open>p\<close> prime.\<close>
haftmann@65416
   395
haftmann@65416
   396
lemma mod_nat_int_pow_eq:
haftmann@65416
   397
  fixes n :: nat and p a :: int
wenzelm@66305
   398
  shows "a \<ge> 0 \<Longrightarrow> p \<ge> 0 \<Longrightarrow> (nat a ^ n) mod (nat p) = nat ((a ^ n) mod p)"
haftmann@65416
   399
  by (simp add: int_one_le_iff_zero_less nat_mod_distrib order_less_imp_le nat_power_eq[symmetric])
haftmann@65416
   400
haftmann@65416
   401
theorem residue_prime_mult_group_has_gen :
haftmann@65416
   402
 fixes p :: nat
haftmann@65416
   403
 assumes prime_p : "prime p"
haftmann@65416
   404
 shows "\<exists>a \<in> {1 .. p - 1}. {1 .. p - 1} = {a^i mod p|i . i \<in> UNIV}"
haftmann@65416
   405
proof -
wenzelm@66305
   406
  have "p \<ge> 2"
wenzelm@66305
   407
    using prime_gt_1_nat[OF prime_p] by simp
wenzelm@66305
   408
  interpret R: residues_prime p "residue_ring p"
wenzelm@66305
   409
    by (simp add: residues_prime_def prime_p)
wenzelm@66305
   410
  have car: "carrier (residue_ring (int p)) - {\<zero>\<^bsub>residue_ring (int p)\<^esub>} = {1 .. int p - 1}"
haftmann@65416
   411
    by (auto simp add: R.zero_cong R.res_carrier_eq)
wenzelm@66305
   412
nipkow@67341
   413
  have "x [^]\<^bsub>residue_ring (int p)\<^esub> i = x ^ i mod (int p)"
wenzelm@66305
   414
    if "x \<in> {1 .. int p - 1}" for x and i :: nat
wenzelm@66305
   415
    using that R.pow_cong[of x i] by auto
wenzelm@66305
   416
  moreover
wenzelm@66305
   417
  obtain a where a: "a \<in> {1 .. int p - 1}"
nipkow@67341
   418
    and a_gen: "{1 .. int p - 1} = {a[^]\<^bsub>residue_ring (int p)\<^esub>i|i::nat . i \<in> UNIV}"
wenzelm@66305
   419
    using field.finite_field_mult_group_has_gen[OF R.is_field]
haftmann@65416
   420
    by (auto simp add: car[symmetric] carrier_mult_of)
wenzelm@66305
   421
  moreover
wenzelm@66305
   422
  have "nat ` {1 .. int p - 1} = {1 .. p - 1}" (is "?L = ?R")
haftmann@65416
   423
  proof
wenzelm@66305
   424
    have "n \<in> ?R" if "n \<in> ?L" for n
wenzelm@66305
   425
      using that \<open>p\<ge>2\<close> by force
wenzelm@66305
   426
    then show "?L \<subseteq> ?R" by blast
wenzelm@66305
   427
    have "n \<in> ?L" if "n \<in> ?R" for n
haftmann@66837
   428
      using that \<open>p\<ge>2\<close> by (auto intro: rev_image_eqI [of "int n"])
wenzelm@66305
   429
    then show "?R \<subseteq> ?L" by blast
haftmann@65416
   430
  qed
wenzelm@66305
   431
  moreover
haftmann@65416
   432
  have "nat ` {a^i mod (int p) | i::nat. i \<in> UNIV} = {nat a^i mod p | i . i \<in> UNIV}" (is "?L = ?R")
haftmann@65416
   433
  proof
wenzelm@66305
   434
    have "x \<in> ?R" if "x \<in> ?L" for x
wenzelm@66305
   435
    proof -
wenzelm@66305
   436
      from that obtain i where i: "x = nat (a^i mod (int p))"
wenzelm@66305
   437
        by blast
wenzelm@66305
   438
      then have "x = nat a ^ i mod p"
wenzelm@66305
   439
        using mod_nat_int_pow_eq[of a "int p" i] a \<open>p\<ge>2\<close> by auto
wenzelm@66305
   440
      with i show ?thesis by blast
wenzelm@66305
   441
    qed
wenzelm@66305
   442
    then show "?L \<subseteq> ?R" by blast
wenzelm@66305
   443
    have "x \<in> ?L" if "x \<in> ?R" for x
wenzelm@66305
   444
    proof -
wenzelm@66305
   445
      from that obtain i where i: "x = nat a^i mod p"
wenzelm@66305
   446
        by blast
wenzelm@66305
   447
      with mod_nat_int_pow_eq[of a "int p" i] a \<open>p\<ge>2\<close> show ?thesis
wenzelm@66305
   448
        by auto
wenzelm@66305
   449
    qed
wenzelm@66305
   450
    then show "?R \<subseteq> ?L" by blast
haftmann@65416
   451
  qed
wenzelm@66305
   452
  ultimately have "{1 .. p - 1} = {nat a^i mod p | i. i \<in> UNIV}"
wenzelm@66305
   453
    by presburger
wenzelm@66305
   454
  moreover from a have "nat a \<in> {1 .. p - 1}" by force
haftmann@65416
   455
  ultimately show ?thesis ..
haftmann@65416
   456
qed
haftmann@65416
   457
nipkow@31719
   458
end