src/HOL/Number_Theory/Residues.thy
author wenzelm
Fri Jun 19 23:40:46 2015 +0200 (2015-06-19)
changeset 60527 eb431a5651fe
parent 60526 fad653acf58f
child 60528 190b4a7d8b87
permissions -rw-r--r--
tuned proofs;
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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 UniqueFactorization MiscAlgebra
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begin
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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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     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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  apply (insert m_gt_one)
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  apply (rule abelian_groupI)
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  apply (unfold R_def residue_ring_def)
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  apply (auto simp add: mod_add_right_eq [symmetric] ac_simps)
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  apply (case_tac "x = 0")
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  apply force
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  apply (subgoal_tac "(x + (m - x)) mod m = 0")
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  apply (erule bexI)
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  apply auto
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  done
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lemma comm_monoid: "comm_monoid R"
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  apply (insert m_gt_one)
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  apply (unfold R_def residue_ring_def)
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  apply (rule comm_monoidI)
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  apply auto
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  apply (subgoal_tac "x * y mod m * z mod m = z * (x * y mod m) mod m")
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  apply (erule ssubst)
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  apply (subst mod_mult_right_eq [symmetric])+
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  apply (simp_all only: ac_simps)
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  done
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lemma cring: "cring R"
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  apply (rule cringI)
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  apply (rule abelian_group)
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  apply (rule comm_monoid)
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  apply (unfold R_def residue_ring_def, auto)
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  apply (subst mod_add_eq [symmetric])
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  apply (subst mult.commute)
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  apply (subst mod_mult_right_eq [symmetric])
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  apply (simp add: field_simps)
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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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  unfolding R_def residue_ring_def by auto
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lemma res_add_eq: "x \<oplus> y = (x + y) mod m"
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  unfolding R_def residue_ring_def by auto
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lemma res_mult_eq: "x \<otimes> y = (x * y) mod m"
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  unfolding R_def residue_ring_def by auto
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lemma res_zero_eq: "\<zero> = 0"
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  unfolding R_def residue_ring_def by auto
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lemma res_one_eq: "\<one> = 1"
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  unfolding R_def residue_ring_def units_of_def by auto
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lemma res_units_eq: "Units R = {x. 0 < x \<and> x < m \<and> coprime x m}"
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  apply (insert m_gt_one)
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  apply (unfold Units_def R_def residue_ring_def)
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  apply auto
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  apply (subgoal_tac "x \<noteq> 0")
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  apply auto
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  apply (metis invertible_coprime_int)
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  apply (subst (asm) coprime_iff_invertible'_int)
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  apply (auto simp add: cong_int_def mult.commute)
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  done
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lemma res_neg_eq: "\<ominus> x = (- x) mod m"
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  apply (insert m_gt_one)
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  apply (unfold R_def a_inv_def m_inv_def residue_ring_def)
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  apply auto
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  apply (rule the_equality)
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  apply auto
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  apply (subst mod_add_right_eq [symmetric])
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  apply auto
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  apply (subst mod_add_left_eq [symmetric])
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  apply auto
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  apply (subgoal_tac "y mod m = - x mod m")
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  apply simp
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  apply (metis minus_add_cancel mod_mult_self1 mult.commute)
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  done
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lemma finite [iff]: "finite (carrier R)"
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  by (subst res_carrier_eq) auto
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lemma finite_Units [iff]: "finite (Units R)"
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  by (subst res_units_eq) auto
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text \<open>
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  The function @{text "a \<mapsto> a mod m"} 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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  unfolding R_def residue_ring_def
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  apply auto
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  apply presburger
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  done
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lemma mult_cong: "(x mod m) \<otimes> (y mod m) = (x * y) mod m"
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  unfolding R_def residue_ring_def
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  by auto (metis mod_mult_eq)
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lemma zero_cong: "\<zero> = 0"
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  unfolding R_def residue_ring_def by auto
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lemma one_cong: "\<one> = 1 mod m"
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  using m_gt_one unfolding R_def residue_ring_def by auto
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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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  apply (insert 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: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: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]: "1 < m \<Longrightarrow> coprime a m \<Longrightarrow> a mod m \<in> Units R"
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  apply (subst res_units_eq)
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  apply auto
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  apply (insert pos_mod_sign [of m a])
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  apply (subgoal_tac "a mod m \<noteq> 0")
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  apply arith
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  apply auto
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  apply (metis gcd_int.commute gcd_red_int)
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  done
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lemma res_eq_to_cong: "((a mod m) = (b mod m)) = [a = b] (mod (m::int))"
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  unfolding cong_int_def by auto
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text \<open>Simplifying with these will translate a ring equation in R to a
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   congruence.\<close>
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lemmas res_to_cong_simps = 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 and R (structure)
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  assumes p_prime [intro]: "prime p"
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  defines "R \<equiv> residue_ring p"
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sublocale residues_prime < residues p
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  apply (unfold R_def residues_def)
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  using p_prime apply auto
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  apply (metis (full_types) int_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 is_field: "field R"
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  apply (rule cring.field_intro2)
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  apply (rule cring)
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  apply (auto simp add: res_carrier_eq res_one_eq res_zero_eq res_units_eq)
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  apply (rule classical)
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  apply (erule notE)
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  apply (subst gcd_commute_int)
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  apply (rule prime_imp_coprime_int)
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  apply (rule p_prime)
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  apply (rule notI)
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  apply (frule zdvd_imp_le)
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  apply auto
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  done
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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
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  apply (subst gcd_commute_int)
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  apply (auto simp add: p_prime prime_imp_coprime_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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text \<open>The definition of the phi function.\<close>
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definition phi :: "int \<Rightarrow> nat"
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  where "phi m = card {x. 0 < x \<and> x < m \<and> gcd x m = 1}"
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lemma phi_def_nat: "phi m = card {x. 0 < x \<and> x < nat m \<and> gcd x (nat m) = 1}"
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  apply (simp add: phi_def)
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  apply (rule bij_betw_same_card [of nat])
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  apply (auto simp add: inj_on_def bij_betw_def image_def)
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  apply (metis dual_order.irrefl dual_order.strict_trans leI nat_1 transfer_nat_int_gcd(1))
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  apply (metis One_nat_def int_0 int_1 int_less_0_conv int_nat_eq nat_int
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    transfer_int_nat_gcd(1) zless_int)
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  done
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lemma prime_phi:
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  assumes "2 \<le> p" "phi p = p - 1"
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  shows "prime p"
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proof -
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  have "{x. 0 < x \<and> x < p \<and> coprime x p} = {1..p - 1}"
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    using assms unfolding phi_def_nat
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    by (intro card_seteq) fastforce+
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  then have cop: "\<And>x::nat. x \<in> {1..p - 1} \<Longrightarrow> coprime x p"
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    by blast
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  have False if *: "1 < x" "x < p" and "x dvd p" for x :: nat
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  proof -
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    have "coprime x p"
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      apply (rule cop)
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      using * apply auto
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      done
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    with \<open>x dvd p\<close> \<open>1 < x\<close> show ?thesis
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      by auto
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  qed
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  then show ?thesis
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    using \<open>2 \<le> p\<close>
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    by (simp add: prime_def)
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       (metis One_nat_def dvd_pos_nat nat_dvd_not_less nat_neq_iff not_gr0
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              not_numeral_le_zero one_dvd)
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qed
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lemma phi_zero [simp]: "phi 0 = 0"
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  unfolding phi_def
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(* Auto hangs here. Once again, where is the simplification rule
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   1 \<equiv> Suc 0 coming from? *)
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  apply (auto simp add: card_eq_0_iff)
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(* Add card_eq_0_iff as a simp rule? delete card_empty_imp? *)
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  done
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lemma phi_one [simp]: "phi 1 = 0"
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  by (auto simp add: phi_def card_eq_0_iff)
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lemma (in residues) phi_eq: "phi m = card (Units R)"
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  by (simp add: phi_def res_units_eq)
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lemma (in residues) euler_theorem1:
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  assumes a: "gcd a m = 1"
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  shows "[a^phi m = 1] (mod m)"
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proof -
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  from a m_gt_one have [simp]: "a mod m \<in> Units R"
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    by (intro mod_in_res_units)
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  from phi_eq have "(a mod m) (^) (phi m) = (a mod m) (^) (card (Units R))"
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    by simp
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  also have "\<dots> = \<one>"
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    by (intro units_power_order_eq_one) auto
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  finally show ?thesis
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    by (simp add: res_to_cong_simps)
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qed
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(* In fact, there is a two line proof!
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lemma (in residues) euler_theorem1:
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  assumes a: "gcd a m = 1"
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  shows "[a^phi m = 1] (mod m)"
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proof -
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  have "(a mod m) (^) (phi m) = \<one>"
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    by (simp add: phi_eq units_power_order_eq_one a m_gt_one)
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  then show ?thesis
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    by (simp add: res_to_cong_simps)
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qed
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*)
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(* outside the locale, we can relax the restriction m > 1 *)
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lemma euler_theorem:
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  assumes "m \<ge> 0"
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    and "gcd a m = 1"
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  shows "[a^phi m = 1] (mod m)"
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proof (cases "m = 0 | 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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    by (intro residues.euler_theorem1, unfold residues_def, auto)
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qed
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lemma (in residues_prime) phi_prime: "phi p = nat p - 1"
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  apply (subst phi_eq)
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  apply (subst res_prime_units_eq)
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  apply auto
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  done
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lemma phi_prime: "prime p \<Longrightarrow> phi p = nat p - 1"
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  apply (rule residues_prime.phi_prime)
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  apply (erule residues_prime.intro)
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  done
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lemma fermat_theorem:
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  fixes a :: int
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   355
  assumes "prime p"
wenzelm@60527
   356
    and "\<not> p dvd a"
lp15@55242
   357
  shows "[a^(p - 1) = 1] (mod p)"
nipkow@31719
   358
proof -
wenzelm@60527
   359
  from assms have "[a ^ phi p = 1] (mod p)"
nipkow@31719
   360
    apply (intro euler_theorem)
lp15@55242
   361
    apply (metis of_nat_0_le_iff)
lp15@55242
   362
    apply (metis gcd_int.commute prime_imp_coprime_int)
nipkow@31719
   363
    done
nipkow@31719
   364
  also have "phi p = nat p - 1"
wenzelm@60527
   365
    by (rule phi_prime) (rule assms)
lp15@55242
   366
  finally show ?thesis
lp15@59667
   367
    by (metis nat_int)
nipkow@31719
   368
qed
nipkow@31719
   369
lp15@55227
   370
lemma fermat_theorem_nat:
wenzelm@60527
   371
  assumes "prime p" and "\<not> p dvd a"
wenzelm@60527
   372
  shows "[a ^ (p - 1) = 1] (mod p)"
wenzelm@60527
   373
  using fermat_theorem [of p a] assms
wenzelm@60527
   374
  by (metis int_1 of_nat_power transfer_int_nat_cong zdvd_int)
lp15@55227
   375
nipkow@31719
   376
wenzelm@60526
   377
subsection \<open>Wilson's theorem\<close>
nipkow@31719
   378
wenzelm@60527
   379
lemma (in field) inv_pair_lemma: "x \<in> Units R \<Longrightarrow> y \<in> Units R \<Longrightarrow>
wenzelm@60527
   380
    {x, inv x} \<noteq> {y, inv y} \<Longrightarrow> {x, inv x} \<inter> {y, inv y} = {}"
nipkow@31719
   381
  apply auto
lp15@55352
   382
  apply (metis Units_inv_inv)+
wenzelm@41541
   383
  done
nipkow@31719
   384
nipkow@31719
   385
lemma (in residues_prime) wilson_theorem1:
nipkow@31719
   386
  assumes a: "p > 2"
lp15@59730
   387
  shows "[fact (p - 1) = (-1::int)] (mod p)"
nipkow@31719
   388
proof -
wenzelm@60527
   389
  let ?Inverse_Pairs = "{{x, inv x}| x. x \<in> Units R - {\<one>, \<ominus> \<one>}}"
wenzelm@60527
   390
  have UR: "Units R = {\<one>, \<ominus> \<one>} \<union> \<Union>?Inverse_Pairs"
nipkow@31719
   391
    by auto
wenzelm@60527
   392
  have "(\<Otimes>i\<in>Units R. i) = (\<Otimes>i\<in>{\<one>, \<ominus> \<one>}. i) \<otimes> (\<Otimes>i\<in>\<Union>?Inverse_Pairs. i)"
nipkow@31732
   393
    apply (subst UR)
nipkow@31719
   394
    apply (subst finprod_Un_disjoint)
lp15@55352
   395
    apply (auto intro: funcsetI)
wenzelm@60527
   396
    using inv_one apply auto[1]
wenzelm@60527
   397
    using inv_eq_neg_one_eq apply auto
nipkow@31719
   398
    done
wenzelm@60527
   399
  also have "(\<Otimes>i\<in>{\<one>, \<ominus> \<one>}. i) = \<ominus> \<one>"
nipkow@31719
   400
    apply (subst finprod_insert)
nipkow@31719
   401
    apply auto
nipkow@31719
   402
    apply (frule one_eq_neg_one)
wenzelm@60527
   403
    using a apply force
nipkow@31719
   404
    done
wenzelm@60527
   405
  also have "(\<Otimes>i\<in>(\<Union>?Inverse_Pairs). i) = (\<Otimes>A\<in>?Inverse_Pairs. (\<Otimes>y\<in>A. y))"
wenzelm@60527
   406
    apply (subst finprod_Union_disjoint)
wenzelm@60527
   407
    apply auto
lp15@55352
   408
    apply (metis Units_inv_inv)+
nipkow@31719
   409
    done
nipkow@31719
   410
  also have "\<dots> = \<one>"
wenzelm@60527
   411
    apply (rule finprod_one)
wenzelm@60527
   412
    apply auto
wenzelm@60527
   413
    apply (subst finprod_insert)
wenzelm@60527
   414
    apply auto
lp15@55352
   415
    apply (metis inv_eq_self)
nipkow@31719
   416
    done
wenzelm@60527
   417
  finally have "(\<Otimes>i\<in>Units R. i) = \<ominus> \<one>"
nipkow@31719
   418
    by simp
wenzelm@60527
   419
  also have "(\<Otimes>i\<in>Units R. i) = (\<Otimes>i\<in>Units R. i mod p)"
nipkow@31719
   420
    apply (rule finprod_cong')
wenzelm@60527
   421
    apply auto
nipkow@31719
   422
    apply (subst (asm) res_prime_units_eq)
nipkow@31719
   423
    apply auto
nipkow@31719
   424
    done
wenzelm@60527
   425
  also have "\<dots> = (\<Prod>i\<in>Units R. i) mod p"
nipkow@31719
   426
    apply (rule prod_cong)
nipkow@31719
   427
    apply auto
nipkow@31719
   428
    done
nipkow@31719
   429
  also have "\<dots> = fact (p - 1) mod p"
lp15@55242
   430
    apply (subst fact_altdef_nat)
lp15@55242
   431
    apply (insert assms)
lp15@55242
   432
    apply (subst res_prime_units_eq)
lp15@55242
   433
    apply (simp add: int_setprod zmod_int setprod_int_eq)
nipkow@31719
   434
    done
wenzelm@60527
   435
  finally have "fact (p - 1) mod p = \<ominus> \<one>" .
wenzelm@60527
   436
  then show ?thesis
lp15@59730
   437
    by (metis of_nat_fact Divides.transfer_int_nat_functions(2) cong_int_def res_neg_eq res_one_eq)
nipkow@31719
   438
qed
nipkow@31719
   439
lp15@55352
   440
lemma wilson_theorem:
wenzelm@60527
   441
  assumes "prime p"
wenzelm@60527
   442
  shows "[fact (p - 1) = - 1] (mod p)"
lp15@55352
   443
proof (cases "p = 2")
lp15@59667
   444
  case True
lp15@55352
   445
  then show ?thesis
lp15@55352
   446
    by (simp add: cong_int_def fact_altdef_nat)
lp15@55352
   447
next
lp15@55352
   448
  case False
lp15@55352
   449
  then show ?thesis
lp15@55352
   450
    using assms prime_ge_2_nat
lp15@55352
   451
    by (metis residues_prime.wilson_theorem1 residues_prime.intro le_eq_less_or_eq)
lp15@55352
   452
qed
nipkow@31719
   453
nipkow@31719
   454
end