Theory Residues

theory Residues
imports UniqueFactorization Binomial MiscAlgebra
(*  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.
*)


header {* Residue rings *}

theory Residues
imports
UniqueFactorization
Binomial
MiscAlgebra
begin


(*

A locale for residue rings

*)


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


locale residues =
fixes m :: int and R (structure)
assumes m_gt_one: "m > 1"
defines "R == residue_ring m"

context residues
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] add_ac)
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: mult_ac)
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

(* 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: "\<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 & 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 (rule invertible_coprime_int)
apply (subgoal_tac "x ~= 0")
apply auto
apply (subst (asm) coprime_iff_invertible'_int)
apply (rule m_gt_one)
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 (subst zmod_eq_dvd_iff)
apply auto
done

lemma finite [iff]: "finite (carrier R)"
by (subst res_carrier_eq, auto)

declare [[simproc del: finite_Collect]]
lemma finite_Units [iff]: "finite (Units R)"
by (subst res_units_eq) auto
declare [[simproc add: finite_Collect]]

(* The function 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"
apply (unfold res_carrier_eq)
apply (insert m_gt_one, auto)
done

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"
apply (unfold R_def residue_ring_def, auto)
apply (subst mod_mult_right_eq [symmetric])
apply (subst mult_commute)
apply (subst mod_mult_right_eq [symmetric])
apply (subst mult_commute)
apply auto
done

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

(* 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 (subst mult_commute)
apply (rule mult_cong)
done

lemma neg_cong: "\<ominus> (x mod m) = (- x) mod m"
apply (rule sym)
apply (rule sum_zero_eq_neg)
apply auto
apply (subst add_cong)
apply (subst zero_cong)
apply auto
done

lemma (in residues) prod_cong:
"finite A ==> (\<Otimes> i:A. (f i) mod m) = (PROD i:A. f i) mod m"
apply (induct set: finite)
apply (auto simp: one_cong mult_cong)
done

lemma (in residues) sum_cong:
"finite A ==> (\<Oplus> i:A. (f i) mod m) = (SUM i: A. f i) mod m"
apply (induct set: finite)
apply (auto simp: zero_cong add_cong)
done

lemma mod_in_res_units [simp]: "1 < m ==> coprime a m ==>
a mod m : Units R"

apply (subst res_units_eq, auto)
apply (insert pos_mod_sign [of m a])
apply (subgoal_tac "a mod m ~= 0")
apply arith
apply auto
apply (subst (asm) gcd_red_int)
apply (subst gcd_commute_int, assumption)
done

lemma res_eq_to_cong: "((a mod m) = (b mod m)) = [a = b] (mod (m::int))"
unfolding cong_int_def by auto

(* 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

(* Other useful facts about the residue ring *)

lemma one_eq_neg_one: "\<one> = \<ominus> \<one> ==> m = 2"
apply (simp add: res_one_eq res_neg_eq)
apply (insert m_gt_one)
apply (subgoal_tac "~(m > 2)")
apply arith
apply (rule notI)
apply (subgoal_tac "-1 mod m = m - 1")
apply force
apply (subst mod_add_self2 [symmetric])
apply (subst mod_pos_pos_trivial)
apply auto
done

end


(* prime residues *)

locale residues_prime =
fixes p :: int 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
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_int)
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_int)
apply (rule prime_imp_coprime_int)
apply (rule p_prime)
apply (rule zdvd_not_zless)
apply auto
done

end

sublocale residues_prime < field
by (rule is_field)


(*
Test cases: Euler's theorem and Wilson's theorem.
*)



subsection{* Euler's theorem *}

(* 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_zero [simp]: "phi 0 = 0"
apply (subst 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 "… = \<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

*)


(* outside the locale, we can relax the restriction m > 1 *)

lemma euler_theorem:
assumes "m >= 0" and "gcd a m = 1"
shows "[a^phi m = 1] (mod m)"
proof (cases)
assume "m = 0 | m = 1"
then show ?thesis by auto
next
assume "~(m = 0 | m = 1)"
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:
assumes "prime p" and "~ (p dvd a)"
shows "[a^(nat p - 1) = 1] (mod p)"
proof -
from assms have "[a^phi p = 1] (mod p)"
apply (intro euler_theorem)
(* auto should get this next part. matching across
substitutions is needed. *)

apply (frule prime_gt_1_int, arith)
apply (subst gcd_commute_int, erule prime_imp_coprime_int, assumption)
done
also have "phi p = nat p - 1"
by (rule phi_prime, rule assms)
finally show ?thesis .
qed


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} Int {y, inv y} = {}"

apply auto
apply (erule notE)
apply (erule inv_eq_imp_eq)
apply auto
apply (erule notE)
apply (erule inv_eq_imp_eq)
apply auto
done

lemma (in residues_prime) wilson_theorem1:
assumes a: "p > 2"
shows "[fact (p - 1) = - 1] (mod p)"
proof -
let ?InversePairs = "{ {x, inv x} | x. x : Units R - {\<one>, \<ominus> \<one>}}"
have UR: "Units R = {\<one>, \<ominus> \<one>} Un (Union ?InversePairs)"
by auto
have "(\<Otimes>i: Units R. i) =
(\<Otimes>i: {\<one>, \<ominus> \<one>}. i) ⊗ (\<Otimes>i: Union ?InversePairs. i)"

apply (subst UR)
apply (subst finprod_Un_disjoint)
apply (auto intro:funcsetI)
apply (drule sym, subst (asm) inv_eq_one_eq)
apply auto
apply (drule sym, subst (asm) inv_eq_neg_one_eq)
apply auto
done
also have "(\<Otimes>i: {\<one>, \<ominus> \<one>}. i) = \<ominus> \<one>"
apply (subst finprod_insert)
apply auto
apply (frule one_eq_neg_one)
apply (insert a, force)
done
also have "(\<Otimes>i:(Union ?InversePairs). i) =
(\<Otimes>A: ?InversePairs. (\<Otimes>y:A. y))"

apply (subst finprod_Union_disjoint)
apply force
apply force
apply clarify
apply (rule inv_pair_lemma)
apply auto
done
also have "… = \<one>"
apply (rule finprod_one)
apply auto
apply (subst finprod_insert)
apply auto
apply (frule inv_eq_self)
apply (auto)
done
finally have "(\<Otimes>i: Units R. i) = \<ominus> \<one>"
by simp
also have "(\<Otimes>i: Units R. i) = (\<Otimes>i: Units R. i mod p)"
apply (rule finprod_cong')
apply (auto)
apply (subst (asm) res_prime_units_eq)
apply auto
done
also have "… = (PROD i: Units R. i) mod p"
apply (rule prod_cong)
apply auto
done
also have "… = fact (p - 1) mod p"
apply (subst fact_altdef_int)
apply (insert assms, force)
apply (subst res_prime_units_eq, rule refl)
done
finally have "fact (p - 1) mod p = \<ominus> \<one>".
then show ?thesis by (simp add: res_to_cong_simps)
qed

lemma wilson_theorem: "prime (p::int) ==> [fact (p - 1) = - 1] (mod p)"
apply (frule prime_gt_1_int)
apply (case_tac "p = 2")
apply (subst fact_altdef_int, simp)
apply (subst cong_int_def)
apply simp
apply (rule residues_prime.wilson_theorem1)
apply (rule residues_prime.intro)
apply auto
done

end