--- a/src/HOL/NewNumberTheory/Residues.thy Tue Sep 01 14:13:34 2009 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,466 +0,0 @@
-(* Title: HOL/Library/Residues.thy
- ID:
- 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
-
-*)
-
-constdefs
- residue_ring :: "int => int ring"
- "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_pos_pos_trivial 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 zmod_zmult1_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 zmod_zmult1_eq [symmetric])
- apply (simp add: ring_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}"
- by (unfold R_def residue_ring_def, auto)
-
-lemma res_add_eq: "x \<oplus> y = (x + y) mod m"
- by (unfold R_def residue_ring_def, auto)
-
-lemma res_mult_eq: "x \<otimes> y = (x * y) mod m"
- by (unfold R_def residue_ring_def, auto)
-
-lemma res_zero_eq: "\<zero> = 0"
- by (unfold R_def residue_ring_def, auto)
-
-lemma res_one_eq: "\<one> = 1"
- by (unfold R_def residue_ring_def units_of_def residue_ring_def, 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)
-
-lemma finite_Units [iff]: "finite(Units R)"
- by (subst res_units_eq, auto)
-
-(* 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) \<oplus> (y mod m) = (x + y) mod m"
- by (unfold R_def residue_ring_def, auto, arith)
-
-lemma mult_cong: "(x mod m) \<otimes> (y mod m) = (x * y) mod m"
- apply (unfold R_def residue_ring_def, auto)
- apply (subst zmod_zmult1_eq [symmetric])
- apply (subst mult_commute)
- apply (subst zmod_zmult1_eq [symmetric])
- apply (subst mult_commute)
- apply auto
-done
-
-lemma zero_cong: "\<zero> = 0"
- apply (unfold R_def residue_ring_def, auto)
-done
-
-lemma one_cong: "\<one> = 1 mod m"
- apply (insert m_gt_one)
- apply (unfold R_def residue_ring_def, auto)
-done
-
-(* 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 One_nat_def)
- 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 \<Longrightarrow> (\<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 \<Longrightarrow> (\<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 \<Longrightarrow> coprime a m \<Longrightarrow>
- 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> \<Longrightarrow> 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 *)
-
-constdefs
- phi :: "int => nat"
- "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"
- apply (auto simp add: phi_def card_eq_0_iff)
-done
-
-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 "\<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)
- thus ?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"
- thus ?thesis by auto
-next
- assume "~(m = 0 | m = 1)"
- with prems 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 \<Longrightarrow> 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 prems 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 prems)
- finally show ?thesis .
-qed
-
-
-subsection {* Wilson's theorem *}
-
-lemma (in field) inv_pair_lemma: "x : Units R \<Longrightarrow> y : Units R \<Longrightarrow>
- {x, inv x} ~= {y, inv y} \<Longrightarrow> {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> (\<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 "\<dots> = \<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 "\<dots> = (PROD i: Units R. i) mod p"
- apply (rule prod_cong)
- apply auto
- done
- also have "\<dots> = fact (p - 1) mod p"
- apply (subst fact_altdef_int)
- apply (insert prems, force)
- apply (subst res_prime_units_eq, rule refl)
- done
- finally have "fact (p - 1) mod p = \<ominus> \<one>".
- thus ?thesis
- by (simp add: res_to_cong_simps)
-qed
-
-lemma wilson_theorem: "prime (p::int) \<Longrightarrow> [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