author | hoelzl |
Thu, 12 Nov 2009 17:21:43 +0100 | |
changeset 33638 | 548a34929e98 |
parent 32479 | 521cc9bf2958 |
child 35416 | d8d7d1b785af |
permissions | -rw-r--r-- |
31719 | 1 |
(* Title: HOL/Library/Residues.thy |
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ID: |
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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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header {* Residue rings *} |
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theory Residues |
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imports |
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UniqueFactorization |
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Binomial |
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MiscAlgebra |
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begin |
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(* |
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A locale for residue rings |
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*) |
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constdefs |
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residue_ring :: "int => int ring" |
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"residue_ring m == (| |
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carrier = {0..m - 1}, |
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mult = (%x y. (x * y) mod m), |
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one = 1, |
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zero = 0, |
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add = (%x y. (x + y) mod m) |)" |
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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 == residue_ring m" |
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context residues 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_pos_pos_trivial mod_add_right_eq [symmetric] |
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add_ac) |
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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 zmod_zmult1_eq [symmetric])+ |
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apply (simp_all only: mult_ac) |
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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 zmod_zmult1_eq [symmetric]) |
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apply (simp add: ring_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 begin |
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(* These lemmas translate back and forth between internal and |
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external concepts *) |
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lemma res_carrier_eq: "carrier R = {0..m - 1}" |
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by (unfold R_def residue_ring_def, auto) |
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lemma res_add_eq: "x \<oplus> y = (x + y) mod m" |
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by (unfold R_def residue_ring_def, auto) |
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lemma res_mult_eq: "x \<otimes> y = (x * y) mod m" |
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by (unfold R_def residue_ring_def, auto) |
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lemma res_zero_eq: "\<zero> = 0" |
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by (unfold R_def residue_ring_def, auto) |
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lemma res_one_eq: "\<one> = 1" |
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by (unfold R_def residue_ring_def units_of_def residue_ring_def, auto) |
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lemma res_units_eq: "Units R = { x. 0 < x & x < m & 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 ~= 0") |
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apply auto |
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apply (rule invertible_coprime_int) |
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apply (subgoal_tac "x ~= 0") |
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apply auto |
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apply (subst (asm) coprime_iff_invertible'_int) |
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apply (rule m_gt_one) |
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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 (subst zmod_eq_dvd_iff) |
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apply auto |
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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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(* The function a -> 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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lemma mod_in_carrier [iff]: "a mod m : carrier R" |
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apply (unfold res_carrier_eq) |
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apply (insert m_gt_one, auto) |
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done |
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lemma add_cong: "(x mod m) \<oplus> (y mod m) = (x + y) mod m" |
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by (unfold R_def residue_ring_def, auto, arith) |
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lemma mult_cong: "(x mod m) \<otimes> (y mod m) = (x * y) mod m" |
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apply (unfold R_def residue_ring_def, auto) |
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apply (subst zmod_zmult1_eq [symmetric]) |
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apply (subst mult_commute) |
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apply (subst zmod_zmult1_eq [symmetric]) |
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apply (subst mult_commute) |
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apply auto |
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done |
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lemma zero_cong: "\<zero> = 0" |
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apply (unfold R_def residue_ring_def, auto) |
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done |
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lemma one_cong: "\<one> = 1 mod m" |
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apply (insert m_gt_one) |
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apply (unfold R_def residue_ring_def, auto) |
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done |
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(* 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 One_nat_def) |
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apply (subst mult_commute) |
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apply (rule 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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apply (rule sym) |
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apply (rule sum_zero_eq_neg) |
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apply auto |
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apply (subst add_cong) |
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apply (subst zero_cong) |
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apply auto |
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done |
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lemma (in residues) prod_cong: |
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"finite A \<Longrightarrow> (\<Otimes> i:A. (f i) mod m) = (PROD i:A. f i) mod m" |
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apply (induct set: finite) |
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apply (auto simp: one_cong mult_cong) |
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done |
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lemma (in residues) sum_cong: |
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"finite A \<Longrightarrow> (\<Oplus> i:A. (f i) mod m) = (SUM i: A. f i) mod m" |
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apply (induct set: finite) |
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apply (auto simp: zero_cong add_cong) |
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done |
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lemma mod_in_res_units [simp]: "1 < m \<Longrightarrow> coprime a m \<Longrightarrow> |
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a mod m : Units R" |
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apply (subst res_units_eq, auto) |
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apply (insert pos_mod_sign [of m a]) |
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apply (subgoal_tac "a mod m ~= 0") |
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apply arith |
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apply auto |
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apply (subst (asm) gcd_red_int) |
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apply (subst gcd_commute_int, assumption) |
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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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(* Simplifying with these will translate a ring equation in R to a |
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congruence. *) |
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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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(* Other useful facts about the residue ring *) |
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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 (insert m_gt_one) |
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apply (subgoal_tac "~(m > 2)") |
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apply arith |
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apply (rule notI) |
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apply (subgoal_tac "-1 mod m = m - 1") |
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apply force |
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apply (subst mod_add_self2 [symmetric]) |
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apply (subst mod_pos_pos_trivial) |
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apply auto |
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done |
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end |
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(* prime residues *) |
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locale residues_prime = |
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fixes p :: int and R (structure) |
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assumes p_prime [intro]: "prime p" |
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defines "R == 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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done |
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context residues_prime 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 |
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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 (rule prime_imp_coprime_int) |
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apply (rule p_prime) |
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apply (rule zdvd_not_zless) |
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apply auto |
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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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(* |
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Test cases: Euler's theorem and Wilson's theorem. |
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*) |
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subsection{* Euler's theorem *} |
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(* the definition of the phi function *) |
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constdefs |
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phi :: "int => nat" |
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"phi m == card({ x. 0 < x & x < m & gcd x m = 1})" |
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lemma phi_zero [simp]: "phi 0 = 0" |
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apply (subst phi_def) |
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(* Auto hangs here. Once again, where is the simplification rule |
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1 == 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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apply (auto simp add: phi_def card_eq_0_iff) |
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done |
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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 : 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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thus ?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 >= 0" and "gcd a m = 1" |
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shows "[a^phi m = 1] (mod m)" |
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proof (cases) |
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assume "m = 0 | m = 1" |
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thus ?thesis by auto |
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next |
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assume "~(m = 0 | m = 1)" |
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with prems 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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assumes "prime p" and "~ (p dvd a)" |
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shows "[a^(nat p - 1) = 1] (mod p)" |
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proof - |
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from prems have "[a^phi p = 1] (mod p)" |
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apply (intro euler_theorem) |
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(* auto should get this next part. matching across |
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substitutions is needed. *) |
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apply (frule prime_gt_1_int, arith) |
40501bb2d57c
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apply (subst gcd_commute_int, erule prime_imp_coprime_int, assumption) |
31719 | 372 |
done |
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also have "phi p = nat p - 1" |
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by (rule phi_prime, rule prems) |
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finally show ?thesis . |
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qed |
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378 |
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subsection {* Wilson's theorem *} |
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lemma (in field) inv_pair_lemma: "x : Units R \<Longrightarrow> y : Units R \<Longrightarrow> |
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{x, inv x} ~= {y, inv y} \<Longrightarrow> {x, inv x} Int {y, inv y} = {}" |
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apply auto |
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apply (erule notE) |
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apply (erule inv_eq_imp_eq) |
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apply auto |
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apply (erule notE) |
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apply (erule inv_eq_imp_eq) |
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apply auto |
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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] (mod p)" |
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proof - |
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let ?InversePairs = "{ {x, inv x} | x. x : Units R - {\<one>, \<ominus> \<one>}}" |
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31732 | 397 |
have UR: "Units R = {\<one>, \<ominus> \<one>} Un (Union ?InversePairs)" |
31719 | 398 |
by auto |
31732 | 399 |
have "(\<Otimes>i: Units R. i) = |
31719 | 400 |
(\<Otimes>i: {\<one>, \<ominus> \<one>}. i) \<otimes> (\<Otimes>i: Union ?InversePairs. i)" |
31732 | 401 |
apply (subst UR) |
31719 | 402 |
apply (subst finprod_Un_disjoint) |
31732 | 403 |
apply (auto intro:funcsetI) |
31719 | 404 |
apply (drule sym, subst (asm) inv_eq_one_eq) |
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apply auto |
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apply (drule sym, subst (asm) inv_eq_neg_one_eq) |
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apply auto |
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done |
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also have "(\<Otimes>i: {\<one>, \<ominus> \<one>}. i) = \<ominus> \<one>" |
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apply (subst finprod_insert) |
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apply auto |
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apply (frule one_eq_neg_one) |
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apply (insert a, force) |
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done |
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also have "(\<Otimes>i:(Union ?InversePairs). i) = |
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(\<Otimes> A: ?InversePairs. (\<Otimes> y:A. y))" |
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apply (subst finprod_Union_disjoint) |
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apply force |
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apply force |
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apply clarify |
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apply (rule inv_pair_lemma) |
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apply auto |
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done |
|
424 |
also have "\<dots> = \<one>" |
|
425 |
apply (rule finprod_one) |
|
426 |
apply auto |
|
427 |
apply (subst finprod_insert) |
|
428 |
apply auto |
|
429 |
apply (frule inv_eq_self) |
|
31732 | 430 |
apply (auto) |
31719 | 431 |
done |
432 |
finally have "(\<Otimes>i: Units R. i) = \<ominus> \<one>" |
|
433 |
by simp |
|
434 |
also have "(\<Otimes>i: Units R. i) = (\<Otimes>i: Units R. i mod p)" |
|
435 |
apply (rule finprod_cong') |
|
31732 | 436 |
apply (auto) |
31719 | 437 |
apply (subst (asm) res_prime_units_eq) |
438 |
apply auto |
|
439 |
done |
|
440 |
also have "\<dots> = (PROD i: Units R. i) mod p" |
|
441 |
apply (rule prod_cong) |
|
442 |
apply auto |
|
443 |
done |
|
444 |
also have "\<dots> = fact (p - 1) mod p" |
|
31952
40501bb2d57c
renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents:
31798
diff
changeset
|
445 |
apply (subst fact_altdef_int) |
31719 | 446 |
apply (insert prems, force) |
447 |
apply (subst res_prime_units_eq, rule refl) |
|
448 |
done |
|
449 |
finally have "fact (p - 1) mod p = \<ominus> \<one>". |
|
450 |
thus ?thesis |
|
451 |
by (simp add: res_to_cong_simps) |
|
452 |
qed |
|
453 |
||
454 |
lemma wilson_theorem: "prime (p::int) \<Longrightarrow> [fact (p - 1) = - 1] (mod p)" |
|
31952
40501bb2d57c
renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents:
31798
diff
changeset
|
455 |
apply (frule prime_gt_1_int) |
31719 | 456 |
apply (case_tac "p = 2") |
31952
40501bb2d57c
renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents:
31798
diff
changeset
|
457 |
apply (subst fact_altdef_int, simp) |
31719 | 458 |
apply (subst cong_int_def) |
459 |
apply simp |
|
460 |
apply (rule residues_prime.wilson_theorem1) |
|
461 |
apply (rule residues_prime.intro) |
|
462 |
apply auto |
|
463 |
done |
|
464 |
||
465 |
||
466 |
end |