src/HOL/Number_Theory/Residues.thy
author paulson <lp15@cam.ac.uk>
Tue, 28 Feb 2017 15:17:57 +0000
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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 Cong MiscAlgebra
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begin
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definition QuadRes :: "int \<Rightarrow> int \<Rightarrow> bool" where
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  "QuadRes p a = (\<exists>y. ([y^2 = a] (mod p)))"
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definition Legendre :: "int \<Rightarrow> int \<Rightarrow> int" where
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  "Legendre a p = (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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     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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  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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  using m_gt_one
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  unfolding 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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  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 (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 \<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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  unfolding R_def residue_ring_def
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  by (auto simp add: mod_simps)
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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 simp add: mod_simps)
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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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  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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diff changeset
   154
  apply (metis mult.commute mult_cong)
41541
1fa4725c4656 eliminated global prems;
wenzelm
parents: 36350
diff changeset
   155
  done
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   156
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   157
lemma neg_cong: "\<ominus> (x mod m) = (- x) mod m"
55352
paulson <lp15@cam.ac.uk>
parents: 55262
diff changeset
   158
  by (metis mod_minus_eq res_neg_eq)
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   159
60528
wenzelm
parents: 60527
diff changeset
   160
lemma (in residues) prod_cong: "finite A \<Longrightarrow> (\<Otimes>i\<in>A. (f i) mod m) = (\<Prod>i\<in>A. f i) mod m"
55352
paulson <lp15@cam.ac.uk>
parents: 55262
diff changeset
   161
  by (induct set: finite) (auto simp: one_cong mult_cong)
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   162
60528
wenzelm
parents: 60527
diff changeset
   163
lemma (in residues) sum_cong: "finite A \<Longrightarrow> (\<Oplus>i\<in>A. (f i) mod m) = (\<Sum>i\<in>A. f i) mod m"
55352
paulson <lp15@cam.ac.uk>
parents: 55262
diff changeset
   164
  by (induct set: finite) (auto simp: zero_cong add_cong)
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   165
60688
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60528
diff changeset
   166
lemma mod_in_res_units [simp]:
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60528
diff changeset
   167
  assumes "1 < m" and "coprime a m"
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60528
diff changeset
   168
  shows "a mod m \<in> Units R"
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60528
diff changeset
   169
proof (cases "a mod m = 0")
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60528
diff changeset
   170
  case True with assms show ?thesis
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60528
diff changeset
   171
    by (auto simp add: res_units_eq gcd_red_int [symmetric])
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60528
diff changeset
   172
next
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60528
diff changeset
   173
  case False
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60528
diff changeset
   174
  from assms have "0 < m" by simp
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60528
diff changeset
   175
  with pos_mod_sign [of m a] have "0 \<le> a mod m" .
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60528
diff changeset
   176
  with False have "0 < a mod m" by simp
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60528
diff changeset
   177
  with assms show ?thesis
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60528
diff changeset
   178
    by (auto simp add: res_units_eq gcd_red_int [symmetric] ac_simps)
01488b559910 avoid explicit definition of the relation of associated elements in a ring -- prefer explicit normalization instead
haftmann
parents: 60528
diff changeset
   179
qed
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   180
60528
wenzelm
parents: 60527
diff changeset
   181
lemma res_eq_to_cong: "(a mod m) = (b mod m) \<longleftrightarrow> [a = b] (mod m)"
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   182
  unfolding cong_int_def by auto
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   183
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   184
60528
wenzelm
parents: 60527
diff changeset
   185
text \<open>Simplifying with these will translate a ring equation in R to a congruence.\<close>
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   186
lemmas res_to_cong_simps = add_cong mult_cong pow_cong one_cong
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   187
    prod_cong sum_cong neg_cong res_eq_to_cong
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   188
60527
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   189
text \<open>Other useful facts about the residue ring.\<close>
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   190
lemma one_eq_neg_one: "\<one> = \<ominus> \<one> \<Longrightarrow> m = 2"
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   191
  apply (simp add: res_one_eq res_neg_eq)
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 55352
diff changeset
   192
  apply (metis add.commute add_diff_cancel mod_mod_trivial one_add_one uminus_add_conv_diff
60528
wenzelm
parents: 60527
diff changeset
   193
    zero_neq_one zmod_zminus1_eq_if)
41541
1fa4725c4656 eliminated global prems;
wenzelm
parents: 36350
diff changeset
   194
  done
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   195
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   196
end
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   197
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   198
60527
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   199
subsection \<open>Prime residues\<close>
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   200
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   201
locale residues_prime =
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63417
diff changeset
   202
  fixes p :: nat and R (structure)
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   203
  assumes p_prime [intro]: "prime p"
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63417
diff changeset
   204
  defines "R \<equiv> residue_ring (int p)"
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   205
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   206
sublocale residues_prime < residues p
65066
c64d778a593a tidied some messy proofs
paulson <lp15@cam.ac.uk>
parents: 64593
diff changeset
   207
  unfolding R_def residues_def
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   208
  using p_prime apply auto
62348
9a5f43dac883 dropped various legacy fact bindings
haftmann
parents: 60688
diff changeset
   209
  apply (metis (full_types) of_nat_1 of_nat_less_iff prime_gt_1_nat)
41541
1fa4725c4656 eliminated global prems;
wenzelm
parents: 36350
diff changeset
   210
  done
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   211
44872
a98ef45122f3 misc tuning;
wenzelm
parents: 41959
diff changeset
   212
context residues_prime
a98ef45122f3 misc tuning;
wenzelm
parents: 41959
diff changeset
   213
begin
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   214
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   215
lemma is_field: "field R"
65066
c64d778a593a tidied some messy proofs
paulson <lp15@cam.ac.uk>
parents: 64593
diff changeset
   216
proof -
c64d778a593a tidied some messy proofs
paulson <lp15@cam.ac.uk>
parents: 64593
diff changeset
   217
  have "\<And>x. \<lbrakk>gcd x (int p) \<noteq> 1; 0 \<le> x; x < int p\<rbrakk> \<Longrightarrow> x = 0"
c64d778a593a tidied some messy proofs
paulson <lp15@cam.ac.uk>
parents: 64593
diff changeset
   218
    by (metis dual_order.order_iff_strict gcd.commute less_le_not_le p_prime prime_imp_coprime prime_nat_int_transfer zdvd_imp_le)
c64d778a593a tidied some messy proofs
paulson <lp15@cam.ac.uk>
parents: 64593
diff changeset
   219
  then show ?thesis
c64d778a593a tidied some messy proofs
paulson <lp15@cam.ac.uk>
parents: 64593
diff changeset
   220
  apply (intro cring.field_intro2 cring)
44872
a98ef45122f3 misc tuning;
wenzelm
parents: 41959
diff changeset
   221
  apply (auto simp add: res_carrier_eq res_one_eq res_zero_eq res_units_eq)
65066
c64d778a593a tidied some messy proofs
paulson <lp15@cam.ac.uk>
parents: 64593
diff changeset
   222
    done
c64d778a593a tidied some messy proofs
paulson <lp15@cam.ac.uk>
parents: 64593
diff changeset
   223
qed
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   224
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   225
lemma res_prime_units_eq: "Units R = {1..p - 1}"
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   226
  apply (subst res_units_eq)
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   227
  apply auto
62348
9a5f43dac883 dropped various legacy fact bindings
haftmann
parents: 60688
diff changeset
   228
  apply (subst gcd.commute)
55352
paulson <lp15@cam.ac.uk>
parents: 55262
diff changeset
   229
  apply (auto simp add: p_prime prime_imp_coprime_int zdvd_not_zless)
41541
1fa4725c4656 eliminated global prems;
wenzelm
parents: 36350
diff changeset
   230
  done
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   231
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   232
end
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   233
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   234
sublocale residues_prime < field
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   235
  by (rule is_field)
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   236
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   237
60527
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   238
section \<open>Test cases: Euler's theorem and Wilson's theorem\<close>
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   239
60527
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   240
subsection \<open>Euler's theorem\<close>
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   241
65066
c64d778a593a tidied some messy proofs
paulson <lp15@cam.ac.uk>
parents: 64593
diff changeset
   242
text \<open>The definition of the totient function.\<close>
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   243
60527
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   244
definition phi :: "int \<Rightarrow> nat"
65066
c64d778a593a tidied some messy proofs
paulson <lp15@cam.ac.uk>
parents: 64593
diff changeset
   245
  where "phi m = card {x. 0 < x \<and> x < m \<and> coprime x m}"
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   246
65066
c64d778a593a tidied some messy proofs
paulson <lp15@cam.ac.uk>
parents: 64593
diff changeset
   247
lemma phi_def_nat: "phi m = card {x. 0 < x \<and> x < nat m \<and> coprime x (nat m)}"
c64d778a593a tidied some messy proofs
paulson <lp15@cam.ac.uk>
parents: 64593
diff changeset
   248
  unfolding phi_def
c64d778a593a tidied some messy proofs
paulson <lp15@cam.ac.uk>
parents: 64593
diff changeset
   249
proof (rule bij_betw_same_card [of nat])
c64d778a593a tidied some messy proofs
paulson <lp15@cam.ac.uk>
parents: 64593
diff changeset
   250
  show "bij_betw nat {x. 0 < x \<and> x < m \<and> coprime x m} {x. 0 < x \<and> x < nat m \<and> coprime x (nat m)}"
c64d778a593a tidied some messy proofs
paulson <lp15@cam.ac.uk>
parents: 64593
diff changeset
   251
    apply (auto simp add: inj_on_def bij_betw_def image_def)
c64d778a593a tidied some messy proofs
paulson <lp15@cam.ac.uk>
parents: 64593
diff changeset
   252
     apply (metis dual_order.irrefl dual_order.strict_trans leI nat_1 transfer_nat_int_gcd(1))
c64d778a593a tidied some messy proofs
paulson <lp15@cam.ac.uk>
parents: 64593
diff changeset
   253
    apply (metis One_nat_def of_nat_0 of_nat_1 of_nat_less_0_iff int_nat_eq nat_int
c64d778a593a tidied some messy proofs
paulson <lp15@cam.ac.uk>
parents: 64593
diff changeset
   254
        transfer_int_nat_gcd(1) of_nat_less_iff)
c64d778a593a tidied some messy proofs
paulson <lp15@cam.ac.uk>
parents: 64593
diff changeset
   255
    done
c64d778a593a tidied some messy proofs
paulson <lp15@cam.ac.uk>
parents: 64593
diff changeset
   256
qed
c64d778a593a tidied some messy proofs
paulson <lp15@cam.ac.uk>
parents: 64593
diff changeset
   257
  
55261
ad3604df6bc6 new lemmas involving phi from Lehmer AFP entry
paulson <lp15@cam.ac.uk>
parents: 55242
diff changeset
   258
lemma prime_phi:
60527
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   259
  assumes "2 \<le> p" "phi p = p - 1"
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   260
  shows "prime p"
55261
ad3604df6bc6 new lemmas involving phi from Lehmer AFP entry
paulson <lp15@cam.ac.uk>
parents: 55242
diff changeset
   261
proof -
60528
wenzelm
parents: 60527
diff changeset
   262
  have *: "{x. 0 < x \<and> x < p \<and> coprime x p} = {1..p - 1}"
55261
ad3604df6bc6 new lemmas involving phi from Lehmer AFP entry
paulson <lp15@cam.ac.uk>
parents: 55242
diff changeset
   263
    using assms unfolding phi_def_nat
ad3604df6bc6 new lemmas involving phi from Lehmer AFP entry
paulson <lp15@cam.ac.uk>
parents: 55242
diff changeset
   264
    by (intro card_seteq) fastforce+
60528
wenzelm
parents: 60527
diff changeset
   265
  have False if **: "1 < x" "x < p" and "x dvd p" for x :: nat
60527
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   266
  proof -
60528
wenzelm
parents: 60527
diff changeset
   267
    from * have cop: "x \<in> {1..p - 1} \<Longrightarrow> coprime x p"
wenzelm
parents: 60527
diff changeset
   268
      by blast
59667
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   269
    have "coprime x p"
55261
ad3604df6bc6 new lemmas involving phi from Lehmer AFP entry
paulson <lp15@cam.ac.uk>
parents: 55242
diff changeset
   270
      apply (rule cop)
60528
wenzelm
parents: 60527
diff changeset
   271
      using ** apply auto
55261
ad3604df6bc6 new lemmas involving phi from Lehmer AFP entry
paulson <lp15@cam.ac.uk>
parents: 55242
diff changeset
   272
      done
60527
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   273
    with \<open>x dvd p\<close> \<open>1 < x\<close> show ?thesis
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   274
      by auto
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   275
  qed
59667
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   276
  then show ?thesis
60526
fad653acf58f isabelle update_cartouches;
wenzelm
parents: 59730
diff changeset
   277
    using \<open>2 \<le> p\<close>
63633
2accfb71e33b is_prime -> prime
eberlm <eberlm@in.tum.de>
parents: 63537
diff changeset
   278
    by (simp add: prime_nat_iff)
59667
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   279
       (metis One_nat_def dvd_pos_nat nat_dvd_not_less nat_neq_iff not_gr0
55352
paulson <lp15@cam.ac.uk>
parents: 55262
diff changeset
   280
              not_numeral_le_zero one_dvd)
55261
ad3604df6bc6 new lemmas involving phi from Lehmer AFP entry
paulson <lp15@cam.ac.uk>
parents: 55242
diff changeset
   281
qed
ad3604df6bc6 new lemmas involving phi from Lehmer AFP entry
paulson <lp15@cam.ac.uk>
parents: 55242
diff changeset
   282
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   283
lemma phi_zero [simp]: "phi 0 = 0"
65066
c64d778a593a tidied some messy proofs
paulson <lp15@cam.ac.uk>
parents: 64593
diff changeset
   284
  unfolding phi_def by (auto simp add: card_eq_0_iff)
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   285
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   286
lemma phi_one [simp]: "phi 1 = 0"
44872
a98ef45122f3 misc tuning;
wenzelm
parents: 41959
diff changeset
   287
  by (auto simp add: phi_def card_eq_0_iff)
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   288
60527
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   289
lemma (in residues) phi_eq: "phi m = card (Units R)"
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   290
  by (simp add: phi_def res_units_eq)
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   291
44872
a98ef45122f3 misc tuning;
wenzelm
parents: 41959
diff changeset
   292
lemma (in residues) euler_theorem1:
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   293
  assumes a: "gcd a m = 1"
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   294
  shows "[a^phi m = 1] (mod m)"
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   295
proof -
65066
c64d778a593a tidied some messy proofs
paulson <lp15@cam.ac.uk>
parents: 64593
diff changeset
   296
  have "a ^ phi m mod m = 1 mod m"
c64d778a593a tidied some messy proofs
paulson <lp15@cam.ac.uk>
parents: 64593
diff changeset
   297
    by (metis assms finite_Units m_gt_one mod_in_res_units one_cong phi_eq pow_cong units_power_order_eq_one)
c64d778a593a tidied some messy proofs
paulson <lp15@cam.ac.uk>
parents: 64593
diff changeset
   298
  then show ?thesis
c64d778a593a tidied some messy proofs
paulson <lp15@cam.ac.uk>
parents: 64593
diff changeset
   299
    using res_eq_to_cong by blast
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   300
qed
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   301
63167
0909deb8059b isabelle update_cartouches -c -t;
wenzelm
parents: 62348
diff changeset
   302
text \<open>Outside the locale, we can relax the restriction \<open>m > 1\<close>.\<close>
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   303
lemma euler_theorem:
60527
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   304
  assumes "m \<ge> 0"
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   305
    and "gcd a m = 1"
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   306
  shows "[a^phi m = 1] (mod m)"
60527
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   307
proof (cases "m = 0 | m = 1")
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   308
  case True
44872
a98ef45122f3 misc tuning;
wenzelm
parents: 41959
diff changeset
   309
  then show ?thesis by auto
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   310
next
60527
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   311
  case False
41541
1fa4725c4656 eliminated global prems;
wenzelm
parents: 36350
diff changeset
   312
  with assms show ?thesis
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   313
    by (intro residues.euler_theorem1, unfold residues_def, auto)
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   314
qed
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   315
60527
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   316
lemma (in residues_prime) phi_prime: "phi p = nat p - 1"
65066
c64d778a593a tidied some messy proofs
paulson <lp15@cam.ac.uk>
parents: 64593
diff changeset
   317
  by (simp add: residues.phi_eq residues_axioms residues_prime.res_prime_units_eq residues_prime_axioms)
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   318
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63417
diff changeset
   319
lemma phi_prime: "prime (int p) \<Longrightarrow> phi p = nat p - 1"
65066
c64d778a593a tidied some messy proofs
paulson <lp15@cam.ac.uk>
parents: 64593
diff changeset
   320
  by (simp add: residues_prime.intro residues_prime.phi_prime)
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   321
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   322
lemma fermat_theorem:
60527
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   323
  fixes a :: int
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63417
diff changeset
   324
  assumes "prime (int p)"
60527
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   325
    and "\<not> p dvd a"
55242
413ec965f95d Number_Theory: prime is no longer overloaded, but only for nat. Automatic coercion to int enabled.
paulson <lp15@cam.ac.uk>
parents: 55227
diff changeset
   326
  shows "[a^(p - 1) = 1] (mod p)"
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   327
proof -
60527
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   328
  from assms have "[a ^ phi p = 1] (mod p)"
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63417
diff changeset
   329
    by (auto intro!: euler_theorem intro!: prime_imp_coprime_int[of p]
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63417
diff changeset
   330
             simp: gcd.commute[of _ "int p"])
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   331
  also have "phi p = nat p - 1"
60527
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   332
    by (rule phi_prime) (rule assms)
55242
413ec965f95d Number_Theory: prime is no longer overloaded, but only for nat. Automatic coercion to int enabled.
paulson <lp15@cam.ac.uk>
parents: 55227
diff changeset
   333
  finally show ?thesis
59667
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   334
    by (metis nat_int)
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   335
qed
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   336
55227
653de351d21c version of Fermat's Theorem for type nat
paulson <lp15@cam.ac.uk>
parents: 55172
diff changeset
   337
lemma fermat_theorem_nat:
63534
523b488b15c9 Overhaul of prime/multiplicity/prime_factors
eberlm <eberlm@in.tum.de>
parents: 63417
diff changeset
   338
  assumes "prime (int p)" and "\<not> p dvd a"
60527
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   339
  shows "[a ^ (p - 1) = 1] (mod p)"
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   340
  using fermat_theorem [of p a] assms
62348
9a5f43dac883 dropped various legacy fact bindings
haftmann
parents: 60688
diff changeset
   341
  by (metis of_nat_1 of_nat_power transfer_int_nat_cong zdvd_int)
55227
653de351d21c version of Fermat's Theorem for type nat
paulson <lp15@cam.ac.uk>
parents: 55172
diff changeset
   342
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   343
60526
fad653acf58f isabelle update_cartouches;
wenzelm
parents: 59730
diff changeset
   344
subsection \<open>Wilson's theorem\<close>
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   345
60527
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   346
lemma (in field) inv_pair_lemma: "x \<in> Units R \<Longrightarrow> y \<in> Units R \<Longrightarrow>
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   347
    {x, inv x} \<noteq> {y, inv y} \<Longrightarrow> {x, inv x} \<inter> {y, inv y} = {}"
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   348
  apply auto
55352
paulson <lp15@cam.ac.uk>
parents: 55262
diff changeset
   349
  apply (metis Units_inv_inv)+
41541
1fa4725c4656 eliminated global prems;
wenzelm
parents: 36350
diff changeset
   350
  done
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   351
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   352
lemma (in residues_prime) wilson_theorem1:
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   353
  assumes a: "p > 2"
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59667
diff changeset
   354
  shows "[fact (p - 1) = (-1::int)] (mod p)"
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   355
proof -
60527
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   356
  let ?Inverse_Pairs = "{{x, inv x}| x. x \<in> Units R - {\<one>, \<ominus> \<one>}}"
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   357
  have UR: "Units R = {\<one>, \<ominus> \<one>} \<union> \<Union>?Inverse_Pairs"
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   358
    by auto
60527
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   359
  have "(\<Otimes>i\<in>Units R. i) = (\<Otimes>i\<in>{\<one>, \<ominus> \<one>}. i) \<otimes> (\<Otimes>i\<in>\<Union>?Inverse_Pairs. i)"
31732
052399f580cf fixed proof
nipkow
parents: 31727
diff changeset
   360
    apply (subst UR)
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   361
    apply (subst finprod_Un_disjoint)
55352
paulson <lp15@cam.ac.uk>
parents: 55262
diff changeset
   362
    apply (auto intro: funcsetI)
60527
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   363
    using inv_one apply auto[1]
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   364
    using inv_eq_neg_one_eq apply auto
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   365
    done
60527
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   366
  also have "(\<Otimes>i\<in>{\<one>, \<ominus> \<one>}. i) = \<ominus> \<one>"
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   367
    apply (subst finprod_insert)
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   368
    apply auto
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   369
    apply (frule one_eq_neg_one)
60527
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   370
    using a apply force
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   371
    done
60527
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   372
  also have "(\<Otimes>i\<in>(\<Union>?Inverse_Pairs). i) = (\<Otimes>A\<in>?Inverse_Pairs. (\<Otimes>y\<in>A. y))"
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   373
    apply (subst finprod_Union_disjoint)
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   374
    apply auto
55352
paulson <lp15@cam.ac.uk>
parents: 55262
diff changeset
   375
    apply (metis Units_inv_inv)+
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   376
    done
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   377
  also have "\<dots> = \<one>"
60527
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   378
    apply (rule finprod_one)
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   379
    apply auto
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   380
    apply (subst finprod_insert)
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   381
    apply auto
55352
paulson <lp15@cam.ac.uk>
parents: 55262
diff changeset
   382
    apply (metis inv_eq_self)
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   383
    done
60527
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   384
  finally have "(\<Otimes>i\<in>Units R. i) = \<ominus> \<one>"
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   385
    by simp
60527
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   386
  also have "(\<Otimes>i\<in>Units R. i) = (\<Otimes>i\<in>Units R. i mod p)"
65066
c64d778a593a tidied some messy proofs
paulson <lp15@cam.ac.uk>
parents: 64593
diff changeset
   387
    by (rule finprod_cong') (auto simp: res_units_eq)
60527
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   388
  also have "\<dots> = (\<Prod>i\<in>Units R. i) mod p"
65066
c64d778a593a tidied some messy proofs
paulson <lp15@cam.ac.uk>
parents: 64593
diff changeset
   389
    by (rule prod_cong) auto
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   390
  also have "\<dots> = fact (p - 1) mod p"
64272
f76b6dda2e56 setprod -> prod
nipkow
parents: 63633
diff changeset
   391
    apply (simp add: fact_prod)
65066
c64d778a593a tidied some messy proofs
paulson <lp15@cam.ac.uk>
parents: 64593
diff changeset
   392
    using assms
55242
413ec965f95d Number_Theory: prime is no longer overloaded, but only for nat. Automatic coercion to int enabled.
paulson <lp15@cam.ac.uk>
parents: 55227
diff changeset
   393
    apply (subst res_prime_units_eq)
64272
f76b6dda2e56 setprod -> prod
nipkow
parents: 63633
diff changeset
   394
    apply (simp add: int_prod zmod_int prod_int_eq)
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   395
    done
60527
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   396
  finally have "fact (p - 1) mod p = \<ominus> \<one>" .
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   397
  then show ?thesis
60528
wenzelm
parents: 60527
diff changeset
   398
    by (metis of_nat_fact Divides.transfer_int_nat_functions(2)
wenzelm
parents: 60527
diff changeset
   399
      cong_int_def res_neg_eq res_one_eq)
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   400
qed
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   401
55352
paulson <lp15@cam.ac.uk>
parents: 55262
diff changeset
   402
lemma wilson_theorem:
60527
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   403
  assumes "prime p"
eb431a5651fe tuned proofs;
wenzelm
parents: 60526
diff changeset
   404
  shows "[fact (p - 1) = - 1] (mod p)"
55352
paulson <lp15@cam.ac.uk>
parents: 55262
diff changeset
   405
proof (cases "p = 2")
59667
651ea265d568 Removal of the file HOL/Number_Theory/Binomial!! And class field_char_0 now declared in Int.thy
paulson <lp15@cam.ac.uk>
parents: 58889
diff changeset
   406
  case True
55352
paulson <lp15@cam.ac.uk>
parents: 55262
diff changeset
   407
  then show ?thesis
64272
f76b6dda2e56 setprod -> prod
nipkow
parents: 63633
diff changeset
   408
    by (simp add: cong_int_def fact_prod)
55352
paulson <lp15@cam.ac.uk>
parents: 55262
diff changeset
   409
next
paulson <lp15@cam.ac.uk>
parents: 55262
diff changeset
   410
  case False
paulson <lp15@cam.ac.uk>
parents: 55262
diff changeset
   411
  then show ?thesis
paulson <lp15@cam.ac.uk>
parents: 55262
diff changeset
   412
    using assms prime_ge_2_nat
paulson <lp15@cam.ac.uk>
parents: 55262
diff changeset
   413
    by (metis residues_prime.wilson_theorem1 residues_prime.intro le_eq_less_or_eq)
paulson <lp15@cam.ac.uk>
parents: 55262
diff changeset
   414
qed
31719
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   415
29f5b20e8ee8 Added NewNumberTheory by Jeremy Avigad
nipkow
parents:
diff changeset
   416
end