(* Authors: Jeremy Avigad, David Gray, and Adam Kramer
Ported by lcp but unfinished
*)
header {* Gauss' Lemma *}
theory Gauss
imports Residues
begin
lemma cong_prime_prod_zero_nat:
fixes a::nat
shows "\<lbrakk>[a * b = 0] (mod p); prime p\<rbrakk> \<Longrightarrow> [a = 0] (mod p) | [b = 0] (mod p)"
by (auto simp add: cong_altdef_nat)
lemma cong_prime_prod_zero_int:
fixes a::int
shows "\<lbrakk>[a * b = 0] (mod p); prime p\<rbrakk> \<Longrightarrow> [a = 0] (mod p) | [b = 0] (mod p)"
by (auto simp add: cong_altdef_int)
locale GAUSS =
fixes p :: "nat"
fixes a :: "int"
assumes p_prime: "prime p"
assumes p_ge_2: "2 < p"
assumes p_a_relprime: "[a \<noteq> 0](mod p)"
assumes a_nonzero: "0 < a"
begin
definition "A = {0::int <.. ((int p - 1) div 2)}"
definition "B = (\<lambda>x. x * a) ` A"
definition "C = (\<lambda>x. x mod p) ` B"
definition "D = C \<inter> {.. (int p - 1) div 2}"
definition "E = C \<inter> {(int p - 1) div 2 <..}"
definition "F = (\<lambda>x. (int p - x)) ` E"
subsection {* Basic properties of p *}
lemma odd_p: "odd p"
by (metis p_prime p_ge_2 prime_odd_nat)
lemma p_minus_one_l: "(int p - 1) div 2 < p"
proof -
have "(p - 1) div 2 \<le> (p - 1) div 1"
by (metis div_by_1 div_le_dividend)
also have "\<dots> = p - 1" by simp
finally show ?thesis using p_ge_2 by arith
qed
lemma p_eq2: "int p = (2 * ((int p - 1) div 2)) + 1"
using odd_p p_ge_2 div_mult_self1_is_id [of 2 "p - 1"]
by auto presburger
lemma p_odd_int: obtains z::int where "int p = 2*z+1" "0<z"
using odd_p p_ge_2
by (auto simp add: even_iff_mod_2_eq_zero) (metis p_eq2)
subsection {* Basic Properties of the Gauss Sets *}
lemma finite_A: "finite (A)"
by (auto simp add: A_def)
lemma finite_B: "finite (B)"
by (auto simp add: B_def finite_A)
lemma finite_C: "finite (C)"
by (auto simp add: C_def finite_B)
lemma finite_D: "finite (D)"
by (auto simp add: D_def finite_C)
lemma finite_E: "finite (E)"
by (auto simp add: E_def finite_C)
lemma finite_F: "finite (F)"
by (auto simp add: F_def finite_E)
lemma C_eq: "C = D \<union> E"
by (auto simp add: C_def D_def E_def)
lemma A_card_eq: "card A = nat ((int p - 1) div 2)"
by (auto simp add: A_def)
lemma inj_on_xa_A: "inj_on (\<lambda>x. x * a) A"
using a_nonzero by (simp add: A_def inj_on_def)
definition ResSet :: "int => int set => bool"
where "ResSet m X = (\<forall>y1 y2. (y1 \<in> X & y2 \<in> X & [y1 = y2] (mod m) --> y1 = y2))"
lemma ResSet_image:
"\<lbrakk> 0 < m; ResSet m A; \<forall>x \<in> A. \<forall>y \<in> A. ([f x = f y](mod m) --> x = y) \<rbrakk> \<Longrightarrow>
ResSet m (f ` A)"
by (auto simp add: ResSet_def)
lemma A_res: "ResSet p A"
using p_ge_2
by (auto simp add: A_def ResSet_def intro!: cong_less_imp_eq_int)
lemma B_res: "ResSet p B"
proof -
{fix x fix y
assume a: "[x * a = y * a] (mod p)"
assume b: "0 < x"
assume c: "x \<le> (int p - 1) div 2"
assume d: "0 < y"
assume e: "y \<le> (int p - 1) div 2"
from a p_a_relprime p_prime a_nonzero cong_mult_rcancel_int [of _ a x y]
have "[x = y](mod p)"
by (metis comm_monoid_mult_class.mult.left_neutral cong_dvd_modulus_int cong_mult_rcancel_int
cong_mult_self_int gcd_int.commute prime_imp_coprime_int)
with cong_less_imp_eq_int [of x y p] p_minus_one_l
order_le_less_trans [of x "(int p - 1) div 2" p]
order_le_less_trans [of y "(int p - 1) div 2" p]
have "x = y"
by (metis b c cong_less_imp_eq_int d e zero_less_imp_eq_int zero_zle_int)
} note xy = this
show ?thesis
apply (insert p_ge_2 p_a_relprime p_minus_one_l)
apply (auto simp add: B_def)
apply (rule ResSet_image)
apply (auto simp add: A_res)
apply (auto simp add: A_def xy)
done
qed
lemma SR_B_inj: "inj_on (\<lambda>x. x mod p) B"
proof -
{ fix x fix y
assume a: "x * a mod p = y * a mod p"
assume b: "0 < x"
assume c: "x \<le> (int p - 1) div 2"
assume d: "0 < y"
assume e: "y \<le> (int p - 1) div 2"
assume f: "x \<noteq> y"
from a have "[x * a = y * a](mod p)"
by (metis cong_int_def)
with p_a_relprime p_prime cong_mult_rcancel_int [of a p x y]
have "[x = y](mod p)"
by (metis cong_mult_self_int dvd_div_mult_self gcd_commute_int prime_imp_coprime_int)
with cong_less_imp_eq_int [of x y p] p_minus_one_l
order_le_less_trans [of x "(int p - 1) div 2" p]
order_le_less_trans [of y "(int p - 1) div 2" p]
have "x = y"
by (metis b c cong_less_imp_eq_int d e zero_less_imp_eq_int zero_zle_int)
then have False
by (simp add: f)}
then show ?thesis
by (auto simp add: B_def inj_on_def A_def) metis
qed
lemma inj_on_pminusx_E: "inj_on (\<lambda>x. p - x) E"
apply (auto simp add: E_def C_def B_def A_def)
apply (rule_tac g = "(op - (int p))" in inj_on_inverseI)
apply auto
done
lemma nonzero_mod_p:
fixes x::int shows "\<lbrakk>0 < x; x < int p\<rbrakk> \<Longrightarrow> [x \<noteq> 0](mod p)"
by (metis Nat_Transfer.transfer_nat_int_function_closures(9) cong_less_imp_eq_int
inf.semilattice_strict_iff_order int_less_0_conv le_numeral_extra(3) zero_less_imp_eq_int)
lemma A_ncong_p: "x \<in> A \<Longrightarrow> [x \<noteq> 0](mod p)"
by (rule nonzero_mod_p) (auto simp add: A_def)
lemma A_greater_zero: "x \<in> A \<Longrightarrow> 0 < x"
by (auto simp add: A_def)
lemma B_ncong_p: "x \<in> B \<Longrightarrow> [x \<noteq> 0](mod p)"
by (auto simp add: B_def) (metis cong_prime_prod_zero_int A_ncong_p p_a_relprime p_prime)
lemma B_greater_zero: "x \<in> B \<Longrightarrow> 0 < x"
using a_nonzero by (auto simp add: B_def A_greater_zero)
lemma C_greater_zero: "y \<in> C \<Longrightarrow> 0 < y"
proof (auto simp add: C_def)
fix x :: int
assume a1: "x \<in> B"
have f2: "\<And>x\<^sub>1. int x\<^sub>1 = 0 \<or> 0 < int x\<^sub>1" by linarith
have "x mod int p \<noteq> 0" using a1 B_ncong_p cong_int_def by simp
thus "0 < x mod int p" using a1 f2
by (metis (no_types) B_greater_zero Divides.transfer_int_nat_functions(2) zero_less_imp_eq_int)
qed
lemma F_subset: "F \<subseteq> {x. 0 < x & x \<le> ((int p - 1) div 2)}"
apply (auto simp add: F_def E_def C_def)
apply (metis p_ge_2 Divides.pos_mod_bound less_diff_eq nat_int plus_int_code(2) zless_nat_conj)
apply (auto intro: p_odd_int)
done
lemma D_subset: "D \<subseteq> {x. 0 < x & x \<le> ((p - 1) div 2)}"
by (auto simp add: D_def C_greater_zero)
lemma F_eq: "F = {x. \<exists>y \<in> A. ( x = p - ((y*a) mod p) & (int p - 1) div 2 < (y*a) mod p)}"
by (auto simp add: F_def E_def D_def C_def B_def A_def)
lemma D_eq: "D = {x. \<exists>y \<in> A. ( x = (y*a) mod p & (y*a) mod p \<le> (int p - 1) div 2)}"
by (auto simp add: D_def C_def B_def A_def)
lemma all_A_relprime: assumes "x \<in> A" shows "gcd x p = 1"
using p_prime A_ncong_p [OF assms]
by (simp add: cong_altdef_int) (metis gcd_int.commute prime_imp_coprime_int)
lemma A_prod_relprime: "gcd (setprod id A) p = 1"
by (metis id_def all_A_relprime setprod_coprime_int)
subsection {* Relationships Between Gauss Sets *}
lemma StandardRes_inj_on_ResSet: "ResSet m X \<Longrightarrow> (inj_on (\<lambda>b. b mod m) X)"
by (auto simp add: ResSet_def inj_on_def cong_int_def)
lemma B_card_eq_A: "card B = card A"
using finite_A by (simp add: finite_A B_def inj_on_xa_A card_image)
lemma B_card_eq: "card B = nat ((int p - 1) div 2)"
by (simp add: B_card_eq_A A_card_eq)
lemma F_card_eq_E: "card F = card E"
using finite_E
by (simp add: F_def inj_on_pminusx_E card_image)
lemma C_card_eq_B: "card C = card B"
proof -
have "inj_on (\<lambda>x. x mod p) B"
by (metis SR_B_inj)
then show ?thesis
by (metis C_def card_image)
qed
lemma D_E_disj: "D \<inter> E = {}"
by (auto simp add: D_def E_def)
lemma C_card_eq_D_plus_E: "card C = card D + card E"
by (auto simp add: C_eq card_Un_disjoint D_E_disj finite_D finite_E)
lemma C_prod_eq_D_times_E: "setprod id E * setprod id D = setprod id C"
by (metis C_eq D_E_disj finite_D finite_E inf_commute setprod.union_disjoint sup_commute)
lemma C_B_zcong_prod: "[setprod id C = setprod id B] (mod p)"
apply (auto simp add: C_def)
apply (insert finite_B SR_B_inj)
apply (drule setprod.reindex [of "\<lambda>x. x mod int p" B id])
apply auto
apply (rule cong_setprod_int)
apply (auto simp add: cong_int_def)
done
lemma F_Un_D_subset: "(F \<union> D) \<subseteq> A"
apply (intro Un_least subset_trans [OF F_subset] subset_trans [OF D_subset])
apply (auto simp add: A_def)
done
lemma F_D_disj: "(F \<inter> D) = {}"
proof (auto simp add: F_eq D_eq)
fix y::int and z::int
assume "p - (y*a) mod p = (z*a) mod p"
then have "[(y*a) mod p + (z*a) mod p = 0] (mod p)"
by (metis add.commute diff_eq_eq dvd_refl cong_int_def dvd_eq_mod_eq_0 mod_0)
moreover have "[y * a = (y*a) mod p] (mod p)"
by (metis cong_int_def mod_mod_trivial)
ultimately have "[a * (y + z) = 0] (mod p)"
by (metis cong_int_def mod_add_left_eq mod_add_right_eq mult.commute ring_class.ring_distribs(1))
with p_prime a_nonzero p_a_relprime
have a: "[y + z = 0] (mod p)"
by (metis cong_prime_prod_zero_int)
assume b: "y \<in> A" and c: "z \<in> A"
with A_def have "0 < y + z"
by auto
moreover from b c p_eq2 A_def have "y + z < p"
by auto
ultimately show False
by (metis a nonzero_mod_p)
qed
lemma F_Un_D_card: "card (F \<union> D) = nat ((p - 1) div 2)"
proof -
have "card (F \<union> D) = card E + card D"
by (auto simp add: finite_F finite_D F_D_disj card_Un_disjoint F_card_eq_E)
then have "card (F \<union> D) = card C"
by (simp add: C_card_eq_D_plus_E)
then show "card (F \<union> D) = nat ((p - 1) div 2)"
by (simp add: C_card_eq_B B_card_eq)
qed
lemma F_Un_D_eq_A: "F \<union> D = A"
using finite_A F_Un_D_subset A_card_eq F_Un_D_card
by (auto simp add: card_seteq)
lemma prod_D_F_eq_prod_A: "(setprod id D) * (setprod id F) = setprod id A"
by (metis F_D_disj F_Un_D_eq_A Int_commute Un_commute finite_D finite_F setprod.union_disjoint)
lemma prod_F_zcong: "[setprod id F = ((-1) ^ (card E)) * (setprod id E)] (mod p)"
proof -
have FE: "setprod id F = setprod (op - p) E"
apply (auto simp add: F_def)
apply (insert finite_E inj_on_pminusx_E)
apply (drule setprod.reindex, auto)
done
then have "\<forall>x \<in> E. [(p-x) mod p = - x](mod p)"
by (metis cong_int_def minus_mod_self1 mod_mod_trivial)
then have "[setprod ((\<lambda>x. x mod p) o (op - p)) E = setprod (uminus) E](mod p)"
using finite_E p_ge_2
cong_setprod_int [of E "(\<lambda>x. x mod p) o (op - p)" uminus p]
by auto
then have two: "[setprod id F = setprod (uminus) E](mod p)"
by (metis FE cong_cong_mod_int cong_refl_int cong_setprod_int minus_mod_self1)
have "setprod uminus E = (-1) ^ (card E) * (setprod id E)"
using finite_E by (induct set: finite) auto
with two show ?thesis
by simp
qed
subsection {* Gauss' Lemma *}
lemma aux: "setprod id A * (- 1) ^ card E * a ^ card A * (- 1) ^ card E = setprod id A * a ^ card A"
by (metis (no_types) minus_minus mult.commute mult.left_commute power_minus power_one)
theorem pre_gauss_lemma:
"[a ^ nat((int p - 1) div 2) = (-1) ^ (card E)] (mod p)"
proof -
have "[setprod id A = setprod id F * setprod id D](mod p)"
by (auto simp add: prod_D_F_eq_prod_A mult.commute cong del:setprod.cong)
then have "[setprod id A = ((-1)^(card E) * setprod id E) * setprod id D] (mod p)"
apply (rule cong_trans_int)
apply (metis cong_scalar_int prod_F_zcong)
done
then have "[setprod id A = ((-1)^(card E) * setprod id C)] (mod p)"
by (metis C_prod_eq_D_times_E mult.commute mult.left_commute)
then have "[setprod id A = ((-1)^(card E) * setprod id B)] (mod p)"
by (rule cong_trans_int) (metis C_B_zcong_prod cong_scalar2_int)
then have "[setprod id A = ((-1)^(card E) *
(setprod id ((\<lambda>x. x * a) ` A)))] (mod p)"
by (simp add: B_def)
then have "[setprod id A = ((-1)^(card E) * (setprod (\<lambda>x. x * a) A))]
(mod p)"
by (simp add: inj_on_xa_A setprod.reindex)
moreover have "setprod (\<lambda>x. x * a) A =
setprod (\<lambda>x. a) A * setprod id A"
using finite_A by (induct set: finite) auto
ultimately have "[setprod id A = ((-1)^(card E) * (setprod (\<lambda>x. a) A *
setprod id A))] (mod p)"
by simp
then have "[setprod id A = ((-1)^(card E) * a^(card A) *
setprod id A)](mod p)"
apply (rule cong_trans_int)
apply (simp add: cong_scalar2_int cong_scalar_int finite_A setprod_constant mult.assoc)
done
then have a: "[setprod id A * (-1)^(card E) =
((-1)^(card E) * a^(card A) * setprod id A * (-1)^(card E))](mod p)"
by (rule cong_scalar_int)
then have "[setprod id A * (-1)^(card E) = setprod id A *
(-1)^(card E) * a^(card A) * (-1)^(card E)](mod p)"
apply (rule cong_trans_int)
apply (simp add: a mult.commute mult.left_commute)
done
then have "[setprod id A * (-1)^(card E) = setprod id A * a^(card A)](mod p)"
apply (rule cong_trans_int)
apply (simp add: aux cong del:setprod.cong)
done
with A_prod_relprime have "[(- 1) ^ card E = a ^ card A](mod p)"
by (metis cong_mult_lcancel_int)
then show ?thesis
by (simp add: A_card_eq cong_sym_int)
qed
(*NOT WORKING. Old_Number_Theory/Euler.thy needs to be translated, but it's
quite a mess and should better be completely redone.
theorem gauss_lemma: "(Legendre a p) = (-1) ^ (card E)"
proof -
from Euler_Criterion p_prime p_ge_2 have
"[(Legendre a p) = a^(nat (((p) - 1) div 2))] (mod p)"
by auto
moreover note pre_gauss_lemma
ultimately have "[(Legendre a p) = (-1) ^ (card E)] (mod p)"
by (rule cong_trans_int)
moreover from p_a_relprime have "(Legendre a p) = 1 | (Legendre a p) = (-1)"
by (auto simp add: Legendre_def)
moreover have "(-1::int) ^ (card E) = 1 | (-1::int) ^ (card E) = -1"
by (rule neg_one_power)
ultimately show ?thesis
by (auto simp add: p_ge_2 one_not_neg_one_mod_m zcong_sym)
qed
*)
end
end