src/HOL/NumberTheory/Gauss.thy
author haftmann
Wed Sep 26 20:27:55 2007 +0200 (2007-09-26)
changeset 24728 e2b3a1065676
parent 22274 ce1459004c8d
child 26289 9d2c375e242b
permissions -rw-r--r--
moved Finite_Set before Datatype
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(*  Title:      HOL/Quadratic_Reciprocity/Gauss.thy
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    ID:         $Id$
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    Authors:    Jeremy Avigad, David Gray, and Adam Kramer)
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*)
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header {* Gauss' Lemma *}
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theory Gauss imports Euler begin
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locale GAUSS =
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  fixes p :: "int"
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  fixes a :: "int"
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  assumes p_prime: "zprime p"
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  assumes p_g_2: "2 < p"
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  assumes p_a_relprime: "~[a = 0](mod p)"
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  assumes a_nonzero:    "0 < a"
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begin
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definition
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  A :: "int set" where
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  "A = {(x::int). 0 < x & x \<le> ((p - 1) div 2)}"
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definition
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  B :: "int set" where
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  "B = (%x. x * a) ` A"
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definition
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  C :: "int set" where
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  "C = StandardRes p ` B"
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definition
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  D :: "int set" where
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  "D = C \<inter> {x. x \<le> ((p - 1) div 2)}"
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definition
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  E :: "int set" where
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  "E = C \<inter> {x. ((p - 1) div 2) < x}"
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definition
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  F :: "int set" where
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  "F = (%x. (p - x)) ` E"
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subsection {* Basic properties of p *}
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lemma p_odd: "p \<in> zOdd"
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  by (auto simp add: p_prime p_g_2 zprime_zOdd_eq_grt_2)
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lemma p_g_0: "0 < p"
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  using p_g_2 by auto
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lemma int_nat: "int (nat ((p - 1) div 2)) = (p - 1) div 2"
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  using insert p_g_2 by (auto simp add: pos_imp_zdiv_nonneg_iff)
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lemma p_minus_one_l: "(p - 1) div 2 < p"
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proof -
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  have "(p - 1) div 2 \<le> (p - 1) div 1"
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    by (rule zdiv_mono2) (auto simp add: p_g_0)
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  also have "\<dots> = p - 1" by simp
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  finally show ?thesis by simp
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qed
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lemma p_eq: "p = (2 * (p - 1) div 2) + 1"
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  using zdiv_zmult_self2 [of 2 "p - 1"] by auto
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lemma (in -) zodd_imp_zdiv_eq: "x \<in> zOdd ==> 2 * (x - 1) div 2 = 2 * ((x - 1) div 2)"
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  apply (frule odd_minus_one_even)
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  apply (simp add: zEven_def)
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  apply (subgoal_tac "2 \<noteq> 0")
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  apply (frule_tac b = "2 :: int" and a = "x - 1" in zdiv_zmult_self2)
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  apply (auto simp add: even_div_2_prop2)
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  done
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lemma p_eq2: "p = (2 * ((p - 1) div 2)) + 1"
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  apply (insert p_eq p_prime p_g_2 zprime_zOdd_eq_grt_2 [of p], auto)
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  apply (frule zodd_imp_zdiv_eq, auto)
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  done
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subsection {* Basic Properties of the Gauss Sets *}
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lemma finite_A: "finite (A)"
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  apply (auto simp add: A_def)
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  apply (subgoal_tac "{x. 0 < x & x \<le> (p - 1) div 2} \<subseteq> {x. 0 \<le> x & x < 1 + (p - 1) div 2}")
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  apply (auto simp add: bdd_int_set_l_finite finite_subset)
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  done
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lemma finite_B: "finite (B)"
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  by (auto simp add: B_def finite_A finite_imageI)
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lemma finite_C: "finite (C)"
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  by (auto simp add: C_def finite_B finite_imageI)
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lemma finite_D: "finite (D)"
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  by (auto simp add: D_def finite_Int finite_C)
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lemma finite_E: "finite (E)"
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  by (auto simp add: E_def finite_Int finite_C)
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lemma finite_F: "finite (F)"
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  by (auto simp add: F_def finite_E finite_imageI)
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lemma C_eq: "C = D \<union> E"
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  by (auto simp add: C_def D_def E_def)
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lemma A_card_eq: "card A = nat ((p - 1) div 2)"
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  apply (auto simp add: A_def)
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  apply (insert int_nat)
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  apply (erule subst)
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  apply (auto simp add: card_bdd_int_set_l_le)
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  done
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lemma inj_on_xa_A: "inj_on (%x. x * a) A"
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  using a_nonzero by (simp add: A_def inj_on_def)
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lemma A_res: "ResSet p A"
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  apply (auto simp add: A_def ResSet_def)
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  apply (rule_tac m = p in zcong_less_eq)
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  apply (insert p_g_2, auto)
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  done
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lemma B_res: "ResSet p B"
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  apply (insert p_g_2 p_a_relprime p_minus_one_l)
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  apply (auto simp add: B_def)
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  apply (rule ResSet_image)
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  apply (auto simp add: A_res)
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  apply (auto simp add: A_def)
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proof -
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  fix x fix y
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  assume a: "[x * a = y * a] (mod p)"
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  assume b: "0 < x"
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  assume c: "x \<le> (p - 1) div 2"
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  assume d: "0 < y"
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  assume e: "y \<le> (p - 1) div 2"
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  from a p_a_relprime p_prime a_nonzero zcong_cancel [of p a x y]
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  have "[x = y](mod p)"
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    by (simp add: zprime_imp_zrelprime zcong_def p_g_0 order_le_less)
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  with zcong_less_eq [of x y p] p_minus_one_l
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      order_le_less_trans [of x "(p - 1) div 2" p]
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      order_le_less_trans [of y "(p - 1) div 2" p] show "x = y"
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    by (simp add: prems p_minus_one_l p_g_0)
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qed
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lemma SR_B_inj: "inj_on (StandardRes p) B"
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  apply (auto simp add: B_def StandardRes_def inj_on_def A_def prems)
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proof -
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  fix x fix y
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  assume a: "x * a mod p = y * a mod p"
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  assume b: "0 < x"
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  assume c: "x \<le> (p - 1) div 2"
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  assume d: "0 < y"
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  assume e: "y \<le> (p - 1) div 2"
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  assume f: "x \<noteq> y"
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  from a have "[x * a = y * a](mod p)"
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    by (simp add: zcong_zmod_eq p_g_0)
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  with p_a_relprime p_prime a_nonzero zcong_cancel [of p a x y]
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  have "[x = y](mod p)"
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    by (simp add: zprime_imp_zrelprime zcong_def p_g_0 order_le_less)
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  with zcong_less_eq [of x y p] p_minus_one_l
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    order_le_less_trans [of x "(p - 1) div 2" p]
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    order_le_less_trans [of y "(p - 1) div 2" p] have "x = y"
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    by (simp add: prems p_minus_one_l p_g_0)
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  then have False
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    by (simp add: f)
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  then show "a = 0"
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    by simp
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qed
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lemma inj_on_pminusx_E: "inj_on (%x. p - x) E"
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  apply (auto simp add: E_def C_def B_def A_def)
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  apply (rule_tac g = "%x. -1 * (x - p)" in inj_on_inverseI)
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  apply auto
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  done
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lemma A_ncong_p: "x \<in> A ==> ~[x = 0](mod p)"
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  apply (auto simp add: A_def)
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  apply (frule_tac m = p in zcong_not_zero)
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  apply (insert p_minus_one_l)
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  apply auto
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  done
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lemma A_greater_zero: "x \<in> A ==> 0 < x"
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  by (auto simp add: A_def)
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lemma B_ncong_p: "x \<in> B ==> ~[x = 0](mod p)"
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  apply (auto simp add: B_def)
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  apply (frule A_ncong_p)
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  apply (insert p_a_relprime p_prime a_nonzero)
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  apply (frule_tac a = x and b = a in zcong_zprime_prod_zero_contra)
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  apply (auto simp add: A_greater_zero)
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  done
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lemma B_greater_zero: "x \<in> B ==> 0 < x"
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  using a_nonzero by (auto simp add: B_def mult_pos_pos A_greater_zero)
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lemma C_ncong_p: "x \<in> C ==>  ~[x = 0](mod p)"
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  apply (auto simp add: C_def)
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  apply (frule B_ncong_p)
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  apply (subgoal_tac "[x = StandardRes p x](mod p)")
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  defer apply (simp add: StandardRes_prop1)
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  apply (frule_tac a = x and b = "StandardRes p x" and c = 0 in zcong_trans)
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  apply auto
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  done
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lemma C_greater_zero: "y \<in> C ==> 0 < y"
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  apply (auto simp add: C_def)
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proof -
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  fix x
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  assume a: "x \<in> B"
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  from p_g_0 have "0 \<le> StandardRes p x"
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    by (simp add: StandardRes_lbound)
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  moreover have "~[x = 0] (mod p)"
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    by (simp add: a B_ncong_p)
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  then have "StandardRes p x \<noteq> 0"
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    by (simp add: StandardRes_prop3)
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  ultimately show "0 < StandardRes p x"
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    by (simp add: order_le_less)
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qed
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lemma D_ncong_p: "x \<in> D ==> ~[x = 0](mod p)"
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  by (auto simp add: D_def C_ncong_p)
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lemma E_ncong_p: "x \<in> E ==> ~[x = 0](mod p)"
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  by (auto simp add: E_def C_ncong_p)
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lemma F_ncong_p: "x \<in> F ==> ~[x = 0](mod p)"
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  apply (auto simp add: F_def)
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proof -
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  fix x assume a: "x \<in> E" assume b: "[p - x = 0] (mod p)"
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  from E_ncong_p have "~[x = 0] (mod p)"
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    by (simp add: a)
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  moreover from a have "0 < x"
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    by (simp add: a E_def C_greater_zero)
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  moreover from a have "x < p"
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    by (auto simp add: E_def C_def p_g_0 StandardRes_ubound)
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  ultimately have "~[p - x = 0] (mod p)"
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    by (simp add: zcong_not_zero)
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  from this show False by (simp add: b)
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qed
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lemma F_subset: "F \<subseteq> {x. 0 < x & x \<le> ((p - 1) div 2)}"
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  apply (auto simp add: F_def E_def)
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  apply (insert p_g_0)
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  apply (frule_tac x = xa in StandardRes_ubound)
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  apply (frule_tac x = x in StandardRes_ubound)
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  apply (subgoal_tac "xa = StandardRes p xa")
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  apply (auto simp add: C_def StandardRes_prop2 StandardRes_prop1)
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proof -
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  from zodd_imp_zdiv_eq p_prime p_g_2 zprime_zOdd_eq_grt_2 have
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    "2 * (p - 1) div 2 = 2 * ((p - 1) div 2)"
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    by simp
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  with p_eq2 show " !!x. [| (p - 1) div 2 < StandardRes p x; x \<in> B |]
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      ==> p - StandardRes p x \<le> (p - 1) div 2"
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    by simp
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qed
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lemma D_subset: "D \<subseteq> {x. 0 < x & x \<le> ((p - 1) div 2)}"
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  by (auto simp add: D_def C_greater_zero)
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lemma F_eq: "F = {x. \<exists>y \<in> A. ( x = p - (StandardRes p (y*a)) & (p - 1) div 2 < StandardRes p (y*a))}"
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  by (auto simp add: F_def E_def D_def C_def B_def A_def)
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lemma D_eq: "D = {x. \<exists>y \<in> A. ( x = StandardRes p (y*a) & StandardRes p (y*a) \<le> (p - 1) div 2)}"
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  by (auto simp add: D_def C_def B_def A_def)
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lemma D_leq: "x \<in> D ==> x \<le> (p - 1) div 2"
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  by (auto simp add: D_eq)
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lemma F_ge: "x \<in> F ==> x \<le> (p - 1) div 2"
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  apply (auto simp add: F_eq A_def)
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proof -
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  fix y
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  assume "(p - 1) div 2 < StandardRes p (y * a)"
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  then have "p - StandardRes p (y * a) < p - ((p - 1) div 2)"
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    by arith
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  also from p_eq2 have "... = 2 * ((p - 1) div 2) + 1 - ((p - 1) div 2)"
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    by auto
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  also have "2 * ((p - 1) div 2) + 1 - (p - 1) div 2 = (p - 1) div 2 + 1"
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    by arith
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  finally show "p - StandardRes p (y * a) \<le> (p - 1) div 2"
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    using zless_add1_eq [of "p - StandardRes p (y * a)" "(p - 1) div 2"] by auto
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qed
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lemma all_A_relprime: "\<forall>x \<in> A. zgcd(x, p) = 1"
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  using p_prime p_minus_one_l by (auto simp add: A_def zless_zprime_imp_zrelprime)
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lemma A_prod_relprime: "zgcd((setprod id A),p) = 1"
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  using all_A_relprime finite_A by (simp add: all_relprime_prod_relprime)
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subsection {* Relationships Between Gauss Sets *}
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lemma B_card_eq_A: "card B = card A"
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  using finite_A by (simp add: finite_A B_def inj_on_xa_A card_image)
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lemma B_card_eq: "card B = nat ((p - 1) div 2)"
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  by (simp add: B_card_eq_A A_card_eq)
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lemma F_card_eq_E: "card F = card E"
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  using finite_E by (simp add: F_def inj_on_pminusx_E card_image)
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lemma C_card_eq_B: "card C = card B"
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  apply (insert finite_B)
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  apply (subgoal_tac "inj_on (StandardRes p) B")
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  apply (simp add: B_def C_def card_image)
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  apply (rule StandardRes_inj_on_ResSet)
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  apply (simp add: B_res)
wenzelm@18369
   311
  done
paulson@13871
   312
wenzelm@21233
   313
lemma D_E_disj: "D \<inter> E = {}"
paulson@13871
   314
  by (auto simp add: D_def E_def)
paulson@13871
   315
wenzelm@21233
   316
lemma C_card_eq_D_plus_E: "card C = card D + card E"
paulson@13871
   317
  by (auto simp add: C_eq card_Un_disjoint D_E_disj finite_D finite_E)
paulson@13871
   318
wenzelm@21233
   319
lemma C_prod_eq_D_times_E: "setprod id E * setprod id D = setprod id C"
paulson@13871
   320
  apply (insert D_E_disj finite_D finite_E C_eq)
nipkow@15392
   321
  apply (frule setprod_Un_disjoint [of D E id])
wenzelm@18369
   322
  apply auto
wenzelm@18369
   323
  done
paulson@13871
   324
wenzelm@21233
   325
lemma C_B_zcong_prod: "[setprod id C = setprod id B] (mod p)"
paulson@13871
   326
  apply (auto simp add: C_def)
wenzelm@18369
   327
  apply (insert finite_B SR_B_inj)
wenzelm@20898
   328
  apply (frule_tac f = "StandardRes p" in setprod_reindex_id [symmetric], auto)
nipkow@15392
   329
  apply (rule setprod_same_function_zcong)
wenzelm@18369
   330
  apply (auto simp add: StandardRes_prop1 zcong_sym p_g_0)
wenzelm@18369
   331
  done
paulson@13871
   332
wenzelm@21233
   333
lemma F_Un_D_subset: "(F \<union> D) \<subseteq> A"
paulson@13871
   334
  apply (rule Un_least)
wenzelm@18369
   335
  apply (auto simp add: A_def F_subset D_subset)
wenzelm@18369
   336
  done
paulson@13871
   337
wenzelm@21233
   338
lemma F_D_disj: "(F \<inter> D) = {}"
paulson@13871
   339
  apply (simp add: F_eq D_eq)
paulson@13871
   340
  apply (auto simp add: F_eq D_eq)
wenzelm@18369
   341
proof -
wenzelm@18369
   342
  fix y fix ya
wenzelm@18369
   343
  assume "p - StandardRes p (y * a) = StandardRes p (ya * a)"
wenzelm@18369
   344
  then have "p = StandardRes p (y * a) + StandardRes p (ya * a)"
wenzelm@18369
   345
    by arith
wenzelm@18369
   346
  moreover have "p dvd p"
wenzelm@18369
   347
    by auto
wenzelm@18369
   348
  ultimately have "p dvd (StandardRes p (y * a) + StandardRes p (ya * a))"
wenzelm@18369
   349
    by auto
wenzelm@18369
   350
  then have a: "[StandardRes p (y * a) + StandardRes p (ya * a) = 0] (mod p)"
wenzelm@18369
   351
    by (auto simp add: zcong_def)
wenzelm@18369
   352
  have "[y * a = StandardRes p (y * a)] (mod p)"
wenzelm@18369
   353
    by (simp only: zcong_sym StandardRes_prop1)
wenzelm@18369
   354
  moreover have "[ya * a = StandardRes p (ya * a)] (mod p)"
wenzelm@18369
   355
    by (simp only: zcong_sym StandardRes_prop1)
wenzelm@18369
   356
  ultimately have "[y * a + ya * a =
wenzelm@18369
   357
    StandardRes p (y * a) + StandardRes p (ya * a)] (mod p)"
wenzelm@18369
   358
    by (rule zcong_zadd)
wenzelm@18369
   359
  with a have "[y * a + ya * a = 0] (mod p)"
wenzelm@18369
   360
    apply (elim zcong_trans)
wenzelm@18369
   361
    by (simp only: zcong_refl)
wenzelm@18369
   362
  also have "y * a + ya * a = a * (y + ya)"
wenzelm@18369
   363
    by (simp add: zadd_zmult_distrib2 zmult_commute)
wenzelm@18369
   364
  finally have "[a * (y + ya) = 0] (mod p)" .
wenzelm@18369
   365
  with p_prime a_nonzero zcong_zprime_prod_zero [of p a "y + ya"]
wenzelm@18369
   366
    p_a_relprime
wenzelm@18369
   367
  have a: "[y + ya = 0] (mod p)"
wenzelm@18369
   368
    by auto
wenzelm@18369
   369
  assume b: "y \<in> A" and c: "ya: A"
wenzelm@18369
   370
  with A_def have "0 < y + ya"
wenzelm@18369
   371
    by auto
wenzelm@18369
   372
  moreover from b c A_def have "y + ya \<le> (p - 1) div 2 + (p - 1) div 2"
wenzelm@18369
   373
    by auto
wenzelm@18369
   374
  moreover from b c p_eq2 A_def have "y + ya < p"
wenzelm@18369
   375
    by auto
wenzelm@18369
   376
  ultimately show False
wenzelm@18369
   377
    apply simp
wenzelm@18369
   378
    apply (frule_tac m = p in zcong_not_zero)
wenzelm@18369
   379
    apply (auto simp add: a)
wenzelm@18369
   380
    done
wenzelm@18369
   381
qed
paulson@13871
   382
wenzelm@21233
   383
lemma F_Un_D_card: "card (F \<union> D) = nat ((p - 1) div 2)"
wenzelm@18369
   384
proof -
wenzelm@18369
   385
  have "card (F \<union> D) = card E + card D"
wenzelm@18369
   386
    by (auto simp add: finite_F finite_D F_D_disj
wenzelm@18369
   387
      card_Un_disjoint F_card_eq_E)
wenzelm@18369
   388
  then have "card (F \<union> D) = card C"
wenzelm@18369
   389
    by (simp add: C_card_eq_D_plus_E)
wenzelm@18369
   390
  from this show "card (F \<union> D) = nat ((p - 1) div 2)"
wenzelm@18369
   391
    by (simp add: C_card_eq_B B_card_eq)
wenzelm@18369
   392
qed
paulson@13871
   393
wenzelm@21233
   394
lemma F_Un_D_eq_A: "F \<union> D = A"
wenzelm@18369
   395
  using finite_A F_Un_D_subset A_card_eq F_Un_D_card by (auto simp add: card_seteq)
paulson@13871
   396
wenzelm@21233
   397
lemma prod_D_F_eq_prod_A:
wenzelm@18369
   398
    "(setprod id D) * (setprod id F) = setprod id A"
paulson@13871
   399
  apply (insert F_D_disj finite_D finite_F)
nipkow@15392
   400
  apply (frule setprod_Un_disjoint [of F D id])
wenzelm@18369
   401
  apply (auto simp add: F_Un_D_eq_A)
wenzelm@18369
   402
  done
paulson@13871
   403
wenzelm@21233
   404
lemma prod_F_zcong:
wenzelm@18369
   405
  "[setprod id F = ((-1) ^ (card E)) * (setprod id E)] (mod p)"
wenzelm@18369
   406
proof -
wenzelm@18369
   407
  have "setprod id F = setprod id (op - p ` E)"
wenzelm@18369
   408
    by (auto simp add: F_def)
wenzelm@18369
   409
  then have "setprod id F = setprod (op - p) E"
wenzelm@18369
   410
    apply simp
wenzelm@18369
   411
    apply (insert finite_E inj_on_pminusx_E)
wenzelm@18369
   412
    apply (frule_tac f = "op - p" in setprod_reindex_id, auto)
wenzelm@18369
   413
    done
wenzelm@18369
   414
  then have one:
wenzelm@18369
   415
    "[setprod id F = setprod (StandardRes p o (op - p)) E] (mod p)"
wenzelm@18369
   416
    apply simp
wenzelm@18369
   417
    apply (insert p_g_0 finite_E)
wenzelm@18369
   418
    by (auto simp add: StandardRes_prod)
wenzelm@18369
   419
  moreover have a: "\<forall>x \<in> E. [p - x = 0 - x] (mod p)"
wenzelm@18369
   420
    apply clarify
wenzelm@18369
   421
    apply (insert zcong_id [of p])
wenzelm@18369
   422
    apply (rule_tac a = p and m = p and c = x and d = x in zcong_zdiff, auto)
wenzelm@18369
   423
    done
wenzelm@18369
   424
  moreover have b: "\<forall>x \<in> E. [StandardRes p (p - x) = p - x](mod p)"
wenzelm@18369
   425
    apply clarify
wenzelm@18369
   426
    apply (simp add: StandardRes_prop1 zcong_sym)
wenzelm@18369
   427
    done
wenzelm@18369
   428
  moreover have "\<forall>x \<in> E. [StandardRes p (p - x) = - x](mod p)"
wenzelm@18369
   429
    apply clarify
wenzelm@18369
   430
    apply (insert a b)
wenzelm@18369
   431
    apply (rule_tac b = "p - x" in zcong_trans, auto)
wenzelm@18369
   432
    done
wenzelm@18369
   433
  ultimately have c:
wenzelm@18369
   434
    "[setprod (StandardRes p o (op - p)) E = setprod (uminus) E](mod p)"
wenzelm@18369
   435
    apply simp
wenzelm@18369
   436
    apply (insert finite_E p_g_0)
wenzelm@18369
   437
    apply (rule setprod_same_function_zcong
wenzelm@18369
   438
      [of E "StandardRes p o (op - p)" uminus p], auto)
wenzelm@18369
   439
    done
wenzelm@18369
   440
  then have two: "[setprod id F = setprod (uminus) E](mod p)"
wenzelm@18369
   441
    apply (insert one c)
wenzelm@18369
   442
    apply (rule zcong_trans [of "setprod id F"
nipkow@15392
   443
                               "setprod (StandardRes p o op - p) E" p
wenzelm@18369
   444
                               "setprod uminus E"], auto)
wenzelm@18369
   445
    done
wenzelm@18369
   446
  also have "setprod uminus E = (setprod id E) * (-1)^(card E)"
berghofe@22274
   447
    using finite_E by (induct set: finite) auto
wenzelm@18369
   448
  then have "setprod uminus E = (-1) ^ (card E) * (setprod id E)"
wenzelm@18369
   449
    by (simp add: zmult_commute)
wenzelm@18369
   450
  with two show ?thesis
wenzelm@18369
   451
    by simp
nipkow@15392
   452
qed
paulson@13871
   453
wenzelm@21233
   454
paulson@13871
   455
subsection {* Gauss' Lemma *}
paulson@13871
   456
wenzelm@21233
   457
lemma aux: "setprod id A * -1 ^ card E * a ^ card A * -1 ^ card E = setprod id A * a ^ card A"
paulson@13871
   458
  by (auto simp add: finite_E neg_one_special)
paulson@13871
   459
wenzelm@21233
   460
theorem pre_gauss_lemma:
wenzelm@18369
   461
  "[a ^ nat((p - 1) div 2) = (-1) ^ (card E)] (mod p)"
wenzelm@18369
   462
proof -
wenzelm@18369
   463
  have "[setprod id A = setprod id F * setprod id D](mod p)"
wenzelm@18369
   464
    by (auto simp add: prod_D_F_eq_prod_A zmult_commute)
wenzelm@18369
   465
  then have "[setprod id A = ((-1)^(card E) * setprod id E) *
wenzelm@18369
   466
      setprod id D] (mod p)"
wenzelm@18369
   467
    apply (rule zcong_trans)
wenzelm@18369
   468
    apply (auto simp add: prod_F_zcong zcong_scalar)
wenzelm@18369
   469
    done
wenzelm@18369
   470
  then have "[setprod id A = ((-1)^(card E) * setprod id C)] (mod p)"
wenzelm@18369
   471
    apply (rule zcong_trans)
wenzelm@18369
   472
    apply (insert C_prod_eq_D_times_E, erule subst)
wenzelm@18369
   473
    apply (subst zmult_assoc, auto)
wenzelm@18369
   474
    done
wenzelm@18369
   475
  then have "[setprod id A = ((-1)^(card E) * setprod id B)] (mod p)"
wenzelm@18369
   476
    apply (rule zcong_trans)
wenzelm@18369
   477
    apply (simp add: C_B_zcong_prod zcong_scalar2)
wenzelm@18369
   478
    done
wenzelm@18369
   479
  then have "[setprod id A = ((-1)^(card E) *
wenzelm@18369
   480
    (setprod id ((%x. x * a) ` A)))] (mod p)"
wenzelm@18369
   481
    by (simp add: B_def)
wenzelm@18369
   482
  then have "[setprod id A = ((-1)^(card E) * (setprod (%x. x * a) A))]
wenzelm@18369
   483
    (mod p)"
wenzelm@18369
   484
    by (simp add:finite_A inj_on_xa_A setprod_reindex_id[symmetric])
wenzelm@18369
   485
  moreover have "setprod (%x. x * a) A =
wenzelm@18369
   486
    setprod (%x. a) A * setprod id A"
berghofe@22274
   487
    using finite_A by (induct set: finite) auto
wenzelm@18369
   488
  ultimately have "[setprod id A = ((-1)^(card E) * (setprod (%x. a) A *
wenzelm@18369
   489
    setprod id A))] (mod p)"
wenzelm@18369
   490
    by simp
wenzelm@18369
   491
  then have "[setprod id A = ((-1)^(card E) * a^(card A) *
wenzelm@18369
   492
      setprod id A)](mod p)"
wenzelm@18369
   493
    apply (rule zcong_trans)
wenzelm@18369
   494
    apply (simp add: zcong_scalar2 zcong_scalar finite_A setprod_constant zmult_assoc)
wenzelm@18369
   495
    done
wenzelm@18369
   496
  then have a: "[setprod id A * (-1)^(card E) =
wenzelm@18369
   497
      ((-1)^(card E) * a^(card A) * setprod id A * (-1)^(card E))](mod p)"
wenzelm@18369
   498
    by (rule zcong_scalar)
wenzelm@18369
   499
  then have "[setprod id A * (-1)^(card E) = setprod id A *
wenzelm@18369
   500
      (-1)^(card E) * a^(card A) * (-1)^(card E)](mod p)"
wenzelm@18369
   501
    apply (rule zcong_trans)
wenzelm@18369
   502
    apply (simp add: a mult_commute mult_left_commute)
wenzelm@18369
   503
    done
wenzelm@18369
   504
  then have "[setprod id A * (-1)^(card E) = setprod id A *
wenzelm@18369
   505
      a^(card A)](mod p)"
wenzelm@18369
   506
    apply (rule zcong_trans)
wenzelm@18369
   507
    apply (simp add: aux)
wenzelm@18369
   508
    done
wenzelm@18369
   509
  with this zcong_cancel2 [of p "setprod id A" "-1 ^ card E" "a ^ card A"]
wenzelm@18369
   510
      p_g_0 A_prod_relprime have "[-1 ^ card E = a ^ card A](mod p)"
wenzelm@18369
   511
    by (simp add: order_less_imp_le)
wenzelm@18369
   512
  from this show ?thesis
wenzelm@18369
   513
    by (simp add: A_card_eq zcong_sym)
nipkow@15392
   514
qed
paulson@13871
   515
wenzelm@21233
   516
theorem gauss_lemma: "(Legendre a p) = (-1) ^ (card E)"
nipkow@15392
   517
proof -
paulson@13871
   518
  from Euler_Criterion p_prime p_g_2 have
wenzelm@18369
   519
      "[(Legendre a p) = a^(nat (((p) - 1) div 2))] (mod p)"
paulson@13871
   520
    by auto
nipkow@15392
   521
  moreover note pre_gauss_lemma
nipkow@15392
   522
  ultimately have "[(Legendre a p) = (-1) ^ (card E)] (mod p)"
paulson@13871
   523
    by (rule zcong_trans)
nipkow@15392
   524
  moreover from p_a_relprime have "(Legendre a p) = 1 | (Legendre a p) = (-1)"
paulson@13871
   525
    by (auto simp add: Legendre_def)
nipkow@15392
   526
  moreover have "(-1::int) ^ (card E) = 1 | (-1::int) ^ (card E) = -1"
paulson@13871
   527
    by (rule neg_one_power)
nipkow@15392
   528
  ultimately show ?thesis
paulson@13871
   529
    by (auto simp add: p_g_2 one_not_neg_one_mod_m zcong_sym)
nipkow@15392
   530
qed
paulson@13871
   531
avigad@16775
   532
end
wenzelm@21233
   533
wenzelm@21233
   534
end