--- a/src/HOL/Old_Number_Theory/Gauss.thy Tue Oct 18 07:04:08 2016 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,504 +0,0 @@
-(* Title: HOL/Old_Number_Theory/Gauss.thy
- Authors: Jeremy Avigad, David Gray, and Adam Kramer
-*)
-
-section \<open>Gauss' Lemma\<close>
-
-theory Gauss
-imports Euler
-begin
-
-locale GAUSS =
- fixes p :: "int"
- fixes a :: "int"
-
- assumes p_prime: "zprime p"
- assumes p_g_2: "2 < p"
- assumes p_a_relprime: "~[a = 0](mod p)"
- assumes a_nonzero: "0 < a"
-begin
-
-definition "A = {(x::int). 0 < x & x \<le> ((p - 1) div 2)}"
-definition "B = (%x. x * a) ` A"
-definition "C = StandardRes p ` B"
-definition "D = C \<inter> {x. x \<le> ((p - 1) div 2)}"
-definition "E = C \<inter> {x. ((p - 1) div 2) < x}"
-definition "F = (%x. (p - x)) ` E"
-
-
-subsection \<open>Basic properties of p\<close>
-
-lemma p_odd: "p \<in> zOdd"
- by (auto simp add: p_prime p_g_2 zprime_zOdd_eq_grt_2)
-
-lemma p_g_0: "0 < p"
- using p_g_2 by auto
-
-lemma int_nat: "int (nat ((p - 1) div 2)) = (p - 1) div 2"
- using ListMem.insert p_g_2 by (auto simp add: pos_imp_zdiv_nonneg_iff)
-
-lemma p_minus_one_l: "(p - 1) div 2 < p"
-proof -
- have "(p - 1) div 2 \<le> (p - 1) div 1"
- by (rule zdiv_mono2) (auto simp add: p_g_0)
- also have "\<dots> = p - 1" by simp
- finally show ?thesis by simp
-qed
-
-lemma p_eq: "p = (2 * (p - 1) div 2) + 1"
- using nonzero_mult_div_cancel_left [of 2 "p - 1"] by auto
-
-
-lemma (in -) zodd_imp_zdiv_eq: "x \<in> zOdd ==> 2 * (x - 1) div 2 = 2 * ((x - 1) div 2)"
- by (simp add: in_zEven_zOdd_iff)
-
-
-lemma p_eq2: "p = (2 * ((p - 1) div 2)) + 1"
- apply (insert p_eq p_prime p_g_2 zprime_zOdd_eq_grt_2 [of p], auto)
- apply (frule zodd_imp_zdiv_eq, auto)
- done
-
-
-subsection \<open>Basic Properties of the Gauss Sets\<close>
-
-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 ((p - 1) div 2)"
- apply (auto simp add: A_def)
- apply (insert int_nat)
- apply (erule subst)
- apply (auto simp add: card_bdd_int_set_l_le)
- done
-
-lemma inj_on_xa_A: "inj_on (%x. x * a) A"
- using a_nonzero by (simp add: A_def inj_on_def)
-
-lemma A_res: "ResSet p A"
- apply (auto simp add: A_def ResSet_def)
- apply (rule_tac m = p in zcong_less_eq)
- apply (insert p_g_2, auto)
- done
-
-lemma B_res: "ResSet p B"
- apply (insert p_g_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)
-proof -
- fix x fix y
- assume a: "[x * a = y * a] (mod p)"
- assume b: "0 < x"
- assume c: "x \<le> (p - 1) div 2"
- assume d: "0 < y"
- assume e: "y \<le> (p - 1) div 2"
- from a p_a_relprime p_prime a_nonzero zcong_cancel [of p a x y]
- have "[x = y](mod p)"
- by (simp add: zprime_imp_zrelprime zcong_def p_g_0 order_le_less)
- with zcong_less_eq [of x y p] p_minus_one_l
- order_le_less_trans [of x "(p - 1) div 2" p]
- order_le_less_trans [of y "(p - 1) div 2" p] show "x = y"
- by (simp add: b c d e p_minus_one_l p_g_0)
-qed
-
-lemma SR_B_inj: "inj_on (StandardRes p) B"
- apply (auto simp add: B_def StandardRes_def inj_on_def A_def)
-proof -
- fix x fix y
- assume a: "x * a mod p = y * a mod p"
- assume b: "0 < x"
- assume c: "x \<le> (p - 1) div 2"
- assume d: "0 < y"
- assume e: "y \<le> (p - 1) div 2"
- assume f: "x \<noteq> y"
- from a have "[x * a = y * a](mod p)"
- by (simp add: zcong_zmod_eq p_g_0)
- with p_a_relprime p_prime a_nonzero zcong_cancel [of p a x y]
- have "[x = y](mod p)"
- by (simp add: zprime_imp_zrelprime zcong_def p_g_0 order_le_less)
- with zcong_less_eq [of x y p] p_minus_one_l
- order_le_less_trans [of x "(p - 1) div 2" p]
- order_le_less_trans [of y "(p - 1) div 2" p] have "x = y"
- by (simp add: b c d e p_minus_one_l p_g_0)
- then have False
- by (simp add: f)
- then show "a = 0"
- by simp
-qed
-
-lemma inj_on_pminusx_E: "inj_on (%x. p - x) E"
- apply (auto simp add: E_def C_def B_def A_def)
- apply (rule_tac g = "%x. -1 * (x - p)" in inj_on_inverseI)
- apply auto
- done
-
-lemma A_ncong_p: "x \<in> A ==> ~[x = 0](mod p)"
- apply (auto simp add: A_def)
- apply (frule_tac m = p in zcong_not_zero)
- apply (insert p_minus_one_l)
- apply auto
- done
-
-lemma A_greater_zero: "x \<in> A ==> 0 < x"
- by (auto simp add: A_def)
-
-lemma B_ncong_p: "x \<in> B ==> ~[x = 0](mod p)"
- apply (auto simp add: B_def)
- apply (frule A_ncong_p)
- apply (insert p_a_relprime p_prime a_nonzero)
- apply (frule_tac a = xa and b = a in zcong_zprime_prod_zero_contra)
- apply (auto simp add: A_greater_zero)
- done
-
-lemma B_greater_zero: "x \<in> B ==> 0 < x"
- using a_nonzero by (auto simp add: B_def A_greater_zero)
-
-lemma C_ncong_p: "x \<in> C ==> ~[x = 0](mod p)"
- apply (auto simp add: C_def)
- apply (frule B_ncong_p)
- apply (subgoal_tac "[xa = StandardRes p xa](mod p)")
- defer apply (simp add: StandardRes_prop1)
- apply (frule_tac a = xa and b = "StandardRes p xa" and c = 0 in zcong_trans)
- apply auto
- done
-
-lemma C_greater_zero: "y \<in> C ==> 0 < y"
- apply (auto simp add: C_def)
-proof -
- fix x
- assume a: "x \<in> B"
- from p_g_0 have "0 \<le> StandardRes p x"
- by (simp add: StandardRes_lbound)
- moreover have "~[x = 0] (mod p)"
- by (simp add: a B_ncong_p)
- then have "StandardRes p x \<noteq> 0"
- by (simp add: StandardRes_prop3)
- ultimately show "0 < StandardRes p x"
- by (simp add: order_le_less)
-qed
-
-lemma D_ncong_p: "x \<in> D ==> ~[x = 0](mod p)"
- by (auto simp add: D_def C_ncong_p)
-
-lemma E_ncong_p: "x \<in> E ==> ~[x = 0](mod p)"
- by (auto simp add: E_def C_ncong_p)
-
-lemma F_ncong_p: "x \<in> F ==> ~[x = 0](mod p)"
- apply (auto simp add: F_def)
-proof -
- fix x assume a: "x \<in> E" assume b: "[p - x = 0] (mod p)"
- from E_ncong_p have "~[x = 0] (mod p)"
- by (simp add: a)
- moreover from a have "0 < x"
- by (simp add: a E_def C_greater_zero)
- moreover from a have "x < p"
- by (auto simp add: E_def C_def p_g_0 StandardRes_ubound)
- ultimately have "~[p - x = 0] (mod p)"
- by (simp add: zcong_not_zero)
- from this show False by (simp add: b)
-qed
-
-lemma F_subset: "F \<subseteq> {x. 0 < x & x \<le> ((p - 1) div 2)}"
- apply (auto simp add: F_def E_def)
- apply (insert p_g_0)
- apply (frule_tac x = xa in StandardRes_ubound)
- apply (frule_tac x = x in StandardRes_ubound)
- apply (subgoal_tac "xa = StandardRes p xa")
- apply (auto simp add: C_def StandardRes_prop2 StandardRes_prop1)
-proof -
- from zodd_imp_zdiv_eq p_prime p_g_2 zprime_zOdd_eq_grt_2 have
- "2 * (p - 1) div 2 = 2 * ((p - 1) div 2)"
- by simp
- with p_eq2 show " !!x. [| (p - 1) div 2 < StandardRes p x; x \<in> B |]
- ==> p - StandardRes p x \<le> (p - 1) div 2"
- by simp
-qed
-
-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 - (StandardRes p (y*a)) & (p - 1) div 2 < StandardRes p (y*a))}"
- 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 = StandardRes p (y*a) & StandardRes p (y*a) \<le> (p - 1) div 2)}"
- by (auto simp add: D_def C_def B_def A_def)
-
-lemma D_leq: "x \<in> D ==> x \<le> (p - 1) div 2"
- by (auto simp add: D_eq)
-
-lemma F_ge: "x \<in> F ==> x \<le> (p - 1) div 2"
- apply (auto simp add: F_eq A_def)
-proof -
- fix y
- assume "(p - 1) div 2 < StandardRes p (y * a)"
- then have "p - StandardRes p (y * a) < p - ((p - 1) div 2)"
- by arith
- also from p_eq2 have "... = 2 * ((p - 1) div 2) + 1 - ((p - 1) div 2)"
- by auto
- also have "2 * ((p - 1) div 2) + 1 - (p - 1) div 2 = (p - 1) div 2 + 1"
- by arith
- finally show "p - StandardRes p (y * a) \<le> (p - 1) div 2"
- using zless_add1_eq [of "p - StandardRes p (y * a)" "(p - 1) div 2"] by auto
-qed
-
-lemma all_A_relprime: "\<forall>x \<in> A. zgcd x p = 1"
- using p_prime p_minus_one_l by (auto simp add: A_def zless_zprime_imp_zrelprime)
-
-lemma A_prod_relprime: "zgcd (prod id A) p = 1"
-by(rule all_relprime_prod_relprime[OF finite_A all_A_relprime])
-
-
-subsection \<open>Relationships Between Gauss Sets\<close>
-
-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 ((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"
- apply (insert finite_B)
- apply (subgoal_tac "inj_on (StandardRes p) B")
- apply (simp add: B_def C_def card_image)
- apply (rule StandardRes_inj_on_ResSet)
- apply (simp add: B_res)
- done
-
-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: "prod id E * prod id D = prod id C"
- apply (insert D_E_disj finite_D finite_E C_eq)
- apply (frule prod.union_disjoint [of D E id])
- apply auto
- done
-
-lemma C_B_zcong_prod: "[prod id C = prod id B] (mod p)"
- apply (auto simp add: C_def)
- apply (insert finite_B SR_B_inj)
- apply (frule prod.reindex [of "StandardRes p" B id])
- apply auto
- apply (rule prod_same_function_zcong)
- apply (auto simp add: StandardRes_prop1 zcong_sym p_g_0)
- done
-
-lemma F_Un_D_subset: "(F \<union> D) \<subseteq> A"
- apply (rule Un_least)
- apply (auto simp add: A_def F_subset D_subset)
- done
-
-lemma F_D_disj: "(F \<inter> D) = {}"
- apply (simp add: F_eq D_eq)
- apply (auto simp add: F_eq D_eq)
-proof -
- fix y fix ya
- assume "p - StandardRes p (y * a) = StandardRes p (ya * a)"
- then have "p = StandardRes p (y * a) + StandardRes p (ya * a)"
- by arith
- moreover have "p dvd p"
- by auto
- ultimately have "p dvd (StandardRes p (y * a) + StandardRes p (ya * a))"
- by auto
- then have a: "[StandardRes p (y * a) + StandardRes p (ya * a) = 0] (mod p)"
- by (auto simp add: zcong_def)
- have "[y * a = StandardRes p (y * a)] (mod p)"
- by (simp only: zcong_sym StandardRes_prop1)
- moreover have "[ya * a = StandardRes p (ya * a)] (mod p)"
- by (simp only: zcong_sym StandardRes_prop1)
- ultimately have "[y * a + ya * a =
- StandardRes p (y * a) + StandardRes p (ya * a)] (mod p)"
- by (rule zcong_zadd)
- with a have "[y * a + ya * a = 0] (mod p)"
- apply (elim zcong_trans)
- by (simp only: zcong_refl)
- also have "y * a + ya * a = a * (y + ya)"
- by (simp add: distrib_left mult.commute)
- finally have "[a * (y + ya) = 0] (mod p)" .
- with p_prime a_nonzero zcong_zprime_prod_zero [of p a "y + ya"]
- p_a_relprime
- have a: "[y + ya = 0] (mod p)"
- by auto
- assume b: "y \<in> A" and c: "ya: A"
- with A_def have "0 < y + ya"
- by auto
- moreover from b c A_def have "y + ya \<le> (p - 1) div 2 + (p - 1) div 2"
- by auto
- moreover from b c p_eq2 A_def have "y + ya < p"
- by auto
- ultimately show False
- apply simp
- apply (frule_tac m = p in zcong_not_zero)
- apply (auto simp add: a)
- done
-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)
- from this 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:
- "(prod id D) * (prod id F) = prod id A"
- apply (insert F_D_disj finite_D finite_F)
- apply (frule prod.union_disjoint [of F D id])
- apply (auto simp add: F_Un_D_eq_A)
- done
-
-lemma prod_F_zcong:
- "[prod id F = ((-1) ^ (card E)) * (prod id E)] (mod p)"
-proof -
- have "prod id F = prod id (op - p ` E)"
- by (auto simp add: F_def)
- then have "prod id F = prod (op - p) E"
- apply simp
- apply (insert finite_E inj_on_pminusx_E)
- apply (frule prod.reindex [of "minus p" E id])
- apply auto
- done
- then have one:
- "[prod id F = prod (StandardRes p o (op - p)) E] (mod p)"
- apply simp
- apply (insert p_g_0 finite_E StandardRes_prod)
- by (auto)
- moreover have a: "\<forall>x \<in> E. [p - x = 0 - x] (mod p)"
- apply clarify
- apply (insert zcong_id [of p])
- apply (rule_tac a = p and m = p and c = x and d = x in zcong_zdiff, auto)
- done
- moreover have b: "\<forall>x \<in> E. [StandardRes p (p - x) = p - x](mod p)"
- apply clarify
- apply (simp add: StandardRes_prop1 zcong_sym)
- done
- moreover have "\<forall>x \<in> E. [StandardRes p (p - x) = - x](mod p)"
- apply clarify
- apply (insert a b)
- apply (rule_tac b = "p - x" in zcong_trans, auto)
- done
- ultimately have c:
- "[prod (StandardRes p o (op - p)) E = prod (uminus) E](mod p)"
- apply simp
- using finite_E p_g_0
- prod_same_function_zcong [of E "StandardRes p o (op - p)" uminus p]
- by auto
- then have two: "[prod id F = prod (uminus) E](mod p)"
- apply (insert one c)
- apply (rule zcong_trans [of "prod id F"
- "prod (StandardRes p o op - p) E" p
- "prod uminus E"], auto)
- done
- also have "prod uminus E = (prod id E) * (-1)^(card E)"
- using finite_E by (induct set: finite) auto
- then have "prod uminus E = (-1) ^ (card E) * (prod id E)"
- by (simp add: mult.commute)
- with two show ?thesis
- by simp
-qed
-
-
-subsection \<open>Gauss' Lemma\<close>
-
-lemma aux: "prod id A * (- 1) ^ card E * a ^ card A * (- 1) ^ card E = prod id A * a ^ card A"
- by (auto simp add: finite_E neg_one_special)
-
-theorem pre_gauss_lemma:
- "[a ^ nat((p - 1) div 2) = (-1) ^ (card E)] (mod p)"
-proof -
- have "[prod id A = prod id F * prod id D](mod p)"
- by (auto simp add: prod_D_F_eq_prod_A mult.commute cong del: prod.strong_cong)
- then have "[prod id A = ((-1)^(card E) * prod id E) *
- prod id D] (mod p)"
- by (rule zcong_trans) (auto simp add: prod_F_zcong zcong_scalar cong del: prod.strong_cong)
- then have "[prod id A = ((-1)^(card E) * prod id C)] (mod p)"
- apply (rule zcong_trans)
- apply (insert C_prod_eq_D_times_E, erule subst)
- apply (subst mult.assoc)
- apply auto
- done
- then have "[prod id A = ((-1)^(card E) * prod id B)] (mod p)"
- apply (rule zcong_trans)
- apply (simp add: C_B_zcong_prod zcong_scalar2 cong del: prod.strong_cong)
- done
- then have "[prod id A = ((-1)^(card E) *
- (prod id ((%x. x * a) ` A)))] (mod p)"
- by (simp add: B_def)
- then have "[prod id A = ((-1)^(card E) * (prod (%x. x * a) A))]
- (mod p)"
- by (simp add:finite_A inj_on_xa_A prod.reindex cong del: prod.strong_cong)
- moreover have "prod (%x. x * a) A =
- prod (%x. a) A * prod id A"
- using finite_A by (induct set: finite) auto
- ultimately have "[prod id A = ((-1)^(card E) * (prod (%x. a) A *
- prod id A))] (mod p)"
- by simp
- then have "[prod id A = ((-1)^(card E) * a^(card A) *
- prod id A)](mod p)"
- by (rule zcong_trans) (simp add: zcong_scalar2 zcong_scalar finite_A prod_constant mult.assoc)
- then have a: "[prod id A * (-1)^(card E) =
- ((-1)^(card E) * a^(card A) * prod id A * (-1)^(card E))](mod p)"
- by (rule zcong_scalar)
- then have "[prod id A * (-1)^(card E) = prod id A *
- (-1)^(card E) * a^(card A) * (-1)^(card E)](mod p)"
- by (rule zcong_trans) (simp add: a mult.commute mult.left_commute)
- then have "[prod id A * (-1)^(card E) = prod id A *
- a^(card A)](mod p)"
- by (rule zcong_trans) (simp add: aux cong del: prod.strong_cong)
- with this zcong_cancel2 [of p "prod id A" "(- 1) ^ card E" "a ^ card A"]
- p_g_0 A_prod_relprime have "[(- 1) ^ card E = a ^ card A](mod p)"
- by (simp add: order_less_imp_le)
- from this show ?thesis
- by (simp add: A_card_eq zcong_sym)
-qed
-
-theorem gauss_lemma: "(Legendre a p) = (-1) ^ (card E)"
-proof -
- from Euler_Criterion p_prime p_g_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 zcong_trans)
- 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_g_2 one_not_neg_one_mod_m zcong_sym)
-qed
-
-end
-
-end