(* Title: HOL/Quadratic_Reciprocity/Gauss.thy ID: $Id: Gauss.thy,v 1.18 2007/02/07 16:54:00 berghofe Exp $ Authors: Jeremy Avigad, David Gray, and Adam Kramer) *) header {* Gauss' Lemma *} 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 :: "int set" where "A = {(x::int). 0 < x & x ≤ ((p - 1) div 2)}" definition B :: "int set" where "B = (%x. x * a) ` A" definition C :: "int set" where "C = StandardRes p ` B" definition D :: "int set" where "D = C ∩ {x. x ≤ ((p - 1) div 2)}" definition E :: "int set" where "E = C ∩ {x. ((p - 1) div 2) < x}" definition F :: "int set" where "F = (%x. (p - x)) ` E" subsection {* Basic properties of p *} lemma p_odd: "p ∈ 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 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 ≤ (p - 1) div 1" by (rule zdiv_mono2) (auto simp add: p_g_0) also have "… = p - 1" by simp finally show ?thesis by simp qed lemma p_eq: "p = (2 * (p - 1) div 2) + 1" using zdiv_zmult_self2 [of 2 "p - 1"] by auto lemma (in -) zodd_imp_zdiv_eq: "x ∈ zOdd ==> 2 * (x - 1) div 2 = 2 * ((x - 1) div 2)" apply (frule odd_minus_one_even) apply (simp add: zEven_def) apply (subgoal_tac "2 ≠ 0") apply (frule_tac b = "2 :: int" and a = "x - 1" in zdiv_zmult_self2) apply (auto simp add: even_div_2_prop2) done 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 {* Basic Properties of the Gauss Sets *} lemma finite_A: "finite (A)" apply (auto simp add: A_def) apply (subgoal_tac "{x. 0 < x & x ≤ (p - 1) div 2} ⊆ {x. 0 ≤ x & x < 1 + (p - 1) div 2}") apply (auto simp add: bdd_int_set_l_finite finite_subset) done lemma finite_B: "finite (B)" by (auto simp add: B_def finite_A finite_imageI) lemma finite_C: "finite (C)" by (auto simp add: C_def finite_B finite_imageI) lemma finite_D: "finite (D)" by (auto simp add: D_def finite_Int finite_C) lemma finite_E: "finite (E)" by (auto simp add: E_def finite_Int finite_C) lemma finite_F: "finite (F)" by (auto simp add: F_def finite_E finite_imageI) lemma C_eq: "C = D ∪ 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 ≤ (p - 1) div 2" assume d: "0 < y" assume e: "y ≤ (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: prems 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 prems) proof - fix x fix y assume a: "x * a mod p = y * a mod p" assume b: "0 < x" assume c: "x ≤ (p - 1) div 2" assume d: "0 < y" assume e: "y ≤ (p - 1) div 2" assume f: "x ≠ 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: prems 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 ∈ 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 ∈ A ==> 0 < x" by (auto simp add: A_def) lemma B_ncong_p: "x ∈ 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 = x and b = a in zcong_zprime_prod_zero_contra) apply (auto simp add: A_greater_zero) done lemma B_greater_zero: "x ∈ B ==> 0 < x" using a_nonzero by (auto simp add: B_def mult_pos_pos A_greater_zero) lemma C_ncong_p: "x ∈ C ==> ~[x = 0](mod p)" apply (auto simp add: C_def) apply (frule B_ncong_p) apply (subgoal_tac "[x = StandardRes p x](mod p)") defer apply (simp add: StandardRes_prop1) apply (frule_tac a = x and b = "StandardRes p x" and c = 0 in zcong_trans) apply auto done lemma C_greater_zero: "y ∈ C ==> 0 < y" apply (auto simp add: C_def) proof - fix x assume a: "x ∈ B" from p_g_0 have "0 ≤ 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 ≠ 0" by (simp add: StandardRes_prop3) ultimately show "0 < StandardRes p x" by (simp add: order_le_less) qed lemma D_ncong_p: "x ∈ D ==> ~[x = 0](mod p)" by (auto simp add: D_def C_ncong_p) lemma E_ncong_p: "x ∈ E ==> ~[x = 0](mod p)" by (auto simp add: E_def C_ncong_p) lemma F_ncong_p: "x ∈ F ==> ~[x = 0](mod p)" apply (auto simp add: F_def) proof - fix x assume a: "x ∈ 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 ⊆ {x. 0 < x & x ≤ ((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 ∈ B |] ==> p - StandardRes p x ≤ (p - 1) div 2" by simp qed lemma D_subset: "D ⊆ {x. 0 < x & x ≤ ((p - 1) div 2)}" by (auto simp add: D_def C_greater_zero) lemma F_eq: "F = {x. ∃y ∈ 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. ∃y ∈ A. ( x = StandardRes p (y*a) & StandardRes p (y*a) ≤ (p - 1) div 2)}" by (auto simp add: D_def C_def B_def A_def) lemma D_leq: "x ∈ D ==> x ≤ (p - 1) div 2" by (auto simp add: D_eq) lemma F_ge: "x ∈ F ==> x ≤ (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) ≤ (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: "∀x ∈ 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((setprod id A),p) = 1" using all_A_relprime finite_A by (simp add: all_relprime_prod_relprime) subsection {* Relationships Between Gauss Sets *} 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 ∩ 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" apply (insert D_E_disj finite_D finite_E C_eq) apply (frule setprod_Un_disjoint [of D E id]) apply auto done 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 (frule_tac f = "StandardRes p" in setprod_reindex_id [symmetric], auto) apply (rule setprod_same_function_zcong) apply (auto simp add: StandardRes_prop1 zcong_sym p_g_0) done lemma F_Un_D_subset: "(F ∪ D) ⊆ A" apply (rule Un_least) apply (auto simp add: A_def F_subset D_subset) done lemma F_D_disj: "(F ∩ 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: zadd_zmult_distrib2 zmult_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 ∈ A" and c: "ya: A" with A_def have "0 < y + ya" by auto moreover from b c A_def have "y + ya ≤ (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 ∪ D) = nat ((p - 1) div 2)" proof - have "card (F ∪ 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 ∪ D) = card C" by (simp add: C_card_eq_D_plus_E) from this show "card (F ∪ D) = nat ((p - 1) div 2)" by (simp add: C_card_eq_B B_card_eq) qed lemma F_Un_D_eq_A: "F ∪ 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" apply (insert F_D_disj finite_D finite_F) apply (frule setprod_Un_disjoint [of F D id]) apply (auto simp add: F_Un_D_eq_A) done lemma prod_F_zcong: "[setprod id F = ((-1) ^ (card E)) * (setprod id E)] (mod p)" proof - have "setprod id F = setprod id (op - p ` E)" by (auto simp add: F_def) then have "setprod id F = setprod (op - p) E" apply simp apply (insert finite_E inj_on_pminusx_E) apply (frule_tac f = "op - p" in setprod_reindex_id, auto) done then have one: "[setprod id F = setprod (StandardRes p o (op - p)) E] (mod p)" apply simp apply (insert p_g_0 finite_E) by (auto simp add: StandardRes_prod) moreover have a: "∀x ∈ 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: "∀x ∈ E. [StandardRes p (p - x) = p - x](mod p)" apply clarify apply (simp add: StandardRes_prop1 zcong_sym) done moreover have "∀x ∈ 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: "[setprod (StandardRes p o (op - p)) E = setprod (uminus) E](mod p)" apply simp apply (insert finite_E p_g_0) apply (rule setprod_same_function_zcong [of E "StandardRes p o (op - p)" uminus p], auto) done then have two: "[setprod id F = setprod (uminus) E](mod p)" apply (insert one c) apply (rule zcong_trans [of "setprod id F" "setprod (StandardRes p o op - p) E" p "setprod uminus E"], auto) done also have "setprod uminus E = (setprod id E) * (-1)^(card E)" using finite_E by (induct set: finite) auto then have "setprod uminus E = (-1) ^ (card E) * (setprod id E)" by (simp add: zmult_commute) 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 (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 "[setprod id A = setprod id F * setprod id D](mod p)" by (auto simp add: prod_D_F_eq_prod_A zmult_commute) then have "[setprod id A = ((-1)^(card E) * setprod id E) * setprod id D] (mod p)" apply (rule zcong_trans) apply (auto simp add: prod_F_zcong zcong_scalar) done then have "[setprod id A = ((-1)^(card E) * setprod id C)] (mod p)" apply (rule zcong_trans) apply (insert C_prod_eq_D_times_E, erule subst) apply (subst zmult_assoc, auto) done then have "[setprod id A = ((-1)^(card E) * setprod id B)] (mod p)" apply (rule zcong_trans) apply (simp add: C_B_zcong_prod zcong_scalar2) done then have "[setprod id A = ((-1)^(card E) * (setprod id ((%x. x * a) ` A)))] (mod p)" by (simp add: B_def) then have "[setprod id A = ((-1)^(card E) * (setprod (%x. x * a) A))] (mod p)" by (simp add:finite_A inj_on_xa_A setprod_reindex_id[symmetric]) moreover have "setprod (%x. x * a) A = setprod (%x. a) A * setprod id A" using finite_A by (induct set: finite) auto ultimately have "[setprod id A = ((-1)^(card E) * (setprod (%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 zcong_trans) apply (simp add: zcong_scalar2 zcong_scalar finite_A setprod_constant zmult_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 zcong_scalar) then have "[setprod id A * (-1)^(card E) = setprod id A * (-1)^(card E) * a^(card A) * (-1)^(card E)](mod p)" apply (rule zcong_trans) 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 zcong_trans) apply (simp add: aux) done with this zcong_cancel2 [of p "setprod 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
lemma p_odd:
p ∈ zOdd
lemma p_g_0:
0 < p
lemma int_nat:
int (nat ((p - 1) div 2)) = (p - 1) div 2
lemma p_minus_one_l:
(p - 1) div 2 < p
lemma p_eq:
p = 2 * (p - 1) div 2 + 1
lemma zodd_imp_zdiv_eq:
x ∈ zOdd ==> 2 * (x - 1) div 2 = 2 * ((x - 1) div 2)
lemma p_eq2:
p = 2 * ((p - 1) div 2) + 1
lemma finite_A:
finite A
lemma finite_B:
finite B
lemma finite_C:
finite C
lemma finite_D:
finite D
lemma finite_E:
finite E
lemma finite_F:
finite F
lemma C_eq:
C = D ∪ E
lemma A_card_eq:
card A = nat ((p - 1) div 2)
lemma inj_on_xa_A:
inj_on (λx. x * a) A
lemma A_res:
ResSet p A
lemma B_res:
ResSet p B
lemma SR_B_inj:
inj_on (StandardRes p) B
lemma inj_on_pminusx_E:
inj_on (op - p) E
lemma A_ncong_p:
x ∈ A ==> ¬ [x = 0] (mod p)
lemma A_greater_zero:
x ∈ A ==> 0 < x
lemma B_ncong_p:
x ∈ B ==> ¬ [x = 0] (mod p)
lemma B_greater_zero:
x ∈ B ==> 0 < x
lemma C_ncong_p:
x ∈ C ==> ¬ [x = 0] (mod p)
lemma C_greater_zero:
y ∈ C ==> 0 < y
lemma D_ncong_p:
x ∈ D ==> ¬ [x = 0] (mod p)
lemma E_ncong_p:
x ∈ E ==> ¬ [x = 0] (mod p)
lemma F_ncong_p:
x ∈ F ==> ¬ [x = 0] (mod p)
lemma F_subset:
F ⊆ {x. 0 < x ∧ x ≤ (p - 1) div 2}
lemma D_subset:
D ⊆ {x. 0 < x ∧ x ≤ (p - 1) div 2}
lemma F_eq:
F = {x. ∃y∈A. x = p - StandardRes p (y * a) ∧
(p - 1) div 2 < StandardRes p (y * a)}
lemma D_eq:
D = {x. ∃y∈A. x = StandardRes p (y * a) ∧ StandardRes p (y * a) ≤ (p - 1) div 2}
lemma D_leq:
x ∈ D ==> x ≤ (p - 1) div 2
lemma F_ge:
x ∈ F ==> x ≤ (p - 1) div 2
lemma all_A_relprime:
∀x∈A. zgcd (x, p) = 1
lemma A_prod_relprime:
zgcd (setprod id A, p) = 1
lemma B_card_eq_A:
card B = card A
lemma B_card_eq:
card B = nat ((p - 1) div 2)
lemma F_card_eq_E:
card F = card E
lemma C_card_eq_B:
card C = card B
lemma D_E_disj:
D ∩ E = {}
lemma C_card_eq_D_plus_E:
card C = card D + card E
lemma C_prod_eq_D_times_E:
setprod id E * setprod id D = setprod id C
lemma C_B_zcong_prod:
[setprod id C = setprod id B] (mod p)
lemma F_Un_D_subset:
F ∪ D ⊆ A
lemma F_D_disj:
F ∩ D = {}
lemma F_Un_D_card:
card (F ∪ D) = nat ((p - 1) div 2)
lemma F_Un_D_eq_A:
F ∪ D = A
lemma prod_D_F_eq_prod_A:
setprod id D * setprod id F = setprod id A
lemma prod_F_zcong:
[setprod id F = -1 ^ card E * setprod id E] (mod p)
lemma aux:
setprod id A * -1 ^ card E * a ^ card A * -1 ^ card E =
setprod id A * a ^ card A
theorem pre_gauss_lemma:
[a ^ nat ((p - 1) div 2) = -1 ^ card E] (mod p)
theorem gauss_lemma:
Legendre a p = -1 ^ card E