--- a/src/HOL/NumberTheory/Euler.thy Tue Sep 01 14:13:34 2009 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,304 +0,0 @@
-(* Title: HOL/Quadratic_Reciprocity/Euler.thy
- ID: $Id$
- Authors: Jeremy Avigad, David Gray, and Adam Kramer
-*)
-
-header {* Euler's criterion *}
-
-theory Euler imports Residues EvenOdd begin
-
-definition
- MultInvPair :: "int => int => int => int set" where
- "MultInvPair a p j = {StandardRes p j, StandardRes p (a * (MultInv p j))}"
-
-definition
- SetS :: "int => int => int set set" where
- "SetS a p = (MultInvPair a p ` SRStar p)"
-
-
-subsection {* Property for MultInvPair *}
-
-lemma MultInvPair_prop1a:
- "[| zprime p; 2 < p; ~([a = 0](mod p));
- X \<in> (SetS a p); Y \<in> (SetS a p);
- ~((X \<inter> Y) = {}) |] ==> X = Y"
- apply (auto simp add: SetS_def)
- apply (drule StandardRes_SRStar_prop1a)+ defer 1
- apply (drule StandardRes_SRStar_prop1a)+
- apply (auto simp add: MultInvPair_def StandardRes_prop2 zcong_sym)
- apply (drule notE, rule MultInv_zcong_prop1, auto)[]
- apply (drule notE, rule MultInv_zcong_prop2, auto simp add: zcong_sym)[]
- apply (drule MultInv_zcong_prop2, auto simp add: zcong_sym)[]
- apply (drule MultInv_zcong_prop3, auto simp add: zcong_sym)[]
- apply (drule MultInv_zcong_prop1, auto)[]
- apply (drule MultInv_zcong_prop2, auto simp add: zcong_sym)[]
- apply (drule MultInv_zcong_prop2, auto simp add: zcong_sym)[]
- apply (drule MultInv_zcong_prop3, auto simp add: zcong_sym)[]
- done
-
-lemma MultInvPair_prop1b:
- "[| zprime p; 2 < p; ~([a = 0](mod p));
- X \<in> (SetS a p); Y \<in> (SetS a p);
- X \<noteq> Y |] ==> X \<inter> Y = {}"
- apply (rule notnotD)
- apply (rule notI)
- apply (drule MultInvPair_prop1a, auto)
- done
-
-lemma MultInvPair_prop1c: "[| zprime p; 2 < p; ~([a = 0](mod p)) |] ==>
- \<forall>X \<in> SetS a p. \<forall>Y \<in> SetS a p. X \<noteq> Y --> X\<inter>Y = {}"
- by (auto simp add: MultInvPair_prop1b)
-
-lemma MultInvPair_prop2: "[| zprime p; 2 < p; ~([a = 0](mod p)) |] ==>
- Union ( SetS a p) = SRStar p"
- apply (auto simp add: SetS_def MultInvPair_def StandardRes_SRStar_prop4
- SRStar_mult_prop2)
- apply (frule StandardRes_SRStar_prop3)
- apply (rule bexI, auto)
- done
-
-lemma MultInvPair_distinct: "[| zprime p; 2 < p; ~([a = 0] (mod p));
- ~([j = 0] (mod p));
- ~(QuadRes p a) |] ==>
- ~([j = a * MultInv p j] (mod p))"
-proof
- assume "zprime p" and "2 < p" and "~([a = 0] (mod p))" and
- "~([j = 0] (mod p))" and "~(QuadRes p a)"
- assume "[j = a * MultInv p j] (mod p)"
- then have "[j * j = (a * MultInv p j) * j] (mod p)"
- by (auto simp add: zcong_scalar)
- then have a:"[j * j = a * (MultInv p j * j)] (mod p)"
- by (auto simp add: zmult_ac)
- have "[j * j = a] (mod p)"
- proof -
- from prems have b: "[MultInv p j * j = 1] (mod p)"
- by (simp add: MultInv_prop2a)
- from b a show ?thesis
- by (auto simp add: zcong_zmult_prop2)
- qed
- then have "[j^2 = a] (mod p)"
- by (metis number_of_is_id power2_eq_square succ_bin_simps)
- with prems show False
- by (simp add: QuadRes_def)
-qed
-
-lemma MultInvPair_card_two: "[| zprime p; 2 < p; ~([a = 0] (mod p));
- ~(QuadRes p a); ~([j = 0] (mod p)) |] ==>
- card (MultInvPair a p j) = 2"
- apply (auto simp add: MultInvPair_def)
- apply (subgoal_tac "~ (StandardRes p j = StandardRes p (a * MultInv p j))")
- apply auto
- apply (metis MultInvPair_distinct Pls_def StandardRes_def aux number_of_is_id one_is_num_one)
- done
-
-
-subsection {* Properties of SetS *}
-
-lemma SetS_finite: "2 < p ==> finite (SetS a p)"
- by (auto simp add: SetS_def SRStar_finite [of p] finite_imageI)
-
-lemma SetS_elems_finite: "\<forall>X \<in> SetS a p. finite X"
- by (auto simp add: SetS_def MultInvPair_def)
-
-lemma SetS_elems_card: "[| zprime p; 2 < p; ~([a = 0] (mod p));
- ~(QuadRes p a) |] ==>
- \<forall>X \<in> SetS a p. card X = 2"
- apply (auto simp add: SetS_def)
- apply (frule StandardRes_SRStar_prop1a)
- apply (rule MultInvPair_card_two, auto)
- done
-
-lemma Union_SetS_finite: "2 < p ==> finite (Union (SetS a p))"
- by (auto simp add: SetS_finite SetS_elems_finite finite_Union)
-
-lemma card_setsum_aux: "[| finite S; \<forall>X \<in> S. finite (X::int set);
- \<forall>X \<in> S. card X = n |] ==> setsum card S = setsum (%x. n) S"
- by (induct set: finite) auto
-
-lemma SetS_card: "[| zprime p; 2 < p; ~([a = 0] (mod p)); ~(QuadRes p a) |] ==>
- int(card(SetS a p)) = (p - 1) div 2"
-proof -
- assume "zprime p" and "2 < p" and "~([a = 0] (mod p))" and "~(QuadRes p a)"
- then have "(p - 1) = 2 * int(card(SetS a p))"
- proof -
- have "p - 1 = int(card(Union (SetS a p)))"
- by (auto simp add: prems MultInvPair_prop2 SRStar_card)
- also have "... = int (setsum card (SetS a p))"
- by (auto simp add: prems SetS_finite SetS_elems_finite
- MultInvPair_prop1c [of p a] card_Union_disjoint)
- also have "... = int(setsum (%x.2) (SetS a p))"
- using prems
- by (auto simp add: SetS_elems_card SetS_finite SetS_elems_finite
- card_setsum_aux simp del: setsum_constant)
- also have "... = 2 * int(card( SetS a p))"
- by (auto simp add: prems SetS_finite setsum_const2)
- finally show ?thesis .
- qed
- from this show ?thesis
- by auto
-qed
-
-lemma SetS_setprod_prop: "[| zprime p; 2 < p; ~([a = 0] (mod p));
- ~(QuadRes p a); x \<in> (SetS a p) |] ==>
- [\<Prod>x = a] (mod p)"
- apply (auto simp add: SetS_def MultInvPair_def)
- apply (frule StandardRes_SRStar_prop1a)
- apply (subgoal_tac "StandardRes p x \<noteq> StandardRes p (a * MultInv p x)")
- apply (auto simp add: StandardRes_prop2 MultInvPair_distinct)
- apply (frule_tac m = p and x = x and y = "(a * MultInv p x)" in
- StandardRes_prop4)
- apply (subgoal_tac "[x * (a * MultInv p x) = a * (x * MultInv p x)] (mod p)")
- apply (drule_tac a = "StandardRes p x * StandardRes p (a * MultInv p x)" and
- b = "x * (a * MultInv p x)" and
- c = "a * (x * MultInv p x)" in zcong_trans, force)
- apply (frule_tac p = p and x = x in MultInv_prop2, auto)
-apply (metis StandardRes_SRStar_prop3 mult_1_right mult_commute zcong_sym zcong_zmult_prop1)
- apply (auto simp add: zmult_ac)
- done
-
-lemma aux1: "[| 0 < x; (x::int) < a; x \<noteq> (a - 1) |] ==> x < a - 1"
- by arith
-
-lemma aux2: "[| (a::int) < c; b < c |] ==> (a \<le> b | b \<le> a)"
- by auto
-
-lemma SRStar_d22set_prop: "2 < p \<Longrightarrow> (SRStar p) = {1} \<union> (d22set (p - 1))"
- apply (induct p rule: d22set.induct)
- apply auto
- apply (simp add: SRStar_def d22set.simps)
- apply (simp add: SRStar_def d22set.simps, clarify)
- apply (frule aux1)
- apply (frule aux2, auto)
- apply (simp_all add: SRStar_def)
- apply (simp add: d22set.simps)
- apply (frule d22set_le)
- apply (frule d22set_g_1, auto)
- done
-
-lemma Union_SetS_setprod_prop1: "[| zprime p; 2 < p; ~([a = 0] (mod p)); ~(QuadRes p a) |] ==>
- [\<Prod>(Union (SetS a p)) = a ^ nat ((p - 1) div 2)] (mod p)"
-proof -
- assume "zprime p" and "2 < p" and "~([a = 0] (mod p))" and "~(QuadRes p a)"
- then have "[\<Prod>(Union (SetS a p)) =
- setprod (setprod (%x. x)) (SetS a p)] (mod p)"
- by (auto simp add: SetS_finite SetS_elems_finite
- MultInvPair_prop1c setprod_Union_disjoint)
- also have "[setprod (setprod (%x. x)) (SetS a p) =
- setprod (%x. a) (SetS a p)] (mod p)"
- by (rule setprod_same_function_zcong)
- (auto simp add: prems SetS_setprod_prop SetS_finite)
- also (zcong_trans) have "[setprod (%x. a) (SetS a p) =
- a^(card (SetS a p))] (mod p)"
- by (auto simp add: prems SetS_finite setprod_constant)
- finally (zcong_trans) show ?thesis
- apply (rule zcong_trans)
- apply (subgoal_tac "card(SetS a p) = nat((p - 1) div 2)", auto)
- apply (subgoal_tac "nat(int(card(SetS a p))) = nat((p - 1) div 2)", force)
- apply (auto simp add: prems SetS_card)
- done
-qed
-
-lemma Union_SetS_setprod_prop2: "[| zprime p; 2 < p; ~([a = 0](mod p)) |] ==>
- \<Prod>(Union (SetS a p)) = zfact (p - 1)"
-proof -
- assume "zprime p" and "2 < p" and "~([a = 0](mod p))"
- then have "\<Prod>(Union (SetS a p)) = \<Prod>(SRStar p)"
- by (auto simp add: MultInvPair_prop2)
- also have "... = \<Prod>({1} \<union> (d22set (p - 1)))"
- by (auto simp add: prems SRStar_d22set_prop)
- also have "... = zfact(p - 1)"
- proof -
- have "~(1 \<in> d22set (p - 1)) & finite( d22set (p - 1))"
- by (metis d22set_fin d22set_g_1 linorder_neq_iff)
- then have "\<Prod>({1} \<union> (d22set (p - 1))) = \<Prod>(d22set (p - 1))"
- by auto
- then show ?thesis
- by (auto simp add: d22set_prod_zfact)
- qed
- finally show ?thesis .
-qed
-
-lemma zfact_prop: "[| zprime p; 2 < p; ~([a = 0] (mod p)); ~(QuadRes p a) |] ==>
- [zfact (p - 1) = a ^ nat ((p - 1) div 2)] (mod p)"
- apply (frule Union_SetS_setprod_prop1)
- apply (auto simp add: Union_SetS_setprod_prop2)
- done
-
-text {* \medskip Prove the first part of Euler's Criterion: *}
-
-lemma Euler_part1: "[| 2 < p; zprime p; ~([x = 0](mod p));
- ~(QuadRes p x) |] ==>
- [x^(nat (((p) - 1) div 2)) = -1](mod p)"
- by (metis Wilson_Russ number_of_is_id zcong_sym zcong_trans zfact_prop)
-
-text {* \medskip Prove another part of Euler Criterion: *}
-
-lemma aux_1: "0 < p ==> (a::int) ^ nat (p) = a * a ^ (nat (p) - 1)"
-proof -
- assume "0 < p"
- then have "a ^ (nat p) = a ^ (1 + (nat p - 1))"
- by (auto simp add: diff_add_assoc)
- also have "... = (a ^ 1) * a ^ (nat(p) - 1)"
- by (simp only: zpower_zadd_distrib)
- also have "... = a * a ^ (nat(p) - 1)"
- by auto
- finally show ?thesis .
-qed
-
-lemma aux_2: "[| (2::int) < p; p \<in> zOdd |] ==> 0 < ((p - 1) div 2)"
-proof -
- assume "2 < p" and "p \<in> zOdd"
- then have "(p - 1):zEven"
- by (auto simp add: zEven_def zOdd_def)
- then have aux_1: "2 * ((p - 1) div 2) = (p - 1)"
- by (auto simp add: even_div_2_prop2)
- with `2 < p` have "1 < (p - 1)"
- by auto
- then have " 1 < (2 * ((p - 1) div 2))"
- by (auto simp add: aux_1)
- then have "0 < (2 * ((p - 1) div 2)) div 2"
- by auto
- then show ?thesis by auto
-qed
-
-lemma Euler_part2:
- "[| 2 < p; zprime p; [a = 0] (mod p) |] ==> [0 = a ^ nat ((p - 1) div 2)] (mod p)"
- apply (frule zprime_zOdd_eq_grt_2)
- apply (frule aux_2, auto)
- apply (frule_tac a = a in aux_1, auto)
- apply (frule zcong_zmult_prop1, auto)
- done
-
-text {* \medskip Prove the final part of Euler's Criterion: *}
-
-lemma aux__1: "[| ~([x = 0] (mod p)); [y ^ 2 = x] (mod p)|] ==> ~(p dvd y)"
- by (metis dvdI power2_eq_square zcong_sym zcong_trans zcong_zero_equiv_div dvd_trans)
-
-lemma aux__2: "2 * nat((p - 1) div 2) = nat (2 * ((p - 1) div 2))"
- by (auto simp add: nat_mult_distrib)
-
-lemma Euler_part3: "[| 2 < p; zprime p; ~([x = 0](mod p)); QuadRes p x |] ==>
- [x^(nat (((p) - 1) div 2)) = 1](mod p)"
- apply (subgoal_tac "p \<in> zOdd")
- apply (auto simp add: QuadRes_def)
- prefer 2
- apply (metis number_of_is_id numeral_1_eq_1 zprime_zOdd_eq_grt_2)
- apply (frule aux__1, auto)
- apply (drule_tac z = "nat ((p - 1) div 2)" in zcong_zpower)
- apply (auto simp add: zpower_zpower)
- apply (rule zcong_trans)
- apply (auto simp add: zcong_sym [of "x ^ nat ((p - 1) div 2)"])
- apply (metis Little_Fermat even_div_2_prop2 mult_Bit0 number_of_is_id odd_minus_one_even one_is_num_one zmult_1 aux__2)
- done
-
-
-text {* \medskip Finally show Euler's Criterion: *}
-
-theorem Euler_Criterion: "[| 2 < p; zprime p |] ==> [(Legendre a p) =
- a^(nat (((p) - 1) div 2))] (mod p)"
- apply (auto simp add: Legendre_def Euler_part2)
- apply (frule Euler_part3, auto simp add: zcong_sym)[]
- apply (frule Euler_part1, auto simp add: zcong_sym)[]
- done
-
-end