--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/NumberTheory/Euler.thy Thu Mar 20 15:58:25 2003 +0100
@@ -0,0 +1,345 @@
+(* Title: HOL/Quadratic_Reciprocity/Euler.thy
+ Authors: Jeremy Avigad, David Gray, and Adam Kramer
+ License: GPL (GNU GENERAL PUBLIC LICENSE)
+*)
+
+header {* Euler's criterion *}
+
+theory Euler = Residues + EvenOdd:;
+
+constdefs
+ MultInvPair :: "int => int => int => int set"
+ "MultInvPair a p j == {StandardRes p j, StandardRes p (a * (MultInv p j))}"
+ SetS :: "int => int => int set set"
+ "SetS a p == ((MultInvPair a p) ` (SRStar p))";
+
+(****************************************************************)
+(* *)
+(* Property for MultInvPair *)
+(* *)
+(****************************************************************)
+
+lemma MultInvPair_prop1a: "[| p \<in> zprime; 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)
+ apply (drule MultInv_zcong_prop2, auto)
+ apply (drule MultInv_zcong_prop3, auto simp add: zcong_sym)
+ apply (drule MultInv_zcong_prop1, auto)
+ apply (drule MultInv_zcong_prop2, auto)
+ apply (drule MultInv_zcong_prop2, auto)
+ apply (drule MultInv_zcong_prop3, auto simp add: zcong_sym)
+done
+
+lemma MultInvPair_prop1b: "[| p \<in> zprime; 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: "[| p \<in> zprime; 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: "[| p \<in> zprime; 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: "[| p \<in> zprime; 2 < p; ~([a = 0] (mod p));
+ ~([j = 0] (mod p));
+ ~(QuadRes p a) |] ==>
+ ~([j = a * MultInv p j] (mod p))";
+ apply auto
+proof -;
+ assume "p \<in> zprime" 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)";
+ apply(subgoal_tac "2 = Suc(Suc(0))");
+ apply (erule ssubst)
+ apply (auto simp only: power_Suc power_0)
+ by auto
+ with prems show False;
+ by (simp add: QuadRes_def)
+qed;
+
+lemma MultInvPair_card_two: "[| p \<in> zprime; 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 (simp only: StandardRes_prop2)
+ apply (drule MultInvPair_distinct)
+by auto
+
+(****************************************************************)
+(* *)
+(* 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: "[| p \<in> zprime; 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_finite_subsets)
+
+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: Finites, auto)
+
+lemma SetS_card: "[| p \<in> zprime; 2 < p; ~([a = 0] (mod p)); ~(QuadRes p a) |] ==>
+ int(card(SetS a p)) = (p - 1) div 2";
+proof -;
+ assume "p \<in> zprime" 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_sets)
+ also have "... = int(setsum (%x.2) (SetS a p))";
+ apply (insert prems)
+ apply (auto simp add: SetS_elems_card SetS_finite SetS_elems_finite
+ card_setsum_aux)
+ done
+ 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: "[| p \<in> zprime; 2 < p; ~([a = 0] (mod p));
+ ~(QuadRes p a); x \<in> (SetS a p) |] ==>
+ [setprod 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 (drule_tac a = "x * MultInv p x" and b = 1 in zcong_zmult_prop2)
+ 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 [rule_format]: "2 < p --> (SRStar p) = {1} \<union>
+ (d22set (p - 1))";
+ apply (induct p rule: d22set.induct, auto)
+ apply (simp add: SRStar_def d22set.simps, arith)
+ 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: "[| p \<in> zprime; 2 < p; ~([a = 0] (mod p)); ~(QuadRes p a) |] ==>
+ [setprod (Union (SetS a p)) = a ^ nat ((p - 1) div 2)] (mod p)";
+proof -;
+ assume "p \<in> zprime" and "2 < p" and "~([a = 0] (mod p))" and "~(QuadRes p a)";
+ then have "[setprod (Union (SetS a p)) =
+ gsetprod setprod (SetS a p)] (mod p)";
+ by (auto simp add: SetS_finite SetS_elems_finite
+ MultInvPair_prop1c setprod_disj_sets)
+ also; have "[gsetprod setprod (SetS a p) =
+ gsetprod (%x. a) (SetS a p)] (mod p)";
+ apply (rule gsetprod_same_function_zcong)
+ by (auto simp add: prems SetS_setprod_prop SetS_finite)
+ also (zcong_trans) have "[gsetprod (%x. a) (SetS a p) =
+ a^(card (SetS a p))] (mod p)";
+ by (auto simp add: prems SetS_finite gsetprod_const)
+ 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: "[| p \<in> zprime; 2 < p; ~([a = 0](mod p)) |] ==>
+ setprod (Union (SetS a p)) = zfact (p - 1)";
+proof -;
+ assume "p \<in> zprime" and "2 < p" and "~([a = 0](mod p))";
+ then have "setprod (Union (SetS a p)) = setprod (SRStar p)";
+ by (auto simp add: MultInvPair_prop2)
+ also have "... = setprod ({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))";
+ apply (insert prems, auto)
+ apply (drule d22set_g_1)
+ apply (auto simp add: d22set_fin)
+ done
+ then have "setprod({1} \<union> (d22set (p - 1))) = setprod (d22set (p - 1))";
+ by auto
+ then show ?thesis
+ by (auto simp add: d22set_prod_zfact)
+ qed;
+ finally show ?thesis .;
+qed;
+
+lemma zfact_prop: "[| p \<in> zprime; 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
+
+(****************************************************************)
+(* *)
+(* Prove the first part of Euler's Criterion: *)
+(* ~(QuadRes p x) |] ==> *)
+(* [x^(nat (((p) - 1) div 2)) = -1](mod p) *)
+(* *)
+(****************************************************************)
+
+lemma Euler_part1: "[| 2 < p; p \<in> zprime; ~([x = 0](mod p));
+ ~(QuadRes p x) |] ==>
+ [x^(nat (((p) - 1) div 2)) = -1](mod p)";
+ apply (frule zfact_prop, auto)
+ apply (frule Wilson_Russ)
+ apply (auto simp add: zcong_sym)
+ apply (rule zcong_trans, auto)
+done
+
+(********************************************************************)
+(* *)
+(* Prove another part of Euler Criterion: *)
+(* [a = 0] (mod p) ==> [0 = a ^ nat ((p - 1) div 2)] (mod p) *)
+(* *)
+(********************************************************************)
+
+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)
+ then 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; p \<in> zprime; [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
+
+(****************************************************************)
+(* *)
+(* Prove the final part of Euler's Criterion: *)
+(* QuadRes p x |] ==> *)
+(* [x^(nat (((p) - 1) div 2)) = 1](mod p) *)
+(* *)
+(****************************************************************)
+
+lemma aux__1: "[| ~([x = 0] (mod p)); [y ^ 2 = x] (mod p)|] ==> ~(p dvd y)";
+ apply (subgoal_tac "[| ~([x = 0] (mod p)); [y ^ 2 = x] (mod p)|] ==>
+ ~([y ^ 2 = 0] (mod p))");
+ apply (auto simp add: zcong_sym [of "y^2" x p] intro: zcong_trans)
+ apply (auto simp add: zcong_eq_zdvd_prop intro: zpower_zdvd_prop1)
+done
+
+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; p \<in> zprime; ~([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)
+ 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 (simp add: aux__2)
+ apply (frule odd_minus_one_even)
+ apply (frule even_div_2_prop2)
+ apply (auto intro: Little_Fermat simp add: zprime_zOdd_eq_grt_2)
+done
+
+(********************************************************************)
+(* *)
+(* Finally show Euler's Criterion *)
+(* *)
+(********************************************************************)
+
+theorem Euler_Criterion: "[| 2 < p; p \<in> zprime |] ==> [(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;
\ No newline at end of file