(* 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 simp del: setsum_constant_nat)
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