# Theory Euler

theory Euler
imports Residues EvenOdd
```(*  Title:      HOL/Old_Number_Theory/Euler.thy
*)

section ‹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 ∈ (SetS a p); Y ∈ (SetS a p);
~((X ∩ Y) = {}) |] ==> X = Y"
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 ∈ (SetS a p); Y ∈ (SetS a p);
X ≠ Y |] ==> X ∩ Y = {}"
apply (rule notnotD)
apply (rule notI)
apply (drule MultInvPair_prop1a, auto)
done

lemma MultInvPair_prop1c: "[| zprime p; 2 < p; ~([a = 0](mod p)) |] ==>
∀X ∈ SetS a p. ∀Y ∈ SetS a p. X ≠ Y --> X∩Y = {}"

lemma MultInvPair_prop2: "[| zprime p; 2 < p; ~([a = 0](mod p)) |] ==>
⋃(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:
assumes "zprime p" and "2 < p" and
"~([a = 0] (mod p))" and
"~([j = 0] (mod p))" and
shows "~([j = a * MultInv p j] (mod p))"
proof
assume "[j = a * MultInv p j] (mod p)"
then have "[j * j = (a * MultInv p j) * j] (mod p)"
then have a:"[j * j = a * (MultInv p j * j)] (mod p)"
have "[j * j = a] (mod p)"
proof -
from assms(1,2,4) have "[MultInv p j * j = 1] (mod p)"
from this and a show ?thesis
qed
then have "[j⇧2 = a] (mod p)" by (simp add: power2_eq_square)
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 (subgoal_tac "~ (StandardRes p j = StandardRes p (a * MultInv p j))")
apply auto
apply (metis MultInvPair_distinct StandardRes_def aux)
done

subsection ‹Properties of SetS›

lemma SetS_finite: "2 < p ==> finite (SetS a p)"
by (auto simp add: SetS_def SRStar_finite [of p])

lemma SetS_elems_finite: "∀X ∈ 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));
∀X ∈ SetS a p. card X = 2"
apply (frule StandardRes_SRStar_prop1a)
apply (rule MultInvPair_card_two, auto)
done

lemma Union_SetS_finite: "2 < p ==> finite (⋃(SetS a p))"
by (auto simp add: SetS_finite SetS_elems_finite)

lemma card_setsum_aux: "[| finite S; ∀X ∈ S. finite (X::int set);
∀X ∈ S. card X = n |] ==> setsum card S = setsum (%x. n) S"
by (induct set: finite) auto

lemma SetS_card:
assumes "zprime p" and "2 < p" and "~([a = 0] (mod p))" and "~(QuadRes p a)"
shows "int(card(SetS a p)) = (p - 1) div 2"
proof -
have "(p - 1) = 2 * int(card(SetS a p))"
proof -
have "p - 1 = int(card(⋃(SetS a p)))"
by (auto simp add: assms MultInvPair_prop2 SRStar_card)
also have "... = int (setsum card (SetS a p))"
by (auto simp add: assms SetS_finite SetS_elems_finite
MultInvPair_prop1c [of p a] card_Union_disjoint)
also have "... = int(setsum (%x.2) (SetS a p))"
using assms 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: assms SetS_finite setsum_const2)
finally show ?thesis .
qed
then show ?thesis by auto
qed

lemma SetS_setprod_prop: "[| zprime p; 2 < p; ~([a = 0] (mod p));
~(QuadRes p a); x ∈ (SetS a p) |] ==>
[∏x = a] (mod p)"
apply (auto simp add: SetS_def MultInvPair_def)
apply (frule StandardRes_SRStar_prop1a)
apply hypsubst_thin
apply (subgoal_tac "StandardRes p x ≠ 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)
done

lemma aux1: "[| 0 < x; (x::int) < a; x ≠ (a - 1) |] ==> x < a - 1"
by arith

lemma aux2: "[| (a::int) < c; b < c |] ==> (a ≤ b | b ≤ a)"
by auto

lemma d22set_induct_old: "(⋀a::int. 1 < a ⟶ P (a - 1) ⟹ P a) ⟹ P x"
using d22set.induct by blast

lemma SRStar_d22set_prop: "2 < p ⟹ (SRStar p) = {1} ∪ (d22set (p - 1))"
apply (induct p rule: d22set_induct_old)
apply auto
apply (simp add: SRStar_def d22set.simps, clarify)
apply (frule aux1)
apply (frule aux2, auto)
apply (frule d22set_le)
apply (frule d22set_g_1, auto)
done

lemma Union_SetS_setprod_prop1:
assumes "zprime p" and "2 < p" and "~([a = 0] (mod p))" and
shows "[∏(⋃(SetS a p)) = a ^ nat ((p - 1) div 2)] (mod p)"
proof -
from assms have "[∏(⋃(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: assms 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: assms 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: assms SetS_card)
done
qed

lemma Union_SetS_setprod_prop2:
assumes "zprime p" and "2 < p" and "~([a = 0](mod p))"
shows "∏(⋃(SetS a p)) = zfact (p - 1)"
proof -
from assms have "∏(⋃(SetS a p)) = ∏(SRStar p)"
also have "... = ∏({1} ∪ (d22set (p - 1)))"
by (auto simp add: assms SRStar_d22set_prop)
also have "... = zfact(p - 1)"
proof -
have "~(1 ∈ d22set (p - 1)) & finite( d22set (p - 1))"
by (metis d22set_fin d22set_g_1 linorder_neq_iff)
then have "∏({1} ∪ (d22set (p - 1))) = ∏(d22set (p - 1))"
by auto
then show ?thesis
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)
done

text ‹\medskip Prove the first part of Euler's Criterion:›

lemma Euler_part1: "[| 2 < p; zprime p; ~([x = 0](mod p));
[x^(nat (((p) - 1) div 2)) = -1](mod p)"
by (metis Wilson_Russ 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))"
also have "... = (a ^ 1) * a ^ (nat(p) - 1)"
also have "... = a * a ^ (nat(p) - 1)"
by auto
finally show ?thesis .
qed

lemma aux_2: "[| (2::int) < p; p ∈ zOdd |] ==> 0 < ((p - 1) div 2)"
proof -
assume "2 < p" and "p ∈ 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)"
with ‹2 < p› have "1 < (p - 1)"
by auto
then have " 1 < (2 * ((p - 1) div 2))"
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))"

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 ∈ zOdd")
prefer 2
apply (metis 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: power_mult [symmetric])
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 odd_minus_one_even mult_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
```