# Theory EvenOdd

theory EvenOdd
imports Int2
```(*  Title:      HOL/Old_Number_Theory/EvenOdd.thy
*)

section ‹Parity: Even and Odd Integers›

theory EvenOdd
imports Int2
begin

definition zOdd :: "int set"
where "zOdd = {x. ∃k. x = 2 * k + 1}"

definition zEven :: "int set"
where "zEven = {x. ∃k. x = 2 * k}"

subsection ‹Some useful properties about even and odd›

lemma zOddI [intro?]: "x = 2 * k + 1 ⟹ x ∈ zOdd"
and zOddE [elim?]: "x ∈ zOdd ⟹ (!!k. x = 2 * k + 1 ⟹ C) ⟹ C"

lemma zEvenI [intro?]: "x = 2 * k ⟹ x ∈ zEven"
and zEvenE [elim?]: "x ∈ zEven ⟹ (!!k. x = 2 * k ⟹ C) ⟹ C"

lemma one_not_even: "~(1 ∈ zEven)"
proof
assume "1 ∈ zEven"
then obtain k :: int where "1 = 2 * k" ..
then show False by arith
qed

lemma even_odd_conj: "~(x ∈ zOdd & x ∈ zEven)"
proof -
{
fix a b
assume "2 * (a::int) = 2 * (b::int) + 1"
then have "2 * (a::int) - 2 * (b :: int) = 1"
by arith
then have "2 * (a - b) = 1"
moreover have "(2 * (a - b)):zEven"
by (auto simp only: zEven_def)
ultimately have False
}
then show ?thesis
by (auto simp add: zOdd_def zEven_def)
qed

lemma even_odd_disj: "(x ∈ zOdd | x ∈ zEven)"
by (simp add: zOdd_def zEven_def) arith

lemma not_odd_impl_even: "~(x ∈ zOdd) ==> x ∈ zEven"
using even_odd_disj by auto

lemma odd_mult_odd_prop: "(x*y):zOdd ==> x ∈ zOdd"
proof (rule classical)
assume "¬ ?thesis"
then have "x ∈ zEven" by (rule not_odd_impl_even)
then obtain a where a: "x = 2 * a" ..
assume "x * y : zOdd"
then obtain b where "x * y = 2 * b + 1" ..
with a have "2 * a * y = 2 * b + 1" by simp
then have "2 * a * y - 2 * b = 1"
by arith
then have "2 * (a * y - b) = 1"
moreover have "(2 * (a * y - b)):zEven"
by (auto simp only: zEven_def)
ultimately have False
then show ?thesis ..
qed

lemma odd_minus_one_even: "x ∈ zOdd ==> (x - 1):zEven"
by (auto simp add: zOdd_def zEven_def)

lemma even_div_2_prop1: "x ∈ zEven ==> (x mod 2) = 0"

lemma even_div_2_prop2: "x ∈ zEven ==> (2 * (x div 2)) = x"

lemma even_plus_even: "[| x ∈ zEven; y ∈ zEven |] ==> x + y ∈ zEven"
apply (auto simp only: distrib_left [symmetric])
done

lemma even_times_either: "x ∈ zEven ==> x * y ∈ zEven"

lemma even_minus_even: "[| x ∈ zEven; y ∈ zEven |] ==> x - y ∈ zEven"
apply (auto simp only: right_diff_distrib [symmetric])
done

lemma odd_minus_odd: "[| x ∈ zOdd; y ∈ zOdd |] ==> x - y ∈ zEven"
apply (auto simp add: zOdd_def zEven_def)
apply (auto simp only: right_diff_distrib [symmetric])
done

lemma even_minus_odd: "[| x ∈ zEven; y ∈ zOdd |] ==> x - y ∈ zOdd"
apply (auto simp add: zOdd_def zEven_def)
apply (rule_tac x = "k - ka - 1" in exI)
apply auto
done

lemma odd_minus_even: "[| x ∈ zOdd; y ∈ zEven |] ==> x - y ∈ zOdd"
apply (auto simp add: zOdd_def zEven_def)
apply (auto simp only: right_diff_distrib [symmetric])
done

lemma odd_times_odd: "[| x ∈ zOdd;  y ∈ zOdd |] ==> x * y ∈ zOdd"
apply (auto simp add: zOdd_def distrib_right distrib_left)
apply (rule_tac x = "2 * ka * k + ka + k" in exI)
done

lemma odd_iff_not_even: "(x ∈ zOdd) = (~ (x ∈ zEven))"
using even_odd_conj even_odd_disj by auto

lemma even_product: "x * y ∈ zEven ==> x ∈ zEven | y ∈ zEven"
using odd_iff_not_even odd_times_odd by auto

lemma even_diff: "x - y ∈ zEven = ((x ∈ zEven) = (y ∈ zEven))"
proof
assume xy: "x - y ∈ zEven"
{
assume x: "x ∈ zEven"
have "y ∈ zEven"
proof (rule classical)
assume "¬ ?thesis"
then have "y ∈ zOdd"
with x have "x - y ∈ zOdd"
with xy have False
then show ?thesis ..
qed
} moreover {
assume y: "y ∈ zEven"
have "x ∈ zEven"
proof (rule classical)
assume "¬ ?thesis"
then have "x ∈ zOdd"
with y have "x - y ∈ zOdd"
with xy have False
then show ?thesis ..
qed
}
ultimately show "(x ∈ zEven) = (y ∈ zEven)"
by (auto simp add: odd_iff_not_even even_minus_even odd_minus_odd
even_minus_odd odd_minus_even)
next
assume "(x ∈ zEven) = (y ∈ zEven)"
then show "x - y ∈ zEven"
by (auto simp add: odd_iff_not_even even_minus_even odd_minus_odd
even_minus_odd odd_minus_even)
qed

lemma neg_one_even_power: "[| x ∈ zEven; 0 ≤ x |] ==> (-1::int)^(nat x) = 1"
proof -
assume "x ∈ zEven" and "0 ≤ x"
from ‹x ∈ zEven› obtain a where "x = 2 * a" ..
with ‹0 ≤ x› have "0 ≤ a" by simp
from ‹0 ≤ x› and ‹x = 2 * a› have "nat x = nat (2 * a)"
by simp
also from ‹x = 2 * a› have "nat (2 * a) = 2 * nat a"
finally have "(-1::int)^nat x = (-1)^(2 * nat a)"
by simp
also have "... = (-1::int)⇧2 ^ nat a"
also have "(-1::int)⇧2 = 1"
by simp
finally show ?thesis
by simp
qed

lemma neg_one_odd_power: "[| x ∈ zOdd; 0 ≤ x |] ==> (-1::int)^(nat x) = -1"
proof -
assume "x ∈ zOdd" and "0 ≤ x"
from ‹x ∈ zOdd› obtain a where "x = 2 * a + 1" ..
with ‹0 ≤ x› have a: "0 ≤ a" by simp
with ‹0 ≤ x› and ‹x = 2 * a + 1› have "nat x = nat (2 * a + 1)"
by simp
also from a have "nat (2 * a + 1) = 2 * nat a + 1"
finally have "(-1::int) ^ nat x = (-1)^(2 * nat a + 1)"
by simp
also have "... = ((-1::int)⇧2) ^ nat a * (-1)^1"
also have "(-1::int)⇧2 = 1"
by simp
finally show ?thesis
by simp
qed

lemma neg_one_power_parity: "[| 0 ≤ x; 0 ≤ y; (x ∈ zEven) = (y ∈ zEven) |] ==>
(-1::int)^(nat x) = (-1::int)^(nat y)"
using even_odd_disj [of x] even_odd_disj [of y]
by (auto simp add: neg_one_even_power neg_one_odd_power)

lemma one_not_neg_one_mod_m: "2 < m ==> ~([1 = -1] (mod m))"
by (auto simp add: zcong_def zdvd_not_zless)

lemma even_div_2_l: "[| y ∈ zEven; x < y |] ==> x div 2 < y div 2"
proof -
assume "y ∈ zEven" and "x < y"
from ‹y ∈ zEven› obtain k where k: "y = 2 * k" ..
with ‹x < y› have "x < 2 * k" by simp
then have "x div 2 < k" by (auto simp add: div_prop1)
also have "k = (2 * k) div 2" by simp
finally have "x div 2 < 2 * k div 2" by simp
with k show ?thesis by simp
qed

lemma even_sum_div_2: "[| x ∈ zEven; y ∈ zEven |] ==> (x + y) div 2 = x div 2 + y div 2"

lemma even_prod_div_2: "[| x ∈ zEven |] ==> (x * y) div 2 = (x div 2) * y"

(* An odd prime is greater than 2 *)

lemma zprime_zOdd_eq_grt_2: "zprime p ==> (p ∈ zOdd) = (2 < p)"
apply (auto simp add: zOdd_def zprime_def)
apply (drule_tac x = 2 in allE)
using odd_iff_not_even [of p]
apply (auto simp add: zOdd_def zEven_def)
done

(* Powers of -1 and parity *)

lemma neg_one_special: "finite A ==>
((- 1) ^ card A) * ((- 1) ^ card A) = (1 :: int)"