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"
by (auto simp add: zOdd_def)
lemma zEvenI [intro?]: "x = 2 * k ⟹ x ∈ zEven"
and zEvenE [elim?]: "x ∈ zEven ⟹ (!!k. x = 2 * k ⟹ C) ⟹ C"
by (auto simp add: zEven_def)
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"
by (auto simp add: left_diff_distrib)
moreover have "(2 * (a - b)):zEven"
by (auto simp only: zEven_def)
ultimately have False
by (auto simp add: one_not_even)
}
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"
by (auto simp add: left_diff_distrib)
moreover have "(2 * (a * y - b)):zEven"
by (auto simp only: zEven_def)
ultimately have False
by (auto simp add: one_not_even)
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"
by (auto simp add: zEven_def)
lemma even_div_2_prop2: "x ∈ zEven ==> (2 * (x div 2)) = x"
by (auto simp add: zEven_def)
lemma even_plus_even: "[| x ∈ zEven; y ∈ zEven |] ==> x + y ∈ zEven"
apply (auto simp add: zEven_def)
apply (auto simp only: distrib_left [symmetric])
done
lemma even_times_either: "x ∈ zEven ==> x * y ∈ zEven"
by (auto simp add: zEven_def)
lemma even_minus_even: "[| x ∈ zEven; y ∈ zEven |] ==> x - y ∈ zEven"
apply (auto simp add: zEven_def)
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)
apply (auto simp add: distrib_right)
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"
by (simp add: odd_iff_not_even)
with x have "x - y ∈ zOdd"
by (simp add: even_minus_odd)
with xy have False
by (auto simp add: odd_iff_not_even)
then show ?thesis ..
qed
} moreover {
assume y: "y ∈ zEven"
have "x ∈ zEven"
proof (rule classical)
assume "¬ ?thesis"
then have "x ∈ zOdd"
by (auto simp add: odd_iff_not_even)
with y have "x - y ∈ zOdd"
by (simp add: odd_minus_even)
with xy have False
by (auto simp add: odd_iff_not_even)
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"
by (simp add: nat_mult_distrib)
finally have "(-1::int)^nat x = (-1)^(2 * nat a)"
by simp
also have "... = (-1::int)⇧2 ^ nat a"
by (simp add: power_mult)
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"
by (auto simp add: nat_mult_distrib nat_add_distrib)
finally have "(-1::int) ^ nat x = (-1)^(2 * nat a + 1)"
by simp
also have "... = ((-1::int)⇧2) ^ nat a * (-1)^1"
by (auto simp add: power_mult power_add)
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"
by (auto simp add: zEven_def)
lemma even_prod_div_2: "[| x ∈ zEven |] ==> (x * y) div 2 = (x div 2) * y"
by (auto simp add: zEven_def)
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
lemma neg_one_special: "finite A ==>
((- 1) ^ card A) * ((- 1) ^ card A) = (1 :: int)"
unfolding power_add [symmetric] by simp
lemma neg_one_power: "(-1::int)^n = 1 | (-1::int)^n = -1"
by (induct n) auto
lemma neg_one_power_eq_mod_m: "[| 2 < m; [(-1::int)^j = (-1::int)^k] (mod m) |]
==> ((-1::int)^j = (-1::int)^k)"
using neg_one_power [of j] and ListMem.insert neg_one_power [of k]
by (auto simp add: one_not_neg_one_mod_m zcong_sym)
end