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``````use core::num::Wrapping;

/// Defines an additive identity element for `Self`.
///
/// # Laws
///
/// ```text
/// a + 0 = a       ∀ a ∈ Self
/// 0 + a = a       ∀ a ∈ Self
/// ```
pub trait Zero: Sized + Add<Self, Output = Self> {
/// Returns the additive identity element of `Self`, `0`.
/// # Purity
///
/// This function should return the same result at all times regardless of
/// external mutable state, for example values stored in TLS or in
/// `static mut`s.
// This cannot be an associated constant, because of bignums.
fn zero() -> Self;

/// Sets `self` to the additive identity element of `Self`, `0`.
fn set_zero(&mut self) {
*self = Zero::zero();
}

/// Returns `true` if `self` is equal to the additive identity.
fn is_zero(&self) -> bool;
}

/// Defines an associated constant representing the additive identity element
/// for `Self`.
pub trait ConstZero: Zero {
/// The additive identity element of `Self`, `0`.
const ZERO: Self;
}

macro_rules! zero_impl {
(\$t:ty, \$v:expr) => {
impl Zero for \$t {
#[inline]
fn zero() -> \$t {
\$v
}
#[inline]
fn is_zero(&self) -> bool {
*self == \$v
}
}

impl ConstZero for \$t {
const ZERO: Self = \$v;
}
};
}

zero_impl!(usize, 0);
zero_impl!(u8, 0);
zero_impl!(u16, 0);
zero_impl!(u32, 0);
zero_impl!(u64, 0);
zero_impl!(u128, 0);

zero_impl!(isize, 0);
zero_impl!(i8, 0);
zero_impl!(i16, 0);
zero_impl!(i32, 0);
zero_impl!(i64, 0);
zero_impl!(i128, 0);

zero_impl!(f32, 0.0);
zero_impl!(f64, 0.0);

impl<T: Zero> Zero for Wrapping<T>
where
{
fn is_zero(&self) -> bool {
self.0.is_zero()
}

fn set_zero(&mut self) {
self.0.set_zero();
}

fn zero() -> Self {
Wrapping(T::zero())
}
}

impl<T: ConstZero> ConstZero for Wrapping<T>
where
{
const ZERO: Self = Wrapping(T::ZERO);
}

/// Defines a multiplicative identity element for `Self`.
///
/// # Laws
///
/// ```text
/// a * 1 = a       ∀ a ∈ Self
/// 1 * a = a       ∀ a ∈ Self
/// ```
pub trait One: Sized + Mul<Self, Output = Self> {
/// Returns the multiplicative identity element of `Self`, `1`.
///
/// # Purity
///
/// This function should return the same result at all times regardless of
/// external mutable state, for example values stored in TLS or in
/// `static mut`s.
// This cannot be an associated constant, because of bignums.
fn one() -> Self;

/// Sets `self` to the multiplicative identity element of `Self`, `1`.
fn set_one(&mut self) {
*self = One::one();
}

/// Returns `true` if `self` is equal to the multiplicative identity.
///
/// For performance reasons, it's best to implement this manually.
/// After a semver bump, this method will be required, and the
/// `where Self: PartialEq` bound will be removed.
#[inline]
fn is_one(&self) -> bool
where
Self: PartialEq,
{
*self == Self::one()
}
}

/// Defines an associated constant representing the multiplicative identity
/// element for `Self`.
pub trait ConstOne: One {
/// The multiplicative identity element of `Self`, `1`.
const ONE: Self;
}

macro_rules! one_impl {
(\$t:ty, \$v:expr) => {
impl One for \$t {
#[inline]
fn one() -> \$t {
\$v
}
#[inline]
fn is_one(&self) -> bool {
*self == \$v
}
}

impl ConstOne for \$t {
const ONE: Self = \$v;
}
};
}

one_impl!(usize, 1);
one_impl!(u8, 1);
one_impl!(u16, 1);
one_impl!(u32, 1);
one_impl!(u64, 1);
one_impl!(u128, 1);

one_impl!(isize, 1);
one_impl!(i8, 1);
one_impl!(i16, 1);
one_impl!(i32, 1);
one_impl!(i64, 1);
one_impl!(i128, 1);

one_impl!(f32, 1.0);
one_impl!(f64, 1.0);

impl<T: One> One for Wrapping<T>
where
Wrapping<T>: Mul<Output = Wrapping<T>>,
{
fn set_one(&mut self) {
self.0.set_one();
}

fn one() -> Self {
Wrapping(T::one())
}
}

impl<T: ConstOne> ConstOne for Wrapping<T>
where
Wrapping<T>: Mul<Output = Wrapping<T>>,
{
const ONE: Self = Wrapping(T::ONE);
}

// Some helper functions provided for backwards compatibility.

/// Returns the additive identity, `0`.
#[inline(always)]
pub fn zero<T: Zero>() -> T {
Zero::zero()
}

/// Returns the multiplicative identity, `1`.
#[inline(always)]
pub fn one<T: One>() -> T {
One::one()
}

#[test]
fn wrapping_identities() {
macro_rules! test_wrapping_identities {
(\$(\$t:ty)+) => {
\$(
assert_eq!(zero::<\$t>(), zero::<Wrapping<\$t>>().0);
assert_eq!(one::<\$t>(), one::<Wrapping<\$t>>().0);
assert_eq!((0 as \$t).is_zero(), Wrapping(0 as \$t).is_zero());
assert_eq!((1 as \$t).is_zero(), Wrapping(1 as \$t).is_zero());
)+
};
}

test_wrapping_identities!(isize i8 i16 i32 i64 usize u8 u16 u32 u64);
}

#[test]
fn wrapping_is_zero() {
fn require_zero<T: Zero>(_: &T) {}
require_zero(&Wrapping(42));
}
#[test]
fn wrapping_is_one() {
fn require_one<T: One>(_: &T) {}
require_one(&Wrapping(42));
}
``````