fp_library/types/

pair.rs

use crate::{
	Apply,
	brands::{PairBrand, PairWithFirstBrand, PairWithSecondBrand},
	classes::{
		applicative::Applicative, apply_first::ApplyFirst, apply_second::ApplySecond,
		bifunctor::Bifunctor, cloneable_fn::CloneableFn, foldable::Foldable, functor::Functor,
		lift::Lift, monoid::Monoid, par_foldable::ParFoldable, pointed::Pointed,
		semiapplicative::Semiapplicative, semigroup::Semigroup, semimonad::Semimonad,
		send_cloneable_fn::SendCloneableFn, traversable::Traversable,
	},
	impl_kind,
	kinds::*,
};

/// Wraps two values.
///
/// A simple tuple struct that holds two values of potentially different types.
///
/// ### Type Parameters
///
/// * `First`: The type of the first value.
/// * `Second`: The type of the second value.
///
/// ### Fields
///
/// * `0`: The first value.
/// * `1`: The second value.
///
/// ### Examples
///
/// ```
/// use fp_library::types::*;
///
/// let p = Pair(1, "hello");
/// assert_eq!(p.0, 1);
/// assert_eq!(p.1, "hello");
/// ```
#[derive(Clone, Copy, Debug, Default, Eq, Hash, Ord, PartialEq, PartialOrd)]
pub struct Pair<First, Second>(pub First, pub Second);

impl_kind! {
	for PairBrand {
		type Of<A, B> = Pair<A, B>;
	}
}

impl_kind! {
	for PairBrand {
		type Of<'a, A: 'a, B: 'a>: 'a = Pair<A, B>;
	}
}

impl Bifunctor for PairBrand {
	/// Maps functions over the values in the pair.
	///
	/// This method applies one function to the first value and another to the second value.
	///
	/// ### Type Signature
	///
	/// `forall a b c d. Bifunctor Pair => (a -> b, c -> d, Pair a c) -> Pair b d`
	///
	/// ### Type Parameters
	///
	/// * `A`: The type of the first value.
	/// * `B`: The type of the mapped first value.
	/// * `C`: The type of the second value.
	/// * `D`: The type of the mapped second value.
	/// * `F`: The type of the function to apply to the first value.
	/// * `G`: The type of the function to apply to the second value.
	///
	/// ### Parameters
	///
	/// * `f`: The function to apply to the first value.
	/// * `g`: The function to apply to the second value.
	/// * `p`: The pair to map over.
	///
	/// ### Returns
	///
	/// A new pair containing the mapped values.
	///
	/// ### Examples
	///
	/// ```
	/// use fp_library::{brands::*, classes::bifunctor::*, functions::*, types::*};
	///
	/// let x = Pair(1, 5);
	/// assert_eq!(bimap::<PairBrand, _, _, _, _, _, _>(|a| a + 1, |b| b * 2, x), Pair(2, 10));
	/// ```
	fn bimap<'a, A: 'a, B: 'a, C: 'a, D: 'a, F, G>(
		f: F,
		g: G,
		p: Apply!(<Self as Kind!( type Of<'a, A: 'a, B: 'a>: 'a; )>::Of<'a, A, C>),
	) -> Apply!(<Self as Kind!( type Of<'a, A: 'a, B: 'a>: 'a; )>::Of<'a, B, D>)
	where
		F: Fn(A) -> B + 'a,
		G: Fn(C) -> D + 'a,
	{
		let Pair(a, c) = p;
		Pair(f(a), g(c))
	}
}

// PairWithFirstBrand<First> (Functor over Second)

impl_kind! {
	impl<First: 'static> for PairWithFirstBrand<First> {
		type Of<'a, A: 'a>: 'a = Pair<First, A>;
	}
}

impl<First: 'static> Functor for PairWithFirstBrand<First> {
	/// Maps a function over the second value in the pair.
	///
	/// This method applies a function to the second value inside the pair, producing a new pair with the transformed second value. The first value remains unchanged.
	///
	/// ### Type Signature
	///
	/// `forall t b a. Functor (Pair t) => (a -> b, Pair t a) -> Pair t b`
	///
	/// ### Type Parameters
	///
	/// * `B`: The type of the result of applying the function.
	/// * `A`: The type of the second value.
	/// * `F`: The type of the function to apply.
	///
	/// ### Parameters
	///
	/// * `f`: The function to apply to the second value.
	/// * `fa`: The pair to map over.
	///
	/// ### Returns
	///
	/// A new pair containing the result of applying the function to the second value.
	///
	/// ### Examples
	///
	/// ```
	/// use fp_library::{brands::*, functions::*, types::*};
	///
	/// assert_eq!(map::<PairWithFirstBrand<_>, _, _, _>(|x: i32| x * 2, Pair(1, 5)), Pair(1, 10));
	/// ```
	fn map<'a, B: 'a, A: 'a, F>(
		f: F,
		fa: Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, A>),
	) -> Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, B>)
	where
		F: Fn(A) -> B + 'a,
	{
		Pair(fa.0, f(fa.1))
	}
}

impl<First: Clone + 'static> Lift for PairWithFirstBrand<First>
where
	First: Semigroup,
{
	/// Lifts a binary function into the pair context (over second).
	///
	/// This method lifts a binary function to operate on the second values within the pair context. The first values are combined using their `Semigroup` implementation.
	///
	/// ### Type Signature
	///
	/// `forall t c a b. (Lift (Pair t), Semigroup t) => ((a, b) -> c, Pair t a, Pair t b) -> Pair t c`
	///
	/// ### Type Parameters
	///
	/// * `C`: The type of the result second value.
	/// * `A`: The type of the first second value.
	/// * `B`: The type of the second second value.
	/// * `F`: The type of the binary function.
	///
	/// ### Parameters
	///
	/// * `f`: The binary function to apply to the second values.
	/// * `fa`: The first pair.
	/// * `fb`: The second pair.
	///
	/// ### Returns
	///
	/// A new pair where the first values are combined using `Semigroup::append` and the second values are combined using `f`.
	///
	/// ### Examples
	///
	/// ```
	/// use fp_library::{brands::*, functions::*, types::*};
	///
	/// assert_eq!(
	///     lift2::<PairWithFirstBrand<String>, _, _, _, _>(|x, y| x + y, Pair("a".to_string(), 1), Pair("b".to_string(), 2)),
	///     Pair("ab".to_string(), 3)
	/// );
	/// ```
	fn lift2<'a, C, A, B, F>(
		f: F,
		fa: Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, A>),
		fb: Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, B>),
	) -> Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, C>)
	where
		F: Fn(A, B) -> C + 'a,
		A: Clone + 'a,
		B: Clone + 'a,
		C: 'a,
	{
		Pair(Semigroup::append(fa.0, fb.0), f(fa.1, fb.1))
	}
}

impl<First: Clone + 'static> Pointed for PairWithFirstBrand<First>
where
	First: Monoid,
{
	/// Wraps a value in a pair (with empty first).
	///
	/// This method wraps a value in a pair, using the `Monoid::empty()` value for the first element.
	///
	/// ### Type Signature
	///
	/// `forall t a. (Pointed (Pair t), Monoid t) => a -> Pair t a`
	///
	/// ### Type Parameters
	///
	/// * `A`: The type of the value to wrap.
	///
	/// ### Parameters
	///
	/// * `a`: The value to wrap.
	///
	/// ### Returns
	///
	/// A pair containing the empty value of the first type and `a`.
	///
	/// ### Examples
	///
	/// ```
	/// use fp_library::{brands::*, functions::*, types::*};
	///
	/// assert_eq!(pure::<PairWithFirstBrand<String>, _>(5), Pair("".to_string(), 5));
	/// ```
	fn pure<'a, A: 'a>(a: A) -> Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, A>) {
		Pair(Monoid::empty(), a)
	}
}

impl<First: Clone + Semigroup + 'static> ApplyFirst for PairWithFirstBrand<First> {}
impl<First: Clone + Semigroup + 'static> ApplySecond for PairWithFirstBrand<First> {}

impl<First: Clone + 'static> Semiapplicative for PairWithFirstBrand<First>
where
	First: Semigroup,
{
	/// Applies a wrapped function to a wrapped value (over second).
	///
	/// This method applies a function wrapped in a pair to a value wrapped in a pair. The first values are combined using their `Semigroup` implementation.
	///
	/// ### Type Signature
	///
	/// `forall fn_brand t b a. (Semiapplicative (Pair t), Semigroup t) => (Pair t (fn_brand a b), Pair t a) -> Pair t b`
	///
	/// ### Type Parameters
	///
	/// * `FnBrand`: The brand of the cloneable function wrapper.
	/// * `B`: The type of the output value.
	/// * `A`: The type of the input value.
	///
	/// ### Parameters
	///
	/// * `ff`: The pair containing the function.
	/// * `fa`: The pair containing the value.
	///
	/// ### Returns
	///
	/// A new pair where the first values are combined and the function is applied to the second value.
	///
	/// ### Examples
	///
	/// ```
	/// use fp_library::{brands::*, functions::*, types::*};
	///
	/// let f = Pair("a".to_string(), cloneable_fn_new::<RcFnBrand, _, _>(|x: i32| x * 2));
	/// assert_eq!(apply::<RcFnBrand, PairWithFirstBrand<String>, _, _>(f, Pair("b".to_string(), 5)), Pair("ab".to_string(), 10));
	/// ```
	fn apply<'a, FnBrand: 'a + CloneableFn, B: 'a, A: 'a + Clone>(
		ff: Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, <FnBrand as CloneableFn>::Of<'a, A, B>>),
		fa: Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, A>),
	) -> Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, B>) {
		Pair(Semigroup::append(ff.0, fa.0), ff.1(fa.1))
	}
}

impl<First: Clone + 'static> Semimonad for PairWithFirstBrand<First>
where
	First: Semigroup,
{
	/// Chains pair computations (over second).
	///
	/// This method chains two computations, where the second computation depends on the result of the first. The first values are combined using their `Semigroup` implementation.
	///
	/// ### Type Signature
	///
	/// `forall t b a. (Semimonad (Pair t), Semigroup t) => (Pair t a, a -> Pair t b) -> Pair t b`
	///
	/// ### Type Parameters
	///
	/// * `B`: The type of the result of the second computation.
	/// * `A`: The type of the result of the first computation.
	/// * `F`: The type of the function to apply.
	///
	/// ### Parameters
	///
	/// * `ma`: The first pair.
	/// * `f`: The function to apply to the second value.
	///
	/// ### Returns
	///
	/// A new pair where the first values are combined.
	///
	/// ### Examples
	///
	/// ```
	/// use fp_library::{brands::*, functions::*, types::*};
	///
	/// assert_eq!(
	///     bind::<PairWithFirstBrand<String>, _, _, _>(Pair("a".to_string(), 5), |x| Pair("b".to_string(), x * 2)),
	///     Pair("ab".to_string(), 10)
	/// );
	/// ```
	fn bind<'a, B: 'a, A: 'a, F>(
		ma: Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, A>),
		f: F,
	) -> Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, B>)
	where
		F: Fn(A) -> Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, B>) + 'a,
	{
		let Pair(first, second) = ma;
		let Pair(next_first, next_second) = f(second);
		Pair(Semigroup::append(first, next_first), next_second)
	}
}

impl<First: 'static> Foldable for PairWithFirstBrand<First> {
	/// Folds the pair from the right (over second).
	///
	/// This method performs a right-associative fold of the pair (over second).
	///
	/// ### Type Signature
	///
	/// `forall t b a. Foldable (Pair t) => ((a, b) -> b, b, Pair t a) -> b`
	///
	/// ### Type Parameters
	///
	/// * `FnBrand`: The brand of the cloneable function to use.
	/// * `B`: The type of the accumulator.
	/// * `A`: The type of the elements in the structure.
	/// * `Func`: The type of the folding function.
	///
	/// ### Parameters
	///
	/// * `func`: The folding function.
	/// * `initial`: The initial value.
	/// * `fa`: The pair to fold.
	///
	/// ### Returns
	///
	/// `func(a, initial)`.
	///
	/// ### Examples
	///
	/// ```
	/// use fp_library::{brands::*, functions::*, types::*};
	///
	/// assert_eq!(fold_right::<RcFnBrand, PairWithFirstBrand<()>, _, _, _>(|x, acc| x + acc, 0, Pair((), 5)), 5);
	/// ```
	fn fold_right<'a, FnBrand, B: 'a, A: 'a, Func>(
		func: Func,
		initial: B,
		fa: Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, A>),
	) -> B
	where
		Func: Fn(A, B) -> B + 'a,
		FnBrand: CloneableFn + 'a,
	{
		func(fa.1, initial)
	}

	/// Folds the pair from the left (over second).
	///
	/// This method performs a left-associative fold of the pair (over second).
	///
	/// ### Type Signature
	///
	/// `forall t b a. Foldable (Pair t) => ((b, a) -> b, b, Pair t a) -> b`
	///
	/// ### Type Parameters
	///
	/// * `FnBrand`: The brand of the cloneable function to use.
	/// * `B`: The type of the accumulator.
	/// * `A`: The type of the elements in the structure.
	/// * `Func`: The type of the folding function.
	///
	/// ### Parameters
	///
	/// * `func`: The function to apply to the accumulator and each element.
	/// * `initial`: The initial value of the accumulator.
	/// * `fa`: The identity to fold.
	///
	/// ### Returns
	///
	/// `func(initial, a)`.
	///
	/// ### Examples
	///
	/// ```
	/// use fp_library::{brands::*, functions::*, types::*};
	///
	/// assert_eq!(fold_left::<RcFnBrand, PairWithFirstBrand<()>, _, _, _>(|acc, x| acc + x, 0, Pair((), 5)), 5);
	/// ```
	fn fold_left<'a, FnBrand, B: 'a, A: 'a, Func>(
		func: Func,
		initial: B,
		fa: Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, A>),
	) -> B
	where
		Func: Fn(B, A) -> B + 'a,
		FnBrand: CloneableFn + 'a,
	{
		func(initial, fa.1)
	}

	/// Maps the value to a monoid and returns it (over second).
	///
	/// This method maps the element of the pair to a monoid and then returns it (over second).
	///
	/// ### Type Signature
	///
	/// `forall t m a. (Foldable (Pair t), Monoid m) => ((a) -> m, Pair t a) -> m`
	///
	/// ### Type Parameters
	///
	/// * `FnBrand`: The brand of the cloneable function to use.
	/// * `M`: The type of the monoid.
	/// * `A`: The type of the elements in the structure.
	/// * `Func`: The type of the mapping function.
	///
	/// ### Parameters
	///
	/// * `func`: The mapping function.
	/// * `fa`: The pair to fold.
	///
	/// ### Returns
	///
	/// `func(a)`.
	///
	/// ### Examples
	///
	/// ```
	/// use fp_library::{brands::*, functions::*, types::*};
	///
	/// assert_eq!(
	///     fold_map::<RcFnBrand, PairWithFirstBrand<()>, _, _, _>(|x: i32| x.to_string(), Pair((), 5)),
	///     "5".to_string()
	/// );
	/// ```
	fn fold_map<'a, FnBrand, M, A: 'a, Func>(
		func: Func,
		fa: Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, A>),
	) -> M
	where
		M: Monoid + 'a,
		Func: Fn(A) -> M + 'a,
		FnBrand: CloneableFn + 'a,
	{
		func(fa.1)
	}
}

impl<First: Clone + 'static> Traversable for PairWithFirstBrand<First> {
	/// Traverses the pair with an applicative function (over second).
	///
	/// This method maps the element of the pair to a computation, evaluates it, and combines the result into an applicative context (over second).
	///
	/// ### Type Signature
	///
	/// `forall t f b a. (Traversable (Pair t), Applicative f) => (a -> f b, Pair t a) -> f (Pair t b)`
	///
	/// ### Type Parameters
	///
	/// * `F`: The applicative context.
	/// * `B`: The type of the elements in the resulting traversable structure.
	/// * `A`: The type of the elements in the traversable structure.
	/// * `Func`: The type of the function to apply.
	///
	/// ### Parameters
	///
	/// * `func`: The function to apply to each element, returning a value in an applicative context.
	/// * `ta`: The pair to traverse.
	///
	/// ### Returns
	///
	/// The pair wrapped in the applicative context.
	///
	/// ### Examples
	///
	/// ```
	/// use fp_library::{brands::*, functions::*, types::*};
	///
	/// assert_eq!(
	///     traverse::<PairWithFirstBrand<()>, OptionBrand, _, _, _>(|x| Some(x * 2), Pair((), 5)),
	///     Some(Pair((), 10))
	/// );
	/// ```
	fn traverse<'a, F: Applicative, B: 'a + Clone, A: 'a + Clone, Func>(
		func: Func,
		ta: Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, A>),
	) -> Apply!(<F as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, B>)>)
	where
		Func: Fn(A) -> Apply!(<F as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, B>) + 'a,
		Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, B>): Clone,
	{
		let Pair(first, second) = ta;
		F::map(move |b| Pair(first.clone(), b), func(second))
	}
	/// Sequences a pair of applicative (over second).
	///
	/// This method evaluates the computation inside the pair and accumulates the result into an applicative context (over second).
	///
	/// ### Type Signature
	///
	/// `forall t f a. (Traversable (Pair t), Applicative f) => (Pair t (f a)) -> f (Pair t a)`
	///
	/// ### Type Parameters
	///
	/// * `F`: The applicative context.
	/// * `A`: The type of the elements in the traversable structure.
	///
	/// ### Parameters
	///
	/// * `ta`: The pair containing the applicative value.
	///
	/// ### Returns
	///
	/// The pair wrapped in the applicative context.
	///
	/// ### Examples
	///
	/// ```
	/// use fp_library::{brands::*, functions::*, types::*};
	///
	/// assert_eq!(
	///     sequence::<PairWithFirstBrand<()>, OptionBrand, _>(Pair((), Some(5))),
	///     Some(Pair((), 5))
	/// );
	/// ```
	fn sequence<'a, F: Applicative, A: 'a + Clone>(
		ta: Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, Apply!(<F as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, A>)>)
	) -> Apply!(<F as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, A>)>)
	where
		Apply!(<F as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, A>): Clone,
		Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, A>): Clone,
	{
		let Pair(first, second) = ta;
		F::map(move |a| Pair(first.clone(), a), second)
	}
}

impl<First: 'static, FnBrand: SendCloneableFn> ParFoldable<FnBrand> for PairWithFirstBrand<First> {
	/// Maps the value to a monoid and returns it in parallel (over second).
	///
	/// This method maps the element of the pair to a monoid and then returns it (over second). The mapping operation may be executed in parallel.
	///
	/// ### Type Signature
	///
	/// `forall fn_brand t m a. (ParFoldable (Pair t), Monoid m, Send m, Sync m) => (fn_brand a m, Pair t a) -> m`
	///
	/// ### Type Parameters
	///
	/// * `M`: The monoid type (must be `Send + Sync`).
	/// * `A`: The element type (must be `Send + Sync`).
	///
	/// ### Parameters
	///
	/// * `func`: The thread-safe function to map each element to a monoid.
	/// * `fa`: The pair to fold.
	///
	/// ### Returns
	///
	/// The combined monoid value.
	///
	/// ### Examples
	///
	/// ```
	/// use fp_library::{brands::*, functions::*, types::*};
	///
	/// let x = Pair("a".to_string(), 1);
	/// let f = send_cloneable_fn_new::<ArcFnBrand, _, _>(|x: i32| x.to_string());
	/// assert_eq!(
	///     par_fold_map::<ArcFnBrand, PairWithFirstBrand<String>, _, _>(f, x),
	///     "1".to_string()
	/// );
	/// ```
	fn par_fold_map<'a, M, A>(
		func: <FnBrand as SendCloneableFn>::SendOf<'a, A, M>,
		fa: Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, A>),
	) -> M
	where
		A: 'a + Clone + Send + Sync,
		M: Monoid + Send + Sync + 'a,
	{
		func(fa.1)
	}

	/// Folds the pair from the right in parallel (over second).
	///
	/// This method folds the pair by applying a function from right to left, potentially in parallel (over second).
	///
	/// ### Type Signature
	///
	/// `forall fn_brand t b a. ParFoldable (Pair t) => (fn_brand (a, b) b, b, Pair t a) -> b`
	///
	/// ### Type Parameters
	///
	/// * `B`: The accumulator type (must be `Send + Sync`).
	/// * `A`: The element type (must be `Send + Sync`).
	///
	/// ### Parameters
	///
	/// * `func`: The thread-safe function to apply to each element and the accumulator.
	/// * `initial`: The initial value.
	/// * `fa`: The pair to fold.
	///
	/// ### Returns
	///
	/// The final accumulator value.
	///
	/// ### Examples
	///
	/// ```
	/// use fp_library::{brands::*, functions::*, types::*};
	///
	/// let x = Pair("a".to_string(), 1);
	/// let f = send_cloneable_fn_new::<ArcFnBrand, _, _>(|(a, b): (i32, i32)| a + b);
	/// assert_eq!(par_fold_right::<ArcFnBrand, PairWithFirstBrand<String>, _, _>(f, 10, x), 11);
	/// ```
	fn par_fold_right<'a, B, A>(
		func: <FnBrand as SendCloneableFn>::SendOf<'a, (A, B), B>,
		initial: B,
		fa: Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, A>),
	) -> B
	where
		A: 'a + Clone + Send + Sync,
		B: Send + Sync + 'a,
	{
		func((fa.1, initial))
	}
}
// PairWithSecondBrand<Second> (Functor over First)

impl_kind! {
	impl<Second: 'static> for PairWithSecondBrand<Second> {
		type Of<'a, A: 'a>: 'a = Pair<A, Second>;
	}
}

impl<Second: 'static> Functor for PairWithSecondBrand<Second> {
	/// Maps a function over the first value in the pair.
	///
	/// This method applies a function to the first value inside the pair, producing a new pair with the transformed first value. The second value remains unchanged.
	///
	/// ### Type Signature
	///
	/// `forall t b a. Functor (PairWithSecond t) => (a -> b, Pair a t) -> Pair b t`
	///
	/// ### Type Parameters
	///
	/// * `B`: The type of the result of applying the function.
	/// * `A`: The type of the first value.
	/// * `F`: The type of the function to apply.
	///
	/// ### Parameters
	///
	/// * `f`: The function to apply to the first value.
	/// * `fa`: The pair to map over.
	///
	/// ### Returns
	///
	/// A new pair containing the result of applying the function to the first value.
	///
	/// ### Examples
	///
	/// ```
	/// use fp_library::{brands::*, functions::*, types::*};
	///
	/// assert_eq!(map::<PairWithSecondBrand<_>, _, _, _>(|x: i32| x * 2, Pair(5, 1)), Pair(10, 1));
	/// ```
	fn map<'a, B: 'a, A: 'a, F>(
		f: F,
		fa: Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, A>),
	) -> Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, B>)
	where
		F: Fn(A) -> B + 'a,
	{
		Pair(f(fa.0), fa.1)
	}
}

impl<Second: Clone + 'static> Lift for PairWithSecondBrand<Second>
where
	Second: Semigroup,
{
	/// Lifts a binary function into the pair context (over first).
	///
	/// This method lifts a binary function to operate on the first values within the pair context. The second values are combined using their `Semigroup` implementation.
	///
	/// ### Type Signature
	///
	/// `forall t c a b. (Lift (PairWithSecond t), Semigroup t) => ((a, b) -> c, Pair a t, Pair b t) -> Pair c t`
	///
	/// ### Type Parameters
	///
	/// * `C`: The type of the result first value.
	/// * `A`: The type of the first first value.
	/// * `B`: The type of the second first value.
	/// * `F`: The type of the binary function.
	///
	/// ### Parameters
	///
	/// * `f`: The binary function to apply to the first values.
	/// * `fa`: The first pair.
	/// * `fb`: The second pair.
	///
	/// ### Returns
	///
	/// A new pair where the first values are combined using `f` and the second values are combined using `Semigroup::append`.
	///
	/// ### Examples
	///
	/// ```
	/// use fp_library::{brands::*, functions::*, types::*};
	///
	/// assert_eq!(
	///     lift2::<PairWithSecondBrand<String>, _, _, _, _>(|x, y| x + y, Pair(1, "a".to_string()), Pair(2, "b".to_string())),
	///     Pair(3, "ab".to_string())
	/// );
	/// ```
	fn lift2<'a, C, A, B, F>(
		f: F,
		fa: Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, A>),
		fb: Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, B>),
	) -> Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, C>)
	where
		F: Fn(A, B) -> C + 'a,
		A: Clone + 'a,
		B: Clone + 'a,
		C: 'a,
	{
		Pair(f(fa.0, fb.0), Semigroup::append(fa.1, fb.1))
	}
}

impl<Second: Clone + 'static> Pointed for PairWithSecondBrand<Second>
where
	Second: Monoid,
{
	/// Wraps a value in a pair (with empty second).
	///
	/// This method wraps a value in a pair, using the `Monoid::empty()` value for the second element.
	///
	/// ### Type Signature
	///
	/// `forall t a. (Pointed (PairWithSecond t), Monoid t) => a -> Pair a t`
	///
	/// ### Type Parameters
	///
	/// * `A`: The type of the value to wrap.
	///
	/// ### Parameters
	///
	/// * `a`: The value to wrap.
	///
	/// ### Returns
	///
	/// A pair containing `a` and the empty value of the second type.
	///
	/// ### Examples
	///
	/// ```
	/// use fp_library::{brands::*, functions::*, types::*};
	///
	/// assert_eq!(pure::<PairWithSecondBrand<String>, _>(5), Pair(5, "".to_string()));
	/// ```
	fn pure<'a, A: 'a>(a: A) -> Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, A>) {
		Pair(a, Monoid::empty())
	}
}

impl<Second: Clone + Semigroup + 'static> ApplyFirst for PairWithSecondBrand<Second> {}
impl<Second: Clone + Semigroup + 'static> ApplySecond for PairWithSecondBrand<Second> {}

impl<Second: Clone + 'static> Semiapplicative for PairWithSecondBrand<Second>
where
	Second: Semigroup,
{
	/// Applies a wrapped function to a wrapped value (over first).
	///
	/// This method applies a function wrapped in a result (as error) to a value wrapped in a result (as error).
	///
	/// ### Type Signature
	///
	/// `forall fn_brand t b a. (Semiapplicative (PairWithSecond t), Semigroup t) => (Pair (fn_brand a b) t, Pair a t) -> Pair b t`
	///
	/// ### Type Parameters
	///
	/// * `FnBrand`: The brand of the cloneable function wrapper.
	/// * `B`: The type of the output value.
	/// * `A`: The type of the input value.
	///
	/// ### Parameters
	///
	/// * `ff`: The pair containing the function (in Err).
	/// * `fa`: The pair containing the value (in Err).
	///
	/// ### Returns
	///
	/// `Err(f(a))` if both are `Err`, otherwise the first success encountered.
	///
	/// ### Examples
	///
	/// ```
	/// use fp_library::{brands::*, functions::*, types::*};
	///
	/// let f = Pair(cloneable_fn_new::<RcFnBrand, _, _>(|x: i32| x * 2), "a".to_string());
	/// assert_eq!(apply::<RcFnBrand, PairWithSecondBrand<String>, _, _>(f, Pair(5, "b".to_string())), Pair(10, "ab".to_string()));
	/// ```
	fn apply<'a, FnBrand: 'a + CloneableFn, B: 'a, A: 'a + Clone>(
		ff: Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, <FnBrand as CloneableFn>::Of<'a, A, B>>),
		fa: Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, A>),
	) -> Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, B>) {
		Pair(ff.0(fa.0), Semigroup::append(ff.1, fa.1))
	}
}

impl<Second: Clone + 'static> Semimonad for PairWithSecondBrand<Second>
where
	Second: Semigroup,
{
	/// Chains pair computations (over first).
	///
	/// This method chains two computations, where the second computation depends on the result of the first (over error).
	///
	/// ### Type Signature
	///
	/// `forall t b a. (Semimonad (PairWithSecond t), Semigroup t) => (Pair a t, a -> Pair b t) -> Pair b t`
	///
	/// ### Type Parameters
	///
	/// * `B`: The type of the result of the second computation.
	/// * `A`: The type of the result of the first computation.
	/// * `F`: The type of the function to apply.
	///
	/// ### Parameters
	///
	/// * `ma`: The first result.
	/// * `f`: The function to apply to the error value.
	///
	/// ### Returns
	///
	/// The result of applying `f` to the error if `ma` is `Err`, otherwise the original success.
	///
	/// ### Examples
	///
	/// ```
	/// use fp_library::{brands::*, functions::*, types::*};
	///
	/// assert_eq!(
	///     bind::<PairWithSecondBrand<String>, _, _, _>(Pair(5, "a".to_string()), |x| Pair(x * 2, "b".to_string())),
	///     Pair(10, "ab".to_string())
	/// );
	/// ```
	fn bind<'a, B: 'a, A: 'a, F>(
		ma: Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, A>),
		f: F,
	) -> Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, B>)
	where
		F: Fn(A) -> Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, B>) + 'a,
	{
		let Pair(first, second) = ma;
		let Pair(next_first, next_second) = f(first);
		Pair(next_first, Semigroup::append(second, next_second))
	}
}

impl<Second: 'static> Foldable for PairWithSecondBrand<Second> {
	/// Folds the pair from the right (over first).
	///
	/// This method performs a right-associative fold of the result (over error).
	///
	/// ### Type Signature
	///
	/// `forall t b a. Foldable (PairWithSecond t) => ((a, b) -> b, b, Pair a t) -> b`
	///
	/// ### Type Parameters
	///
	/// * `FnBrand`: The brand of the cloneable function to use.
	/// * `B`: The type of the accumulator.
	/// * `A`: The type of the elements in the structure.
	/// * `F`: The type of the folding function.
	///
	/// ### Parameters
	///
	/// * `func`: The folding function.
	/// * `initial`: The initial value.
	/// * `fa`: The result to fold.
	///
	/// ### Returns
	///
	/// `func(a, initial)` if `fa` is `Err(a)`, otherwise `initial`.
	///
	/// ### Examples
	///
	/// ```
	/// use fp_library::{brands::*, functions::*, types::*};
	///
	/// assert_eq!(fold_right::<RcFnBrand, PairWithSecondBrand<()>, _, _, _>(|x, acc| x + acc, 0, Pair(5, ())), 5);
	/// ```
	fn fold_right<'a, FnBrand, B: 'a, A: 'a, F>(
		func: F,
		initial: B,
		fa: Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, A>),
	) -> B
	where
		F: Fn(A, B) -> B + 'a,
		FnBrand: CloneableFn + 'a,
	{
		func(fa.0, initial)
	}

	/// Folds the pair from the left (over first).
	///
	/// This method performs a left-associative fold of the result (over error).
	///
	/// ### Type Signature
	///
	/// `forall t b a. Foldable (PairWithSecond t) => ((b, a) -> b, b, Pair a t) -> b`
	///
	/// ### Type Parameters
	///
	/// * `FnBrand`: The brand of the cloneable function to use.
	/// * `B`: The type of the accumulator.
	/// * `A`: The type of the elements in the structure.
	/// * `F`: The type of the folding function.
	///
	/// ### Parameters
	///
	/// * `func`: The folding function.
	/// * `initial`: The initial value.
	/// * `fa`: The result to fold.
	///
	/// ### Returns
	///
	/// `func(initial, a)` if `fa` is `Err(a)`, otherwise `initial`.
	///
	/// ### Examples
	///
	/// ```
	/// use fp_library::{brands::*, functions::*, types::*};
	///
	/// assert_eq!(fold_left::<RcFnBrand, PairWithSecondBrand<()>, _, _, _>(|acc, x| acc + x, 0, Pair(5, ())), 5);
	/// ```
	fn fold_left<'a, FnBrand, B: 'a, A: 'a, F>(
		func: F,
		initial: B,
		fa: Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, A>),
	) -> B
	where
		F: Fn(B, A) -> B + 'a,
		FnBrand: CloneableFn + 'a,
	{
		func(initial, fa.0)
	}

	/// Maps the value to a monoid and returns it (over first).
	///
	/// This method maps the element of the result to a monoid and then returns it (over error).
	///
	/// ### Type Signature
	///
	/// `forall t m a. (Foldable (PairWithSecond t), Monoid m) => ((a) -> m, Pair a t) -> m`
	///
	/// ### Type Parameters
	///
	/// * `FnBrand`: The brand of the cloneable function to use.
	/// * `M`: The type of the monoid.
	/// * `A`: The type of the elements in the structure.
	/// * `Func`: The type of the mapping function.
	///
	/// ### Parameters
	///
	/// * `func`: The mapping function.
	/// * `fa`: The result to fold.
	///
	/// ### Returns
	///
	/// `func(a)` if `fa` is `Err(a)`, otherwise `M::empty()`.
	///
	/// ### Examples
	///
	/// ```
	/// use fp_library::{brands::*, functions::*, types::*};
	///
	/// assert_eq!(
	///     fold_map::<RcFnBrand, PairWithSecondBrand<()>, _, _, _>(|x: i32| x.to_string(), Pair(5, ())),
	///     "5".to_string()
	/// );
	/// ```
	fn fold_map<'a, FnBrand, M, A: 'a, Func>(
		func: Func,
		fa: Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, A>),
	) -> M
	where
		M: Monoid + 'a,
		Func: Fn(A) -> M + 'a,
		FnBrand: CloneableFn + 'a,
	{
		func(fa.0)
	}
}

impl<Second: Clone + 'static> Traversable for PairWithSecondBrand<Second> {
	/// Traverses the pair with an applicative function (over first).
	///
	/// This method maps the element of the result to a computation, evaluates it, and combines the result into an applicative context (over error).
	///
	/// ### Type Signature
	///
	/// `forall t f b a. (Traversable (PairWithSecond t), Applicative f) => (a -> f b, Pair a t) -> f (Pair b t)`
	///
	/// ### Type Parameters
	///
	/// * `F`: The applicative context.
	/// * `B`: The type of the elements in the resulting traversable structure.
	/// * `A`: The type of the elements in the traversable structure.
	/// * `Func`: The type of the function to apply.
	///
	/// ### Parameters
	///
	/// * `func`: The function to apply.
	/// * `ta`: The result to traverse.
	///
	/// ### Returns
	///
	/// The result wrapped in the applicative context.
	///
	/// ### Examples
	///
	/// ```
	/// use fp_library::{brands::*, functions::*, types::*};
	///
	/// assert_eq!(
	///     traverse::<PairWithSecondBrand<()>, OptionBrand, _, _, _>(|x| Some(x * 2), Pair(5, ())),
	///     Some(Pair(10, ()))
	/// );
	/// ```
	fn traverse<'a, F: Applicative, B: 'a + Clone, A: 'a + Clone, Func>(
		func: Func,
		ta: Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, A>),
	) -> Apply!(<F as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, B>)>)
	where
		Func: Fn(A) -> Apply!(<F as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, B>) + 'a,
		Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, B>): Clone,
	{
		let Pair(first, second) = ta;
		F::map(move |b| Pair(b, second.clone()), func(first))
	}

	/// Sequences a pair of applicative (over first).
	///
	/// This method evaluates the computation inside the result and accumulates the result into an applicative context (over error).
	///
	/// ### Type Signature
	///
	/// `forall t f a. (Traversable (PairWithSecond t), Applicative f) => (Pair (f a) t) -> f (Pair a t)`
	///
	/// ### Type Parameters
	///
	/// * `F`: The applicative context.
	/// * `A`: The type of the elements in the traversable structure.
	///
	/// ### Parameters
	///
	/// * `ta`: The result containing the applicative value.
	///
	/// ### Returns
	///
	/// The result wrapped in the applicative context.
	///
	/// ### Examples
	///
	/// ```
	/// use fp_library::{brands::*, functions::*, types::*};
	///
	/// assert_eq!(
	///     sequence::<PairWithSecondBrand<()>, OptionBrand, _>(Pair(Some(5), ())),
	///     Some(Pair(5, ()))
	/// );
	/// ```
	fn sequence<'a, F: Applicative, A: 'a + Clone>(
		ta: Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, Apply!(<F as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, A>)>)
	) -> Apply!(<F as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, A>)>)
	where
		Apply!(<F as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, A>): Clone,
		Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, A>): Clone,
	{
		let Pair(first, second) = ta;
		F::map(move |a| Pair(a, second.clone()), first)
	}
}

impl<Second: 'static, FnBrand: SendCloneableFn> ParFoldable<FnBrand>
	for PairWithSecondBrand<Second>
{
	/// Maps the value to a monoid and returns it in parallel (over first).
	///
	/// This method maps the element of the pair to a monoid and then returns it (over first). The mapping operation may be executed in parallel.
	///
	/// ### Type Signature
	///
	/// `forall fn_brand t m a. (ParFoldable (PairWithSecond t), Monoid m, Send m, Sync m) => (fn_brand a m, Pair a t) -> m`
	///
	/// ### Type Parameters
	///
	/// * `M`: The monoid type (must be `Send + Sync`).
	/// * `A`: The element type (must be `Send + Sync`).
	///
	/// ### Parameters
	///
	/// * `func`: The thread-safe function to map each element to a monoid.
	/// * `fa`: The pair to fold.
	///
	/// ### Returns
	///
	/// The combined monoid value.
	///
	/// ### Examples
	///
	/// ```
	/// use fp_library::{brands::*, functions::*, types::*};
	///
	/// let x = Pair(1, "a".to_string());
	/// let f = send_cloneable_fn_new::<ArcFnBrand, _, _>(|x: i32| x.to_string());
	/// assert_eq!(
	///     par_fold_map::<ArcFnBrand, PairWithSecondBrand<String>, _, _>(f, x),
	///     "1".to_string()
	/// );
	/// ```
	fn par_fold_map<'a, M, A>(
		func: <FnBrand as SendCloneableFn>::SendOf<'a, A, M>,
		fa: Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, A>),
	) -> M
	where
		A: 'a + Clone + Send + Sync,
		M: Monoid + Send + Sync + 'a,
	{
		func(fa.0)
	}

	/// Folds the pair from the right in parallel (over first).
	///
	/// This method folds the pair by applying a function from right to left, potentially in parallel (over first).
	///
	/// ### Type Signature
	///
	/// `forall fn_brand t b a. ParFoldable (PairWithSecond t) => (fn_brand (a, b) b, b, Pair a t) -> b`
	///
	/// ### Type Parameters
	///
	/// * `B`: The accumulator type (must be `Send + Sync`).
	/// * `A`: The element type (must be `Send + Sync`).
	///
	/// ### Parameters
	///
	/// * `func`: The thread-safe function to apply to each element and the accumulator.
	/// * `initial`: The initial value.
	/// * `fa`: The pair to fold.
	///
	/// ### Returns
	///
	/// The final accumulator value.
	///
	/// ### Examples
	///
	/// ```
	/// use fp_library::{brands::*, functions::*, types::*};
	///
	/// let x = Pair(1, "a".to_string());
	/// let f = send_cloneable_fn_new::<ArcFnBrand, _, _>(|(a, b): (i32, i32)| a + b);
	/// assert_eq!(par_fold_right::<ArcFnBrand, PairWithSecondBrand<String>, _, _>(f, 10, x), 11);
	/// ```
	fn par_fold_right<'a, B, A>(
		func: <FnBrand as SendCloneableFn>::SendOf<'a, (A, B), B>,
		initial: B,
		fa: Apply!(<Self as Kind!( type Of<'a, T: 'a>: 'a; )>::Of<'a, A>),
	) -> B
	where
		A: 'a + Clone + Send + Sync,
		B: Send + Sync + 'a,
	{
		func((fa.0, initial))
	}
}

#[cfg(test)]
mod tests {
	use super::*;
	use crate::{brands::*, classes::bifunctor::*, functions::*};
	use quickcheck_macros::quickcheck;

	// Bifunctor Tests

	/// Tests `bimap` on `Pair`.
	#[test]
	fn test_bimap() {
		let x = Pair(1, 5);
		assert_eq!(bimap::<PairBrand, _, _, _, _, _, _>(|a| a + 1, |b| b * 2, x), Pair(2, 10));
	}

	// Bifunctor Laws

	/// Tests the identity law for Bifunctor.
	#[quickcheck]
	fn bifunctor_identity(
		first: String,
		second: i32,
	) -> bool {
		let x = Pair(first, second);
		bimap::<PairBrand, _, _, _, _, _, _>(identity, identity, x.clone()) == x
	}

	/// Tests the composition law for Bifunctor.
	#[quickcheck]
	fn bifunctor_composition(
		first: i32,
		second: i32,
	) -> bool {
		let x = Pair(first, second);
		let f = |x: i32| x.wrapping_add(1);
		let g = |x: i32| x.wrapping_mul(2);
		let h = |x: i32| x.wrapping_sub(1);
		let i = |x: i32| if x == 0 { 0 } else { x.wrapping_div(2) };

		bimap::<PairBrand, _, _, _, _, _, _>(compose(f, g), compose(h, i), x.clone())
			== bimap::<PairBrand, _, _, _, _, _, _>(
				f,
				h,
				bimap::<PairBrand, _, _, _, _, _, _>(g, i, x),
			)
	}

	// Functor Laws

	/// Tests the identity law for Functor.
	#[quickcheck]
	fn functor_identity(
		first: String,
		second: i32,
	) -> bool {
		let x = Pair(first, second);
		map::<PairWithFirstBrand<String>, _, _, _>(identity, x.clone()) == x
	}

	/// Tests the composition law for Functor.
	#[quickcheck]
	fn functor_composition(
		first: String,
		second: i32,
	) -> bool {
		let x = Pair(first, second);
		let f = |x: i32| x.wrapping_add(1);
		let g = |x: i32| x.wrapping_mul(2);
		map::<PairWithFirstBrand<String>, _, _, _>(compose(f, g), x.clone())
			== map::<PairWithFirstBrand<String>, _, _, _>(
				f,
				map::<PairWithFirstBrand<String>, _, _, _>(g, x),
			)
	}

	// Applicative Laws

	/// Tests the identity law for Applicative.
	#[quickcheck]
	fn applicative_identity(
		first: String,
		second: i32,
	) -> bool {
		let v = Pair(first, second);
		apply::<RcFnBrand, PairWithFirstBrand<String>, _, _>(
			pure::<PairWithFirstBrand<String>, _>(<RcFnBrand as CloneableFn>::new(identity)),
			v.clone(),
		) == v
	}

	/// Tests the homomorphism law for Applicative.
	#[quickcheck]
	fn applicative_homomorphism(x: i32) -> bool {
		let f = |x: i32| x.wrapping_mul(2);
		apply::<RcFnBrand, PairWithFirstBrand<String>, _, _>(
			pure::<PairWithFirstBrand<String>, _>(<RcFnBrand as CloneableFn>::new(f)),
			pure::<PairWithFirstBrand<String>, _>(x),
		) == pure::<PairWithFirstBrand<String>, _>(f(x))
	}

	/// Tests the composition law for Applicative.
	#[quickcheck]
	fn applicative_composition(
		w_first: String,
		w_second: i32,
		u_seed: i32,
		v_seed: i32,
	) -> bool {
		let w = Pair(w_first, w_second);

		let u_fn = <RcFnBrand as CloneableFn>::new(move |x: i32| x.wrapping_add(u_seed));
		let u = pure::<PairWithFirstBrand<String>, _>(u_fn);

		let v_fn = <RcFnBrand as CloneableFn>::new(move |x: i32| x.wrapping_mul(v_seed));
		let v = pure::<PairWithFirstBrand<String>, _>(v_fn);

		// RHS: u <*> (v <*> w)
		let vw = apply::<RcFnBrand, PairWithFirstBrand<String>, _, _>(v.clone(), w.clone());
		let rhs = apply::<RcFnBrand, PairWithFirstBrand<String>, _, _>(u.clone(), vw);

		// LHS: pure(compose) <*> u <*> v <*> w
		let compose_fn = <RcFnBrand as CloneableFn>::new(|f: std::rc::Rc<dyn Fn(i32) -> i32>| {
			let f = f.clone();
			<RcFnBrand as CloneableFn>::new(move |g: std::rc::Rc<dyn Fn(i32) -> i32>| {
				let f = f.clone();
				let g = g.clone();
				<RcFnBrand as CloneableFn>::new(move |x| f(g(x)))
			})
		});

		let pure_compose = pure::<PairWithFirstBrand<String>, _>(compose_fn);
		let u_applied = apply::<RcFnBrand, PairWithFirstBrand<String>, _, _>(pure_compose, u);
		let uv = apply::<RcFnBrand, PairWithFirstBrand<String>, _, _>(u_applied, v);
		let lhs = apply::<RcFnBrand, PairWithFirstBrand<String>, _, _>(uv, w);

		lhs == rhs
	}

	/// Tests the interchange law for Applicative.
	#[quickcheck]
	fn applicative_interchange(
		y: i32,
		u_seed: i32,
	) -> bool {
		// u <*> pure y = pure ($ y) <*> u
		let f = move |x: i32| x.wrapping_mul(u_seed);
		let u = pure::<PairWithFirstBrand<String>, _>(<RcFnBrand as CloneableFn>::new(f));

		let lhs = apply::<RcFnBrand, PairWithFirstBrand<String>, _, _>(
			u.clone(),
			pure::<PairWithFirstBrand<String>, _>(y),
		);

		let rhs_fn =
			<RcFnBrand as CloneableFn>::new(move |f: std::rc::Rc<dyn Fn(i32) -> i32>| f(y));
		let rhs = apply::<RcFnBrand, PairWithFirstBrand<String>, _, _>(
			pure::<PairWithFirstBrand<String>, _>(rhs_fn),
			u,
		);

		lhs == rhs
	}

	// Monad Laws

	/// Tests the left identity law for Monad.
	#[quickcheck]
	fn monad_left_identity(a: i32) -> bool {
		let f = |x: i32| Pair("f".to_string(), x.wrapping_mul(2));
		bind::<PairWithFirstBrand<String>, _, _, _>(pure::<PairWithFirstBrand<String>, _>(a), f)
			== f(a)
	}

	/// Tests the right identity law for Monad.
	#[quickcheck]
	fn monad_right_identity(
		first: String,
		second: i32,
	) -> bool {
		let m = Pair(first, second);
		bind::<PairWithFirstBrand<String>, _, _, _>(
			m.clone(),
			pure::<PairWithFirstBrand<String>, _>,
		) == m
	}

	/// Tests the associativity law for Monad.
	#[quickcheck]
	fn monad_associativity(
		first: String,
		second: i32,
	) -> bool {
		let m = Pair(first, second);
		let f = |x: i32| Pair("f".to_string(), x.wrapping_mul(2));
		let g = |x: i32| Pair("g".to_string(), x.wrapping_add(1));
		bind::<PairWithFirstBrand<String>, _, _, _>(
			bind::<PairWithFirstBrand<String>, _, _, _>(m.clone(), f),
			g,
		) == bind::<PairWithFirstBrand<String>, _, _, _>(m, |x| {
			bind::<PairWithFirstBrand<String>, _, _, _>(f(x), g)
		})
	}

	// ParFoldable Tests for PairWithFirstBrand (Functor over Second)

	/// Tests `par_fold_map` on `PairWithFirstBrand`.
	#[test]
	fn par_fold_map_pair_with_first() {
		let x = Pair("a".to_string(), 1);
		let f = send_cloneable_fn_new::<ArcFnBrand, _, _>(|x: i32| x.to_string());
		assert_eq!(
			par_fold_map::<ArcFnBrand, PairWithFirstBrand<String>, _, _>(f, x),
			"1".to_string()
		);
	}

	/// Tests `par_fold_right` on `PairWithFirstBrand`.
	#[test]
	fn par_fold_right_pair_with_first() {
		let x = Pair("a".to_string(), 1);
		let f = send_cloneable_fn_new::<ArcFnBrand, _, _>(|(a, b): (i32, i32)| a + b);
		assert_eq!(par_fold_right::<ArcFnBrand, PairWithFirstBrand<String>, _, _>(f, 10, x), 11);
	}

	// ParFoldable Tests for PairWithSecondBrand (Functor over First)

	/// Tests `par_fold_map` on `PairWithSecondBrand`.
	#[test]
	fn par_fold_map_pair_with_second() {
		let x = Pair(1, "a".to_string());
		let f = send_cloneable_fn_new::<ArcFnBrand, _, _>(|x: i32| x.to_string());
		assert_eq!(
			par_fold_map::<ArcFnBrand, PairWithSecondBrand<String>, _, _>(f, x),
			"1".to_string()
		);
	}

	/// Tests `par_fold_right` on `PairWithSecondBrand`.
	#[test]
	fn par_fold_right_pair_with_second() {
		let x = Pair(1, "a".to_string());
		let f = send_cloneable_fn_new::<ArcFnBrand, _, _>(|(a, b): (i32, i32)| a + b);
		assert_eq!(par_fold_right::<ArcFnBrand, PairWithSecondBrand<String>, _, _>(f, 10, x), 11);
	}
}