Rust’s type system is a powerhouse, and I’ve been fascinated by how we can push it to its limits. Type-level integer arithmetic is one of those mind-bending concepts that opens up new possibilities for compile-time computations. Let’s explore this together.

At its core, type-level integer arithmetic allows us to perform calculations during compilation, resulting in zero runtime overhead. It’s like having a mini-computer within the compiler itself. This technique is particularly useful for creating highly optimized code, ensuring type safety, and implementing complex algorithms that run at compile-time.

To get started, we need to understand how to represent integers at the type level. In Rust, we can use zero-sized types to encode numerical values. Here’s a simple example:

struct Zero;
struct Succ<T>;

type One = Succ<Zero>;
type Two = Succ<One>;
type Three = Succ<Two>;

In this representation, Zero is our base case, and Succ<T> represents the successor of a number. We can build up any natural number using this approach.

Now, let’s implement some basic arithmetic operations. We’ll start with addition:

trait Add<B> {
    type Output;
}

impl<B> Add<B> for Zero {
    type Output = B;
}

impl<A, B> Add<B> for Succ<A>
where
    A: Add<B>,
{
    type Output = Succ<<A as Add<B>>::Output>;
}

This trait allows us to add two type-level integers. The implementation recursively breaks down the addition until we reach the base case of adding zero to a number.

We can use this to perform compile-time addition:

type Sum = <Two as Add<Three>>::Output;

Sum will be equivalent to Five at compile-time.

Multiplication can be implemented in a similar way:

trait Mul<B> {
    type Output;
}

impl<B> Mul<B> for Zero {
    type Output = Zero;
}

impl<A, B> Mul<B> for Succ<A>
where
    A: Mul<B>,
    B: Add<<A as Mul<B>>::Output>,
{
    type Output = <B as Add<<A as Mul<B>>::Output>>::Output;
}

This implementation recursively breaks down multiplication into repeated addition.

One of the challenges with type-level arithmetic is handling overflow. Since we’re working with the type system, we can’t rely on traditional runtime checks. Instead, we need to design our operations to be safe by construction. One approach is to use a fixed upper bound:

trait Bounded {
    type Max;
}

impl Bounded for Zero {
    type Max = Zero;
}

impl<T: Bounded> Bounded for Succ<T> {
    type Max = Succ<<T as Bounded>::Max>;
}

trait SafeAdd<B>: Bounded {
    type Output;
}

impl<B: Bounded> SafeAdd<B> for Zero {
    type Output = B;
}

impl<A: Bounded, B: Bounded> SafeAdd<B> for Succ<A>
where
    A: SafeAdd<B>,
    <A as SafeAdd<B>>::Output: Bounded,
{
    type Output = Succ<<A as SafeAdd<B>>::Output>;
}

This implementation ensures that we can only add numbers up to a certain maximum value, preventing overflow at the type level.

Now, let’s tackle something more complex: generating prime numbers at compile-time. We’ll start by implementing a trait to check if a number is prime:

trait IsPrime {
    const VALUE: bool;
}

impl IsPrime for Zero {
    const VALUE: bool = false;
}

impl IsPrime for Succ<Zero> {
    const VALUE: bool = false;
}

impl<T> IsPrime for Succ<Succ<T>>
where
    Succ<Succ<T>>: CheckPrime<Succ<Succ<Zero>>>,
{
    const VALUE: bool = <Succ<Succ<T>> as CheckPrime<Succ<Succ<Zero>>>>::VALUE;
}

trait CheckPrime<N> {
    const VALUE: bool;
}

impl<T> CheckPrime<Succ<T>> for Succ<Succ<Zero>>
where
    T: Mod<Succ<Succ<Zero>>>,
    <T as Mod<Succ<Succ<Zero>>>>::Output: IsZero,
{
    const VALUE: bool = !<<T as Mod<Succ<Succ<Zero>>>>::Output as IsZero>::VALUE;
}

impl<N, T> CheckPrime<N> for Succ<Succ<T>>
where
    Succ<Succ<T>>: Mod<N>,
    <Succ<Succ<T>> as Mod<N>>::Output: IsZero,
    N: Succ<PrevN>,
    Succ<Succ<T>>: CheckPrime<PrevN>,
{
    const VALUE: bool = !<<Succ<Succ<T>> as Mod<N>>::Output as IsZero>::VALUE
        && <Succ<Succ<T>> as CheckPrime<PrevN>>::VALUE;
}

trait IsZero {
    const VALUE: bool;
}

impl IsZero for Zero {
    const VALUE: bool = true;
}

impl<T> IsZero for Succ<T> {
    const VALUE: bool = false;
}

This implementation checks if a number is prime by testing divisibility up to its square root. We can use it like this:

type Seven = Succ<Succ<Succ<Succ<Succ<Succ<Succ<Zero>>>>>>>;
const IS_SEVEN_PRIME: bool = <Seven as IsPrime>::VALUE;

The compiler will evaluate IS_SEVEN_PRIME to true at compile-time.

Type-level integer arithmetic opens up exciting possibilities for creating compile-time validated data structures. For example, we can implement a fixed-size vector type that checks its length at compile-time:

struct Vec<T, N: Nat>(PhantomData<(T, N)>);

trait Nat {
    const VALUE: usize;
}

impl Nat for Zero {
    const VALUE: usize = 0;
}

impl<T: Nat> Nat for Succ<T> {
    const VALUE: usize = T::VALUE + 1;
}

impl<T, N: Nat> Vec<T, N> {
    fn new() -> Self {
        Vec(PhantomData)
    }

    fn len(&self) -> usize {
        N::VALUE
    }
}

This Vec type ensures that its length is known at compile-time, allowing for more efficient code and preventing out-of-bounds errors.

As we delve deeper into type-level integer arithmetic, we can implement more advanced algorithms. For instance, we could create a compile-time Fibonacci sequence generator:

trait Fibonacci {
    type Output;
}

impl Fibonacci for Zero {
    type Output = Zero;
}

impl Fibonacci for Succ<Zero> {
    type Output = Succ<Zero>;
}

impl<N: Nat> Fibonacci for Succ<Succ<N>>
where
    N: Fibonacci,
    Succ<N>: Fibonacci,
    <N as Fibonacci>::Output: Add<<Succ<N> as Fibonacci>::Output>,
{
    type Output = <<N as Fibonacci>::Output as Add<<Succ<N> as Fibonacci>::Output>>::Output;
}

This implementation calculates Fibonacci numbers recursively at the type level. We can use it to get compile-time Fibonacci numbers:

type Fib5 = <Succ<Succ<Succ<Succ<Succ<Zero>>>>> as Fibonacci>::Output;

The compiler will evaluate Fib5 to the fifth Fibonacci number at compile-time.

One of the most powerful applications of type-level integer arithmetic is in creating domain-specific type systems. For example, we can implement a type-safe matrix multiplication system:

struct Matrix<R, C>(PhantomData<(R, C)>);

trait Mul<B> {
    type Output;
}

impl<R1, C1, C2> Mul<Matrix<C1, C2>> for Matrix<R1, C1>
where
    R1: Nat,
    C1: Nat,
    C2: Nat,
{
    type Output = Matrix<R1, C2>;
}

This implementation ensures that we can only multiply matrices with compatible dimensions, catching errors at compile-time rather than runtime.

As we push the boundaries of type-level integer arithmetic, we encounter some limitations. Rust’s type system is Turing-complete, but in practice, deeply nested types can cause the compiler to struggle. It’s important to balance the power of compile-time computations with practical considerations.

Despite these challenges, the benefits of type-level integer arithmetic are significant. By moving computations to compile-time, we can create more efficient code, catch errors earlier in the development process, and express complex invariants in the type system itself.

In conclusion, type-level integer arithmetic in Rust is a powerful technique that allows us to leverage the full potential of the type system for compile-time computations. It enables us to create safer, more efficient code by pushing the boundaries of what’s possible at compile-time. As we continue to explore this fascinating area, we’ll find new ways to express complex algorithms and invariants in the type system, leading to more robust and performant Rust code.