Type-Level Computation with Rust Generics and Traits

This challenge explores the power of Rust's type system to perform computations at compile time. You will implement a system that mimics type-level arithmetic, demonstrating how generic parameters and trait bounds can be used to encode and execute logic without any runtime overhead. This technique is fundamental to advanced metaprogramming and can be used for compile-time configuration, generating complex data structures, and ensuring properties that would otherwise be difficult to enforce at runtime.

Problem Description

Your task is to implement type-level addition for unsigned integers in Rust. You will define a trait that represents an "addition" operation at the type level, and then instantiate this trait for specific integer types. The goal is to be able to express a computation like Add<TypeA, TypeB>::Output and have the type system resolve Output to the correct type representing the sum of TypeA and TypeB.

Key Requirements:

Define a Type-Level Integer Representation: Create a way to represent unsigned integers using types. For simplicity, let's limit the scope to small, non-negative integers. You can use zero and successor patterns, similar to Peano arithmetic, or a simpler enum-based approach for this challenge.
Define the Add Trait: Create a trait named Add with an associated type Output. This trait should take two type parameters representing the numbers to be added.
Implement Add for Specific Integers: Provide impl blocks for the Add trait, specifying how to add different combinations of your type-level integers.
Demonstrate Type-Level Addition: Show how to use your Add trait to declare a type that represents the sum of two other type-level integers. The compiler should resolve the Output associated type.

Expected Behavior:

When you write code like let sum: <MyTypeA as Add<MyTypeB>>::Output;, the compiler should infer sum to be of the type representing the sum of MyTypeA and MyTypeB.

Edge Cases/Considerations:

Integer Size Limits: Decide on a reasonable upper limit for the integers you will support to keep the trait implementations manageable.
Zero Representation: Ensure your integer representation handles zero correctly.

Examples

Example 1: Adding Zero and One

// Assume we have type-level representations:
// struct Zero;
// struct One;

// We want to achieve:
// <One as Add<Zero>>::Output should resolve to the type representing 1

Example 2: Adding Two and Three

// Assume we have type-level representations:
// struct Two;
// struct Three;

// We want to achieve:
// <Two as Add<Three>>::Output should resolve to the type representing 5

Example 3: Demonstrating Usage in a static variable

// If MyTypeA represents 3 and MyTypeB represents 4,
// then <MyTypeA as Add<MyTypeB>>::Output should resolve to the type representing 7.

// A static variable whose type is determined by type-level addition.
// let result_type: <MyTypeA as Add<MyTypeB>>::Output;
// The type of 'result_type' will be inferred by the compiler.

Constraints

The type-level integers should represent non-negative whole numbers.
The implementation should focus on a limited range of integers (e.g., up to 10) to make the trait implementations feasible.
The solution must rely entirely on Rust's type system (generics, traits, associated types) without any runtime computation.
The Add trait implementation should not require any function bodies that perform runtime logic; the logic is encoded in the trait definitions and implementations themselves.

Notes

Consider using a pattern like the Peano numbers (Zero and Successor) for your type-level integers. For example: struct Zero; struct Succ<N: Integer>(N);.
The core of this challenge is defining the recursive structure of the Add trait implementations. For instance, adding Succ<A> to B might involve adding A to B and then wrapping the result in Succ.
Think about how you would typically define addition for numbers and translate that logic into trait bounds and associated types.
The goal is for the compiler to "compute" the sum during the compilation phase. You won't be running any code to perform addition.