Type-Level Natural Numbers in TypeScript

This challenge focuses on implementing a system of "type-level" natural numbers within TypeScript. This means defining number types that exist at compile-time, allowing for type-safe arithmetic operations and checks. This is useful for enforcing constraints and performing computations that should be validated before runtime, leading to more robust and predictable code.

Problem Description

Your task is to create a set of TypeScript type definitions that represent natural numbers (0 and positive integers) and enable basic arithmetic operations on them purely at the type level. This involves defining types for each number and creating generic types that act as functions for operations like addition and subtraction.

Key Requirements:

1. Zero Type: Define a type representing the number 0.
2. Number<N> Type: Define a generic type Number<N> that represents a natural number. You can use a linked-list-like structure where Number<N> is a "successor" of N. For instance, One could be Number<Zero>, Two could be Number<One>, and so on.
3. Add<A, B> Type: Create a generic type Add<A, B> that takes two type-level numbers A and B and produces a new type representing their sum.
4. Subtract<A, B> Type: Create a generic type Subtract<A, B> that takes two type-level numbers A and B and produces a new type representing their difference. This operation should be defined such that subtracting a larger number from a smaller one results in a type representing zero or an error state if you choose to implement one.
5. IsZero<N> Type: Create a generic type IsZero<N> that returns true if N is the Zero type, and false otherwise.

Expected Behavior:

• Arithmetic operations should produce the correct type-level number representing the result.
• Type checking should correctly infer the results of these operations.

Edge Cases to Consider:

• Subtracting zero from any number.
• Subtracting a number from itself.
• Subtracting a larger number from a smaller number (define how you want to handle this, e.g., return Zero or a specific "negative" indicator if you extend beyond natural numbers). For this challenge, returning Zero is a reasonable approach.

Examples

Example 1: Representing Numbers

// Assuming Number<N> represents N+1, and Zero represents 0

type Zero = /* Your definition for Zero */;
type One = /* Your definition for One, e.g., Number<Zero> */;
type Two = /* Your definition for Two, e.g., Number<One> */;
type Three = /* Your definition for Three, e.g., Number<Two> */;

// Expected type:
// If Zero is represented by { type: 'Zero' }
// Then One would be { type: 'Number', prev: { type: 'Zero' } }
// And Two would be { type: 'Number', prev: { type: 'Number', prev: { type: 'Zero' } } }

Example 2: Addition

// Given the number types from Example 1

type Sum1 = Add<One, Two>; // Should represent the type for 3
type Sum2 = Add<Two, Zero>; // Should represent the type for 2

// Expected type for Sum1: The type representing 3
// Expected type for Sum2: The type representing 2

Example 3: Subtraction

// Given the number types from Example 1

type Diff1 = Subtract<Three, One>; // Should represent the type for 2
type Diff2 = Subtract<One, Two>;  // Should represent the type for 0 (as 1-2 is not a natural number)
type Diff3 = Subtract<Two, Two>;  // Should represent the type for 0

// Expected type for Diff1: The type representing 2
// Expected type for Diff2: The type representing 0
// Expected type for Diff3: The type representing 0

Example 4: IsZero Check

// Given the number types from Example 1

type Check1 = IsZero<Zero>;     // Should be true
type Check2 = IsZero<One>;      // Should be false
type Check3 = IsZero<Subtract<Two, Two>>; // Should be true (because Subtract<Two, Two> results in Zero)

// Expected type for Check1: true
// Expected type for Check2: false
// Expected type for Check3: true

Constraints

• The implementation must solely use TypeScript's type system. No runtime JavaScript code is allowed for the number representation or arithmetic operations.
• The linked-list-like representation for numbers is recommended to avoid exponential type complexity for basic operations.
• The IsZero type should be able to correctly identify your Zero type.
• Performance of type checking should be considered; extremely deep recursion for simple operations can lead to long compilation times. Aim for an efficient recursive structure for your number representation and operations.

Notes

• Consider using conditional types and recursive type definitions.
• Think about how you will represent the "predecessor" of a number to build up your number system.
• For Subtract<A, B>, you'll likely need a way to "count down" from A by removing the "predecessor" relationship B times.
• The IsZero type is a good base case for your recursive operations.