Type-Level Natural Numbers in TypeScript

This challenge involves creating a system of type-level natural numbers in TypeScript. This means defining types that represent numbers, allowing you to perform operations and comparisons on them at compile time, without relying on runtime JavaScript values. This is a fundamental concept in advanced type system programming and can lead to more robust and expressive code.

Problem Description

Your task is to define a type-level representation for natural numbers (0, 1, 2, ...). You will need to create types for:

• Zero: Representing the number 0.
• Successor: A way to construct a number from its predecessor (e.g., representing 3 based on the type representing 2).
• Arithmetic Operations: Implement addition and multiplication for these type-level numbers.
• Comparison: Implement a way to check if one type-level number is less than or equal to another.

Key Requirements:

1. Zero Representation: Define a type Zero that represents the natural number 0.
2. Successor Representation: Define a generic type Succ<T> where T is itself a type-level natural number. Succ<T> should represent the number T + 1.
3. Addition (Add<A, B>): Implement a generic type Add<A, B> that takes two type-level natural numbers A and B and returns a type-level natural number representing A + B.
4. Multiplication (Multiply<A, B>): Implement a generic type Multiply<A, B> that takes two type-level natural numbers A and B and returns a type-level natural number representing A * B.
5. Less Than or Equal To (IsLE<A, B>): Implement a generic type IsLE<A, B> that takes two type-level natural numbers A and B and returns a union type with true if A <= B, and false otherwise. You can use literal types true and false for this.

Expected Behavior:

When you use these types with actual TypeScript types, the compiler should be able to infer the correct type-level result. For example, Add<Succ<Zero>, Succ<Succ<Zero>>> should resolve to a type representing 3.

Edge Cases to Consider:

• Operations involving Zero.
• Deeply nested Succ types.

Examples

Example 1: Basic Number Representation

// Define Zero and Succ
type Zero = /* Your definition for Zero */;
type Succ<T extends /* constraint for type-level numbers */> = /* Your definition for Successor */;

// Representing numbers
type One = Succ<Zero>;
type Two = Succ<One>;
type Three = Succ<Two>;
type Four = Succ<Three>;

// Expected type inferences:
// One should be inferred as representing 1
// Two should be inferred as representing 2
// Three should be inferred as representing 3
// Four should be inferred as representing 4

Example 2: Addition

// Assuming Zero, Succ, One, Two, Three are defined as above

type Sum1 = Add<Two, Three>; // Should infer a type representing 5
type Sum2 = Add<Zero, Four>; // Should infer a type representing 4
type Sum3 = Add<One, Zero>;  // Should infer a type representing 1

Example 3: Multiplication

// Assuming Zero, Succ, One, Two, Three, Four are defined as above

type Prod1 = Multiply<Two, Three>; // Should infer a type representing 6
type Prod2 = Multiply<Four, Zero>; // Should infer a type representing 0
type Prod3 = Multiply<One, Four>;  // Should infer a type representing 4

Example 4: Less Than or Equal To

// Assuming Zero, Succ, One, Two, Three, Four are defined as above

type LE1 = IsLE<Two, Three>; // Should infer the type `true`
type LE2 = IsLE<Three, Two>; // Should infer the type `false`
type LE3 = IsLE<Two, Two>;   // Should infer the type `true`
type LE4 = IsLE<Zero, One>;  // Should infer the type `true`
type LE5 = IsLE<One, Zero>;  // Should infer the type `false`

Constraints

• The solution must be entirely within TypeScript's type system. No runtime JavaScript code is allowed for the core number representation and operations.
• You should aim for a solution that is reasonably performant for common cases. Extremely deep nesting might lead to compiler performance issues, which is an inherent trade-off in complex type-level programming.
• Your types should correctly handle all non-negative integers.

Notes

• This problem is a classic example of "dependent types" or "computational types" within a language that doesn't fully support them. You'll be leveraging TypeScript's conditional types and recursive type definitions.
• Think about how you can "deconstruct" a number (i.e., get its predecessor) to implement operations like addition and multiplication.
• The IsLE comparison is a good building block for more complex comparisons or control flow at the type level.
• Consider how to constrain your generic types to only accept valid type-level natural numbers.