Type-Safe Number Representation and Arithmetic in TypeScript

This challenge focuses on building a robust and type-safe system for representing numbers and performing arithmetic operations on them directly within TypeScript's type system. This exercise will deepen your understanding of advanced TypeScript features like conditional types, template literal types, and type-level manipulation, enabling you to create truly immutable and verifiable numerical constructs at compile time.

Problem Description

Your task is to create a set of TypeScript types that can represent non-negative integers and perform basic arithmetic operations (addition, subtraction, multiplication, and division) on them at the type level. The goal is to achieve type safety, meaning that operations should be constrained by the types themselves, and results should be accurately represented as new types.

Key Requirements:

1. Number Representation: Design a type, let's call it Num<T>, that can represent any non-negative integer. This type should be constructible from a string representation of the number (e.g., Num<'123'>).
2. Addition: Implement a type Add<A, B> that takes two Num types, A and B, and returns a new Num type representing their sum.
3. Subtraction: Implement a type Subtract<A, B> that takes two Num types, A and B, and returns a new Num type representing their difference. If A is less than B, the result should be Num<'0'>.
4. Multiplication: Implement a type Multiply<A, B> that takes two Num types, A and B, and returns a new Num type representing their product.
5. Division: Implement a type Divide<A, B> that takes two Num types, A and B, and returns a new Num type representing their integer division (floor). If B is Num<'0'>, the result should be Num<'0'> to avoid division by zero errors.
6. String Conversion: Implement a type ToString<T> that converts a Num type back into its string literal representation (e.g., ToString<Num<'123'>> should resolve to '123').
7. Type Safety: All operations must be performed exclusively within the TypeScript type system. No runtime JavaScript execution should be involved in the core arithmetic logic.

Expected Behavior:

When you use these types in your TypeScript code, the compiler should be able to infer and resolve the resulting numeric types. For instance, declaring a variable with a type like Add<Num<'5'>, Num<'7'>> should result in that variable having the type Num<'12'>.

Edge Cases to Consider:

• Addition involving zero.
• Subtraction where the subtrahend is greater than or equal to the minuend.
• Multiplication involving zero.
• Division by zero.
• Division where the dividend is less than the divisor.
• Handling of large numbers (within reasonable TypeScript limits).

Examples

Example 1: Addition

type Sum = Add<Num<'123'>, Num<'45'>'>; // Expected: Num<'168'>

Example 2: Subtraction

type Difference1 = Subtract<Num<'100'>, Num<'25'>'>; // Expected: Num<'75'>
type Difference2 = Subtract<Num<'10'>, Num<'15'>'>; // Expected: Num<'0'>

Example 3: Multiplication

type Product = Multiply<Num<'12'>, Num<'5'>'>; // Expected: Num<'60'>

Example 4: Division

type Quotient1 = Divide<Num<'100'>, Num<'10'>'>; // Expected: Num<'10'>
type Quotient2 = Divide<Num<'27'>, Num<'5'>'>; // Expected: Num<'5'>
type Quotient3 = Divide<Num<'10'>, Num<'0'>'>; // Expected: Num<'0'>

Example 5: String Conversion

type NumberAsString = ToString<Add<Num<'9'>, Num<'8'>'>>; // Expected: '17'

Constraints

• Numbers represented will be non-negative integers.
• Input numbers will be provided as string literals (e.g., '123').
• The maximum value representable should be practical within TypeScript's type inference limits. Extremely large numbers (e.g., requiring thousands of recursive steps) might lead to performance issues or exceed compiler limits. Aim for numbers that can be handled within a few hundred recursive calls at most.
• Your solution should not rely on any external libraries or runtime JavaScript code for the type arithmetic logic.

Notes

This challenge is a significant undertaking and requires a deep understanding of TypeScript's advanced type system. You will likely need to leverage:

• Template Literal Types: For parsing and manipulating string representations of numbers.
• Conditional Types: To implement logic based on number comparisons and other conditions.
• Recursive Types: To handle multi-digit numbers and perform arithmetic digit by digit.
• Infer Keyword: To extract parts of types.

Consider breaking down the problem:

1. Single Digit Arithmetic: Start by implementing arithmetic for single digits.
2. Number Parsing: Develop a robust way to parse multi-digit numbers into a more manageable type representation (perhaps an array of digits).
3. Carries and Borrows: For addition and subtraction, you'll need to handle carry-over and borrow mechanics.
4. Multiplication and Division: These are generally more complex and can be built upon the addition and subtraction primitives. Multiplication can often be implemented using repeated addition. Division can be a more involved process of repeated subtraction.
5. String Conversion: This might involve another recursive process to reconstruct the string from your internal numeric representation.

Good luck! This is an excellent opportunity to push the boundaries of what you can achieve with TypeScript types.