Type-Level Programming with TypeScript: A Simple Calculator

This challenge involves creating a type-level programming language within TypeScript. You will implement a system that can perform basic arithmetic operations (addition and subtraction) solely at the type level, meaning the computations are resolved during compile-time rather than runtime. This exercise explores the power of TypeScript's type system for metaprogramming and static analysis.

Problem Description

Your task is to design and implement a type-level calculator that can evaluate simple arithmetic expressions involving non-negative integers and the operations of addition and subtraction. The expressions will be represented using TypeScript's branded types or discriminated unions, allowing you to define a syntax for these operations at the type level.

Key Requirements:

1. Number Representation: Define a type that can represent non-negative integers at the type level. Consider using a recursive approach (e.g., Peano-like numbers) or a more direct representation if feasible within TypeScript's limits.
2. Operation Types: Create distinct types to represent addition and subtraction operations. These types should encapsulate the operands involved.
3. Evaluation Type: Design a central Evaluate type that takes an expression (represented by your type-level syntax) and resolves it to a single numeric type.
4. Type Safety: Ensure that invalid operations or representations are caught at compile time.

Expected Behavior:

The Evaluate type should be able to correctly compute the result of an expression. For instance, if you have a type Add<Num1, Num2> and an Evaluate type that can process it, Evaluate<Add<One, Two>> should resolve to Three. Similarly, Evaluate<Subtract<Five, Two>> should resolve to Three.

Edge Cases:

• Subtraction resulting in negative numbers: The type-level language should handle this. You might choose to represent negative numbers or have specific behavior (e.g., resolving to zero or a dedicated "NegativeResult" type). For simplicity, we'll aim for representing non-negative results. If a subtraction would result in a negative number, it should resolve to Zero.
• Large numbers: Consider the practical limits of TypeScript's type system for recursive types and generic instantiation.

Examples

Example 1: Addition

// Assuming you have Number types like Zero, One, Two, Three, etc.
// and an Add type: type Add<N1, N2> = ...

// Input:
type Expression1 = Add<One, Two>;
type Result1 = Evaluate<Expression1>;

// Expected Output (assuming Evaluate correctly resolves Add):
// Result1 will resolve to the type representing the number 3.

Example 2: Subtraction

// Assuming you have Number types like Two, Five, etc.
// and a Subtract type: type Subtract<N1, N2> = ...

// Input:
type Expression2 = Subtract<Five, Two>;
type Result2 = Evaluate<Expression2>;

// Expected Output (assuming Evaluate correctly resolves Subtract):
// Result2 will resolve to the type representing the number 3.

Example 3: Subtraction resulting in zero

// Input:
type Expression3 = Subtract<Two, Five>;
type Result3 = Evaluate<Expression3>;

// Expected Output (assuming subtraction resulting in negative resolves to Zero):
// Result3 will resolve to the type representing the number 0.

Constraints

• The type-level calculator should only support non-negative integers.
• The supported operations are addition (Add) and subtraction (Subtract).
• The output of subtraction that would result in a negative number must resolve to Zero.
• The complexity of the evaluated expressions should be manageable within TypeScript's typical type instantiation limits. Avoid excessively deep recursion.
• The implementation should primarily leverage conditional types, mapped types, and recursive type definitions.

Notes

• Consider how you will represent numbers. A common approach in type-level programming is to use a recursive structure, such as a Num<N> where N is a type representing the successor of the previous number (e.g., Zero, Succ<Zero>, Succ<Succ<Zero>>).
• Think about how to deconstruct and reconstruct numbers within conditional types to perform calculations.
• You will likely need helper types to determine if one number type is greater than or equal to another for subtraction.
• This challenge is about understanding the expressiveness of TypeScript's type system. Focus on correctness and clarity of your type definitions.