Type-Level Peano Arithmetic in TypeScript

This challenge asks you to implement Peano arithmetic, a foundational system for defining natural numbers using types, entirely within TypeScript's type system. This exercise explores the expressive power of TypeScript's advanced features like generic types, conditional types, and type inference to model mathematical concepts at compile time.

Problem Description

You are tasked with creating a type-level representation of natural numbers using the Peano axioms. In this system, a natural number is either zero or the successor of another natural number. You will define types for Zero and Successor<T> (where T is another natural number type) and then implement type-level operations for addition and multiplication.

Key Requirements:

1. Zero Type: Define a type that represents the number 0.
2. Successor<T> Type: Define a generic type that takes another natural number type T and represents its successor.
3. Add<A, B> Type: Implement a generic type that takes two natural number types, A and B, and returns a new natural number type representing their sum.
4. Multiply<A, B> Type: Implement a generic type that takes two natural number types, A and B, and returns a new natural number type representing their product.
5. Type Inference: Ensure that your implementations can be used with TypeScript's type inference to correctly determine the resulting number types.

Expected Behavior:

Your types should behave according to standard arithmetic rules. For example:

• Add<Zero, Zero> should resolve to Zero.
• Add<Successor<Zero>, Zero> should resolve to Successor<Zero>.
• Add<Successor<Zero>, Successor<Zero>> should resolve to Successor<Successor<Zero>>.
• Multiply<Successor<Zero>, Zero> should resolve to Zero.
• Multiply<Successor<Successor<Zero>>, Successor<Zero>> should resolve to Successor<Successor<Zero>>.

Edge Cases:

• Ensure Add and Multiply correctly handle Zero as an operand.
• Consider the order of operands for Add and Multiply (they should be commutative, though your implementation might naturally enforce this).

Examples

Let's define some basic numbers for demonstration:

type One = Successor<Zero>;
type Two = Successor<One>; // Successor<Successor<Zero>>
type Three = Successor<Two>; // Successor<Successor<Successor<Zero>>>

Example 1: Addition

// Input types representing 2 and 3
type NumA = Two;
type NumB = Three;

// Calculate the sum: 2 + 3
type Sum = Add<NumA, NumB>;

// Expected Output: A type representing 5
// type Sum = Successor<Successor<Successor<Successor<Successor<Zero>>>>>;

Example 2: Multiplication

// Input types representing 3 and 2
type NumC = Three;
type NumD = Two;

// Calculate the product: 3 * 2
type Product = Multiply<NumC, NumD>;

// Expected Output: A type representing 6
// type Product = Successor<Successor<Successor<Successor<Successor<Successor<Zero>>>>>>>;

Example 3: Addition with Zero

// Input types representing 4 and 0
type NumE = Successor<Successor<Successor<Successor<Zero>>>>;
type NumF = Zero;

// Calculate the sum: 4 + 0
type SumZero = Add<NumE, NumF>;

// Expected Output: A type representing 4
// type SumZero = Successor<Successor<Successor<Successor<Zero>>>>;

Constraints

• All arithmetic operations must be implemented purely at the type level using TypeScript's type system.
• No runtime JavaScript code should be used to perform the calculations. The results should be derivable solely by the TypeScript compiler.
• The depth of nested Successor types should not exceed typical compiler limits (e.g., avoid excessively large numbers that might cause stack overflows during type checking).

Notes

• The core of Peano arithmetic lies in recursion. Consider how you can translate recursive mathematical definitions into recursive TypeScript conditional types.
• Think about how to "decrement" a Peano number for the base cases of your recursive functions. A helper type might be useful.
• The Add<A, B> operation can often be defined by repeatedly taking the successor of A, B times.
• The Multiply<A, B> operation can be defined as adding A to itself, B times.
• You'll likely want to use conditional types (infer) to deconstruct and reconstruct your Peano number types.