Type-Level Lambda Calculus in TypeScript

This challenge involves implementing a form of lambda calculus entirely at the TypeScript type level. You will define types that represent lambda abstractions, applications, and variables, and then create a system to evaluate these type-level expressions. This exercise explores the expressive power of TypeScript's type system and its potential for compile-time computation.

Problem Description

Your goal is to create a TypeScript type-level system that can represent and evaluate simple lambda calculus expressions. This means defining types for:

1. Variables: A type to represent a symbolic variable, like x, y, etc.
2. Lambda Abstractions: A type representing a function, like λx. M, where x is the bound variable and M is the body of the function.
3. Applications: A type representing function application, like (F M), where F is a function and M is its argument.

You will need to implement a mechanism for beta-reduction, the fundamental evaluation rule in lambda calculus, at the type level. This involves substituting the argument into the body of the function when a lambda abstraction is applied to an argument.

Key Requirements:

• Define types for Variable, Abstraction, and Application.
• Implement a Reduce type that performs beta-reduction.
• The system should handle simple cases of reduction.

Expected Behavior:

Given a type-level expression representing a lambda calculus term, the Reduce type should, after a sufficient number of reductions, simplify to its normal form (or a representation of it). For this challenge, focus on achieving one level of beta-reduction where possible.

Edge Cases to Consider:

• Applying an abstraction to a variable.
• Applying an abstraction to another abstraction.
• Applying a variable (if your system allows for it, though typically starts with abstractions).
• Terms that cannot be reduced further.

Examples

Example 1: Representing λx. x (the identity function) and applying it to y.

// Representing the identity function: λx. x
type Identity<T> = ...

// Representing the variable 'y'
type Y = ...

// Applying Identity to Y: (λx. x) y
type Result = Reduce<Application<Identity<Var<'x'>>, Var<'y'>>>

Expected Result (simplified representation):

// The type representing 'y'
Var<'y'>

Explanation: The Identity abstraction binds x. When applied to y, the x within the body x is substituted with y, resulting in y.

Example 2: Representing λx. λy. x (a function that returns its first argument) and applying it to a and then to b.

// Representing λx. λy. x
type FuncAB<T> = ...

// Representing variable 'a'
type A = ...

// Representing variable 'b'
type B = ...

// Applying FuncAB to A: (λx. λy. x) a
type IntermediateResult = Reduce<Application<FuncAB<Var<'x'>, Var<'y'>>, Var<'a'>>>

// Applying the result to B: ((λx. λy. x) a) b
type FinalResult = Reduce<Application<IntermediateResult, Var<'b'>>>

Expected FinalResult (simplified representation):

// The type representing 'a'
Var<'a'>

Explanation:

1. Reduce<(λx. λy. x) a>: The a is substituted for x in the body λy. x, resulting in λy. a.
2. Reduce<(λy. a) b>: The b is substituted for y in the body a. Since a does not contain y, the body a remains unchanged.

Example 3: A non-reducible expression.

// Representing a simple variable 'z'
type Z = Var<'z'>

// Applying Reduce to Z
type NonReduced = Reduce<Z>

Expected NonReduced:

// The type remains 'z'
Var<'z'>

Explanation: There is no abstraction to apply, so no reduction occurs.

Constraints

• The types defined should be generic where appropriate to represent any variable name and any term.
• The Reduce type should aim to perform a single step of beta-reduction if applicable. Recursive reduction is a bonus but not strictly required for a passing solution.
• Variable names will be represented by string literals (e.g., 'x', 'y').
• Your solution must compile and type-check successfully within a standard TypeScript environment.

Notes

• Consider how you will represent variable capture and substitution. This is a core challenge in lambda calculus implementation.
• Think about using conditional types (extends) and mapped types to implement the substitution logic.
• You might need helper types to manage the scope of variables.
• The goal is to mimic the behavior of lambda calculus evaluation using TypeScript's type system.