Type-Level Combinators in TypeScript

This challenge explores the fascinating world of type-level programming in TypeScript by implementing classic combinators. Type-level combinators are functions that operate on types themselves, allowing for powerful metaprogramming and the creation of complex type manipulations without runtime code. Successfully implementing these will deepen your understanding of TypeScript's generic system and advanced type inference.

Problem Description

Your task is to implement several fundamental type-level combinators in TypeScript. These combinators are analogous to their lambda calculus counterparts but operate on types instead of values. You will define generic type aliases or interfaces that represent these combinators.

Key Requirements:

Implement the I (Identity) Combinator: This combinator simply returns its input type.
Implement the K (Constant) Combinator: This combinator takes two types and always returns the first one, discarding the second.
Implement the S (Substitution) Combinator: This combinator is more complex. It takes three types, A, B, and C, and produces a new type that represents (A C) (B C). This is the most powerful combinator and can be used to express any computable function.
Implement the B (Composition) Combinator: This combinator takes two types, F and G, and produces a new type representing the composition of F and G, i.e., F G.

Expected Behavior:

Your implementations should be generic types that accept type parameters representing the input types. When used, they should correctly resolve to the type described by their respective combinator logic.

Important Considerations:

You will be working exclusively with TypeScript's type system. No runtime JavaScript code will be involved in the implementation of the combinators themselves.
Think carefully about how to represent functions between types. For instance, (A C) in the S combinator implies a type that takes C and produces a result based on A.

Examples

Example 1: The I Combinator

Input:

type Identity<T> = T;
type TestI = Identity<string>;

Output:

// TestI would be inferred as 'string'

Explanation: The Identity type takes a type T and simply returns T. When string is passed as T, TestI becomes string.

Example 2: The K Combinator

Input:

type Constant<A, B> = A;
type TestK = Constant<number, boolean>;

Output:

// TestK would be inferred as 'number'

Explanation: The Constant type takes two types, A and B, and always returns A. When number and boolean are passed, TestK becomes number, effectively ignoring boolean.

Example 3: The B Combinator (Conceptual)

Let's assume we have types that represent functions. For example, F<X> = X[] and G<Y> = Y | null.

Input:

// Conceptual types for functions (simplified for example)
type F<X> = X[];
type G<Y> = Y | null;

// Implementing B combinator: B<F, G><Z> = F<G<Z>>
type Compose<F, G> = <Z>(gResult: G<Z>) => F<Z>; // This is a type-level function definition

// To use it, we need a way to apply types to type constructors.
// This is where advanced techniques might be needed.
// For this challenge, let's assume we are working with specific types.

// Let's define concrete type-level functions and compose them.
type Stringifier<T> = T extends string ? T : T extends number ? `${T}` : never;
type ArrayWrapper<T> = T[];

// We want to implement B<Stringifier, ArrayWrapper><number> which should be Stringifier<ArrayWrapper<number>>
// This means: Stringifier<number[]> which would be (number[])[] - this is not what S/B combinators mean.
// S/B combinators work on *function application*.
// Let's reframe: B<F, G>(x) = F(G(x))

// Type representing a function from T to U: (T) => U
// In TS types, this is often represented as an interface with a generic index signature, or a mapped type.

// For this challenge, let's focus on composing type constructors themselves, not just concrete types.

// Example using S combinator's spirit: (A C) (B C)
// Let's consider A = number, B = string, C = boolean
// We need a way to represent a "type-level application" like A<C>
// This is typically done with conditional types.

// Let's use a concrete example of B:
// B<F, G> should represent a type that, when applied to X, results in F<G<X>>.
// We can define a type that *is* the composed function type.

type ComposeTwo<FConstructor, GConstructor> = <X>(x: GConstructor<X>) => FConstructor<X>; // This is getting closer to runtime.

// Let's stick to pure type manipulation.
// B<F, G> = Type that represents F applied to G.
// Consider type constructors:
// TypeConstructor1<T> = T[]
// TypeConstructor2<U> = U | null

// B<TypeConstructor1, TypeConstructor2> should be a type that, when given X, becomes TypeConstructor1<TypeConstructor2<X>>

type B_Combinator_Example<FConstructor, GConstructor, T> = FConstructor<GConstructor<T>>;

type TestB_Example1 = B_Combinator_Example<ArrayWrapper, G, number>; // ArrayWrapper<G<number>> which is (number | null)[]

Output:

// TestB_Example1 would be inferred as (number | null)[]

Explanation: This B_Combinator_Example shows the spirit of B. It takes two type constructors (FConstructor, GConstructor) and a type T. It applies GConstructor to T, and then applies FConstructor to the result. This is the type-level equivalent of function composition F(G(x)).

Constraints

All implementations must be purely declarative TypeScript types (type aliases or interfaces).
No runtime JavaScript logic should be used in the combinator definitions.
The solution should be robust and handle various valid type inputs.
Focus on correctness and adherence to the combinator definitions. Performance is not a primary concern for this challenge.

Notes

TypeScript's generic system and conditional types will be your primary tools.
The S combinator is particularly challenging. It might require careful structuring of generic parameters and conditional types to achieve the desired type transformation.
Consider how to represent "type constructors" (types that take other types as arguments) and how to "apply" them.
Think about the difference between a concrete type and a type constructor. For instance, string[] is a concrete type, but T[] is a type constructor that takes T. Your combinators should ideally work with type constructors.
The S combinator's definition (A C) (B C) means: take type A, apply it to type C (resulting in A<C>), then take type B, apply it to type C (resulting in B<C>), and then apply the result of the first operation to the result of the second operation. This requires thinking about types as functions.

Type-Level Combinators in TypeScript

Problem Description

Key Requirements:

Implement the I (Identity) Combinator: This combinator simply returns its input type.
Implement the K (Constant) Combinator: This combinator takes two types and always returns the first one, discarding the second.
Implement the S (Substitution) Combinator: This combinator is more complex. It takes three types, A, B, and C, and produces a new type that represents (A C) (B C). This is the most powerful combinator and can be used to express any computable function.
Implement the B (Composition) Combinator: This combinator takes two types, F and G, and produces a new type representing the composition of F and G, i.e., F G.

Expected Behavior:

Important Considerations:

You will be working exclusively with TypeScript's type system. No runtime JavaScript code will be involved in the implementation of the combinators themselves.
Think carefully about how to represent functions between types. For instance, (A C) in the S combinator implies a type that takes C and produces a result based on A.

Examples

Example 1: The I Combinator

Input:

type Identity<T> = T;
type TestI = Identity<string>;

Output:

// TestI would be inferred as 'string'

Explanation: The Identity type takes a type T and simply returns T. When string is passed as T, TestI becomes string.

Example 2: The K Combinator

Input:

type Constant<A, B> = A;
type TestK = Constant<number, boolean>;

Output:

// TestK would be inferred as 'number'

Explanation: The Constant type takes two types, A and B, and always returns A. When number and boolean are passed, TestK becomes number, effectively ignoring boolean.

Example 3: The B Combinator (Conceptual)

Let's assume we have types that represent functions. For example, F<X> = X[] and G<Y> = Y | null.

Input:

// Conceptual types for functions (simplified for example)
type F<X> = X[];
type G<Y> = Y | null;

// Implementing B combinator: B<F, G><Z> = F<G<Z>>
type Compose<F, G> = <Z>(gResult: G<Z>) => F<Z>; // This is a type-level function definition

// To use it, we need a way to apply types to type constructors.
// This is where advanced techniques might be needed.
// For this challenge, let's assume we are working with specific types.

// Let's define concrete type-level functions and compose them.
type Stringifier<T> = T extends string ? T : T extends number ? `${T}` : never;
type ArrayWrapper<T> = T[];

// We want to implement B<Stringifier, ArrayWrapper><number> which should be Stringifier<ArrayWrapper<number>>
// This means: Stringifier<number[]> which would be (number[])[] - this is not what S/B combinators mean.
// S/B combinators work on *function application*.
// Let's reframe: B<F, G>(x) = F(G(x))

// Type representing a function from T to U: (T) => U
// In TS types, this is often represented as an interface with a generic index signature, or a mapped type.

// For this challenge, let's focus on composing type constructors themselves, not just concrete types.

// Example using S combinator's spirit: (A C) (B C)
// Let's consider A = number, B = string, C = boolean
// We need a way to represent a "type-level application" like A<C>
// This is typically done with conditional types.

// Let's use a concrete example of B:
// B<F, G> should represent a type that, when applied to X, results in F<G<X>>.
// We can define a type that *is* the composed function type.

type ComposeTwo<FConstructor, GConstructor> = <X>(x: GConstructor<X>) => FConstructor<X>; // This is getting closer to runtime.

// Let's stick to pure type manipulation.
// B<F, G> = Type that represents F applied to G.
// Consider type constructors:
// TypeConstructor1<T> = T[]
// TypeConstructor2<U> = U | null

// B<TypeConstructor1, TypeConstructor2> should be a type that, when given X, becomes TypeConstructor1<TypeConstructor2<X>>

type B_Combinator_Example<FConstructor, GConstructor, T> = FConstructor<GConstructor<T>>;

type TestB_Example1 = B_Combinator_Example<ArrayWrapper, G, number>; // ArrayWrapper<G<number>> which is (number | null)[]

Output:

// TestB_Example1 would be inferred as (number | null)[]

Constraints

All implementations must be purely declarative TypeScript types (type aliases or interfaces).
No runtime JavaScript logic should be used in the combinator definitions.
The solution should be robust and handle various valid type inputs.
Focus on correctness and adherence to the combinator definitions. Performance is not a primary concern for this challenge.

Notes

TypeScript's generic system and conditional types will be your primary tools.
The S combinator is particularly challenging. It might require careful structuring of generic parameters and conditional types to achieve the desired type transformation.
Consider how to represent "type constructors" (types that take other types as arguments) and how to "apply" them.
Think about the difference between a concrete type and a type constructor. For instance, string[] is a concrete type, but T[] is a type constructor that takes T. Your combinators should ideally work with type constructors.
The S combinator's definition (A C) (B C) means: take type A, apply it to type C (resulting in A<C>), then take type B, apply it to type C (resulting in B<C>), and then apply the result of the first operation to the result of the second operation. This requires thinking about types as functions.