Implementing Futumorphisms (Type-Level Functions) in TypeScript

Futomorphisms allow you to treat types as data, enabling powerful type-level computations. This challenge asks you to implement basic futumorphisms in TypeScript using conditional types and inference, mimicking the behavior of Haskell's Fmap and Reduce. Successfully completing this challenge demonstrates a strong understanding of advanced TypeScript type system capabilities.

Problem Description

The goal is to implement two fundamental futumorphisms: Fmap and Reduce. These functions operate on types, transforming them based on a provided function.

• Fmap (Type-Level map): Given a type Container<A> and a function f :: A -> B, Fmap should produce a type Container<B>. Essentially, it applies the function f to the type contained within the Container.
• Reduce (Type-Level fold): Given a type List<A>, an initial value B, and a function f :: B -> A -> B, Reduce should produce a type B. It folds the list of type A into a single value of type B using the provided function.

Key Requirements:

• The implementations must work with arbitrary container and list types.
• The implementations should leverage TypeScript's conditional types and type inference capabilities.
• The implementations should be type-safe, meaning the compiler should catch errors if the types don't match the function signatures.

Expected Behavior:

The type-level functions should produce the correct resulting type based on the input types and function. The actual runtime behavior is irrelevant; we are solely concerned with the type transformations.

Edge Cases to Consider:

• Empty lists for Reduce. Consider what the appropriate behavior should be (e.g., return the initial value).
• Nested containers for Fmap. While not explicitly required, consider how your solution might handle them.
• Complex container/list types with multiple type parameters.

Examples

Example 1: Fmap with Maybe

type Maybe<A> = { type: 'Just', value: A } | { type: 'Nothing' };

type Fmap<F extends { type: 'Just' | 'Nothing' }, A, B> =
  F extends { type: 'Just', value: infer V }
    ? { type: 'Just', value: B }
    : { type: 'Nothing' };

type Result = Fmap<{ type: 'Just', value: 10 }, number, string>; // Expected: { type: 'Just', value: string }

Explanation: Fmap applied to a Just containing a number transforms it into a Just containing a string. A Nothing remains a Nothing.

Example 2: Reduce with a simple list

type List<A> = A | { type: 'Nil' };

type Reduce<L extends List<any>, B, F extends (b: B, a: any) => B> =
  L extends A
    ? Reduce<
        { type: 'Nil' },
        B,
        F
      >
    : L extends { type: 'Nil' }
      ? B
      : F extends (b: B, a: any) => infer R
        ? Reduce<R, B, F>
        : never;

type Result2 = Reduce<1 | { type: 'Nil' }, number, (acc: number, val: number) => number>; // Expected: number

Explanation: The list 1 | { type: 'Nil' } is reduced using the identity function, resulting in a number.

Example 3: Reduce with an empty list

type Result3 = Reduce<{ type: 'Nil' }, number, (acc: number, val: number) => number>; // Expected: number

Explanation: An empty list is reduced with an initial value of number, resulting in number.

Constraints

• The implementations must be purely type-level; no runtime code is involved.
• The Fmap and Reduce types must be generic and work with any valid container/list types and functions.
• The code should be reasonably concise and readable, demonstrating a good understanding of TypeScript's type system.
• The solution should not use any external libraries or advanced TypeScript features beyond conditional types, type inference, and basic type manipulation.

Notes

• Think about how to recursively process the container/list types.
• Consider using helper types to simplify the logic.
• The List type is a simplified representation of a list. You don't need to handle all list operations, just enough to demonstrate the Reduce functionality.
• The Maybe type is a common container type used in functional programming. It represents an optional value.
• Focus on the type transformations; runtime behavior is not relevant. The compiler should be able to infer the correct types based on your implementations.
• This is a challenging problem that requires a deep understanding of TypeScript's type system. Don't be afraid to experiment and iterate on your solution.