Type-Level Recursion Schemes in TypeScript

This challenge focuses on implementing recursion schemes at the type level in TypeScript. Recursion schemes are powerful programming patterns that allow for generic traversal and manipulation of recursive data structures. By implementing them at the type level, we can leverage TypeScript's type system to enforce correctness and enable compile-time reasoning about these operations.

Problem Description

Your task is to create a generic type-level mechanism for defining and applying recursion schemes to algebraic data types (ADTs) in TypeScript. Specifically, you need to define types that represent:

1. Base Functor: A type that defines the recursive structure of a data type (e.g., ListF<A, R>, TreeF<A, R>).
2. Recursion Scheme Combinators: Types that embody common recursion patterns like cata (fold), ana (unfold), and hylo (recursion scheme that combines folding and unfolding).

You will then demonstrate the usage of these combinators by creating a type-level sum operation for lists.

Key Requirements:

• Type-Level Functors: Define a generic FunctorF type (or similar) that captures the recursive structure. This functor should take the "recursive parameter" (the type representing the rest of the structure) as an argument.
• Type-Level Recursion Scheme Types: Implement CataF, AnaF, and HyloF types. These types should take a functor and a function to transform the base case into the recursive result, and return the final recursive type.
• Type-Level List Summation: Create a specific functor for lists (ListF) and then use your CataF combinator to define a type-level function Sum<List> that calculates the sum of a list of numbers.

Expected Behavior:

• The ListF type should correctly represent the structure of a linked list at the type level, where the recursive parameter R represents the tail of the list.
• The CataF type should correctly "fold" a recursive structure.
• The Sum<List> type should resolve to the numeric sum of the elements in the input list type.

Edge Cases:

• Empty List: Ensure your Sum type handles an empty list correctly (e.g., resolves to 0).
• Single Element List: Ensure it works for a list with one element.

Examples

Example 1: Basic FunctorF Definition

Let's define a functor for a simple Pair type: Pair<A, B>. The recursive structure would be List<A> which can be Empty or Cons<A, List<A>>. Our functor will represent a single step of the recursion: ListF<A, R> which means Empty or Cons<A, R>.

// Functor for List: represents Empty or Cons<Head, Tail>
type ListF<A, R> = { type: 'Empty' } | { type: 'Cons', head: A, tail: R };

// How to use it:
// If R is 'never' (representing the end of recursion)
type EmptyListF = ListF<number, never>; // { type: 'Empty' }
type ListWithOneElementF = ListF<number, EmptyListF>; // { type: 'Cons', head: number, tail: { type: 'Empty' }}

Example 2: Type-Level Summation (Conceptual)

Let's say we have a list type like [1, 2, 3]. We want Sum<[1, 2, 3]> to resolve to 6.

// Assume we have a way to represent the recursive structure of a list of numbers
// and a CataF combinator.

// Conceptual example (this is what you need to build):
// type Sum<ListType> = CataF<ListF<number, ???>, ???, ListType>;
// Where ??? needs to be defined to represent the "end" of the list and the folding function.

Example 3: Type-Level List Summation (Actual Input/Output)

Input: A type representing a list of numbers, e.g., [1, 2, 3]. We need a way to convert this into a recursive type that ListF can understand. Let's define a helper type List<T>:

type List<T> = { type: 'Empty' } | { type: 'Cons', head: T, tail: List<T> };

// Example List Type:
type MyNumberList = List<number>; // This is not a list of numbers yet, it's the structure.
// To represent [1, 2, 3]:
type MyConcreteList = { type: 'Cons', head: 1, tail: { type: 'Cons', head: 2, tail: { type: 'Cons', head: 3, tail: { type: 'Empty' } } } };

Output: A literal type representing the sum.

// Expected for MyConcreteList:
type ResultingSum = 6; // number literal type

Explanation:

The CataF combinator would take ListF<number, R> (where R eventually resolves to Empty), and a function that specifies how to handle Empty (return 0) and Cons (return head + tail_sum). CataF recursively applies this logic.

Constraints

• All implementations must be purely at the type level. No runtime JavaScript execution should be required for the core logic.
• The solution should be written in TypeScript using its advanced type system features (conditional types, mapped types, infer keyword, etc.).
• The target TypeScript version should be reasonably modern (e.g., 3.8+).
• Performance: While type-level computation can be slow, aim for clarity and correctness. Avoid excessively complex or deeply nested conditional types where simpler alternatives exist.

Notes

• Think about how to represent the "end" of a recursive structure at the type level (often never or a specific "empty" type).
• Consider how to transform the "recursive result" from the inner call to the correct type for the outer call. This is a key aspect of recursion schemes.
• You might need helper types to convert standard array literals (like [1, 2, 3]) into your recursive ADT structure if you want to use those directly as input. However, for the core challenge, defining a concrete recursive type (like MyConcreteList in Example 3) is sufficient.
• ana (unfolding) and hylo are more advanced. Focus on cata first, then consider ana and hylo. The primary goal is to understand the pattern.