Implementing Futumorphisms in TypeScript

This challenge focuses on implementing futumorphisms, a powerful concept in functional programming that allows for generalized recursion over algebraic data types. By implementing futumorphisms, you will gain a deeper understanding of how to abstract and reuse recursive patterns in your TypeScript code, leading to more elegant and maintainable solutions.

Problem Description

A futumorphism is a higher-order function that abstracts a recursive pattern. For algebraic data types, a futumorphism can be seen as a generalized catamorphism (fold) or anamorphism (unfold) that can handle both consuming and producing structures.

Your task is to implement a futumorphism function in TypeScript for common algebraic data types, starting with a List (or array-like) structure and a Tree structure. The futumorphism should take a "blueprint" of how to process the recursive structure and return a function that, when applied to a value of that structure, produces a result.

Key requirements:

• The futumorphism should be generic enough to work with different types of structures and different output types.
• It should abstract away the explicit recursion, allowing the user to define the behavior for the base case and the recursive step.
• Consider how to handle the "future" aspect of futumorphisms, where the recursive calls might be "deferred" or handled in a specific order.

For this challenge, focus on:

1. List Futumorphism: Implement futumorphismList for a generic List type.
2. Tree Futumorphism: Implement futumorphismTree for a generic Tree type (e.g., a binary tree).

Examples

List Futumorphism

Let's define a simple List type:

type List<A> = {
    kind: 'empty';
} | {
    kind: 'cons';
    head: A;
    tail: List<A>;
};

Example 1: Sum of a list

// Input List: Cons(1, Cons(2, Cons(3, Empty)))
// Goal: Calculate the sum of all numbers in the list.

const sumList = (list: List<number>): number => {
    // Your futumorphism implementation would be used here.
    // The blueprint would specify:
    // - How to handle 'empty' (base case): return 0
    // - How to handle 'cons' (recursive step): take head, recursively process tail, and combine (head + result_of_tail)

    // For demonstration, a direct recursive approach would look like this:
    if (list.kind === 'empty') {
        return 0;
    } else {
        return list.head + sumList(list.tail);
    }
};

const inputList1: List<number> = {
    kind: 'cons',
    head: 1,
    tail: {
        kind: 'cons',
        head: 2,
        tail: {
            kind: 'cons',
            head: 3,
            tail: { kind: 'empty' }
        }
    }
};

// Expected Output: 6
// Explanation: 1 + 2 + 3 = 6

Example 2: Reversing a list

// Input List: Cons(1, Cons(2, Cons(3, Empty)))
// Goal: Reverse the list.

// The futumorphism blueprint would specify:
// - How to handle 'empty': return an empty list.
// - How to handle 'cons': take head, recursively process tail, and append head to the *end* of the recursively reversed tail.

// For demonstration, a direct recursive approach might be:
const reverseListRecursive = (list: List<number>): List<number> => {
    const append = (l: List<number>, item: number): List<number> => {
        if (l.kind === 'empty') {
            return { kind: 'cons', head: item, tail: { kind: 'empty' } };
        } else {
            return { kind: 'cons', head: l.head, tail: append(l.tail, item) };
        }
    };

    if (list.kind === 'empty') {
        return { kind: 'empty' };
    } else {
        return append(reverseListRecursive(list.tail), list.head);
    }
};

const inputList2: List<number> = {
    kind: 'cons',
    head: 1,
    tail: {
        kind: 'cons',
        head: 2,
        tail: {
            kind: 'cons',
            head: 3,
            tail: { kind: 'empty' }
        }
    }
};

// Expected Output: Cons(3, Cons(2, Cons(1, Empty)))
// Explanation: The order of elements is reversed.

Tree Futumorphism

Let's define a simple binary Tree type:

type Tree<A> = {
    kind: 'leaf';
    value: A;
} | {
    kind: 'node';
    left: Tree<A>;
    right: Tree<A>;
};

Example 3: Sum of values in a tree

// Input Tree: Node(Leaf(1), Node(Leaf(2), Leaf(3)))
// Goal: Calculate the sum of all leaf values in the tree.

// The futumorphism blueprint would specify:
// - How to handle 'leaf': return its value.
// - How to handle 'node': recursively process left and right subtrees, and combine their results (left_result + right_result).

const sumTree = (tree: Tree<number>): number => {
    // Your futumorphism implementation would be used here.

    // For demonstration, a direct recursive approach would look like this:
    if (tree.kind === 'leaf') {
        return tree.value;
    } else {
        return sumTree(tree.left) + sumTree(tree.right);
    }
};

const inputTree1: Tree<number> = {
    kind: 'node',
    left: { kind: 'leaf', value: 1 },
    right: {
        kind: 'node',
        left: { kind: 'leaf', value: 2 },
        right: { kind: 'leaf', value: 3 }
    }
};

// Expected Output: 6
// Explanation: 1 + 2 + 3 = 6

Constraints

• The implementations should be in TypeScript.
• Focus on type safety and leveraging TypeScript's features.
• The futumorphism implementations should not require explicit recursion in their usage.
• Performance is a consideration; aim for an efficient abstraction that doesn't introduce unnecessary overhead compared to direct recursion.

Notes

• Think about what the "blueprint" for a futumorphism should look like. It needs to specify how to handle each case of the algebraic data type.
• For futumorphisms, the structure of the recursion is abstracted. The user provides the "logic" for each part of the structure.
• Consider how to represent the "future" or deferred computation. This might involve promises or simply returning a value that can be composed. For this challenge, a simpler interpretation of "generalized recursion" where the combining function is provided is sufficient.
• The term "futumorphism" can have subtle variations in definition. For this challenge, consider it a function that "traverses" a structure and "builds" a result, abstracting the traversal logic. A good starting point is to think of it as a fold that can also produce structures.
• For the List futumorphism, you might consider a pattern where the combining function takes the head and the result of the futumorphism applied to the tail.
• For the Tree futumorphism, the combining function might take the results of the futumorphism applied to the left and right children.