Mastering Zygohistomorphic Prepromorphisms in TypeScript

This challenge focuses on implementing zygohistomorphic prepromorphisms, a powerful concept in functional programming that allows for the simultaneous traversal and transformation of nested data structures. Mastering this technique will enhance your ability to work with complex data in a composable and efficient manner, particularly when dealing with tree-like or recursive structures.

Problem Description

Your task is to implement a generic function in TypeScript that represents a zygohistomorphic prepromorphism. This function should take a data structure (which can be recursively defined), two functions for transforming the "left" and "right" sides of a recursive step, and a function for combining the results. The goal is to traverse the input structure, applying these transformations and aggregations to produce a single, aggregated output.

Specifically, you need to create a higher-order function, let's call it zygohistomorphicPrepromorphism, that accepts the following:

1. g: A function that transforms a base case of the data structure. This function takes a value of the base type and returns a result of the desired output type.
2. f: A function that handles the recursive step. This function takes a value representing the "current" level of the structure (e.g., a node in a tree) and an array of results from its children. It should return a result of the desired output type.
3. input: The data structure to traverse.

The zygohistomorphicPrepromorphism function should return a function that, when given an input data structure, traverses it and produces an aggregated output. The traversal should be depth-first, and the results from sub-structures should be combined using the f function.

Key Requirements:

• The solution must be generic enough to work with various recursive data structures.
• It must correctly handle both base cases and recursive steps of the input structure.
• The output should be a single aggregated value.

Expected Behavior:

The function should behave like a sophisticated fold or catamorphism, but with the added capability of operating on structures where elements can be processed independently and then combined. Imagine a tree where you want to sum all the values in the leaves and also count the number of nodes at each level.

Important Edge Cases:

• Empty structures (e.g., an empty list, a null tree).
• Structures with only base cases.
• Deeply nested structures.

Examples

Let's consider a simple binary tree structure for demonstration.

Data Structure Definition (Conceptual):

type BinaryTree<T> = Leaf<T> | Node<T>;

interface Leaf<T> {
  type: 'leaf';
  value: T;
}

interface Node<T> {
  type: 'node';
  left: BinaryTree<T>;
  right: BinaryTree<T>;
}

Example 1: Summing Leaf Values

Input:

const tree1: BinaryTree<number> = {
  type: 'node',
  left: {
    type: 'leaf',
    value: 5
  },
  right: {
    type: 'node',
    left: {
      type: 'leaf',
      value: 10
    },
    right: {
      type: 'leaf',
      value: 15
    }
  }
};

Desired Output (Conceptual):

A function that sums all the leaf values.

Explanation:

• For a Leaf, we simply return its value.
• For a Node, we recursively process its left and right children, then sum their results.

Example 2: Counting Nodes and Summing Leaves

Input:

const tree2: BinaryTree<number> = {
  type: 'node',
  left: {
    type: 'leaf',
    value: 3
  },
  right: {
    type: 'node',
    left: {
      type: 'leaf',
      value: 7
    },
    right: {
      type: 'leaf',
      value: 12
    }
  }
};

Desired Output (Conceptual):

A function that returns an object containing the total sum of leaf values and the total count of nodes (both leaf and internal).

Explanation:

• For a Leaf, return { sum: value, count: 1 }.
• For a Node, recursively process left and right children. Combine their results:
  • totalSum = leftResult.sum + rightResult.sum
  • totalCount = leftResult.count + rightResult.count + 1 (add 1 for the current node)

Example 3: Handling Empty Structures (e.g., a list)

Data Structure Definition (Conceptual):

type MyList<T> = Nil | Cons<T>;

interface Nil {
  type: 'nil';
}

interface Cons<T> {
  type: 'cons';
  head: T;
  tail: MyList<T>;
}

Input:

const list1: MyList<number> = { type: 'nil' };

Desired Output (Conceptual):

A function that calculates the product of all elements in a list, returning 1 for an empty list.

Explanation:

• For Nil, return 1.
• For Cons(head, tail), return head * (recursive_call_on_tail).

Constraints

• The input data structure can be arbitrarily nested.
• The input data structure will conform to a discriminated union pattern (or similar, allowing for clear distinction between base cases and recursive steps).
• The output type R can be any type.
• The functions g and f must be well-typed to ensure type safety.

Notes

Think about how you can model the recursive structure of the input and how to pass the results of recursive calls back up to the parent level. This problem is about abstracting the pattern of recursion and aggregation. Consider how to represent the "structure" itself in a generic way that zygohistomorphicPrepromorphism can understand. This might involve passing the structure definition or using type guards within your implementation. The core idea is to define how to handle a single element (base case) and how to combine results from multiple sub-elements.