Type-Level Heap Sort in TypeScript

This challenge asks you to implement the Heap Sort algorithm entirely within TypeScript's type system. This means that the sorting process and the data structures involved will be defined and manipulated using types, not runtime JavaScript code. This exercise is a deep dive into advanced TypeScript features and a demonstration of the expressive power of its type system for abstract algorithms.

Problem Description

Your task is to create a type-level implementation of Heap Sort in TypeScript. This involves defining types that represent the heap data structure and a series of type manipulations that mimic the heapify and extract-max operations of the algorithm. The final output should be a type that represents the sorted array.

Key Requirements:

1. Heap Representation: Define a type that can represent a binary heap. This could be a tuple, an array-like structure, or another suitable type. You'll need to consider how to represent the parent-child relationships.
2. Heapify Operation: Implement a type that can take an arbitrary array (represented as a tuple) and transform it into a max-heap. This is the core of building the heap.
3. Extract-Max Operation: Implement a type that can remove the maximum element (the root) from a max-heap and then restore the heap property.
4. Heap Sort Algorithm: Combine the heapify and extract-max operations to implement the full Heap Sort algorithm. This will involve repeatedly extracting the maximum element to build the sorted output.
5. Input and Output: The input will be a tuple of numbers (or a generic type that can be compared). The output should be a tuple representing the sorted version of the input.

Expected Behavior:

Given an input tuple, your type should resolve to a tuple containing the same elements but in ascending order.

Edge Cases:

• Empty input tuple.
• Tuple with a single element.
• Tuple with duplicate elements.

Examples

Example 1:

Input: [3, 1, 4, 1, 5, 9, 2, 6]
Output: [1, 1, 2, 3, 4, 5, 6, 9]
Explanation: The type system will simulate building a max-heap from the input array and then repeatedly extracting the largest element until the heap is empty, producing the sorted output.

Example 2:

Input: [10, 8, 6, 4, 2]
Output: [2, 4, 6, 8, 10]
Explanation: This is already a reverse-sorted input, which is a good test case for heapify.

Example 3:

Input: []
Output: []
Explanation: An empty input should result in an empty sorted output.

Example 4:

Input: [5]
Output: [5]
Explanation: A single-element input is already sorted.

Constraints

• The elements in the input tuple will be numbers.
• The input will be a tuple of numbers.
• The maximum length of the input tuple will be limited (e.g., 16 elements) to avoid excessive type instantiation depth and compilation times.
• Your solution should aim for clarity and correctness over extreme optimization within the type system.

Notes

This is a challenging problem that requires a deep understanding of TypeScript's conditional types, infer keywords, tuple manipulation, and recursion at the type level.

Consider breaking down the problem into smaller, manageable type-level functions:

• A type to get the left child index.
• A type to get the right child index.
• A type to swap elements in a tuple.
• A type to heapify a subtree.
• A type to build the initial heap.
• A type to extract the max and re-heapify.

Think about how you can represent the iterative nature of Heap Sort using recursive type definitions. Good luck!