Type-Level Merge Sort in TypeScript

This challenge requires you to implement the merge sort algorithm entirely at the TypeScript type level. This means you will be working with types and type manipulations to achieve sorting, rather than runtime JavaScript execution. This exercise is valuable for understanding the expressive power of TypeScript's advanced type features and how they can be used for complex static analysis and code generation.

Problem Description

Your task is to create a TypeScript type MergeSort<T extends readonly any[]> that takes an array of primitive types (numbers, strings, booleans) as a readonly tuple and returns a new readonly tuple representing the sorted version of the input array.

Key Requirements:

1. Type-Level Implementation: The entire sorting logic must be implemented using TypeScript's conditional types, mapped types, and other type-level manipulation features. No runtime code should be involved in the sorting process.
2. Primitive Types: The input array T will consist of primitive types (numbers, strings, booleans). You will need to define how these types are compared for sorting. For simplicity, assume standard JavaScript comparison logic.
3. Readonly Tuples: The input and output must be readonly tuples.
4. Merge Sort Logic: You must adhere to the merge sort algorithm:
   • Base Case: If the input array has 0 or 1 element, it is already sorted.
   • Recursive Step:
     • Split the array into two halves.
     • Recursively sort each half.
     • Merge the two sorted halves into a single sorted array.
5. Comparison: You'll need a mechanism to compare two types to determine their order.

Expected Behavior:

The MergeSort<T> type should correctly sort any readonly tuple of primitive types according to their natural ordering.

Edge Cases:

• Empty input tuple.
• Input tuple with a single element.
• Input tuple with duplicate elements.
• Input tuple with mixed primitive types (e.g., numbers and strings – define a consistent ordering if necessary, or assume a reasonable default like numbers before strings).

Examples

Example 1:

type Input1 = readonly [3, 1, 4, 1, 5, 9, 2, 6];
type Output1 = MergeSort<Input1>;
// Expected: readonly [1, 1, 2, 3, 4, 5, 6, 9]

Explanation: The MergeSort type will recursively split Input1, sort the halves, and merge them to produce the sorted output.

Example 2:

type Input2 = readonly ["banana", "apple", "cherry", "date"];
type Output2 = MergeSort<Input2>;
// Expected: readonly ["apple", "banana", "cherry", "date"]

Explanation: This example demonstrates sorting strings alphabetically.

Example 3:

type Input3 = readonly [true, false, true];
type Output3 = MergeSort<Input3>;
// Expected: readonly [false, true, true]

Explanation: Boolean values are sorted, with false typically coming before true.

Example 4:

type Input4 = readonly [];
type Output4 = MergeSort<Input4>;
// Expected: readonly []

Explanation: An empty input tuple should result in an empty output tuple.

Constraints

• The input T will be a readonly tuple of primitive types: number, string, or boolean.
• The maximum depth of recursion will be within TypeScript's compiler limits for type instantiation.
• The comparison logic for types should mimic standard JavaScript comparison for primitives. For mixed types (e.g., numbers and strings), you may define a consistent ordering (e.g., numbers < strings).

Notes

• You'll likely need helper types for:
  • Splitting an array into two halves.
  • Merging two already sorted arrays.
  • Comparing two primitive types at the type level.
• Consider how to handle type inference for your comparison logic.
• This challenge is about demonstrating mastery of advanced TypeScript features. Think carefully about how to represent lists and perform operations on them using type-level programming.
• The core of merge sort lies in its divide-and-conquer strategy, which translates well to recursive type definitions.