Merge Sort in Python

from bisect import bisect_left

# Merge Sort in Python
#
# PURPOSE
# Merge sort combines sorted runs into larger runs. The bottom-up form avoids recursion and
# guarantees predictable worst-case performance.
#
# METHOD
# Merge adjacent runs of width one into a temporary buffer, copy back, then double the
# width. Each pass leaves runs of twice the previous width sorted.
#
# COMPLEXITY
# O(n log n) time and O(n) auxiliary space. Stable because ties are taken from the left run.
#
# CONTRACT
# Sorts n elements ascending in place; returns 0. Empty input is valid. Uses a temporary
# buffer.
# Public APIs infer array lengths except in C, which accepts pointer/length pairs. Vertex
# counts and matrix dimensions remain explicit. Arrays must contain signed 32-bit integers.
# Sorted order, correct tree topology, and caller-owned data-structure invariants are
# preconditions; checking them on every query would change query complexity.
# Scalar and shape checks are included. Value-range checks are included where needed to
# prevent invalid indexing or nontermination. Allocation sizes are capped at 4,000,000
# workspace entries. Arithmetic outside Python raises an error on 32-bit overflow; Python
# uses arbitrary precision. Native allocation failures propagate. In-place outputs may be
# partly updated after arithmetic failure. These are single-threaded, in-memory
# implementations.
#
# LANGUAGE DETAILS
# Python 3.9+: automatic lengths, native lists, arbitrary-precision arithmetic, and
# bisect_left for LIS. Invalid scalar/shape inputs raise ValueError. Integer element types
# and sorted/tree invariants remain caller preconditions.
#
# EXAMPLE
# Expected return: 0.
# Expected a after call: [1, 1, 3, 5].
# Run: python example.py
# Tests for this exact source artifact run before publication. The accompanying source
# archive includes unit cases, runtime results, and source hashes.

def alg_require(condition):
    if not condition:
        raise ValueError("invalid algorithm input; see contract")
def alg_size(n):
    alg_require(0 <= n <= 4000000)
    return n

def merge_sort(a: list[int]) -> int:
    alg_require(isinstance(a, list) and len(a) <= 1000000)
    n: int = len(a)
    alg_require(n >= 0 and n <= 1000000)
    alg_require(len(a) == n)
    buf: list[int] = [0] * alg_size(n + 1)
    width: int = 1
    left: int = 0
    mid: int = 0
    right: int = 0
    i: int = 0
    j: int = 0
    k: int = 0
    while width < n:
        left = 0
        while left < n:
            mid = min(left + width, n)
            right = min(left + 2 * width, n)
            i = left
            j = mid
            k = left
            while k < right:
                if i < mid and (j >= right or a[i] <= a[j]):
                    buf[k] = a[i]
                    i += 1
                else:
                    buf[k] = a[j]
                    j += 1
                k += 1
            left += 2 * width
        i = 0
        while i < n:
            a[i] = buf[i]
            i += 1
        width *= 2
    return 0

a = [5, 1, 3, 1]
result = merge_sort(a)
print(result)
print(a)
# Expected a: [1, 1, 3, 5]