Coin Change Combination Count in Python

from bisect import bisect_left

# Coin Change Combination Count in Python
#
# PURPOSE
# This algorithm counts unordered combinations of reusable denominations. Putting the coin
# loop outside the amount loop prevents counting different orders as different combinations.
#
# METHOD
# There is one way to make zero: choose no coins. Process each denomination once and update
# amounts upward so that the same denomination can be reused.
#
# COMPLEXITY
# O(m * amount) time and O(amount) auxiliary space; pseudo-polynomial.
#
# CONTRACT
# Denominations are distinct positive integers, amount >= 0. Counts and array sizes must
# fit. Duplicate denominations would overcount.
# 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: 4.
# 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 coin_combinations(coins: list[int], amount: int) -> int:
    alg_require(isinstance(coins, (list, tuple)) and len(coins) <= 1000000)
    alg_require(type(amount) is int and -2147483648 <= amount <= 2147483647)
    m: int = len(coins)
    alg_require(m >= 0 and m <= 1000000)
    alg_require(len(coins) == m)
    alg_require(amount >= 0 and amount <= 1000000)
    for value in coins:
        alg_require(value > 0)
    dp: list[int] = [0] * alg_size(amount + 1)
    dp[0] = 1
    j: int = 0
    x: int = 0
    while j < m:
        x = coins[j]
        while x <= amount:
            dp[x] += dp[x - coins[j]]
            x += 1
        j += 1
    return dp[amount]

coins = [1, 2, 5]
amount = 5
result = coin_combinations(coins, amount)
print(result)