Grid Path Counting in Python

from bisect import bisect_left

# Grid Path Counting in Python
#
# PURPOSE
# Grid path counting counts ways to reach a cell when each step moves only right or down.
# Every path arrives from the cell above or the cell to the left.
#
# METHOD
# Initialize one path along the first row. Update a single row left to right, adding the
# path count from the left to the retained count from above.
#
# COMPLEXITY
# O(rows * cols) time and O(cols) auxiliary space.
#
# CONTRACT
# rows and cols are at least one. Grid has no obstacles. Counts must fit the integer type.
# 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: 6.
# 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 grid_paths(rows: int, cols: int) -> int:
    alg_require(type(rows) is int and -2147483648 <= rows <= 2147483647)
    alg_require(type(cols) is int and -2147483648 <= cols <= 2147483647)
    alg_require(rows >= 1 and cols >= 1 and (rows <= 1000000) and (cols <= 1000000))
    dp: list[int] = [0] * alg_size(cols)
    c: int = 0
    r: int = 1
    while c < cols:
        dp[c] = 1
        c += 1
    while r < rows:
        c = 1
        while c < cols:
            dp[c] += dp[c - 1]
            c += 1
        r += 1
    return dp[cols - 1]

rows = 3
cols = 3
result = grid_paths(rows, cols)
print(result)