Lower Bound in Python

from bisect import bisect_left

# Lower Bound in Python
#
# PURPOSE
# A boundary search finds an insertion position in a sorted array, including arrays with
# repeated values. This is useful for maintaining sorted collections and counting
# occurrences.
#
# METHOD
# Use a half-open interval [lo, hi). Move lo beyond elements that belong before the
# requested boundary; otherwise move hi to mid. The interval shrinks until both bounds
# coincide.
#
# COMPLEXITY
# O(log n) time, O(1) auxiliary space.
#
# CONTRACT
# Input is sorted ascending. Returns the first index whose value is at least target, or n if
# none exists. Does not change the input.
# 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: 1.
# 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 lower_bound(a: list[int], target: int) -> int:
    alg_require(isinstance(a, (list, tuple)) and len(a) <= 1000000)
    alg_require(type(target) is int and -2147483648 <= target <= 2147483647)
    n: int = len(a)
    alg_require(n >= 0 and n <= 1000000)
    alg_require(len(a) == n)
    lo: int = 0
    hi: int = n
    mid: int = 0
    while lo < hi:
        mid = lo + (hi - lo) // 2
        if a[mid] < target:
            lo = mid + 1
        else:
            hi = mid
    return lo

a = [1, 2, 2, 4]
target = 2
result = lower_bound(a, target)
print(result)