Longest Increasing Subsequence in Python

from bisect import bisect_left

# Longest Increasing Subsequence in Python
#
# PURPOSE
# Patience-style tail tracking finds the length of a longest strictly increasing subsequence
# using binary search. A subsequence may skip elements while preserving order.
#
# METHOD
# Keep the smallest possible tail for each subsequence length. Lower-bound search finds the
# first tail at least as large as the next value. Replace that tail or extend the length.
# Strict increase means equal values replace rather than extend.
#
# COMPLEXITY
# O(n log n) time and O(n) auxiliary space.
#
# CONTRACT
# Returns the length only. Equal values do not extend a strictly increasing subsequence.
# Empty input returns 0.
# 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 lis_length(a: list[int]) -> int:
    alg_require(isinstance(a, (list, tuple)) and len(a) <= 1000000)
    n: int = len(a)
    alg_require(n >= 0 and n <= 1000000)
    alg_require(len(a) == n)
    tails = []
    for value in a:
        at = bisect_left(tails, value)
        if at == len(tails):
            tails.append(value)
        else:
            tails[at] = value
    return len(tails)

a = [10, 9, 2, 5, 3, 7, 101, 18]
result = lis_length(a)
print(result)