def a_subset_b(bits_a, bits_b):
    """Checks if one bitstring encodes a proper subset of another.

    Parameters
    ----------
    bits_a : Tuple[int]
        The first string.
    bits_b : Tuple[int]
        The second string.

    Returns
    -------
    bool
        True if a subset of b.
    """
    
    subset_eq =  all(bits_a[i] <= bits_b[i] for i in range(0, len(bits_a)))

    return subset_eq and (sum(bits_a) < sum(bits_b))