





'''simple docstring'''


def 						__a(SCREAMING_SNAKE_CASE_	:       str     ):



    '''simple docstring'''





    _lowerCAmelCase						      =   0
    _lowerCAmelCase						      =   len(_SCREAMING_SNAKE_CASE     )

    for i in range(n - 1     ):
        for j in range(i + 1		,							_SCREAMING_SNAKE_CASE     ):
            if arr[i] > arr[j]:
                num_inversions += 1

    return num_inversions


def 						__a(SCREAMING_SNAKE_CASE_	:       Any     ):



    '''simple docstring'''





    if len(_SCREAMING_SNAKE_CASE     ) <= 1:
        return arr, 0
    _lowerCAmelCase						      =   len(_SCREAMING_SNAKE_CASE     ) // 2
    _lowerCAmelCase						      =   arr[0:mid]
    _lowerCAmelCase						      =   arr[mid:]

    _lowerCAmelCase      ,      _lowerCAmelCase						      =   count_inversions_recursive(_SCREAMING_SNAKE_CASE     )
    _lowerCAmelCase      ,      _lowerCAmelCase						      =   count_inversions_recursive(_SCREAMING_SNAKE_CASE     )
    _lowerCAmelCase      ,      _lowerCAmelCase						      =   _count_cross_inversions(_SCREAMING_SNAKE_CASE		,							_SCREAMING_SNAKE_CASE     )

    _lowerCAmelCase						      =   inversion_p + inversions_q + cross_inversions
    return c, num_inversions


def 						__a(SCREAMING_SNAKE_CASE_	:       str		,							SCREAMING_SNAKE_CASE_	:       Optional[int]     ):



    '''simple docstring'''





    _lowerCAmelCase						      =   []
    _lowerCAmelCase						      =   _lowerCAmelCase						      =   _lowerCAmelCase						      =   0
    while i < len(_SCREAMING_SNAKE_CASE     ) and j < len(_SCREAMING_SNAKE_CASE     ):
        if p[i] > q[j]:
            # if P[1] > Q[j], then P[k] > Q[k] for all  i < k <= len(P)
            # These are all inversions. The claim emerges from the
            # property that P is sorted.
            num_inversion += len(_SCREAMING_SNAKE_CASE     ) - i
            r.append(q[j]     )
            j += 1
        else:
            r.append(p[i]     )
            i += 1

    if i < len(_SCREAMING_SNAKE_CASE     ):
        r.extend(p[i:]     )
    else:
        r.extend(q[j:]     )

    return r, num_inversion


def 						__a():



    '''simple docstring'''





    _lowerCAmelCase						      =   [10, 2, 1, 5, 5, 2, 11]

    # this arr has 8 inversions:
    # (10, 2), (10, 1), (10, 5), (10, 5), (10, 2), (2, 1), (5, 2), (5, 2)

    _lowerCAmelCase						      =   count_inversions_bf(_SCREAMING_SNAKE_CASE     )
    _lowerCAmelCase      ,      _lowerCAmelCase						      =   count_inversions_recursive(_SCREAMING_SNAKE_CASE     )

    assert num_inversions_bf == num_inversions_recursive == 8

    print("number of inversions = "		,							_SCREAMING_SNAKE_CASE     )

    # testing an array with zero inversion (a sorted arr_1)

    arr_a.sort()
    _lowerCAmelCase						      =   count_inversions_bf(_SCREAMING_SNAKE_CASE     )
    _lowerCAmelCase      ,      _lowerCAmelCase						      =   count_inversions_recursive(_SCREAMING_SNAKE_CASE     )

    assert num_inversions_bf == num_inversions_recursive == 0
    print("number of inversions = "		,							_SCREAMING_SNAKE_CASE     )

    # an empty list should also have zero inversions
    _lowerCAmelCase						      =   []
    _lowerCAmelCase						      =   count_inversions_bf(_SCREAMING_SNAKE_CASE     )
    _lowerCAmelCase      ,      _lowerCAmelCase						      =   count_inversions_recursive(_SCREAMING_SNAKE_CASE     )

    assert num_inversions_bf == num_inversions_recursive == 0
    print("number of inversions = "		,							_SCREAMING_SNAKE_CASE     )


if __name__ == "__main__":
       main()









import numpy as np





def 				__lowerCAmelCase(       _SCREAMING_SNAKE_CASE				)							-> np.array:




    """simple docstring"""




    return (2 / (1 + np.exp(-2 * vector				))) - 1


if __name__ == "__main__":
 import doctest

 doctest.testmod()





