


def 							UpperCamelCase_( lowerCamelCase_					)       ->		Tuple:
    _lowercase			:				Tuple 						=						0
    _lowercase			:				Any 						=						len(lowerCamelCase_					)

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

    return num_inversions



def 							UpperCamelCase_( lowerCamelCase_					)       ->		Dict:
    if len(lowerCamelCase_					) <= 1:
        return arr, 0
    _lowercase			:				Any 						=						len(lowerCamelCase_					) // 2
    _lowercase			:				List[str] 						=						arr[0:mid]
    _lowercase			:				Tuple 						=						arr[mid:]

    _lowercase						,							_lowercase			:				Dict 						=						count_inversions_recursive(lowerCamelCase_					)
    _lowercase						,							_lowercase			:				Optional[int] 						=						count_inversions_recursive(lowerCamelCase_					)
    _lowercase						,							_lowercase			:				List[str] 						=						_count_cross_inversions(lowerCamelCase_				,					lowerCamelCase_					)

    _lowercase			:				List[Any] 						=						inversion_p + inversions_q + cross_inversions
    return c, num_inversions



def 							UpperCamelCase_( lowerCamelCase_				,					lowerCamelCase_					)       ->		Optional[Any]:
    _lowercase			:				Any 						=						[]
    _lowercase			:				Union[str, Any] 						=						0
    while i < len(lowerCamelCase_					) and j < len(lowerCamelCase_					):
        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(lowerCamelCase_					) - i
            r.append(q[j]					)
            j += 1
        else:
            r.append(p[i]					)
            i += 1

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

    return r, num_inversion



def 							UpperCamelCase_( )       ->		Optional[int]:
    _lowercase			:				Union[str, Any] 						=						[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)

    _lowercase			:				Optional[int] 						=						count_inversions_bf(lowerCamelCase_					)
    _lowercase						,							_lowercase			:				Any 						=						count_inversions_recursive(lowerCamelCase_					)

    assert num_inversions_bf == num_inversions_recursive == 8

    print('number of inversions = '				,					lowerCamelCase_					)

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

    arr_a.sort()
    _lowercase			:				List[str] 						=						count_inversions_bf(lowerCamelCase_					)
    _lowercase						,							_lowercase			:				List[str] 						=						count_inversions_recursive(lowerCamelCase_					)

    assert num_inversions_bf == num_inversions_recursive == 0
    print('number of inversions = '				,					lowerCamelCase_					)

    # an empty list should also have zero inversions
    _lowercase			:				Tuple 						=						[]
    _lowercase			:				Tuple 						=						count_inversions_bf(lowerCamelCase_					)
    _lowercase						,							_lowercase			:				List[str] 						=						count_inversions_recursive(lowerCamelCase_					)

    assert num_inversions_bf == num_inversions_recursive == 0
    print('number of inversions = '				,					lowerCamelCase_					)


if __name__ == "__main__":
 main()






def 							UpperCamelCase_( lowerCamelCase_					)       ->		int:

    if not isinstance(lowerCamelCase_				,					lowerCamelCase_					):
        raise ValueError('multiplicative_persistence() only accepts integral values'					)
    if num < 0:
        raise ValueError('multiplicative_persistence() does not accept negative values'					)

    _lowercase			:				List[str] 						=						0
    _lowercase			:				Optional[int] 						=						str(lowerCamelCase_					)

    while len(lowerCamelCase_					) != 1:
        _lowercase			:				Any 						=						[int(lowerCamelCase_					) for i in num_string]

        _lowercase			:				List[Any] 						=						1
        for i in range(0				,					len(lowerCamelCase_					)					):
            total *= numbers[i]

        _lowercase			:				Optional[Any] 						=						str(lowerCamelCase_					)

        steps += 1
    return steps



def 							UpperCamelCase_( lowerCamelCase_					)       ->		int:

    if not isinstance(lowerCamelCase_				,					lowerCamelCase_					):
        raise ValueError('additive_persistence() only accepts integral values'					)
    if num < 0:
        raise ValueError('additive_persistence() does not accept negative values'					)

    _lowercase			:				Optional[int] 						=						0
    _lowercase			:				str 						=						str(lowerCamelCase_					)

    while len(lowerCamelCase_					) != 1:
        _lowercase			:				Dict 						=						[int(lowerCamelCase_					) for i in num_string]

        _lowercase			:				Any 						=						0
        for i in range(0				,					len(lowerCamelCase_					)					):
            total += numbers[i]

        _lowercase			:				Dict 						=						str(lowerCamelCase_					)

        steps += 1
    return steps


if __name__ == "__main__":
 import doctest

 doctest.testmod()


