



"""simple docstring"""






def 			snake_case_    ( A_   :						List[str]		):


       '''simple docstring'''





       _lowerCamelCase		:							Tuple            =					0
       _lowerCamelCase		:							Union[str, Any]            =					len(A_		)

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

       return num_inversions






def 			snake_case_    ( A_   :						List[str]		):


       '''simple docstring'''





       if len(A_		) <= 1:
              return arr, 0
       _lowerCamelCase		:							int            =					len(A_		) // 2
       _lowerCamelCase		:							int            =					arr[0:mid]
       _lowerCamelCase		:							int            =					arr[mid:]

       _lowerCamelCase     ,       _lowerCamelCase		:							str            =					count_inversions_recursive(A_		)
       _lowerCamelCase     ,       _lowerCamelCase		:							int            =					count_inversions_recursive(A_		)
       _lowerCamelCase     ,       _lowerCamelCase		:							Optional[Any]            =					_count_cross_inversions(A_,							A_		)

       _lowerCamelCase		:							Optional[Any]            =					inversion_p + inversions_q + cross_inversions
       return c, num_inversions






def 			snake_case_    ( A_   :						int,							A_   :						Dict		):


       '''simple docstring'''





       _lowerCamelCase		:							Tuple            =					[]
       _lowerCamelCase		:							List[str]            =					0
       while i < len(A_		) and j < len(A_		):
              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(A_		) - i
                     r.append(q[j]		)
                     j += 1
              else:
                     r.append(p[i]		)
                     i += 1

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

       return r, num_inversion






def 			snake_case_    ( ):


       '''simple docstring'''





       _lowerCamelCase		:							Tuple            =					[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		:							List[str]            =					count_inversions_bf(A_		)
       _lowerCamelCase     ,       _lowerCamelCase		:							List[str]            =					count_inversions_recursive(A_		)

       assert num_inversions_bf == num_inversions_recursive == 8

       print('''number of inversions = ''',							A_		)

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

       arr_a.sort()
       _lowerCamelCase		:							List[Any]            =					count_inversions_bf(A_		)
       _lowerCamelCase     ,       _lowerCamelCase		:							List[Any]            =					count_inversions_recursive(A_		)

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

       # an empty list should also have zero inversions
       _lowerCamelCase		:							str            =					[]
       _lowerCamelCase		:							Optional[int]            =					count_inversions_bf(A_		)
       _lowerCamelCase     ,       _lowerCamelCase		:							Any            =					count_inversions_recursive(A_		)

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


if __name__ == "__main__":
   main()







def 	SCREAMING_SNAKE_CASE_							(	_snake_case	:int				)  ->     int:
   if not isinstance(_snake_case	,   _snake_case				):
      raise ValueError('''Input must be an integer'''				)
   if input_num <= 0:
      raise ValueError('''Input must be positive'''				)
   return sum(
       divisor for divisor in range(1	,   input_num // 2 + 1				) if input_num % divisor == 0				)


if __name__ == "__main__":
   import doctest

   doctest.testmod()



