





def       UpperCAmelCase							(     _lowerCamelCase      :  Optional[int]     ):





						'''simple docstring'''


						SCREAMING_SNAKE_CASE__     :				Any   =					0
						SCREAMING_SNAKE_CASE__     :				int   =					len(lowercase_     )

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

						return num_inversions







def       UpperCAmelCase							(     _lowerCamelCase      :  Dict     ):





						'''simple docstring'''


						if len(lowercase_     ) <= 1:
												return arr, 0
						SCREAMING_SNAKE_CASE__     :				Optional[Any]   =					len(lowercase_     ) // 2
						SCREAMING_SNAKE_CASE__     :				Optional[int]   =					arr[0:mid]
						SCREAMING_SNAKE_CASE__     :				int   =					arr[mid:]

						SCREAMING_SNAKE_CASE__     :				List[Any]   =					count_inversions_recursive(lowercase_     )
						SCREAMING_SNAKE_CASE__     :				Dict   =					count_inversions_recursive(lowercase_     )
						SCREAMING_SNAKE_CASE__     :				int   =					_count_cross_inversions(lowercase_			,   lowercase_     )

						SCREAMING_SNAKE_CASE__     :				Union[str, Any]   =					inversion_p + inversions_q + cross_inversions
						return c, num_inversions







def       UpperCAmelCase							(     _lowerCamelCase      :  str			,   _lowerCamelCase      :  Union[str, Any]     ):





						'''simple docstring'''


						SCREAMING_SNAKE_CASE__     :				Optional[int]   =					[]
						SCREAMING_SNAKE_CASE__     :				int   =					0
						while i < len(lowercase_     ) and j < len(lowercase_     ):
												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(lowercase_     ) - i
																		r.append(q[j]     )
																		j += 1
												else:
																		r.append(p[i]     )
																		i += 1

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

						return r, num_inversion







def       UpperCAmelCase							(     ):





						'''simple docstring'''


						SCREAMING_SNAKE_CASE__     :				List[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)

						SCREAMING_SNAKE_CASE__     :				int   =					count_inversions_bf(lowercase_     )
						SCREAMING_SNAKE_CASE__     :				Tuple   =					count_inversions_recursive(lowercase_     )

						assert num_inversions_bf == num_inversions_recursive == 8

						print("number of inversions = "			,   lowercase_     )

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

						arr_a.sort()
						SCREAMING_SNAKE_CASE__     :				List[str]   =					count_inversions_bf(lowercase_     )
						SCREAMING_SNAKE_CASE__     :				Any   =					count_inversions_recursive(lowercase_     )

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

						# an empty list should also have zero inversions
						SCREAMING_SNAKE_CASE__     :				Optional[int]   =					[]
						SCREAMING_SNAKE_CASE__     :				Tuple   =					count_inversions_bf(lowercase_     )
						SCREAMING_SNAKE_CASE__     :				List[str]   =					count_inversions_recursive(lowercase_     )

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


if __name__ == "__main__":
				main()






from __future__ import annotations







def       UpperCAmelCase							(     _lowerCamelCase      :  list			,   _lowerCamelCase      :  int | None = None			,   _lowerCamelCase      :  int | None = None     ):





						'''simple docstring'''


						if start is None:
												SCREAMING_SNAKE_CASE__     :				Union[str, Any]   =					0

						if end is None:
												SCREAMING_SNAKE_CASE__     :				Any   =					len(_lowerCamelCase     ) - 1

						if start >= end:
												return

						SCREAMING_SNAKE_CASE__     :				List[str]   =					(start + end) // 2

						slowsort(_lowerCamelCase			,   _lowerCamelCase			,   _lowerCamelCase     )
						slowsort(_lowerCamelCase			,   mid + 1			,   _lowerCamelCase     )

						if sequence[end] < sequence[mid]:
												SCREAMING_SNAKE_CASE__,							SCREAMING_SNAKE_CASE__     :				Optional[int]   =					sequence[mid], sequence[end]

						slowsort(_lowerCamelCase			,   _lowerCamelCase			,   end - 1     )


if __name__ == "__main__":
				from doctest import testmod

				testmod()