



'''simple docstring'''




def      a      (			lowerCamelCase__   ):
			'''simple docstring'''





			A_     :						List[str]     =					0
			A_     :						str     =					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      a      (			lowerCamelCase__   ):
			'''simple docstring'''





			if len(lowerCamelCase__   ) <= 1:
						return arr, 0
			A_     :						List[Any]     =					len(lowerCamelCase__   ) // 2
			A_     :						str     =					arr[0:mid]
			A_     :						str     =					arr[mid:]

			A_     :						Union[str, Any]     =					count_inversions_recursive(lowerCamelCase__   )
			A_     :						Union[str, Any]     =					count_inversions_recursive(lowerCamelCase__   )
			A_     :						Any     =					_count_cross_inversions(lowerCamelCase__						,				lowerCamelCase__   )

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



def      a      (			lowerCamelCase__						,				lowerCamelCase__   ):
			'''simple docstring'''





			A_     :						int     =					[]
			A_     :						List[str]     =					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      a      (			):
			'''simple docstring'''





			A_     :						List[str]     =					[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)

			A_     :						List[Any]     =					count_inversions_bf(lowerCamelCase__   )
			A_     :						List[str]     =					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()
			A_     :						str     =					count_inversions_bf(lowerCamelCase__   )
			A_     :						Any     =					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
			A_     :						Union[str, Any]     =					[]
			A_     :						str     =					count_inversions_bf(lowerCamelCase__   )
			A_     :						Optional[int]     =					count_inversions_recursive(lowerCamelCase__   )

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


if __name__ == "__main__":
						main()








'''simple docstring'''




from __future__ import annotations

from collections.abc import Callable

lowerCamelCase							:List[Any]       = list[list[float | int]]



def      a      (			lowerCamelCase__						,				lowerCamelCase__   ):
			'''simple docstring'''





			A_     :						int     =					len(lowerCamelCase__   )
			A_     :						Matrix     =					[[0 for _ in range(size + 1   )] for _ in range(lowerCamelCase__   )]
			A_     :						int
			A_     :						int
			A_     :						int
			A_     :						int
			A_     :						int
			A_     :						float

			for row in range(lowerCamelCase__   ):
						for col in range(lowerCamelCase__   ):
									A_     :						List[str]     =					matrix[row][col]

						A_     :						Optional[int]     =					vector[row][0]

			A_     :						Tuple     =					0
			A_     :						Tuple     =					0
			while row < size and col < size:
						# pivoting
						A_     :						Dict     =					max((abs(augmented[rowa][col]   ), rowa) for rowa in range(lowerCamelCase__						,				lowerCamelCase__   )   )[
						    1
						]
						if augmented[pivot_row][col] == 0:
									col += 1
									continue
						else:
									A_,   A_     :						Optional[Any]     =					augmented[pivot_row], augmented[row]

						for rowa in range(row + 1						,				lowerCamelCase__   ):
									A_     :						int     =					augmented[rowa][col] / augmented[row][col]
									A_     :						Union[str, Any]     =					0
									for cola in range(col + 1						,				size + 1   ):
												augmented[rowa][cola] -= augmented[row][cola] * ratio

						row += 1
						col += 1

			# back substitution
			for col in range(1						,				lowerCamelCase__   ):
						for row in range(lowerCamelCase__   ):
									A_     :						Tuple     =					augmented[row][col] / augmented[col][col]
									for cola in range(lowerCamelCase__						,				size + 1   ):
												augmented[row][cola] -= augmented[col][cola] * ratio

    # round to get rid of numbers like 2.000000000000004
			return [
			    [round(augmented[row][size] / augmented[row][row]						,				10   )] for row in range(lowerCamelCase__   )
			]



def      a      (			lowerCamelCase__   ):
			'''simple docstring'''





			A_     :						int     =					len(lowerCamelCase__   )
			A_     :						Matrix     =					[[0 for _ in range(lowerCamelCase__   )] for _ in range(lowerCamelCase__   )]
			A_     :						Matrix     =					[[0] for _ in range(lowerCamelCase__   )]
			A_     :						Matrix
			A_     :						int
			A_     :						int
			A_     :						int

			for x_val, y_val in enumerate(lowerCamelCase__   ):
						for col in range(lowerCamelCase__   ):
									A_     :						Dict     =					(x_val + 1) ** (size - col - 1)
						A_     :						Any     =					y_val

			A_     :						Dict     =					solve(lowerCamelCase__						,				lowerCamelCase__   )

			def interpolated_func(lowerCamelCase__   ) -> int:
						return sum(
						    round(coeffs[x_val][0]   ) * (var ** (size - x_val - 1))
						    for x_val in range(lowerCamelCase__   )   )

			return interpolated_func



def      a      (			lowerCamelCase__   ):
			'''simple docstring'''





			return (
			    1
			    - variable
			    + variable**2
			    - variable**3
			    + variable**4
			    - variable**5
			    + variable**6
			    - variable**7
			    + variable**8
			    - variable**9
			    + variable**10
			)



def      a      (			lowerCamelCase__ = question_function						,				lowerCamelCase__ = 10   ):
			'''simple docstring'''





			A_     :						list[int]     =					[func(lowerCamelCase__   ) for x_val in range(1						,				order + 1   )]

			A_     :						list[Callable[[int], int]]     =					[
			    interpolate(data_points[:max_coeff]   ) for max_coeff in range(1						,				order + 1   )
			]

			A_     :						int     =					0
			A_     :						Callable[[int], int]
			A_     :						int

			for poly in polynomials:
						A_     :						int     =					1
						while func(lowerCamelCase__   ) == poly(lowerCamelCase__   ):
									x_val += 1

						ret += poly(lowerCamelCase__   )

			return ret


if __name__ == "__main__":
						print(F"{solution() = }")