
def 							__A(lowerCAmelCase )		->				Dict:

		"""simple docstring"""






		_UpperCamelCase						    =	0
		_UpperCamelCase						    =	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 )		->				Any:

		"""simple docstring"""






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

		_UpperCamelCase    ,	_UpperCamelCase						    =	count_inversions_recursive(lowerCAmelCase )
		_UpperCamelCase    ,	_UpperCamelCase						    =	count_inversions_recursive(lowerCAmelCase )
		_UpperCamelCase    ,	_UpperCamelCase						    =	_count_cross_inversions(lowerCAmelCase						,  lowerCAmelCase )

		_UpperCamelCase						    =	inversion_p + inversions_q + cross_inversions
		return c, num_inversions


def 							__A(lowerCAmelCase						,  lowerCAmelCase )		->				Any:

		"""simple docstring"""






		_UpperCamelCase						    =	[]
		_UpperCamelCase						    =	_UpperCamelCase						    =	_UpperCamelCase						    =	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()		->				Tuple:

		"""simple docstring"""






		_UpperCamelCase						    =	[1_0, 2, 1, 5, 5, 2, 1_1]

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

		_UpperCamelCase						    =	count_inversions_bf(lowerCAmelCase )
		_UpperCamelCase    ,	_UpperCamelCase						    =	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()
		_UpperCamelCase						    =	count_inversions_bf(lowerCAmelCase )
		_UpperCamelCase    ,	_UpperCamelCase						    =	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
		_UpperCamelCase						    =	[]
		_UpperCamelCase						    =	count_inversions_bf(lowerCAmelCase )
		_UpperCamelCase    ,	_UpperCamelCase						    =	count_inversions_recursive(lowerCAmelCase )

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


if __name__ == "__main__":
		main()





from math import isclose, sqrt


def 							__A(lowerCAmelCase						,  lowerCAmelCase						,  lowerCAmelCase )		->				tuple[float, float, float]:

		"""simple docstring"""






		_UpperCamelCase						    =	point_y / 4 / point_x
		_UpperCamelCase						    =	2 * normal_gradient / (1 + normal_gradient * normal_gradient)
		_UpperCamelCase						    =	(1 - normal_gradient * normal_gradient) / (
		    1 + normal_gradient * normal_gradient
		)
		_UpperCamelCase						    =	(sa - ca * incoming_gradient) / (ca + sa * incoming_gradient)

		# to find the next point, solve the simultaeneous equations:
		# y^2 + 4x^2 = 100
		# y - b = m * (x - a)
		# ==> A x^2 + B x + C = 0
		_UpperCamelCase						    =	outgoing_gradient**2 + 4
		_UpperCamelCase						    =	2 * outgoing_gradient * (point_y - outgoing_gradient * point_x)
		_UpperCamelCase						    =	(point_y - outgoing_gradient * point_x) ** 2 - 1_0_0

		_UpperCamelCase						    =	(
		    -linear_term - sqrt(linear_term**2 - 4 * quadratic_term * constant_term )
		) / (2 * quadratic_term)
		_UpperCamelCase						    =	(
		    -linear_term + sqrt(linear_term**2 - 4 * quadratic_term * constant_term )
		) / (2 * quadratic_term)

		# two solutions, one of which is our input point
		_UpperCamelCase						    =	x_minus if isclose(lowerCAmelCase						,  lowerCAmelCase ) else x_plus
		_UpperCamelCase						    =	point_y + outgoing_gradient * (next_x - point_x)

		return next_x, next_y, outgoing_gradient


def 							__A(lowerCAmelCase = 1.4						,  lowerCAmelCase = -9.6 )		->				int:

		"""simple docstring"""






		_UpperCamelCase						    =	0
		_UpperCamelCase						    =	first_x_coord
		_UpperCamelCase						    =	first_y_coord
		_UpperCamelCase						    =	(10.1 - point_y) / (0.0 - point_x)

		while not (-0.01 <= point_x <= 0.01 and point_y > 0):
				_UpperCamelCase    ,	_UpperCamelCase    ,	_UpperCamelCase						    =	next_point(lowerCAmelCase						,  lowerCAmelCase						,  lowerCAmelCase )
				num_reflections += 1

		return num_reflections


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



