




"""simple docstring"""

from __future__ import annotations

from random import random

class        lowercase      :







				def __init__(							self   :					Optional[Any]      ,     _lowerCamelCase   :					int | None = None       ):



					"""simple docstring"""

					A_  :				Tuple	   =		value
					A_  :				Dict	   =		random()
					A_  :				Node | None	   =		None
					A_  :				Node | None	   =		None







				def __repr__(							self   :					Union[str, Any]       ):



					"""simple docstring"""

					from pprint import pformat

					if self.left is None and self.right is None:
						return F"""'{self.value}: {self.prior:.5}'"""
					else:
						return pformat(
						    {F"""{self.value}: {self.prior:.5}""": (self.left, self.right)}      ,     indent=1       )







				def __str__(							self   :					Optional[int]       ):



					"""simple docstring"""

					A_  :				str	   =		str(self.value       ) + ''' '''
					A_  :				Optional[Any]	   =		str(self.left or ''''''       )
					A_  :				Any	   =		str(self.right or ''''''       )
					return value + left + right

def 		lowercase_      (		_UpperCAmelCase  ,      _UpperCAmelCase	):




	"""simple docstring"""






	if root is None:  # None tree is split into 2 Nones
		return None, None
	elif root.value is None:
		return None, None
	else:
		if value < root.value:
			A_   , A_  :				Union[str, Any]	   =		split(root.left  ,      _UpperCAmelCase	)
			return left, root
		else:
			A_   , A_  :				Optional[int]	   =		split(root.right  ,      _UpperCAmelCase	)
			return root, right

def 		lowercase_      (		_UpperCAmelCase  ,      _UpperCAmelCase	):




	"""simple docstring"""






	if (not left) or (not right):  # If one node is None, return the other
		return left or right
	elif left.prior < right.prior:
		A_  :				int	   =		merge(left.right  ,      _UpperCAmelCase	)
		return left
	else:
		A_  :				Tuple	   =		merge(_UpperCAmelCase  ,      right.left	)
		return right

def 		lowercase_      (		_UpperCAmelCase  ,      _UpperCAmelCase	):




	"""simple docstring"""






	A_  :				List[Any]	   =		Node(_UpperCAmelCase	)
	A_   , A_  :				str	   =		split(_UpperCAmelCase  ,      _UpperCAmelCase	)
	return merge(merge(_UpperCAmelCase  ,      _UpperCAmelCase	)  ,      _UpperCAmelCase	)

def 		lowercase_      (		_UpperCAmelCase  ,      _UpperCAmelCase	):




	"""simple docstring"""






	A_   , A_  :				Tuple	   =		split(_UpperCAmelCase  ,      value - 1	)
	A_   , A_  :				Union[str, Any]	   =		split(_UpperCAmelCase  ,      _UpperCAmelCase	)
	return merge(_UpperCAmelCase  ,      _UpperCAmelCase	)

def 		lowercase_      (		_UpperCAmelCase	):




	"""simple docstring"""






	if not root:  # None
		return
	else:
		inorder(root.left	)
		print(root.value  ,      end=''','''	)
		inorder(root.right	)

def 		lowercase_      (		_UpperCAmelCase  ,      _UpperCAmelCase	):




	"""simple docstring"""






	for arg in args.split():
		if arg[0] == "+":
			A_  :				Optional[Any]	   =		insert(_UpperCAmelCase  ,      int(arg[1:]	)	)

		elif arg[0] == "-":
			A_  :				List[str]	   =		erase(_UpperCAmelCase  ,      int(arg[1:]	)	)

		else:
			print('''Unknown command'''	)

	return root

def 		lowercase_      (		):




	"""simple docstring"""






	A_  :				Tuple	   =		None
	print(
	    '''enter numbers to create a tree, + value to add value into treap, '''
	    '''- value to erase all nodes with value. \'q\' to quit. '''	)

	A_  :				int	   =		input()
	while args != "q":
		A_  :				Union[str, Any]	   =		interact_treap(_UpperCAmelCase  ,      _UpperCAmelCase	)
		print(_UpperCAmelCase	)
		A_  :				int	   =		input()

	print('''good by!'''	)


if __name__ == "__main__":
		import doctest

		doctest.testmod()
		main()











"""simple docstring"""

from typing import Any

import numpy as np

def 		lowercase_      (		_UpperCAmelCase	):




	"""simple docstring"""






	return np.array_equal(_UpperCAmelCase  ,      matrix.conjugate().T	)

def 		lowercase_      (		_UpperCAmelCase  ,      _UpperCAmelCase	):




	"""simple docstring"""






	A_  :				Tuple	   =		v.conjugate().T
	A_  :				Tuple	   =		v_star.dot(_UpperCAmelCase	)
	assert isinstance(_UpperCAmelCase  ,      np.ndarray	)
	return (v_star_dot.dot(_UpperCAmelCase	)) / (v_star.dot(_UpperCAmelCase	))

def 		lowercase_      (		):




	"""simple docstring"""






	A_  :				Optional[Any]	   =		np.array([[2, 2 + 1J, 4], [2 - 1J, 3, 1J], [4, -1J, 1]]	)
	A_  :				str	   =		np.array([[1], [2], [3]]	)
	assert is_hermitian(_UpperCAmelCase	), f"""{a} is not hermitian."""
	print(rayleigh_quotient(_UpperCAmelCase  ,      _UpperCAmelCase	)	)

	A_  :				int	   =		np.array([[1, 2, 4], [2, 3, -1], [4, -1, 1]]	)
	assert is_hermitian(_UpperCAmelCase	), f"""{a} is not hermitian."""
	assert rayleigh_quotient(_UpperCAmelCase  ,      _UpperCAmelCase	) == float(3	)


if __name__ == "__main__":
		import doctest

		doctest.testmod()
		tests()





