

'''simple docstring'''






from __future__ import annotations

from random import random






class 			lowercase_	:




  """simple docstring"""




  def __init__(      self   :     Union[str, Any]					,lowercase__   :     int | None = None			):
   __lowercase							  = value
   __lowercase							  = random()
   __lowercase							  = None
   __lowercase							  = None






  def __repr__(      self   :     List[str]			):
   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   :     List[Any]			):
   __lowercase							  = str(self.value			) + ''' '''
   __lowercase							  = str(self.left or ''''''			)
   __lowercase							  = str(self.right or ''''''			)
   return value + left + right


def 		_A     (    A__     ,				A__      ):






 """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:
   __lowercase  ,	__lowercase							  = split(root.left     ,				A__      )
   return left, root
  else:
   __lowercase  ,	__lowercase							  = split(root.right     ,				A__      )
   return root, right


def 		_A     (    A__     ,				A__      ):






 """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:
  __lowercase							  = merge(left.right     ,				A__      )
  return left
 else:
  __lowercase							  = merge(A__     ,				right.left      )
  return right


def 		_A     (    A__     ,				A__      ):






 """simple docstring"""



 __lowercase							  = Node(A__      )
 __lowercase  ,	__lowercase							  = split(A__     ,				A__      )
 return merge(merge(A__     ,				A__      )     ,				A__      )


def 		_A     (    A__     ,				A__      ):






 """simple docstring"""



 __lowercase  ,	__lowercase							  = split(A__     ,				value - 1      )
 __lowercase  ,	__lowercase							  = split(A__     ,				A__      )
 return merge(A__     ,				A__      )


def 		_A     (    A__      ):






 """simple docstring"""



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


def 		_A     (    A__     ,				A__      ):






 """simple docstring"""



 for arg in args.split():
  if arg[0] == "+":
   __lowercase							  = insert(A__     ,				int(arg[1:]      )      )

  elif arg[0] == "-":
   __lowercase							  = erase(A__     ,				int(arg[1:]      )      )

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

 return root


def 		_A     (    ):






 """simple docstring"""



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

 __lowercase							  = input()
 while args != "q":
  __lowercase							  = interact_treap(A__     ,				A__      )
  print(A__      )
  __lowercase							  = input()

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


if __name__ == "__main__":
 import doctest

 doctest.testmod()
 main()





'''simple docstring'''



from numpy import exp, pi, sqrt



def 			UpperCAmelCase__					(	UpperCAmelCase_    :				Optional[int]      ,    UpperCAmelCase_    :				float = 0.0      ,    UpperCAmelCase_    :				float = 1.0							)     ->		int:
     return 1 / sqrt(2 * pi * sigma**2							) * exp(-((x - mu) ** 2) / (2 * sigma**2)							)


if __name__ == "__main__":
      import doctest

      doctest.testmod()






