

"""simple docstring"""

from collections.abc import Callable






class 			__a					:





       """simple docstring"""


       def __init__(   self			,					snake_case = None   ):





        """simple docstring"""

        lowerCAmelCase__   :				list 			=				[]
        # Stores indexes of each item for supporting updates and deletion.
        lowerCAmelCase__   :				dict 			=				{}
        # Stores current size of heap.
        lowerCAmelCase__   :				List[Any] 			=				0
        # Stores function used to evaluate the score of an item on which basis ordering
        # will be done.
        lowerCAmelCase__   :				Union[str, Any] 			=				key or (lambda snake_case   : x)






       def 			SCREAMING_SNAKE_CASE_	(   self			,					snake_case   ):





        """simple docstring"""

        return int((i - 1) / 2   ) if i > 0 else None






       def 			SCREAMING_SNAKE_CASE_	(   self			,					snake_case   ):





        """simple docstring"""

        lowerCAmelCase__   :				Dict 			=				int(2 * i + 1   )
        return left if 0 < left < self.size else None






       def 			SCREAMING_SNAKE_CASE_	(   self			,					snake_case   ):





        """simple docstring"""

        lowerCAmelCase__   :				List[str] 			=				int(2 * i + 2   )
        return right if 0 < right < self.size else None






       def 			SCREAMING_SNAKE_CASE_	(   self			,					snake_case			,					snake_case   ):





        """simple docstring"""

        lowerCAmelCase__   :				List[str] 			=				(
            self.pos_map[self.arr[j][0]],
            self.pos_map[self.arr[i][0]],
        )
        # Then swap the items in the list.
        lowerCAmelCase__   :				Any 			=				self.arr[j], self.arr[i]






       def 			SCREAMING_SNAKE_CASE_	(   self			,					snake_case			,					snake_case   ):





        """simple docstring"""

        return self.arr[i][1] < self.arr[j][1]






       def 			SCREAMING_SNAKE_CASE_	(   self			,					snake_case   ):





        """simple docstring"""

        lowerCAmelCase__   :				Optional[Any] 			=				self._left(lowercase__   )
        lowerCAmelCase__   :				List[Any] 			=				self._right(lowercase__   )
        lowerCAmelCase__   :				Optional[int] 			=				i

        if left is not None and not self._cmp(lowercase__			,					lowercase__   ):
         lowerCAmelCase__   :				Union[str, Any] 			=				left
        if right is not None and not self._cmp(lowercase__			,					lowercase__   ):
         lowerCAmelCase__   :				int 			=				right

        return valid_parent






       def 			SCREAMING_SNAKE_CASE_	(   self			,					snake_case   ):





        """simple docstring"""

        lowerCAmelCase__   :				Optional[int] 			=				self._parent(lowercase__   )
        while parent is not None and not self._cmp(lowercase__			,					lowercase__   ):
         self._swap(lowercase__			,					lowercase__   )
         lowerCAmelCase__   :				Optional[Any] 			=				parent, self._parent(lowercase__   )






       def 			SCREAMING_SNAKE_CASE_	(   self			,					snake_case   ):





        """simple docstring"""

        lowerCAmelCase__   :				Union[str, Any] 			=				self._get_valid_parent(lowercase__   )
        while valid_parent != index:
         self._swap(lowercase__			,					lowercase__   )
         lowerCAmelCase__   :				Tuple 			=				valid_parent, self._get_valid_parent(lowercase__   )






       def 			SCREAMING_SNAKE_CASE_	(   self			,					snake_case			,					snake_case   ):





        """simple docstring"""

        if item not in self.pos_map:
         return
        lowerCAmelCase__   :				Union[str, Any] 			=				self.pos_map[item]
        lowerCAmelCase__   :				Tuple 			=				[item, self.key(lowercase__   )]
        # Make sure heap is right in both up and down direction.
        # Ideally only one of them will make any change.
        self._heapify_up(lowercase__   )
        self._heapify_down(lowercase__   )






       def 			SCREAMING_SNAKE_CASE_	(   self			,					snake_case   ):





        """simple docstring"""

        if item not in self.pos_map:
         return
        lowerCAmelCase__   :				int 			=				self.pos_map[item]
        del self.pos_map[item]
        lowerCAmelCase__   :				List[Any] 			=				self.arr[self.size - 1]
        lowerCAmelCase__   :				Union[str, Any] 			=				index
        self.size -= 1
        # Make sure heap is right in both up and down direction. Ideally only one
        # of them will make any change- so no performance loss in calling both.
        if self.size > index:
         self._heapify_up(lowercase__   )
         self._heapify_down(lowercase__   )






       def 			SCREAMING_SNAKE_CASE_	(   self			,					snake_case			,					snake_case   ):





        """simple docstring"""

        lowerCAmelCase__   :				Tuple 			=				len(self.arr   )
        if arr_len == self.size:
         self.arr.append([item, self.key(lowercase__   )]   )
        else:
         lowerCAmelCase__   :				int 			=				[item, self.key(lowercase__   )]
        lowerCAmelCase__   :				List[Any] 			=				self.size
        self.size += 1
        self._heapify_up(self.size - 1   )






       def 			SCREAMING_SNAKE_CASE_	(   self   ):





        """simple docstring"""

        return self.arr[0] if self.size else None






       def 			SCREAMING_SNAKE_CASE_	(   self   ):





        """simple docstring"""

        lowerCAmelCase__   :				int 			=				self.get_top()
        if top_item_tuple:
         self.delete_item(top_item_tuple[0]   )
        return top_item_tuple




def       SCREAMING_SNAKE_CASE							( )			->						Optional[int]:
 pass




if __name__ == "__main__":
    import doctest

    doctest.testmod()












'''simple docstring'''


from math import sqrt

import numpy as np
from sympy import symbols

# Coefficient
# Speed of light (m/s)
lowerCAmelCase_      :  List[Any]				     =  2_9979_2458

# Symbols
lowerCAmelCase_     ,   lowerCAmelCase_     ,   lowerCAmelCase_     ,   lowerCAmelCase_      :  Tuple				     =  symbols('ct x y z')



def  _SCREAMING_SNAKE_CASE     (      UpperCamelCase__    :					float				):




     """simple docstring"""


     if velocity > c:
          raise ValueError("""Speed must not exceed light speed 299,792,458 [m/s]!"""				)
     elif velocity < 1:
          # Usually the speed should be much higher than 1 (c order of magnitude)
          raise ValueError("""Speed must be greater than or equal to 1!"""				)

     return velocity / c



def  _SCREAMING_SNAKE_CASE     (      UpperCamelCase__    :					float				):




     """simple docstring"""


     return 1 / sqrt(1 - beta(UpperCamelCase__				) ** 2				)



def  _SCREAMING_SNAKE_CASE     (      UpperCamelCase__    :					float				):




     """simple docstring"""


     return np.array(
         [
             [gamma(UpperCamelCase__				), -gamma(UpperCamelCase__				) * beta(UpperCamelCase__				), 0, 0],
             [-gamma(UpperCamelCase__				) * beta(UpperCamelCase__				), gamma(UpperCamelCase__				), 0, 0],
             [0, 0, 1, 0],
             [0, 0, 0, 1],
         ]				)



def  _SCREAMING_SNAKE_CASE     (      UpperCamelCase__    :					float ,    UpperCamelCase__    :					np.ndarray | None = None				):




     """simple docstring"""


     if event is None:
          a_     :							str     						=  np.array([ct, x, y, z]				)  # Symbolic four vector
     else:
          event[0] *= c  # x0 is ct (speed of light * time)

     return transformation_matrix(UpperCamelCase__				) @ event


if __name__ == "__main__":
 import doctest

 doctest.testmod()

 # Example of symbolic vector:
 lowerCAmelCase_      :  Union[str, Any]				     =  transform(2997_9245)
 print('Example of four vector: ')
 print(f"ct' = {four_vector[0]}")
 print(f"x' = {four_vector[1]}")
 print(f"y' = {four_vector[2]}")
 print(f"z' = {four_vector[3]}")

 # Substitute symbols with numerical values
 lowerCAmelCase_      :  Tuple				     =  {ct: c, x: 1, y: 1, z: 1}
 lowerCAmelCase_      :  Optional[int]				     =  [four_vector[i].subs(sub_dict) for i in range(4)]

 print(f"\n{numerical_vector}")
