






def        _snake_case   (_snake_case				:       int      ,    _snake_case				:       list)					->	Any:
   _enforce_args(_snake_case      ,    _snake_case)
   if n == 0:
      return 0
   _lowercase              =float('-inf')
   for i in range(1      ,    n + 1):
      _lowercase              =max(
          _snake_case      ,    prices[i - 1] + naive_cut_rod_recursive(n - i      ,    _snake_case))

   return max_revue





def        _snake_case   (_snake_case				:       int      ,    _snake_case				:       list)					->	List[str]:
   _enforce_args(_snake_case      ,    _snake_case)
   _lowercase              =[float('-inf') for _ in range(n + 1)]
   return _top_down_cut_rod_recursive(_snake_case      ,    _snake_case      ,    _snake_case)





def        _snake_case   (_snake_case				:       int      ,    _snake_case				:       list      ,    _snake_case				:       list)					->	Tuple:
   if max_rev[n] >= 0:
      return max_rev[n]
   elif n == 0:
      return 0
   else:
      _lowercase              =float('-inf')
      for i in range(1      ,    n + 1):
         _lowercase              =max(
             _snake_case      ,    prices[i - 1] + _top_down_cut_rod_recursive(n - i      ,    _snake_case      ,    _snake_case)      ,    )

      _lowercase              =max_revenue

   return max_rev[n]





def        _snake_case   (_snake_case				:       int      ,    _snake_case				:       list)					->	Dict:
   _enforce_args(_snake_case      ,    _snake_case)

   # length(max_rev) = n + 1, to accommodate for the revenue obtainable from a rod of
   # length 0.
   _lowercase              =[float('-inf') for _ in range(n + 1)]
   _lowercase              =0

   for i in range(1      ,    n + 1):
      _lowercase              =max_rev[i]
      for j in range(1      ,    i + 1):
         _lowercase              =max(_snake_case      ,    prices[j - 1] + max_rev[i - j])

      _lowercase              =max_revenue_i

   return max_rev[n]





def        _snake_case   (_snake_case				:       int      ,    _snake_case				:       list)					->	List[Any]:
   if n < 0:
      _lowercase              =f'''n must be greater than or equal to 0. Got n = {n}'''
      raise ValueError(_snake_case)

   if n > len(_snake_case):
      _lowercase              =(
          'Each integral piece of rod must have a corresponding price. '
          f'''Got n = {n} but length of prices = {len(_snake_case)}'''
      )
      raise ValueError(_snake_case)





def        _snake_case   ()					->	Any:
   _lowercase              =[6, 10, 12, 15, 20, 23]
   _lowercase              =len(_snake_case)

   # the best revenue comes from cutting the rod into 6 pieces, each
   # of length 1 resulting in a revenue of 6 * 6 = 36.
   _lowercase              =36

   _lowercase              =top_down_cut_rod(_snake_case      ,    _snake_case)
   _lowercase              =bottom_up_cut_rod(_snake_case      ,    _snake_case)
   _lowercase              =naive_cut_rod_recursive(_snake_case      ,    _snake_case)

   assert expected_max_revenue == max_rev_top_down
   assert max_rev_top_down == max_rev_bottom_up
   assert max_rev_bottom_up == max_rev_naive


if __name__ == "__main__":
    main()










from math import sqrt

import numpy as np
from sympy import symbols

# Coefficient
# Speed of light (m/s)
_SCREAMING_SNAKE_CASE   =  2_99_79_24_58

# Symbols
_SCREAMING_SNAKE_CASE  ,	_SCREAMING_SNAKE_CASE  ,	_SCREAMING_SNAKE_CASE  ,	_SCREAMING_SNAKE_CASE   =  symbols("ct x y z")





def        _snake_case   (_snake_case				:       float)					->	float:
   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        _snake_case   (_snake_case				:       float)					->	float:
   return 1 / sqrt(1 - beta(_snake_case) ** 2)





def        _snake_case   (_snake_case				:       float)					->	np.ndarray:
   return np.array(
       [
           [gamma(_snake_case), -gamma(_snake_case) * beta(_snake_case), 0, 0],
           [-gamma(_snake_case) * beta(_snake_case), gamma(_snake_case), 0, 0],
           [0, 0, 1, 0],
           [0, 0, 0, 1],
       ])





def        _snake_case   (_snake_case				:       float      ,    _snake_case				:       np.ndarray | None = None)					->	np.ndarray:
   # Ensure event is not empty
   if event is None:
      _lowercase              =np.array([ct, x, y, z])  # Symbolic four vector
   else:
      event[0] *= c  # x0 is ct (speed of light * time)

   return transformation_matrix(_snake_case) @ event


if __name__ == "__main__":
    import doctest

    doctest.testmod()

    # Example of symbolic vector:
    _SCREAMING_SNAKE_CASE   =  transform(29_97_92_45)
    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
    _SCREAMING_SNAKE_CASE   =  {ct: c, x: 1, y: 1, z: 1}
    _SCREAMING_SNAKE_CASE   =  [four_vector[i].subs(sub_dict) for i in range(4)]

    print(f'''\n{numerical_vector}''')


