


'''simple docstring'''







from math import sqrt

import numpy as np
from sympy import symbols

# Coefficient
# Speed of light (m/s)
A				:					Optional[Any] 						=			299792458

# Symbols
A				:					Optional[Any] 						=			symbols('''ct x y z''')






def 				lowerCAmelCase__	(    lowerCamelCase					:      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 				lowerCAmelCase__	(    lowerCamelCase					:      float       ):
       return 1 / sqrt(1 - beta(lowerCamelCase       ) ** 2       )






def 				lowerCAmelCase__	(    lowerCamelCase					:      float       ):
       return np.array(
           [
               [gamma(lowerCamelCase       ), -gamma(lowerCamelCase       ) * beta(lowerCamelCase       ), 0, 0],
               [-gamma(lowerCamelCase       ) * beta(lowerCamelCase       ), gamma(lowerCamelCase       ), 0, 0],
               [0, 0, 1, 0],
               [0, 0, 0, 1],
           ]       )






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

       return transformation_matrix(lowerCamelCase       ) @ event


if __name__ == "__main__":
   import doctest

   doctest.testmod()

   # Example of symbolic vector:
   A				:					List[str] 						=			transform(29979245)
   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
   A				:					Union[str, Any] 						=			{ct: c, x: 1, y: 1, z: 1}
   A				:					Optional[int] 						=			[four_vector[i].subs(sub_dict) for i in range(4)]

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










'''simple docstring'''







def 				lowerCAmelCase__	(    lowerCamelCase					:      int  ,lowerCamelCase					:      list[int]  ,lowerCamelCase					:      int       ):

       def count_of_possible_combinations(lowerCamelCase					:      int       ) -> int:
              if target < 0:
                     return 0
              if target == 0:
                     return 1
              return sum(count_of_possible_combinations(target - item       ) for item in array       )

       return count_of_possible_combinations(lowerCamelCase       )






def 				lowerCAmelCase__	(    lowerCamelCase					:      int  ,lowerCamelCase					:      list[int]  ,lowerCamelCase					:      int       ):

       def count_of_possible_combinations_with_dp_array(
           lowerCamelCase					:      int  ,lowerCamelCase					:      list[int]       ) -> int:
              if target < 0:
                     return 0
              if target == 0:
                     return 1
              if dp_array[target] != -1:
                     return dp_array[target]
              _A					:     Optional[Any]     = sum(
                  count_of_possible_combinations_with_dp_array(target - item  ,lowerCamelCase       )
                  for item in array       )
              _A					:     List[str]     = answer
              return answer

       _A					:     Optional[int]     = [-1] * (target + 1)
       return count_of_possible_combinations_with_dp_array(lowerCamelCase  ,lowerCamelCase       )






def 				lowerCAmelCase__	(    lowerCamelCase					:      int  ,lowerCamelCase					:      list[int]  ,lowerCamelCase					:      int       ):
       _A					:     Dict     = [0] * (target + 1)
       _A					:     List[str]     = 1

       for i in range(1  ,target + 1       ):
              for j in range(lowerCamelCase       ):
                     if i - array[j] >= 0:
                            dp_array[i] += dp_array[i - array[j]]

       return dp_array[target]


if __name__ == "__main__":
   import doctest

   doctest.testmod()
   A				:					Dict    						=			3
   A				:					Union[str, Any]    						=			5
   A				:					Union[str, Any]    						=			[1, 2, 5]
   print(combination_sum_iv(n, array, target))






