"""simple docstring"""





from typing import Callable, Dict, Optional, Tuple

import torch
from torch import nn
from torch.distributions import (
    AffineTransform,
    Distribution,
    Independent,
    NegativeBinomial,
    Normal,
    StudentT,
    TransformedDistribution,
)


class   UpperCAmelCase_			(					_a):


      def __init__(     self	,    a	,    a=None	,    a=None	,    a=0							) ->      Any:
             lowercase__    :	Optional[Any]     					=			1.0 if scale is None else scale
             lowercase__    :	Optional[Any]     					=			0.0 if loc is None else loc

             super().__init__(a	,    [AffineTransform(loc=self.loc	,    scale=self.scale	,    event_dim=a							)]							)


      @property
      def 	_UpperCAmelCase (     self							) ->      List[Any]:
             return self.base_dist.mean * self.scale + self.loc


      @property
      def 	_UpperCAmelCase (     self							) ->      List[str]:
             return self.base_dist.variance * self.scale**2


      @property
      def 	_UpperCAmelCase (     self							) ->      Any:
             return self.variance.sqrt()





class   UpperCAmelCase_			(					nn.Module):


      def __init__(     self	,    a	,    a	,    a	,    **a							) ->      None:
             super().__init__(**a							)
             lowercase__    :	Tuple     					=			args_dim
             lowercase__    :	Optional[int]     					=			nn.ModuleList([nn.Linear(a	,    a							) for dim in args_dim.values()]							)
             lowercase__    :	str     					=			domain_map


      def 	_UpperCAmelCase (     self	,    a							) ->      Tuple[torch.Tensor]:
             lowercase__    :	Union[str, Any]     					=			[proj(a							) for proj in self.proj]

             return self.domain_map(*a							)





class   UpperCAmelCase_			(					nn.Module):


      def __init__(     self	,    a							) ->      int:
             super().__init__()
             lowercase__    :	Any     					=			function


      def 	_UpperCAmelCase (     self	,    a	,    *a							) ->      List[str]:
             return self.function(a	,    *a							)





class   UpperCAmelCase_			:
      lowerCamelCase__	:					type
      lowerCamelCase__	:					int
      lowerCamelCase__	:					Dict[str, int]


      def __init__(     self	,    a = 1							) ->      None:
             lowercase__    :	Optional[int]     					=			dim
             lowercase__    :	Tuple     					=			{k: dim * self.args_dim[k] for k in self.args_dim}


      def 	_UpperCAmelCase (     self	,    a							) ->      List[str]:
             if self.dim == 1:
                    return self.distribution_class(*a							)
             else:
                    return Independent(self.distribution_class(*a							)	,    1							)


      def 	_UpperCAmelCase (     self	,    a	,    a = None	,    a = None	,    ) ->      Distribution:
             lowercase__    :	Optional[int]     					=			self._base_distribution(a							)
             if loc is None and scale is None:
                    return distr
             else:
                    return AffineTransformed(a	,    loc=a	,    scale=a	,    event_dim=self.event_dim							)


      @property
      def 	_UpperCAmelCase (     self							) ->      Tuple:
             return () if self.dim == 1 else (self.dim,)


      @property
      def 	_UpperCAmelCase (     self							) ->      int:
             return len(self.event_shape							)


      @property
      def 	_UpperCAmelCase (     self							) ->      float:
             return 0.0


      def 	_UpperCAmelCase (     self	,    a							) ->      nn.Module:
             return ParameterProjection(
                 in_features=a	,    args_dim=self.args_dim	,    domain_map=LambdaLayer(self.domain_map							)	,    )


      def 	_UpperCAmelCase (     self	,    *a							) ->      Any:
             raise NotImplementedError()


      @staticmethod
      def 	_UpperCAmelCase (     a							) ->      torch.Tensor:
             return (x + torch.sqrt(torch.square(a							) + 4.0							)) / 2.0





class   UpperCAmelCase_			(					_a):
      lowerCamelCase__	:					Dict[str, int]										=						{"df": 1, "loc": 1, "scale": 1}
      lowerCamelCase__	:					type										=						StudentT


      @classmethod
      def 	_UpperCAmelCase (     cls	,    a	,    a	,    a							) ->      str:
             lowercase__    :	str     					=			cls.squareplus(a							).clamp_min(torch.finfo(scale.dtype							).eps							)
             lowercase__    :	Dict     					=			2.0 + cls.squareplus(a							)
             return df.squeeze(-1							), loc.squeeze(-1							), scale.squeeze(-1							)





class   UpperCAmelCase_			(					_a):
      lowerCamelCase__	:					Dict[str, int]										=						{"loc": 1, "scale": 1}
      lowerCamelCase__	:					type										=						Normal


      @classmethod
      def 	_UpperCAmelCase (     cls	,    a	,    a							) ->      Tuple:
             lowercase__    :	Union[str, Any]     					=			cls.squareplus(a							).clamp_min(torch.finfo(scale.dtype							).eps							)
             return loc.squeeze(-1							), scale.squeeze(-1							)





class   UpperCAmelCase_			(					_a):
      lowerCamelCase__	:					Dict[str, int]										=						{"total_count": 1, "logits": 1}
      lowerCamelCase__	:					type										=						NegativeBinomial


      @classmethod
      def 	_UpperCAmelCase (     cls	,    a	,    a							) ->      str:
             lowercase__    :	List[str]     					=			cls.squareplus(a							)
             return total_count.squeeze(-1							), logits.squeeze(-1							)


      def 	_UpperCAmelCase (     self	,    a							) ->      Distribution:
             lowercase__					,   lowercase__    :	Any     					=			distr_args
             if self.dim == 1:
                    return self.distribution_class(total_count=a	,    logits=a							)
             else:
                    return Independent(self.distribution_class(total_count=a	,    logits=a							)	,    1							)


      def 	_UpperCAmelCase (     self	,    a	,    a = None	,    a = None							) ->      Distribution:
             lowercase__					,   lowercase__    :	int     					=			distr_args

             if scale is not None:
                    # See scaling property of Gamma.
                    logits += scale.log()

             return self._base_distribution((total_count, logits)							)


"""simple docstring"""





import math





def        a_  (   _lowerCAmelCase						:				int = 100  ):





       '''simple docstring'''

       lowercase__    :	Union[str, Any]     					=			sum(i * i for i in range(1     ,				n + 1  )  )
       lowercase__    :	str     					=			int(math.pow(sum(range(1     ,				n + 1  )  )     ,				2  )  )
       return square_of_sum - sum_of_squares


if __name__ == "__main__":
       print(f'''{solution() = }''')

