"""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        snake_case (				__UpperCAmelCase       ):




					"""simple docstring"""

					def __init__(       self     :			Union[str, Any]					,lowerCamelCase__     :			Distribution					,lowerCamelCase__     :			Tuple=None					,lowerCamelCase__     :			List[str]=None					,lowerCamelCase__     :			Dict=0    ):
									UpperCAmelCase__    =							1.0 if scale is None else scale
									UpperCAmelCase__    =							0.0 if loc is None else loc

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







					@property
					def 	__lowerCAmelCase		(       self     :			List[str]    ):
									return self.base_dist.mean * self.scale + self.loc







					@property
					def 	__lowerCAmelCase		(       self     :			Dict    ):
									return self.base_dist.variance * self.scale**2







					@property
					def 	__lowerCAmelCase		(       self     :			int    ):
									return self.variance.sqrt()






class        snake_case (				nn.Module       ):




					"""simple docstring"""

					def __init__(       self     :			int					,lowerCamelCase__     :			int					,lowerCamelCase__     :			Dict[str, int]					,lowerCamelCase__     :			Callable[..., Tuple[torch.Tensor]]					,**lowerCamelCase__     :			Optional[int]    ):
									super().__init__(**lowerCamelCase__    )
									UpperCAmelCase__    =							args_dim
									UpperCAmelCase__    =							nn.ModuleList([nn.Linear(lowerCamelCase__					,lowerCamelCase__    ) for dim in args_dim.values()]    )
									UpperCAmelCase__    =							domain_map







					def 	__lowerCAmelCase		(       self     :			List[str]					,lowerCamelCase__     :			torch.Tensor    ):
									UpperCAmelCase__    =							[proj(lowerCamelCase__    ) for proj in self.proj]

									return self.domain_map(*lowerCamelCase__    )






class        snake_case (				nn.Module       ):




					"""simple docstring"""

					def __init__(       self     :			Union[str, Any]					,lowerCamelCase__     :			Optional[int]    ):
									super().__init__()
									UpperCAmelCase__    =							function







					def 	__lowerCAmelCase		(       self     :			str					,lowerCamelCase__     :			int					,*lowerCamelCase__     :			Dict    ):
									return self.function(lowerCamelCase__					,*lowerCamelCase__    )






class        snake_case :




					"""simple docstring"""

					snake_case__       					=				42
					snake_case__       					=				42
					snake_case__       					=				42







					def __init__(       self     :			List[str]					,lowerCamelCase__     :			int = 1    ):
									UpperCAmelCase__    =							dim
									UpperCAmelCase__    =							{k: dim * self.args_dim[k] for k in self.args_dim}







					def 	__lowerCAmelCase		(       self     :			Union[str, Any]					,lowerCamelCase__     :			Optional[int]    ):
									if self.dim == 1:
													return self.distribution_class(*lowerCamelCase__    )
									else:
													return Independent(self.distribution_class(*lowerCamelCase__    )					,1    )







					def 	__lowerCAmelCase		(       self     :			int					,lowerCamelCase__     :			Tuple					,lowerCamelCase__     :			Optional[torch.Tensor] = None					,lowerCamelCase__     :			Optional[torch.Tensor] = None					,):
									UpperCAmelCase__    =							self._base_distribution(lowerCamelCase__    )
									if loc is None and scale is None:
													return distr
									else:
													return AffineTransformed(lowerCamelCase__					,loc=lowerCamelCase__					,scale=lowerCamelCase__					,event_dim=self.event_dim    )







					@property
					def 	__lowerCAmelCase		(       self     :			List[str]    ):
									return () if self.dim == 1 else (self.dim,)







					@property
					def 	__lowerCAmelCase		(       self     :			Tuple    ):
									return len(self.event_shape    )







					@property
					def 	__lowerCAmelCase		(       self     :			Tuple    ):
									return 0.0







					def 	__lowerCAmelCase		(       self     :			Tuple					,lowerCamelCase__     :			int    ):
									return ParameterProjection(
									    in_features=lowerCamelCase__					,args_dim=self.args_dim					,domain_map=LambdaLayer(self.domain_map    )					,)







					def 	__lowerCAmelCase		(       self     :			Optional[Any]					,*lowerCamelCase__     :			torch.Tensor    ):
									raise NotImplementedError()







					@staticmethod
					def 	__lowerCAmelCase		(       lowerCamelCase__     :			torch.Tensor    ):
									return (x + torch.sqrt(torch.square(lowerCamelCase__    ) + 4.0    )) / 2.0






class        snake_case (				__UpperCAmelCase       ):




					"""simple docstring"""

					snake_case__       					=				{"df": 1, "loc": 1, "scale": 1}
					snake_case__       					=				StudentT







					@classmethod
					def 	__lowerCAmelCase		(       cls     :			Any					,lowerCamelCase__     :			torch.Tensor					,lowerCamelCase__     :			torch.Tensor					,lowerCamelCase__     :			torch.Tensor    ):
									UpperCAmelCase__    =							cls.squareplus(lowerCamelCase__    ).clamp_min(torch.finfo(scale.dtype    ).eps    )
									UpperCAmelCase__    =							2.0 + cls.squareplus(lowerCamelCase__    )
									return df.squeeze(-1    ), loc.squeeze(-1    ), scale.squeeze(-1    )






class        snake_case (				__UpperCAmelCase       ):




					"""simple docstring"""

					snake_case__       					=				{"loc": 1, "scale": 1}
					snake_case__       					=				Normal







					@classmethod
					def 	__lowerCAmelCase		(       cls     :			Tuple					,lowerCamelCase__     :			torch.Tensor					,lowerCamelCase__     :			torch.Tensor    ):
									UpperCAmelCase__    =							cls.squareplus(lowerCamelCase__    ).clamp_min(torch.finfo(scale.dtype    ).eps    )
									return loc.squeeze(-1    ), scale.squeeze(-1    )






class        snake_case (				__UpperCAmelCase       ):




					"""simple docstring"""

					snake_case__       					=				{"total_count": 1, "logits": 1}
					snake_case__       					=				NegativeBinomial







					@classmethod
					def 	__lowerCAmelCase		(       cls     :			Any					,lowerCamelCase__     :			torch.Tensor					,lowerCamelCase__     :			torch.Tensor    ):
									UpperCAmelCase__    =							cls.squareplus(lowerCamelCase__    )
									return total_count.squeeze(-1    ), logits.squeeze(-1    )







					def 	__lowerCAmelCase		(       self     :			int					,lowerCamelCase__     :			str    ):
									UpperCAmelCase__   ,			UpperCAmelCase__    =							distr_args
									if self.dim == 1:
													return self.distribution_class(total_count=lowerCamelCase__					,logits=lowerCamelCase__    )
									else:
													return Independent(self.distribution_class(total_count=lowerCamelCase__					,logits=lowerCamelCase__    )					,1    )







					def 	__lowerCAmelCase		(       self     :			List[Any]					,lowerCamelCase__     :			List[Any]					,lowerCamelCase__     :			Optional[torch.Tensor] = None					,lowerCamelCase__     :			Optional[torch.Tensor] = None    ):
									UpperCAmelCase__   ,			UpperCAmelCase__    =							distr_args

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

									return self._base_distribution((total_count, logits)    )




"""simple docstring"""






def 						a_		(      lowerCamelCase      ,    lowerCamelCase ):
				return x if y == 0 else greatest_common_divisor(lowerCamelCase      ,    x % y )



def 						a_		(      lowerCamelCase      ,    lowerCamelCase ):
				return (x * y) // greatest_common_divisor(lowerCamelCase      ,    lowerCamelCase )



def 						a_		(      lowerCamelCase = 2_0 ):
				UpperCAmelCase__    =							1
				for i in range(1      ,    n + 1 ):
								UpperCAmelCase__    =							lcm(lowerCamelCase      ,    lowerCamelCase )
				return g


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



