code: 
  
  
  from __future__ import annotations
  
  import random
  
  from matplotlib import animation
  from matplotlib import pyplot as plt
  
  # Frame rate of the animation
  lowercase_								=    20
  
  # Time between time steps in seconds
  lowercase_								=    INTERVAL / 10_00
  
  
  
  
  class 					_UpperCamelCase :
  
  
  
  
  
  
  
  							'''simple docstring'''
  
  
  
  
  
  
  
  							def __init__(		self  :				Dict		,				SCREAMING_SNAKE_CASE_  :				float		,				SCREAMING_SNAKE_CASE_  :				float		,				SCREAMING_SNAKE_CASE_  :				float		,				SCREAMING_SNAKE_CASE_  :				float		,				SCREAMING_SNAKE_CASE_  :				float = 1.0		,				SCREAMING_SNAKE_CASE_  :				float = 1.0		,				SCREAMING_SNAKE_CASE_  :				str = "blue"		,				):
  													_a     =			position_x
  													_a     =			position_y
  													_a     =			velocity_x
  													_a     =			velocity_y
  													_a     =			mass
  													_a     =			size
  													_a     =			color
  
  
  
  
  
  
  
  							@property
  							def 		_UpperCAmelCase	(		self  :				str		):
  													return self.position_x, self.position_y
  
  
  
  
  
  
  
  							@property
  							def 		_UpperCAmelCase	(		self  :				int		):
  													return self.velocity_x, self.velocity_y
  
  
  
  
  
  
  
  							def 		_UpperCAmelCase	(		self  :				Union[str, Any]		,				SCREAMING_SNAKE_CASE_  :				float		,				SCREAMING_SNAKE_CASE_  :				float		,				SCREAMING_SNAKE_CASE_  :				float		):
  													self.velocity_x += force_x * delta_time
  													self.velocity_y += force_y * delta_time
  
  
  
  
  
  
  
  							def 		_UpperCAmelCase	(		self  :				List[Any]		,				SCREAMING_SNAKE_CASE_  :				float		):
  													self.position_x += self.velocity_x * delta_time
  													self.position_y += self.velocity_y * delta_time
  
  
  
  
  class 					_UpperCamelCase :
  
  
  
  
  
  
  
  							'''simple docstring'''
  
  
  
  
  
  
  
  							def __init__(		self  :				Optional[Any]		,				SCREAMING_SNAKE_CASE_  :				list[Body]		,				SCREAMING_SNAKE_CASE_  :				float = 1.0		,				SCREAMING_SNAKE_CASE_  :				float = 1.0		,				SCREAMING_SNAKE_CASE_  :				float = 0.0		,				):
  													_a     =			bodies
  													_a     =			gravitation_constant
  													_a     =			time_factor
  													_a     =			softening_factor
  
  
  
  
  
  
  
  							def __len__(		self  :				Optional[Any]		):
  													return len(self.bodies		)
  
  
  
  
  
  
  
  							def 		_UpperCAmelCase	(		self  :				str		,				SCREAMING_SNAKE_CASE_  :				float		):
  													for bodya in self.bodies:
  																			_a     =			0.0
  																			_a     =			0.0
  																			for bodya in self.bodies:
  																									if bodya != bodya:
  																															_a     =			bodya.position_x - bodya.position_x
  																															_a     =			bodya.position_y - bodya.position_y
  
  																															# Calculation of the distance using Pythagoras's theorem
  																															# Extra factor due to the softening technique
  																															_a     =			(dif_x**2 + dif_y**2 + self.softening_factor) ** (
  																															    1 / 2
  																															)
  
  																															# Newton's law of universal gravitation.
  																															force_x += (
  																															    self.gravitation_constant * bodya.mass * dif_x / distance**3
  																															)
  																															force_y += (
  																															    self.gravitation_constant * bodya.mass * dif_y / distance**3
  																															)
  
              # Update the body's velocity once all the force components have been added
  																			bodya.update_velocity(SCREAMING_SNAKE_CASE_		,				SCREAMING_SNAKE_CASE_		,				delta_time * self.time_factor		)
  
  													# Update the positions only after all the velocities have been updated
  													for body in self.bodies:
  																			body.update_position(delta_time * self.time_factor		)
  
  
  
  
  def 							SCREAMING_SNAKE_CASE					(  _UpperCAmelCase							,     _UpperCAmelCase							,     _UpperCAmelCase				)    ->  None:
  						body_system.update_system(_UpperCAmelCase				)
  
  						# Update the positions of the patches
  						for patch, body in zip(_UpperCAmelCase							,     body_system.bodies				):
  												_a     =			(body.position_x, body.position_y)
  
  
  def 							SCREAMING_SNAKE_CASE					(  _UpperCAmelCase							,     _UpperCAmelCase							,     _UpperCAmelCase = -1							,     _UpperCAmelCase = 1							,     _UpperCAmelCase = -1							,     _UpperCAmelCase = 1							,     )    ->  None:
  						_a     =			plt.figure()
  						fig.canvas.set_window_title(_UpperCAmelCase				)
  						_a     =			plt.axes(
  						    xlim=(x_start, x_end)							,     ylim=(y_start, y_end)				)  # Set section to be plotted
  						plt.gca().set_aspect('equal'				)  # Fix aspect ratio
  
  						# Each body is drawn as a patch by the plt-function
  						_a     =			[
  						    plt.Circle((body.position_x, body.position_y)							,     body.size							,     fc=body.color				)
  						    for body in body_system.bodies
  						]
  
  						for patch in patches:
  												ax.add_patch(_UpperCAmelCase				)
  
  						# Function called at each step of the animation
  						def update(_UpperCAmelCase				) -> list[plt.Circle]:
  												update_step(_UpperCAmelCase							,     _UpperCAmelCase							,     _UpperCAmelCase				)
  												return patches
  
  						_a     =			animation.FuncAnimation(  # noqa: F841
  						    _UpperCAmelCase							,     _UpperCAmelCase							,     interval=_UpperCAmelCase							,     blit=_UpperCAmelCase				)
  
  						plt.show()
  
  
  def 							SCREAMING_SNAKE_CASE					(  )    ->  BodySystem:
  						_a     =			0.9700436
  						_a     =			-0.24308753
  						_a     =			0.466203685
  						_a     =			0.43236573
  
  						_a     =			[
  						    Body(_UpperCAmelCase							,     _UpperCAmelCase							,     _UpperCAmelCase							,     _UpperCAmelCase							,     size=0.2							,     color='red'				),
  						    Body(-position_x							,     -position_y							,     _UpperCAmelCase							,     _UpperCAmelCase							,     size=0.2							,     color='green'				),
  						    Body(0							,     0							,     -2 * velocity_x							,     -2 * velocity_y							,     size=0.2							,     color='blue'				),
  						]
  						return BodySystem(_UpperCAmelCase							,     time_factor=3				)
  
  
  def 							SCREAMING_SNAKE_CASE					(  )    ->  BodySystem:
  						_a     =			7.3_476e22
  						_a     =			5.972e24
  						_a     =			1022
  						_a     =			384399000
  						_a     =			6.674e-11
  
  						# Calculation of the respective velocities so that total impulse is zero,
  						# i.e. the two bodies together don't move
  						_a     =			earth_mass * velocity_dif / (earth_mass + moon_mass)
  						_a     =			moon_velocity - velocity_dif
  
  						_a     =			Body(-earth_moon_distance							,     0							,     0							,     _UpperCAmelCase							,     _UpperCAmelCase							,     10000000							,     'grey'				)
  						_a     =			Body(0							,     0							,     0							,     _UpperCAmelCase							,     _UpperCAmelCase							,     50000000							,     'blue'				)
  						return BodySystem([earth, moon]							,     _UpperCAmelCase							,     time_factor=1000000				)
  
  
  def 							SCREAMING_SNAKE_CASE					(  )    ->  BodySystem:
  						_a     =			[]
  						for _ in range(10				):
  												_a     =			random.uniform(-0.5							,     0.5				)
  												_a     =			random.uniform(-0.5							,     0.5				)
  
  												# Bodies are created pairwise with opposite velocities so that the
  												# total impulse remains zero
  												bodies.append(
  												    Body(
  												        random.uniform(-0.5							,     0.5				)							,     random.uniform(-0.5							,     0.5				)							,     _UpperCAmelCase							,     _UpperCAmelCase							,     size=0.05							,     )				)
  												bodies.append(
  												    Body(
  												        random.uniform(-0.5							,     0.5				)							,     random.uniform(-0.5							,     0.5				)							,     -velocity_x							,     -velocity_y							,     size=0.05							,     )				)
  						return BodySystem(_UpperCAmelCase							,     0.01							,     10							,     0.1				)
  
  
  if __name__ == "__main__":
  		plot('Figure-8 solution to the 3-body-problem', example_a(), -2, 2, -2, 2)
  		plot(
  		    'Moon\'s orbit around the earth',
  		    example_a(),
  		    -4_30_00_00_00,
  		    4_30_00_00_00,
  		    -4_30_00_00_00,
  		    4_30_00_00_00,
  		)
  		plot('Random system with many bodies', example_a(), -1.5, 1.5, -1.5, 1.5)
  
  
  
style_context: 
  
  
  from typing import List, Union
  
  from ...configuration_utils import PretrainedConfig
  from ...utils import logging
  
  
  lowercase_								=    logging.get_logger(__name__)
  
  lowercase_								=    {
      'allenai/led-base-16384': 'https://huggingface.co/allenai/led-base-16384/resolve/main/config.json',
      # See all LED models at https://huggingface.co/models?filter=led
  }
  
  
  
  
  class 					_UpperCamelCase (      lowerCamelCase__	):
  
  
  
  
  
  
  
  							'''simple docstring'''
  							_A  				=  "led"
  							_A  				=  {
  							    "num_attention_heads": "encoder_attention_heads",
  							    "hidden_size": "d_model",
  							    "attention_probs_dropout_prob": "attention_dropout",
  							    "initializer_range": "init_std",
  							}
  
  
  
  
  
  
  
  							def __init__(		self  :				List[str]		,				SCREAMING_SNAKE_CASE_  :				List[str]=5_0_2_6_5		,				SCREAMING_SNAKE_CASE_  :				str=1_6_3_8_4		,				SCREAMING_SNAKE_CASE_  :				List[Any]=1_0_2_4		,				SCREAMING_SNAKE_CASE_  :				List[str]=1_2		,				SCREAMING_SNAKE_CASE_  :				Any=4_0_9_6		,				SCREAMING_SNAKE_CASE_  :				Optional[Any]=1_6		,				SCREAMING_SNAKE_CASE_  :				str=1_2		,				SCREAMING_SNAKE_CASE_  :				List[Any]=4_0_9_6		,				SCREAMING_SNAKE_CASE_  :				List[str]=1_6		,				SCREAMING_SNAKE_CASE_  :				Optional[int]=0.0		,				SCREAMING_SNAKE_CASE_  :				Union[str, Any]=0.0		,				SCREAMING_SNAKE_CASE_  :				List[Any]=True		,				SCREAMING_SNAKE_CASE_  :				str=True		,				SCREAMING_SNAKE_CASE_  :				List[str]="gelu"		,				SCREAMING_SNAKE_CASE_  :				List[Any]=1_0_2_4		,				SCREAMING_SNAKE_CASE_  :				Tuple=0.1		,				SCREAMING_SNAKE_CASE_  :				Dict=0.0		,				SCREAMING_SNAKE_CASE_  :				Optional[Any]=0.0		,				SCREAMING_SNAKE_CASE_  :				Tuple=0.02		,				SCREAMING_SNAKE_CASE_  :				str=2		,				SCREAMING_SNAKE_CASE_  :				Tuple=0.0		,				SCREAMING_SNAKE_CASE_  :				Optional[int]=1		,				SCREAMING_SNAKE_CASE_  :				str=0		,				SCREAMING_SNAKE_CASE_  :				Union[str, Any]=2		,				SCREAMING_SNAKE_CASE_  :				Union[List[int], int] = 5_1_2		,				**SCREAMING_SNAKE_CASE_  :				Dict		,				):
  													_a     =			vocab_size
  													_a     =			max_encoder_position_embeddings
  													_a     =			max_decoder_position_embeddings
  													_a     =			d_model
  													_a     =			encoder_ffn_dim
  													_a     =			encoder_layers
  													_a     =			encoder_attention_heads
  													_a     =			decoder_ffn_dim
  													_a     =			decoder_layers
  													_a     =			decoder_attention_heads
  													_a     =			dropout
  													_a     =			attention_dropout
  													_a     =			activation_dropout
  													_a     =			activation_function
  													_a     =			init_std
  													_a     =			encoder_layerdrop
  													_a     =			decoder_layerdrop
  													_a     =			classifier_dropout
  													_a     =			use_cache
  													_a     =			encoder_layers
  													_a     =			attention_window
  
  													super().__init__(
  													    pad_token_id=SCREAMING_SNAKE_CASE_		,				bos_token_id=SCREAMING_SNAKE_CASE_		,				eos_token_id=SCREAMING_SNAKE_CASE_		,				is_encoder_decoder=SCREAMING_SNAKE_CASE_		,				decoder_start_token_id=SCREAMING_SNAKE_CASE_		,				**SCREAMING_SNAKE_CASE_		,				)
  
  
  