
"""simple docstring"""




from __future__ import annotations

from PIL import Image

# Define glider example
UpperCAmelCase__      :    List[str]    							= [
    [0, 1, 0, 0, 0, 0, 0, 0],
    [0, 0, 1, 0, 0, 0, 0, 0],
    [1, 1, 1, 0, 0, 0, 0, 0],
    [0, 0, 0, 0, 0, 0, 0, 0],
    [0, 0, 0, 0, 0, 0, 0, 0],
    [0, 0, 0, 0, 0, 0, 0, 0],
    [0, 0, 0, 0, 0, 0, 0, 0],
    [0, 0, 0, 0, 0, 0, 0, 0],
]

# Define blinker example
UpperCAmelCase__      :    Optional[int]    							= [[0, 1, 0], [0, 1, 0], [0, 1, 0]]

def lowercase_  (   _snake_case						):
							SCREAMING_SNAKE_CASE__						:       Union[str, Any]           =   []
							for i in range(len(_snake_case						)						):
														SCREAMING_SNAKE_CASE__						:       Union[str, Any]           =   []
														for j in range(len(cells[i]						)						):
																					# Get the number of live neighbours
																					SCREAMING_SNAKE_CASE__						:       int           =   0
																					if i > 0 and j > 0:
																												neighbour_count += cells[i - 1][j - 1]
																					if i > 0:
																												neighbour_count += cells[i - 1][j]
																					if i > 0 and j < len(cells[i]						) - 1:
																												neighbour_count += cells[i - 1][j + 1]
																					if j > 0:
																												neighbour_count += cells[i][j - 1]
																					if j < len(cells[i]						) - 1:
																												neighbour_count += cells[i][j + 1]
																					if i < len(_snake_case						) - 1 and j > 0:
																												neighbour_count += cells[i + 1][j - 1]
																					if i < len(_snake_case						) - 1:
																												neighbour_count += cells[i + 1][j]
																					if i < len(_snake_case						) - 1 and j < len(cells[i]						) - 1:
																												neighbour_count += cells[i + 1][j + 1]

																					# Rules of the game of life (excerpt from Wikipedia):
																					# 1. Any live cell with two or three live neighbours survives.
																					# 2. Any dead cell with three live neighbours becomes a live cell.
																					# 3. All other live cells die in the next generation.
																					#    Similarly, all other dead cells stay dead.
																					SCREAMING_SNAKE_CASE__						:       Tuple           =   cells[i][j] == 1
																					if (
																					    (alive and 2 <= neighbour_count <= 3)
																					    or not alive
																					    and neighbour_count == 3
																					):
																												next_generation_row.append(1						)
																					else:
																												next_generation_row.append(0						)

														next_generation.append(_snake_case						)
							return next_generation

def lowercase_  (   _snake_case   ,_snake_case						):
							SCREAMING_SNAKE_CASE__						:       Optional[Any]           =   []
							for _ in range(_snake_case						):
														# Create output image
														SCREAMING_SNAKE_CASE__						:       Optional[Any]           =   Image.new("""RGB"""   ,(len(cells[0]						), len(_snake_case						))						)
														SCREAMING_SNAKE_CASE__						:       Union[str, Any]           =   img.load()

														# Save cells to image
														for x in range(len(_snake_case						)						):
																					for y in range(len(cells[0]						)						):
																												SCREAMING_SNAKE_CASE__						:       Optional[int]           =   255 - cells[y][x] * 255
																												SCREAMING_SNAKE_CASE__						:       str           =   (colour, colour, colour)

        # Save image
														images.append(_snake_case						)
														SCREAMING_SNAKE_CASE__						:       Dict           =   new_generation(_snake_case						)
							return images


if __name__ == "__main__":
					UpperCAmelCase__      :    Union[str, Any]    							= generate_images(GLIDER, 1_6)
					images[0].save('out.gif', save_all=True, append_images=images[1:])






"""simple docstring"""




import mpmath  # for roots of unity
import numpy as np



class 	lowerCAmelCase_						:


				"""simple docstring"""



				def __init__(self   ,       SCREAMING_SNAKE_CASE__=None   ,       SCREAMING_SNAKE_CASE__=None    )					->		Dict:



											"""simple docstring"""




											SCREAMING_SNAKE_CASE__						:       Union[str, Any]           =   list(poly_a or [0]    )[:]
											SCREAMING_SNAKE_CASE__						:       Tuple           =   list(poly_b or [0]    )[:]

											# Remove leading zero coefficients
											while self.polyA[-1] == 0:
																		self.polyA.pop()
											SCREAMING_SNAKE_CASE__						:       int           =   len(self.polyA    )

											while self.polyB[-1] == 0:
																		self.polyB.pop()
											SCREAMING_SNAKE_CASE__						:       List[str]           =   len(self.polyB    )

											# Add 0 to make lengths equal a power of 2
											SCREAMING_SNAKE_CASE__						:       Optional[int]           =   int(
											    2 ** np.ceil(np.loga(len(self.polyA    ) + len(self.polyB    ) - 1    )    )    )

											while len(self.polyA    ) < self.c_max_length:
																		self.polyA.append(0    )
											while len(self.polyB    ) < self.c_max_length:
																		self.polyB.append(0    )
											# A complex root used for the fourier transform
											SCREAMING_SNAKE_CASE__						:       List[str]           =   complex(mpmath.root(x=1   ,       n=self.c_max_length   ,       k=1    )    )

											# The product
											SCREAMING_SNAKE_CASE__						:       Tuple           =   self.__multiply()



				def 	__magic_name__    (self   ,       SCREAMING_SNAKE_CASE__    )					->		Union[str, Any]:



											"""simple docstring"""




											SCREAMING_SNAKE_CASE__						:       str           =   [[x] for x in self.polyA] if which == """A""" else [[x] for x in self.polyB]
											# Corner case
											if len(SCREAMING_SNAKE_CASE__    ) <= 1:
																		return dft[0]
											#
											SCREAMING_SNAKE_CASE__						:       Optional[Any]           =   self.c_max_length // 2
											while next_ncol > 0:
																		SCREAMING_SNAKE_CASE__						:       Any           =   [[] for i in range(SCREAMING_SNAKE_CASE__    )]
																		SCREAMING_SNAKE_CASE__						:       Tuple           =   self.root**next_ncol

																		# First half of next step
																		SCREAMING_SNAKE_CASE__						:       str           =   1
																		for j in range(self.c_max_length // (next_ncol * 2)    ):
																									for i in range(SCREAMING_SNAKE_CASE__    ):
																																new_dft[i].append(dft[i][j] + current_root * dft[i + next_ncol][j]    )
																									current_root *= root
																		# Second half of next step
																		SCREAMING_SNAKE_CASE__						:       int           =   1
																		for j in range(self.c_max_length // (next_ncol * 2)    ):
																									for i in range(SCREAMING_SNAKE_CASE__    ):
																																new_dft[i].append(dft[i][j] - current_root * dft[i + next_ncol][j]    )
																									current_root *= root
																		# Update
																		SCREAMING_SNAKE_CASE__						:       Union[str, Any]           =   new_dft
																		SCREAMING_SNAKE_CASE__						:       Tuple           =   next_ncol // 2
											return dft[0]



				def 	__magic_name__    (self    )					->		int:



											"""simple docstring"""




											SCREAMING_SNAKE_CASE__						:       Union[str, Any]           =   self.__dft("""A"""    )
											SCREAMING_SNAKE_CASE__						:       Dict           =   self.__dft("""B"""    )
											SCREAMING_SNAKE_CASE__						:       List[Any]           =   [[dft_a[i] * dft_b[i] for i in range(self.c_max_length    )]]
											del dft_a
											del dft_b

											# Corner Case
											if len(inverce_c[0]    ) <= 1:
																		return inverce_c[0]
											# Inverse DFT
											SCREAMING_SNAKE_CASE__						:       Optional[Any]           =   2
											while next_ncol <= self.c_max_length:
																		SCREAMING_SNAKE_CASE__						:       List[str]           =   [[] for i in range(SCREAMING_SNAKE_CASE__    )]
																		SCREAMING_SNAKE_CASE__						:       Tuple           =   self.root ** (next_ncol // 2)
																		SCREAMING_SNAKE_CASE__						:       Any           =   1
																		# First half of next step
																		for j in range(self.c_max_length // next_ncol    ):
																									for i in range(next_ncol // 2    ):
																																# Even positions
																																new_inverse_c[i].append(
																																    (
																																        inverce_c[i][j]
																																        + inverce_c[i][j + self.c_max_length // next_ncol]
																																    )
																																    / 2    )
																																# Odd positions
																																new_inverse_c[i + next_ncol // 2].append(
																																    (
																																        inverce_c[i][j]
																																        - inverce_c[i][j + self.c_max_length // next_ncol]
																																    )
																																    / (2 * current_root)    )
																									current_root *= root
																		# Update
																		SCREAMING_SNAKE_CASE__						:       Optional[Any]           =   new_inverse_c
																		next_ncol *= 2
											# Unpack
											SCREAMING_SNAKE_CASE__						:       Optional[Any]           =   [round(x[0].real   ,       8    ) + round(x[0].imag   ,       8    ) * 1j for x in inverce_c]

											# Remove leading 0's
											while inverce_c[-1] == 0:
																		inverce_c.pop()
											return inverce_c



				def __str__(self    )					->		List[str]:



											"""simple docstring"""




											SCREAMING_SNAKE_CASE__						:       List[str]           =   """A = """ + """ + """.join(
											    F'''{coef}*x^{i}''' for coef, i in enumerate(self.polyA[: self.len_A]    )    )
											SCREAMING_SNAKE_CASE__						:       Union[str, Any]           =   """B = """ + """ + """.join(
											    F'''{coef}*x^{i}''' for coef, i in enumerate(self.polyB[: self.len_B]    )    )
											SCREAMING_SNAKE_CASE__						:       int           =   """A*B = """ + """ + """.join(
											    F'''{coef}*x^{i}''' for coef, i in enumerate(self.product    )    )

											return F'''{a}\n{b}\n{c}'''


# Unit tests
if __name__ == "__main__":
					import doctest

					doctest.testmod()




