






def   lowerCamelCase      (		a_			)			->    str:
							if len(a_			) == 0 or not all(len(a_			) == 2 for pair in coordinates			):
														raise ValueError('The program cannot work out a fitting polynomial.'			)

							if len({tuple(a_			) for pair in coordinates}			) != len(a_			):
														raise ValueError('The program cannot work out a fitting polynomial.'			)

							lowerCAmelCase_       		=  {x for x, _ in coordinates}
							if len(a_			) == 1:
														return F'''x={coordinates[0][0]}'''

							if len(a_			) != len(a_			):
														raise ValueError('The program cannot work out a fitting polynomial.'			)

							lowerCAmelCase_       		=  len(a_			)

							lowerCAmelCase_       		=  0
							lowerCAmelCase_       		=  []
							# put the x and x to the power values in a matrix
							while count_of_line < x:
														lowerCAmelCase_       		=  0
														lowerCAmelCase_       		=  coordinates[count_of_line][0]
														lowerCAmelCase_       		=  []
														while count_in_line < x:
																					count_line.append(a ** (x - (count_in_line + 1))			)
																					count_in_line += 1
														matrix.append(a_			)
														count_of_line += 1

							lowerCAmelCase_       		=  0
							# put the y values into a vector
							lowerCAmelCase_       		=  []
							while count_of_line < x:
														vector.append(coordinates[count_of_line][1]			)
														count_of_line += 1

							lowerCAmelCase_       		=  0

							while count < x:
														lowerCAmelCase_       		=  0
														while zahlen < x:
																					if count == zahlen:
																												zahlen += 1
																					if zahlen == x:
																												break
																					lowerCAmelCase_       		=  matrix[zahlen][count] / matrix[count][count]
																					for counting_columns, item in enumerate(matrix[count]			):
																												# manipulating all the values in the matrix
																												matrix[zahlen][counting_columns] -= item * bruch
																					# manipulating the values in the vector
																					vector[zahlen] -= vector[count] * bruch
																					zahlen += 1
														count += 1

							lowerCAmelCase_       		=  0
							# make solutions
							lowerCAmelCase_       		=  []
							while count < x:
														solution.append(str(vector[count] / matrix[count][count]			)			)
														count += 1

							lowerCAmelCase_       		=  0
							lowerCAmelCase_       		=  'f(x)='

							while count < x:
														lowerCAmelCase_       		=  solution[count].split('E'			)
														if len(a_			) > 1:
																					lowerCAmelCase_       		=  F'''{remove_e[0]}*10^{remove_e[1]}'''
														solved += F'''x^{x - (count + 1)}*{solution[count]}'''
														if count + 1 != x:
																					solved += "+"
														count += 1

							return solved


if __name__ == "__main__":
		print(points_to_polynomial([]))
		print(points_to_polynomial([[]]))
		print(points_to_polynomial([[1, 0], [2, 0], [3, 0]]))
		print(points_to_polynomial([[1, 1], [2, 1], [3, 1]]))
		print(points_to_polynomial([[1, 3], [2, 3], [3, 3]]))
		print(points_to_polynomial([[1, 1], [2, 2], [3, 3]]))
		print(points_to_polynomial([[1, 1], [2, 4], [3, 9]]))
		print(points_to_polynomial([[1, 3], [2, 6], [3, 1_1]]))
		print(points_to_polynomial([[1, -3], [2, -6], [3, -1_1]]))
		print(points_to_polynomial([[1, 5], [2, 2], [3, 9]]))











from typing import TYPE_CHECKING

from ...utils import OptionalDependencyNotAvailable, _LazyModule, is_torch_available


lowerCamelCase_		  =						{
    """configuration_prophetnet""": ["""PROPHETNET_PRETRAINED_CONFIG_ARCHIVE_MAP""", """ProphetNetConfig"""],
    """tokenization_prophetnet""": ["""ProphetNetTokenizer"""],
}

try:
		if not is_torch_available():
				raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
		pass
else:
		lowerCamelCase_		  =						[
		    """PROPHETNET_PRETRAINED_MODEL_ARCHIVE_LIST""",
		    """ProphetNetDecoder""",
		    """ProphetNetEncoder""",
		    """ProphetNetForCausalLM""",
		    """ProphetNetForConditionalGeneration""",
		    """ProphetNetModel""",
		    """ProphetNetPreTrainedModel""",
		]


if TYPE_CHECKING:
		from .configuration_prophetnet import PROPHETNET_PRETRAINED_CONFIG_ARCHIVE_MAP, ProphetNetConfig
		from .tokenization_prophetnet import ProphetNetTokenizer

		try:
				if not is_torch_available():
						raise OptionalDependencyNotAvailable()
		except OptionalDependencyNotAvailable:
				pass
		else:
				from .modeling_prophetnet import (
				    PROPHETNET_PRETRAINED_MODEL_ARCHIVE_LIST,
				    ProphetNetDecoder,
				    ProphetNetEncoder,
				    ProphetNetForCausalLM,
				    ProphetNetForConditionalGeneration,
				    ProphetNetModel,
				    ProphetNetPreTrainedModel,
				)

else:
		import sys

		lowerCamelCase_		  =						_LazyModule(__name__, globals()["""__file__"""], _import_structure, module_spec=__spec__)



