
def        lowerCamelCase__    (    __lowerCAmelCase  :  list[list[int]]    ):




			"""simple docstring"""


			if len(_lowerCamelCase    ) == 0 or not all(len(_lowerCamelCase    ) == 2 for pair in coordinates    ):
						raise ValueError("The program cannot work out a fitting polynomial."    )

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

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

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

			lowerCAmelCase_      =     len(_lowerCamelCase    )

			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(_lowerCamelCase    )
						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(_lowerCamelCase    ) > 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, 11]]))
			print(points_to_polynomial([[1, -3], [2, -6], [3, -11]]))
			print(points_to_polynomial([[1, 5], [2, 2], [3, 9]]))





from typing import TYPE_CHECKING

from ...utils import OptionalDependencyNotAvailable, _LazyModule, is_torch_available


_A       =					{"configuration_nat": ["NAT_PRETRAINED_CONFIG_ARCHIVE_MAP", "NatConfig"]}


try:
			if not is_torch_available():
						raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
			pass
else:
			_A       =					[
			    "NAT_PRETRAINED_MODEL_ARCHIVE_LIST",
			    "NatForImageClassification",
			    "NatModel",
			    "NatPreTrainedModel",
			    "NatBackbone",
			]

if TYPE_CHECKING:
			from .configuration_nat import NAT_PRETRAINED_CONFIG_ARCHIVE_MAP, NatConfig

			try:
						if not is_torch_available():
									raise OptionalDependencyNotAvailable()
			except OptionalDependencyNotAvailable:
						pass
			else:
						from .modeling_nat import (
						    NAT_PRETRAINED_MODEL_ARCHIVE_LIST,
						    NatBackbone,
						    NatForImageClassification,
						    NatModel,
						    NatPreTrainedModel,
						)

else:
			import sys

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



