



"""simple docstring"""
def 				lowerCAmelCase_(      lowercase_   :							list[list[int]]							)		->				str:

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

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

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

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

				_lowerCamelCase					     = len(SCREAMING_SNAKE_CASE_							)

				_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(SCREAMING_SNAKE_CASE_							)
								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(SCREAMING_SNAKE_CASE_							) > 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]]))



'''simple docstring'''






from ..utils import DummyObject, requires_backends
class 	SCREAMING_SNAKE_CASE__			(				metaclass=_UpperCAmelCase      ):
    _A			    =							["torch", "transformers", "invisible_watermark"]




    def __init__(					self				,     *lowercase__				,     **lowercase__			):






          """simple docstring"""






          requires_backends(self				,     ["torch", "transformers", "invisible_watermark"]			)




    @classmethod
    def   __lowerCamelCase      (					cls				,     *lowercase__				,     **lowercase__			):






          """simple docstring"""






          requires_backends(cls				,     ["torch", "transformers", "invisible_watermark"]			)







    @classmethod
    def   __lowerCamelCase      (					cls				,     *lowercase__				,     **lowercase__			):






          """simple docstring"""






          requires_backends(cls				,     ["torch", "transformers", "invisible_watermark"]			)






class 	SCREAMING_SNAKE_CASE__			(				metaclass=_UpperCAmelCase      ):
    _A			    =							["torch", "transformers", "invisible_watermark"]




    def __init__(					self				,     *lowercase__				,     **lowercase__			):






          """simple docstring"""






          requires_backends(self				,     ["torch", "transformers", "invisible_watermark"]			)




    @classmethod
    def   __lowerCamelCase      (					cls				,     *lowercase__				,     **lowercase__			):






          """simple docstring"""






          requires_backends(cls				,     ["torch", "transformers", "invisible_watermark"]			)







    @classmethod
    def   __lowerCamelCase      (					cls				,     *lowercase__				,     **lowercase__			):






          """simple docstring"""






          requires_backends(cls				,     ["torch", "transformers", "invisible_watermark"]			)

