




"""simple docstring"""




def 					a__				(       snake_case__       )		->      list[list]:
			lowerCamelCase       =      current_set.copy()
			for row_index, row in enumerate(snake_case__       ):
						lowerCamelCase       =      row[0]
						for column_index, column in enumerate(snake_case__       ):
									if magnitude == 0:
												lowerCamelCase       =      column
												continue
									lowerCamelCase       =      column / magnitude
    # Subtract to cancel term
			lowerCamelCase       =      current_set[0]
			lowerCamelCase       =      [first_row]
			lowerCamelCase       =      current_set[1::]
			for row in current_set:
						lowerCamelCase       =      []
						# If first term is 0, it is already in form we want, so we preserve it
						if row[0] == 0:
									final_set.append(snake_case__       )
									continue
						for column_index in range(len(snake_case__       )       ):
									temp_row.append(first_row[column_index] - row[column_index]       )
						final_set.append(snake_case__       )
			# Create next recursion iteration set
			if len(final_set[0]       ) != 3:
						lowerCamelCase       =      final_set[0]
						lowerCamelCase       =      []
						lowerCamelCase       =      []
						for row in final_set[1::]:
									current_first_column.append(row[0]       )
									next_iteration.append(row[1::]       )
						lowerCamelCase       =      simplify(snake_case__       )
						for i in range(len(snake_case__       )       ):
									resultant[i].insert(0 ,				current_first_column[i]       )
						resultant.insert(0 ,				snake_case__       )
						lowerCamelCase       =      resultant
			return final_set



def 					a__				(       snake_case__       )		->      list:
			if len(snake_case__       ) == 0:
						raise IndexError("""solve_simultaneous() requires n lists of length n+1"""       )
			lowerCamelCase       =      len(snake_case__       ) + 1
			if any(len(snake_case__       ) != _length for item in equations       ):
						raise IndexError("""solve_simultaneous() requires n lists of length n+1"""       )
			for row in equations:
						if any(not isinstance(snake_case__ ,				(int, float)       ) for column in row       ):
									raise ValueError("""solve_simultaneous() requires lists of integers"""       )
			if len(snake_case__       ) == 1:
						return [equations[0][-1] / equations[0][0]]
			lowerCamelCase       =      equations.copy()
			if any(0 in row for row in data_set       ):
						lowerCamelCase       =      data_set.copy()
						lowerCamelCase       =      []
						for row_index, row in enumerate(snake_case__       ):
									if 0 not in row:
												lowerCamelCase       =      data_set.pop(snake_case__       )
												break
						if not full_row:
									raise ValueError("""solve_simultaneous() requires at least 1 full equation"""       )
						data_set.insert(0 ,				snake_case__       )
			lowerCamelCase       =      data_set.copy()
			lowerCamelCase       =      simplify(snake_case__       )
			lowerCamelCase       =      simplified[::-1]
			lowerCamelCase       =      []
			for row in simplified:
						lowerCamelCase       =      row[-1]
						if not solutions:
									if row[-2] == 0:
												solutions.append(0       )
												continue
									solutions.append(current_solution / row[-2]       )
									continue
						lowerCamelCase       =      row.copy()[: len(snake_case__       ) - 1 :]
						while temp_row[0] == 0:
									temp_row.pop(0       )
						if len(snake_case__       ) == 0:
									solutions.append(0       )
									continue
						lowerCamelCase       =      temp_row[1::]
						lowerCamelCase       =      temp_row[::-1]
						for column_index, column in enumerate(snake_case__       ):
									current_solution -= column * solutions[column_index]
						solutions.append(snake_case__       )
			lowerCamelCase       =      []
			for item in solutions:
						final.append(float(round(snake_case__ ,				5       )       )       )
			return final[::-1]


if __name__ == "__main__":
					import doctest

					doctest.testmod()
					lowerCAmelCase							:    str  							=							[
					    [2, 1, 1, 1, 1, 4],
					    [1, 2, 1, 1, 1, 5],
					    [1, 1, 2, 1, 1, 6],
					    [1, 1, 1, 2, 1, 7],
					    [1, 1, 1, 1, 2, 8],
					]
					print(solve_simultaneous(eq))
					print(solve_simultaneous([[4, 2]]))









"""simple docstring"""




# A Bipartite Graph is a graph whose vertices can be divided into two independent sets,
# U and V such that every edge (u, v) either connects a vertex from U to V or a vertex
# from V to U. In other words, for every edge (u, v), either u belongs to U and v to V,
# or u belongs to V and v to U. We can also say that there is no edge that connects
# vertices of same set.



def 					a__				(       snake_case__       )		->      Dict:
			lowerCamelCase       =      [False] * len(snake_case__       )
			lowerCamelCase       =      [-1] * len(snake_case__       )

			def dfs(snake_case__ ,				snake_case__       ):
						lowerCamelCase       =      True
						lowerCamelCase       =      c
						for u in graph[v]:
									if not visited[u]:
												dfs(snake_case__ ,				1 - c       )

			for i in range(len(snake_case__       )       ):
						if not visited[i]:
									dfs(snake_case__ ,				0       )

			for i in range(len(snake_case__       )       ):
						for j in graph[i]:
									if color[i] == color[j]:
												return False

			return True


# Adjacency list of graph
lowerCAmelCase							:    str  							=							{0: [1, 3], 1: [0, 2], 2: [1, 3], 3: [0, 2], 4: []}


print(check_bipartite_dfs(graph))



