
"""simple docstring"""





from __future__ import annotations





def 							A_     (__a				,   __a = None				,   __a = None				,   __a = False				,   ):

					'''simple docstring'''





					A_				 =						cipher_alphabet or [chr(__a				) for i in range(97				,   123				)]

					# If the argument is None or the user provided an empty dictionary
					if not frequencies_dict:
										# Frequencies of letters in the english language (how much they show up)
										A_				 =						{
										    "a": 0.08497,
										    "b": 0.01492,
										    "c": 0.02202,
										    "d": 0.04253,
										    "e": 0.11162,
										    "f": 0.02228,
										    "g": 0.02015,
										    "h": 0.06094,
										    "i": 0.07546,
										    "j": 0.00153,
										    "k": 0.01292,
										    "l": 0.04025,
										    "m": 0.02406,
										    "n": 0.06749,
										    "o": 0.07507,
										    "p": 0.01929,
										    "q": 0.00095,
										    "r": 0.07587,
										    "s": 0.06327,
										    "t": 0.09356,
										    "u": 0.02758,
										    "v": 0.00978,
										    "w": 0.02560,
										    "x": 0.00150,
										    "y": 0.01994,
										    "z": 0.00077,
										}
					else:
										# Custom frequencies dictionary
										A_				 =						frequencies_dict

					if not case_sensitive:
										A_				 =						ciphertext.lower()

					# Chi squared statistic values
					A_				 =						{}

					# cycle through all of the shifts
					for shift in range(len(__a				)				):
										A_				 =						""

										# decrypt the message with the shift
										for letter in ciphertext:
															try:
																				# Try to index the letter in the alphabet
																				A_				 =						(alphabet_letters.index(letter.lower()				) - shift) % len(
																				    __a				)
																				decrypted_with_shift += (
																				    alphabet_letters[new_key].upper()
																				    if case_sensitive and letter.isupper()
																				    else alphabet_letters[new_key]
																				)
															except ValueError:
																				# Append the character if it isn't in the alphabet
																				decrypted_with_shift += letter

										A_				 =						0.0

										# Loop through each letter in the decoded message with the shift
										for letter in decrypted_with_shift:
															if case_sensitive:
																				A_				 =						letter.lower()
																				if letter in frequencies:
																									# Get the amount of times the letter occurs in the message
																									A_				 =						decrypted_with_shift.lower().count(__a				)

																									# Get the excepcted amount of times the letter should appear based
																									# on letter frequencies
																									A_				 =						frequencies[letter] * occurrences

																									# Complete the chi squared statistic formula
																									A_				 =						((occurrences - expected) ** 2) / expected

																									# Add the margin of error to the total chi squared statistic
																									chi_squared_statistic += chi_letter_value
															else:
																				if letter.lower() in frequencies:
																									# Get the amount of times the letter occurs in the message
																									A_				 =						decrypted_with_shift.count(__a				)

																									# Get the excepcted amount of times the letter should appear based
																									# on letter frequencies
																									A_				 =						frequencies[letter] * occurrences

																									# Complete the chi squared statistic formula
																									A_				 =						((occurrences - expected) ** 2) / expected

																									# Add the margin of error to the total chi squared statistic
																									chi_squared_statistic += chi_letter_value

        # Add the data to the chi_squared_statistic_values dictionary
										A_				 =						(
										    chi_squared_statistic,
										    decrypted_with_shift,
										)

					# Get the most likely cipher by finding the cipher with the smallest chi squared
					# statistic
					def chi_squared_statistic_values_sorting_key(__a				) -> tuple[float, str]:
										return chi_squared_statistic_values[key]

					A_				 =						min(
					    __a				,   key=__a				,   )

					# Get all the data from the most likely cipher (key, decoded message)
					(
					    (
					    A_    
					) ,    (
					    A_    
					) ,        
					)				 =						chi_squared_statistic_values[most_likely_cipher]

					# Return the data on the most likely shift
					return (
					    most_likely_cipher,
					    most_likely_cipher_chi_squared_value,
					    decoded_most_likely_cipher,
					)






"""simple docstring"""





def 							A_     (__a				):

					'''simple docstring'''





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

					'''simple docstring'''





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


if __name__ == "__main__":
		import doctest

		doctest.testmod()
		UpperCamelCase_					:						List[Any]				= [
		    [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]]))




