




import numpy as np
import skfuzzy as fuzz

if __name__ == "__main__":
				# Create universe of discourse in Python using linspace ()
				__lowercase       :List[str]       =  np.linspace(start=0, stop=75, num=75, endpoint=True, retstep=False)

				# Create two fuzzy sets by defining any membership function
				# (trapmf(), gbellmf(), gaussmf(), etc).
				__lowercase       :Union[str, Any]       =  [0, 25, 50]
				__lowercase       :Optional[int]       =  [25, 50, 75]
				__lowercase       :Union[str, Any]       =  fuzz.membership.trimf(X, abca)
				__lowercase       :int       =  fuzz.membership.trimf(X, abca)

				# Compute the different operations using inbuilt functions.
				__lowercase       :int       =  np.ones(75)
				__lowercase       :List[str]       =  np.zeros((75,))
				# 1. Union = max(µA(x), µB(x))
				__lowercase       :str       =  fuzz.fuzzy_or(X, young, X, middle_aged)[1]
				# 2. Intersection = min(µA(x), µB(x))
				__lowercase       :str       =  fuzz.fuzzy_and(X, young, X, middle_aged)[1]
				# 3. Complement (A) = (1- min(µA(x))
				__lowercase       :List[Any]       =  fuzz.fuzzy_not(young)
				# 4. Difference (A/B) = min(µA(x),(1- µB(x)))
				__lowercase       :str       =  fuzz.fuzzy_and(X, young, X, fuzz.fuzzy_not(middle_aged)[1])[1]
				# 5. Algebraic Sum = [µA(x) + µB(x) – (µA(x) * µB(x))]
				__lowercase       :int       =  young + middle_aged - (young * middle_aged)
				# 6. Algebraic Product = (µA(x) * µB(x))
				__lowercase       :str       =  young * middle_aged
				# 7. Bounded Sum = min[1,(µA(x), µB(x))]
				__lowercase       :List[str]       =  fuzz.fuzzy_and(X, one, X, young + middle_aged)[1]
				# 8. Bounded difference = min[0,(µA(x), µB(x))]
				__lowercase       :Union[str, Any]       =  fuzz.fuzzy_or(X, zero, X, young - middle_aged)[1]

				# max-min composition
				# max-product composition

				# Plot each set A, set B and each operation result using plot() and subplot().
				from matplotlib import pyplot as plt

				plt.figure()

				plt.subplot(4, 3, 1)
				plt.plot(X, young)
				plt.title("Young")
				plt.grid(True)

				plt.subplot(4, 3, 2)
				plt.plot(X, middle_aged)
				plt.title("Middle aged")
				plt.grid(True)

				plt.subplot(4, 3, 3)
				plt.plot(X, union)
				plt.title("union")
				plt.grid(True)

				plt.subplot(4, 3, 4)
				plt.plot(X, intersection)
				plt.title("intersection")
				plt.grid(True)

				plt.subplot(4, 3, 5)
				plt.plot(X, complement_a)
				plt.title("complement_a")
				plt.grid(True)

				plt.subplot(4, 3, 6)
				plt.plot(X, difference)
				plt.title("difference a/b")
				plt.grid(True)

				plt.subplot(4, 3, 7)
				plt.plot(X, alg_sum)
				plt.title("alg_sum")
				plt.grid(True)

				plt.subplot(4, 3, 8)
				plt.plot(X, alg_product)
				plt.title("alg_product")
				plt.grid(True)

				plt.subplot(4, 3, 9)
				plt.plot(X, bdd_sum)
				plt.title("bdd_sum")
				plt.grid(True)

				plt.subplot(4, 3, 10)
				plt.plot(X, bdd_difference)
				plt.title("bdd_difference")
				plt.grid(True)

				plt.subplots_adjust(hspace=0.5)
				plt.show()






def       UpperCAmelCase							(     _lowerCamelCase      :  int			,   _lowerCamelCase      :  bool = False     ):





						'''simple docstring'''


						if n == 2:
												return True
						if not n % 2 or n < 2:
												return False
						if n > 5 and n % 10 not in (1, 3, 7, 9):  # can quickly check last digit
												return False
						if n > 3_317_044_064_679_887_385_961_981 and not allow_probable:
												raise ValueError(
												    "Warning: upper bound of deterministic test is exceeded. "
												    "Pass allow_probable=True to allow probabilistic test. "
												    "A return value of True indicates a probable prime."     )
						# array bounds provided by analysis
						SCREAMING_SNAKE_CASE__     :				List[str]   =					[
						    2_047,
						    1_373_653,
						    25_326_001,
						    3_215_031_751,
						    2_152_302_898_747,
						    3_474_749_660_383,
						    341_550_071_728_321,
						    1,
						    3_825_123_056_546_413_051,
						    1,
						    1,
						    318_665_857_834_031_151_167_461,
						    3_317_044_064_679_887_385_961_981,
						]

						SCREAMING_SNAKE_CASE__     :				Union[str, Any]   =					[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41]
						for idx, _p in enumerate(_lowerCamelCase			,   1     ):
												if n < _p:
																		# then we have our last prime to check
																		SCREAMING_SNAKE_CASE__     :				Dict   =					primes[:idx]
																		break
						SCREAMING_SNAKE_CASE__,							SCREAMING_SNAKE_CASE__     :				Dict   =					n - 1, 0
						# break up n -1 into a power of 2 (s) and
						# remaining odd component
						# essentially, solve for d * 2 ** s == n - 1
						while d % 2 == 0:
												d //= 2
												s += 1
						for prime in plist:
												SCREAMING_SNAKE_CASE__     :				str   =					False
												for r in range(_lowerCamelCase     ):
																		SCREAMING_SNAKE_CASE__     :				Optional[Any]   =					pow(_lowerCamelCase			,   d * 2**r			,   _lowerCamelCase     )
																		# see article for analysis explanation for m
																		if (r == 0 and m == 1) or ((m + 1) % n == 0):
																								SCREAMING_SNAKE_CASE__     :				str   =					True
																								# this loop will not determine compositeness
																								break
												if pr:
																		continue
												# if pr is False, then the above loop never evaluated to true,
												# and the n MUST be composite
												return False
						return True







def       UpperCAmelCase							(     ):





						'''simple docstring'''


						assert not miller_rabin(561     )
						assert miller_rabin(563     )
						# 2047

						assert not miller_rabin(838_201     )
						assert miller_rabin(838_207     )
						# 1_373_653

						assert not miller_rabin(17_316_001     )
						assert miller_rabin(17_316_017     )
						# 25_326_001

						assert not miller_rabin(3_078_386_641     )
						assert miller_rabin(3_078_386_653     )
						# 3_215_031_751

						assert not miller_rabin(1_713_045_574_801     )
						assert miller_rabin(1_713_045_574_819     )
						# 2_152_302_898_747

						assert not miller_rabin(2_779_799_728_307     )
						assert miller_rabin(2_779_799_728_327     )
						# 3_474_749_660_383

						assert not miller_rabin(113_850_023_909_441     )
						assert miller_rabin(113_850_023_909_527     )
						# 341_550_071_728_321

						assert not miller_rabin(1_275_041_018_848_804_351     )
						assert miller_rabin(1_275_041_018_848_804_391     )
						# 3_825_123_056_546_413_051

						assert not miller_rabin(79_666_464_458_507_787_791_867     )
						assert miller_rabin(79_666_464_458_507_787_791_951     )
						# 318_665_857_834_031_151_167_461

						assert not miller_rabin(552_840_677_446_647_897_660_333     )
						assert miller_rabin(552_840_677_446_647_897_660_359     )
						# 3_317_044_064_679_887_385_961_981
						# upper limit for probabilistic test


if __name__ == "__main__":
				test_miller_rabin()