





def 	_a						(   UpperCAmelCase      ,  UpperCAmelCase = False					)    ->    Any:







   """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 > 3317044064679887385961981 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
   lowerCamelCase__       :						int        =							[
       2047,
       1373653,
       25326001,
       3215031751,
       2152302898747,
       3474749660383,
       341550071728321,
       1,
       3825123056546413051,
       1,
       1,
       318665857834031151167461,
       3317044064679887385961981,
   ]

   lowerCamelCase__       :						Optional[Any]        =							[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41]
   for idx, _p in enumerate(UpperCAmelCase      ,  1					):
      if n < _p:
         # then we have our last prime to check
         lowerCamelCase__       :						Union[str, Any]        =							primes[:idx]
         break
   lowerCamelCase__       :						List[Any]        =							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:
      lowerCamelCase__       :						Union[str, Any]        =							False
      for r in range(UpperCAmelCase					):
         lowerCamelCase__       :						Dict        =							pow(UpperCAmelCase      ,  d * 2**r      ,  UpperCAmelCase					)
         # see article for analysis explanation for m
         if (r == 0 and m == 1) or ((m + 1) % n == 0):
            lowerCamelCase__       :						Optional[int]        =							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 	_a						(   )    ->    Union[str, Any]:







   """simple docstring"""
   assert not miller_rabin(561					)
   assert miller_rabin(563					)
   # 2047

   assert not miller_rabin(838201					)
   assert miller_rabin(838207					)
   # 1_373_653

   assert not miller_rabin(17316001					)
   assert miller_rabin(17316017					)
   # 25_326_001

   assert not miller_rabin(3078386641					)
   assert miller_rabin(3078386653					)
   # 3_215_031_751

   assert not miller_rabin(1713045574801					)
   assert miller_rabin(1713045574819					)
   # 2_152_302_898_747

   assert not miller_rabin(2779799728307					)
   assert miller_rabin(2779799728327					)
   # 3_474_749_660_383

   assert not miller_rabin(113850023909441					)
   assert miller_rabin(113850023909527					)
   # 341_550_071_728_321

   assert not miller_rabin(1275041018848804351					)
   assert miller_rabin(1275041018848804391					)
   # 3_825_123_056_546_413_051

   assert not miller_rabin(79666464458507787791867					)
   assert miller_rabin(79666464458507787791951					)
   # 318_665_857_834_031_151_167_461

   assert not miller_rabin(552840677446647897660333					)
   assert miller_rabin(552840677446647897660359					)
   # 3_317_044_064_679_887_385_961_981
   # upper limit for probabilistic test


if __name__ == "__main__":
       test_miller_rabin()













import numpy as np
import skfuzzy as fuzz

if __name__ == "__main__":
       # Create universe of discourse in Python using linspace ()
       _A :   Optional[int]  			=  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).
       _A :   Any  			=  [0, 25, 50]
       _A :   Dict  			=  [25, 50, 75]
       _A :   Any  			=  fuzz.membership.trimf(X, abca)
       _A :   List[Any]  			=  fuzz.membership.trimf(X, abca)

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



