






from transformers import BertTokenizerFast

from .custom_tokenization import CustomTokenizer
class    _UpperCAmelCase      ( A__			):

				"""simple docstring"""




				lowercase__       							= CustomTokenizer
				pass












from math import sqrt






def 						a    (		lowerCamelCase_       ):






				'''simple docstring'''
				assert isinstance(lowerCamelCase_	,     lowerCamelCase_       ) and (
				    number >= 0
				), "'number' must been an int and positive"

				lowercase__     				=    True

				# 0 and 1 are none primes.
				if number <= 1:
								lowercase__     				=    False

				for divisor in range(2	,     int(round(sqrt(lowerCamelCase_       )       )       ) + 1       ):
								# if 'number' divisible by 'divisor' then sets 'status'
								# of false and break up the loop.
								if number % divisor == 0:
												lowercase__     				=    False
												break

    # precondition
				assert isinstance(lowerCamelCase_	,     lowerCamelCase_       ), "'status' must been from type bool"

				return status






def 						a    (		lowerCamelCase_       ):






				'''simple docstring'''
				assert isinstance(lowerCamelCase_	,     lowerCamelCase_       ) and (n > 2), "'N' must been an int and > 2"

				# beginList: contains all natural numbers from 2 up to N
				lowercase__     				=    list(range(2	,     n + 1       )       )

				lowercase__     				=    []  # this list will be returns.

				# actual sieve of erathostenes
				for i in range(len(lowerCamelCase_       )       ):
								for j in range(i + 1	,     len(lowerCamelCase_       )       ):
												if (begin_list[i] != 0) and (begin_list[j] % begin_list[i] == 0):
																lowercase__     				=    0

    # filters actual prime numbers.
				lowercase__     				=    [x for x in begin_list if x != 0]

				# precondition
				assert isinstance(lowerCamelCase_	,     lowerCamelCase_       ), "'ans' must been from type list"

				return ans






def 						a    (		lowerCamelCase_       ):






				'''simple docstring'''
				assert isinstance(lowerCamelCase_	,     lowerCamelCase_       ) and (n > 2), "'N' must been an int and > 2"

				lowercase__     				=    []

				# iterates over all numbers between 2 up to N+1
				# if a number is prime then appends to list 'ans'
				for number in range(2	,     n + 1       ):
								if is_prime(lowerCamelCase_       ):
												ans.append(lowerCamelCase_       )

    # precondition
				assert isinstance(lowerCamelCase_	,     lowerCamelCase_       ), "'ans' must been from type list"

				return ans






def 						a    (		lowerCamelCase_       ):






				'''simple docstring'''
				assert isinstance(lowerCamelCase_	,     lowerCamelCase_       ) and number >= 0, "'number' must been an int and >= 0"

				lowercase__     				=    []  # this list will be returns of the function.

				# potential prime number factors.

				lowercase__     				=    2

				lowercase__     				=    number

				if number == 0 or number == 1:
								ans.append(lowerCamelCase_       )

				# if 'number' not prime then builds the prime factorization of 'number'
				elif not is_prime(lowerCamelCase_       ):
								while quotient != 1:
												if is_prime(lowerCamelCase_       ) and (quotient % factor == 0):
																ans.append(lowerCamelCase_       )
																quotient /= factor
												else:
																factor += 1

				else:
								ans.append(lowerCamelCase_       )

				# precondition
				assert isinstance(lowerCamelCase_	,     lowerCamelCase_       ), "'ans' must been from type list"

				return ans






def 						a    (		lowerCamelCase_       ):






				'''simple docstring'''
				assert isinstance(lowerCamelCase_	,     lowerCamelCase_       ) and (
				    number >= 0
				), "'number' bust been an int and >= 0"

				lowercase__     				=    0

				# prime factorization of 'number'
				lowercase__     				=    prime_factorization(lowerCamelCase_       )

				lowercase__     				=    max(lowerCamelCase_       )

				# precondition
				assert isinstance(lowerCamelCase_	,     lowerCamelCase_       ), "'ans' must been from type int"

				return ans






def 						a    (		lowerCamelCase_       ):






				'''simple docstring'''
				assert isinstance(lowerCamelCase_	,     lowerCamelCase_       ) and (
				    number >= 0
				), "'number' bust been an int and >= 0"

				lowercase__     				=    0

				# prime factorization of 'number'
				lowercase__     				=    prime_factorization(lowerCamelCase_       )

				lowercase__     				=    min(lowerCamelCase_       )

				# precondition
				assert isinstance(lowerCamelCase_	,     lowerCamelCase_       ), "'ans' must been from type int"

				return ans






def 						a    (		lowerCamelCase_       ):






				'''simple docstring'''
				assert isinstance(lowerCamelCase_	,     lowerCamelCase_       ), "'number' must been an int"
				assert isinstance(number % 2 == 0	,     lowerCamelCase_       ), "compare bust been from type bool"

				return number % 2 == 0






def 						a    (		lowerCamelCase_       ):






				'''simple docstring'''
				assert isinstance(lowerCamelCase_	,     lowerCamelCase_       ), "'number' must been an int"
				assert isinstance(number % 2 != 0	,     lowerCamelCase_       ), "compare bust been from type bool"

				return number % 2 != 0






def 						a    (		lowerCamelCase_       ):






				'''simple docstring'''
				assert (
				    isinstance(lowerCamelCase_	,     lowerCamelCase_       ) and (number > 2) and is_even(lowerCamelCase_       )
				), "'number' must been an int, even and > 2"

				lowercase__     				=    []  # this list will returned

				# creates a list of prime numbers between 2 up to 'number'
				lowercase__     				=    get_prime_numbers(lowerCamelCase_       )
				lowercase__     				=    len(lowerCamelCase_       )

				# run variable for while-loops.
				lowercase__     				=    0
				lowercase__     				=    None

				# exit variable. for break up the loops
				lowercase__     				=    True

				while i < len_pn and loop:
								lowercase__     				=    i + 1

								while j < len_pn and loop:
												if prime_numbers[i] + prime_numbers[j] == number:
																lowercase__     				=    False
																ans.append(prime_numbers[i]       )
																ans.append(prime_numbers[j]       )

												j += 1

								i += 1

				# precondition
				assert (
				    isinstance(lowerCamelCase_	,     lowerCamelCase_       )
				    and (len(lowerCamelCase_       ) == 2)
				    and (ans[0] + ans[1] == number)
				    and is_prime(ans[0]       )
				    and is_prime(ans[1]       )
				), "'ans' must contains two primes. And sum of elements must been eq 'number'"

				return ans






def 						a    (		lowerCamelCase_	,     lowerCamelCase_       ):






				'''simple docstring'''
				assert (
				    isinstance(lowerCamelCase_	,     lowerCamelCase_       )
				    and isinstance(lowerCamelCase_	,     lowerCamelCase_       )
				    and (numbera >= 0)
				    and (numbera >= 0)
				), "'number1' and 'number2' must been positive integer."

				lowercase__     				=    0

				while numbera != 0:
								lowercase__     				=    numbera % numbera
								lowercase__     				=    numbera
								lowercase__     				=    rest

				# precondition
				assert isinstance(lowerCamelCase_	,     lowerCamelCase_       ) and (
				    numbera >= 0
				), "'number' must been from type int and positive"

				return numbera






def 						a    (		lowerCamelCase_	,     lowerCamelCase_       ):






				'''simple docstring'''
				assert (
				    isinstance(lowerCamelCase_	,     lowerCamelCase_       )
				    and isinstance(lowerCamelCase_	,     lowerCamelCase_       )
				    and (numbera >= 1)
				    and (numbera >= 1)
				), "'number1' and 'number2' must been positive integer."

				lowercase__     				=    1  # actual answer that will be return.

				# for kgV (x,1)
				if numbera > 1 and numbera > 1:
								# builds the prime factorization of 'number1' and 'number2'
								lowercase__     				=    prime_factorization(lowerCamelCase_       )
								lowercase__     				=    prime_factorization(lowerCamelCase_       )

				elif numbera == 1 or numbera == 1:
								lowercase__     				=    []
								lowercase__     				=    []
								lowercase__     				=    max(lowerCamelCase_	,     lowerCamelCase_       )

				lowercase__     				=    0
				lowercase__     				=    0

				lowercase__     				=    []  # captured numbers int both 'primeFac1' and 'primeFac2'

				# iterates through primeFac1
				for n in prime_fac_a:
								if n not in done:
												if n in prime_fac_a:
																lowercase__     				=    prime_fac_a.count(lowerCamelCase_       )
																lowercase__     				=    prime_fac_a.count(lowerCamelCase_       )

																for _ in range(max(lowerCamelCase_	,     lowerCamelCase_       )       ):
																				ans *= n

												else:
																lowercase__     				=    prime_fac_a.count(lowerCamelCase_       )

																for _ in range(lowerCamelCase_       ):
																				ans *= n

												done.append(lowerCamelCase_       )

    # iterates through primeFac2
				for n in prime_fac_a:
								if n not in done:
												lowercase__     				=    prime_fac_a.count(lowerCamelCase_       )

												for _ in range(lowerCamelCase_       ):
																ans *= n

												done.append(lowerCamelCase_       )

    # precondition
				assert isinstance(lowerCamelCase_	,     lowerCamelCase_       ) and (
				    ans >= 0
				), "'ans' must been from type int and positive"

				return ans






def 						a    (		lowerCamelCase_       ):






				'''simple docstring'''
				assert isinstance(lowerCamelCase_	,     lowerCamelCase_       ) and (n >= 0), "'number' must been a positive int"

				lowercase__     				=    0
				lowercase__     				=    2  # this variable holds the answer

				while index < n:
								index += 1

								ans += 1  # counts to the next number

								# if ans not prime then
								# runs to the next prime number.
								while not is_prime(lowerCamelCase_       ):
												ans += 1

    # precondition
				assert isinstance(lowerCamelCase_	,     lowerCamelCase_       ) and is_prime(
				    lowerCamelCase_       ), "'ans' must been a prime number and from type int"

				return ans






def 						a    (		lowerCamelCase_	,     lowerCamelCase_       ):






				'''simple docstring'''
				assert (
				    is_prime(lowerCamelCase_       ) and is_prime(lowerCamelCase_       ) and (p_number_a < p_number_a)
				), "The arguments must been prime numbers and 'pNumber1' < 'pNumber2'"

				lowercase__     				=    p_number_a + 1  # jump to the next number

				lowercase__     				=    []  # this list will be returns.

				# if number is not prime then
				# fetch the next prime number.
				while not is_prime(lowerCamelCase_       ):
								number += 1

				while number < p_number_a:
								ans.append(lowerCamelCase_       )

								number += 1

								# fetch the next prime number.
								while not is_prime(lowerCamelCase_       ):
												number += 1

    # precondition
				assert (
				    isinstance(lowerCamelCase_	,     lowerCamelCase_       )
				    and ans[0] != p_number_a
				    and ans[len(lowerCamelCase_       ) - 1] != p_number_a
				), "'ans' must been a list without the arguments"

				# 'ans' contains not 'pNumber1' and 'pNumber2' !
				return ans






def 						a    (		lowerCamelCase_       ):






				'''simple docstring'''
				assert isinstance(lowerCamelCase_	,     lowerCamelCase_       ) and (n >= 1), "'n' must been int and >= 1"

				lowercase__     				=    []  # will be returned.

				for divisor in range(1	,     n + 1       ):
								if n % divisor == 0:
												ans.append(lowerCamelCase_       )

    # precondition
				assert ans[0] == 1 and ans[len(lowerCamelCase_       ) - 1] == n, "Error in function getDivisiors(...)"

				return ans






def 						a    (		lowerCamelCase_       ):






				'''simple docstring'''
				assert isinstance(lowerCamelCase_	,     lowerCamelCase_       ) and (
				    number > 1
				), "'number' must been an int and >= 1"

				lowercase__     				=    get_divisors(lowerCamelCase_       )

				# precondition
				assert (
				    isinstance(lowerCamelCase_	,     lowerCamelCase_       )
				    and (divisors[0] == 1)
				    and (divisors[len(lowerCamelCase_       ) - 1] == number)
				), "Error in help-function getDivisiors(...)"

				# summed all divisors up to 'number' (exclusive), hence [:-1]
				return sum(divisors[:-1]       ) == number






def 						a    (		lowerCamelCase_	,     lowerCamelCase_       ):






				'''simple docstring'''
				assert (
				    isinstance(lowerCamelCase_	,     lowerCamelCase_       )
				    and isinstance(lowerCamelCase_	,     lowerCamelCase_       )
				    and (denominator != 0)
				), "The arguments must been from type int and 'denominator' != 0"

				# build the greatest common divisor of numerator and denominator.
				lowercase__     				=    gcd(abs(lowerCamelCase_       )	,     abs(lowerCamelCase_       )       )

				# precondition
				assert (
				    isinstance(lowerCamelCase_	,     lowerCamelCase_       )
				    and (numerator % gcd_of_fraction == 0)
				    and (denominator % gcd_of_fraction == 0)
				), "Error in function gcd(...,...)"

				return (numerator // gcd_of_fraction, denominator // gcd_of_fraction)






def 						a    (		lowerCamelCase_       ):






				'''simple docstring'''
				assert isinstance(lowerCamelCase_	,     lowerCamelCase_       ) and (n >= 0), "'n' must been a int and >= 0"

				lowercase__     				=    1  # this will be return.

				for factor in range(1	,     n + 1       ):
								ans *= factor

				return ans






def 						a    (		lowerCamelCase_       ):






				'''simple docstring'''
				assert isinstance(lowerCamelCase_	,     lowerCamelCase_       ) and (n >= 0), "'n' must been an int and >= 0"

				lowercase__     				=    0
				lowercase__     				=    1
				lowercase__     				=    1  # this will be return

				for _ in range(n - 1       ):
								lowercase__     				=    ans
								ans += fiba
								lowercase__     				=    tmp

				return ans




