


"""simple docstring"""




from math import sqrt

def 			lowercase				(						lowerCAmelCase__					:	int     )					-> Any:
		assert isinstance(snake_case__       ,   snake_case__     ) and (
		    number >= 0
		), "'number' must been an int and positive"

		__a	 =     True

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

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

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

		return status

def 			lowercase				(						lowerCAmelCase__					:	List[str]     )					-> List[str]:
		assert isinstance(snake_case__       ,   snake_case__     ) and (n > 2), "'N' must been an int and > 2"

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

		__a	 =     []  # this list will be returns.

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

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

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

		return ans

def 			lowercase				(						lowerCAmelCase__					:	int     )					-> Any:
		assert isinstance(snake_case__       ,   snake_case__     ) and (n > 2), "'N' must been an int and > 2"

		__a	 =     []

		# 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(snake_case__     ):
						ans.append(snake_case__     )

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

		return ans

def 			lowercase				(						lowerCAmelCase__					:	str     )					-> Any:
		assert isinstance(snake_case__       ,   snake_case__     ) and number >= 0, "'number' must been an int and >= 0"

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

		# potential prime number factors.

		__a	 =     2

		__a	 =     number

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

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

		else:
				ans.append(snake_case__     )

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

		return ans

def 			lowercase				(						lowerCAmelCase__					:	str     )					-> str:
		assert isinstance(snake_case__       ,   snake_case__     ) and (
		    number >= 0
		), "'number' bust been an int and >= 0"

		__a	 =     0

		# prime factorization of 'number'
		__a	 =     prime_factorization(snake_case__     )

		__a	 =     max(snake_case__     )

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

		return ans

def 			lowercase				(						lowerCAmelCase__					:	Tuple     )					-> List[str]:
		assert isinstance(snake_case__       ,   snake_case__     ) and (
		    number >= 0
		), "'number' bust been an int and >= 0"

		__a	 =     0

		# prime factorization of 'number'
		__a	 =     prime_factorization(snake_case__     )

		__a	 =     min(snake_case__     )

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

		return ans

def 			lowercase				(						lowerCAmelCase__					:	Any     )					-> List[str]:
		assert isinstance(snake_case__       ,   snake_case__     ), "'number' must been an int"
		assert isinstance(number % 2 == 0       ,   snake_case__     ), "compare bust been from type bool"

		return number % 2 == 0

def 			lowercase				(						lowerCAmelCase__					:	List[str]     )					-> List[str]:
		assert isinstance(snake_case__       ,   snake_case__     ), "'number' must been an int"
		assert isinstance(number % 2 != 0       ,   snake_case__     ), "compare bust been from type bool"

		return number % 2 != 0

def 			lowercase				(						lowerCAmelCase__					:	int     )					-> Tuple:
		assert (
		    isinstance(snake_case__       ,   snake_case__     ) and (number > 2) and is_even(snake_case__     )
		), "'number' must been an int, even and > 2"

		__a	 =     []  # this list will returned

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

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

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

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

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

						j += 1

				i += 1

		# precondition
		assert (
		    isinstance(snake_case__       ,   snake_case__     )
		    and (len(snake_case__     ) == 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 			lowercase				(						lowerCAmelCase__					:	str       ,   lowerCAmelCase__					:	Union[str, Any]     )					-> List[Any]:
		assert (
		    isinstance(snake_case__       ,   snake_case__     )
		    and isinstance(snake_case__       ,   snake_case__     )
		    and (numbera >= 0)
		    and (numbera >= 0)
		), "'number1' and 'number2' must been positive integer."

		__a	 =     0

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

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

		return numbera

def 			lowercase				(						lowerCAmelCase__					:	Tuple       ,   lowerCAmelCase__					:	List[str]     )					-> Union[str, Any]:
		assert (
		    isinstance(snake_case__       ,   snake_case__     )
		    and isinstance(snake_case__       ,   snake_case__     )
		    and (numbera >= 1)
		    and (numbera >= 1)
		), "'number1' and 'number2' must been positive integer."

		__a	 =     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'
				__a	 =     prime_factorization(snake_case__     )
				__a	 =     prime_factorization(snake_case__     )

		elif numbera == 1 or numbera == 1:
				__a	 =     []
				__a	 =     []
				__a	 =     max(snake_case__       ,   snake_case__     )

		__a	 =     0
		__a	 =     0

		__a	 =     []  # 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:
								__a	 =     prime_fac_a.count(snake_case__     )
								__a	 =     prime_fac_a.count(snake_case__     )

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

						else:
								__a	 =     prime_fac_a.count(snake_case__     )

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

						done.append(snake_case__     )

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

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

						done.append(snake_case__     )

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

		return ans

def 			lowercase				(						lowerCAmelCase__					:	int     )					-> Tuple:
		assert isinstance(snake_case__       ,   snake_case__     ) and (n >= 0), "'number' must been a positive int"

		__a	 =     0
		__a	 =     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(snake_case__     ):
						ans += 1

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

		return ans

def 			lowercase				(						lowerCAmelCase__					:	int       ,   lowerCAmelCase__					:	Any     )					-> Dict:
		assert (
		    is_prime(snake_case__     ) and is_prime(snake_case__     ) and (p_number_a < p_number_a)
		), "The arguments must been prime numbers and 'pNumber1' < 'pNumber2'"

		__a	 =     p_number_a + 1  # jump to the next number

		__a	 =     []  # this list will be returns.

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

		while number < p_number_a:
				ans.append(snake_case__     )

				number += 1

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

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

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

def 			lowercase				(						lowerCAmelCase__					:	int     )					-> Any:
		assert isinstance(snake_case__       ,   snake_case__     ) and (n >= 1), "'n' must been int and >= 1"

		__a	 =     []  # will be returned.

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

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

		return ans

def 			lowercase				(						lowerCAmelCase__					:	Tuple     )					-> int:
		assert isinstance(snake_case__       ,   snake_case__     ) and (
		    number > 1
		), "'number' must been an int and >= 1"

		__a	 =     get_divisors(snake_case__     )

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

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

def 			lowercase				(						lowerCAmelCase__					:	str       ,   lowerCAmelCase__					:	List[str]     )					-> str:
		assert (
		    isinstance(snake_case__       ,   snake_case__     )
		    and isinstance(snake_case__       ,   snake_case__     )
		    and (denominator != 0)
		), "The arguments must been from type int and 'denominator' != 0"

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

		# precondition
		assert (
		    isinstance(snake_case__       ,   snake_case__     )
		    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 			lowercase				(						lowerCAmelCase__					:	Any     )					-> Union[str, Any]:
		assert isinstance(snake_case__       ,   snake_case__     ) and (n >= 0), "'n' must been a int and >= 0"

		__a	 =     1  # this will be return.

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

		return ans




def 			lowercase				(						lowerCAmelCase__					:	List[str]     )					-> Tuple:
		assert isinstance(snake_case__       ,   snake_case__     ) and (n >= 0), "'n' must been an int and >= 0"

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

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

		return ans








"""simple docstring"""


from typing import TYPE_CHECKING

from ...utils import OptionalDependencyNotAvailable, _LazyModule, is_torch_available


_lowercase				 =					{
    '''configuration_xlm_roberta_xl''': [
        '''XLM_ROBERTA_XL_PRETRAINED_CONFIG_ARCHIVE_MAP''',
        '''XLMRobertaXLConfig''',
        '''XLMRobertaXLOnnxConfig''',
    ],
}

try:
		if not is_torch_available():
				raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
		pass
else:
		_lowercase				 =					[
		    '''XLM_ROBERTA_XL_PRETRAINED_MODEL_ARCHIVE_LIST''',
		    '''XLMRobertaXLForCausalLM''',
		    '''XLMRobertaXLForMaskedLM''',
		    '''XLMRobertaXLForMultipleChoice''',
		    '''XLMRobertaXLForQuestionAnswering''',
		    '''XLMRobertaXLForSequenceClassification''',
		    '''XLMRobertaXLForTokenClassification''',
		    '''XLMRobertaXLModel''',
		    '''XLMRobertaXLPreTrainedModel''',
		]

if TYPE_CHECKING:
		from .configuration_xlm_roberta_xl import (
		    XLM_ROBERTA_XL_PRETRAINED_CONFIG_ARCHIVE_MAP,
		    XLMRobertaXLConfig,
		    XLMRobertaXLOnnxConfig,
		)

		try:
				if not is_torch_available():
						raise OptionalDependencyNotAvailable()
		except OptionalDependencyNotAvailable:
				pass
		else:
				from .modeling_xlm_roberta_xl import (
				    XLM_ROBERTA_XL_PRETRAINED_MODEL_ARCHIVE_LIST,
				    XLMRobertaXLForCausalLM,
				    XLMRobertaXLForMaskedLM,
				    XLMRobertaXLForMultipleChoice,
				    XLMRobertaXLForQuestionAnswering,
				    XLMRobertaXLForSequenceClassification,
				    XLMRobertaXLForTokenClassification,
				    XLMRobertaXLModel,
				    XLMRobertaXLPreTrainedModel,
				)

else:
		import sys

		_lowercase				 =					_LazyModule(__name__, globals()['''__file__'''], _import_structure)