



from math import sqrt


def  __lowerCamelCase (       lowerCamelCase__						):

				"""simple docstring"""


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

				lowercase__					:   List[Any]													=					True

				# 0 and 1 are none primes.
				if number <= 1:
								lowercase__					:   str													=					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__					:   Union[str, Any]													=					False
												break

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

				return status


def  __lowerCamelCase (       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[str]													=					list(range(2					,			n + 1						)						)

				lowercase__					:   str													=					[]  # 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__					:   Tuple													=					0

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

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

				return ans


def  __lowerCamelCase (       lowerCamelCase__						):

				"""simple docstring"""


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

				lowercase__					:   List[str]													=					[]

				# 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  __lowerCamelCase (       lowerCamelCase__						):

				"""simple docstring"""


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

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

				# potential prime number factors.

				lowercase__					:   Optional[Any]													=					2

				lowercase__					:   Optional[Any]													=					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  __lowerCamelCase (       lowerCamelCase__						):

				"""simple docstring"""


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

				lowercase__					:   str													=					0

				# prime factorization of 'number'
				lowercase__					:   Any													=					prime_factorization(lowerCamelCase__						)

				lowercase__					:   Optional[int]													=					max(lowerCamelCase__						)

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

				return ans


def  __lowerCamelCase (       lowerCamelCase__						):

				"""simple docstring"""


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

				lowercase__					:   Union[str, Any]													=					0

				# prime factorization of 'number'
				lowercase__					:   Any													=					prime_factorization(lowerCamelCase__						)

				lowercase__					:   Optional[Any]													=					min(lowerCamelCase__						)

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

				return ans


def  __lowerCamelCase (       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  __lowerCamelCase (       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  __lowerCamelCase (       lowerCamelCase__						):

				"""simple docstring"""


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

				lowercase__					:   Dict													=					[]  # this list will returned

				# creates a list of prime numbers between 2 up to 'number'
				lowercase__					:   int													=					get_prime_numbers(lowerCamelCase__						)
				lowercase__					:   Tuple													=					len(lowerCamelCase__						)

				# run variable for while-loops.
				lowercase__					:   Union[str, Any]													=					0
				lowercase__					:   Tuple													=					None

				# exit variable. for break up the loops
				lowercase__					:   List[Any]													=					True

				while i < len_pn and loop:
								lowercase__					:   List[Any]													=					i + 1

								while j < len_pn and loop:
												if prime_numbers[i] + prime_numbers[j] == number:
																lowercase__					:   List[Any]													=					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  __lowerCamelCase (       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__					:   int													=					0

				while numbera != 0:
								lowercase__					:   int													=					numbera % numbera
								lowercase__					:   Dict													=					numbera
								lowercase__					:   Optional[Any]													=					rest

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

				return numbera


def  __lowerCamelCase (       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__					:   Any													=					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__					:   Any													=					prime_factorization(lowerCamelCase__						)
								lowercase__					:   str													=					prime_factorization(lowerCamelCase__						)

				elif numbera == 1 or numbera == 1:
								lowercase__					:   Any													=					[]
								lowercase__					:   Optional[Any]													=					[]
								lowercase__					:   Dict													=					max(lowerCamelCase__					,			lowerCamelCase__						)

				lowercase__					:   Optional[Any]													=					0
				lowercase__					:   str													=					0

				lowercase__					:   Optional[Any]													=					[]  # 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__					:   int													=					prime_fac_a.count(lowerCamelCase__						)
																lowercase__					:   List[Any]													=					prime_fac_a.count(lowerCamelCase__						)

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

												else:
																lowercase__					:   Any													=					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__					:   Optional[Any]													=					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  __lowerCamelCase (       lowerCamelCase__						):

				"""simple docstring"""


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

				lowercase__					:   Any													=					0
				lowercase__					:   Optional[int]													=					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  __lowerCamelCase (       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__					:   Any													=					p_number_a + 1  # jump to the next number

				lowercase__					:   Tuple													=					[]  # 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  __lowerCamelCase (       lowerCamelCase__						):

				"""simple docstring"""


				assert isinstance(lowerCamelCase__					,			lowerCamelCase__						) and (n >= 1), "'n' must been int and >= 1"

				lowercase__					:   int													=					[]  # 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  __lowerCamelCase (       lowerCamelCase__						):

				"""simple docstring"""


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

				lowercase__					:   Optional[int]													=					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  __lowerCamelCase (       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__					:   Union[str, Any]													=					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  __lowerCamelCase (       lowerCamelCase__						):

				"""simple docstring"""


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

				lowercase__					:   Any													=					1  # this will be return.

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

				return ans


def  __lowerCamelCase (       lowerCamelCase__						):

				"""simple docstring"""


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

				lowercase__					:   List[Any]													=					0
				lowercase__					:   int													=					1
				lowercase__					:   Union[str, Any]													=					1  # this will be return

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

				return ans









import unittest

from transformers import is_flax_available
from transformers.testing_utils import require_flax, require_sentencepiece, require_tokenizers, require_torch, slow


if is_flax_available():
				import optax
				from flax.training.common_utils import onehot

				from transformers import AutoTokenizer, FlaxMTaForConditionalGeneration
				from transformers.models.ta.modeling_flax_ta import shift_tokens_right

@require_torch
@require_sentencepiece
@require_tokenizers
@require_flax
class 				snake_case__(unittest.TestCase      ):

					"""simple docstring"""






					@slow
					def 			snake_case					(							self :     Optional[int]					):
									lowercase__					:   Dict													=					FlaxMTaForConditionalGeneration.from_pretrained("google/mt5-small"					)
									lowercase__					:   Dict													=					AutoTokenizer.from_pretrained("google/mt5-small"					)

									lowercase__					:   Optional[Any]													=					tokenizer("Hello there"				,       return_tensors="np"					).input_ids
									lowercase__					:   Optional[Any]													=					tokenizer("Hi I am"				,       return_tensors="np"					).input_ids

									lowercase__					:   int													=					shift_tokens_right(SCREAMING_SNAKE_CASE				,       model.config.pad_token_id				,       model.config.decoder_start_token_id					)

									lowercase__					:   Tuple													=					model(SCREAMING_SNAKE_CASE				,       decoder_input_ids=SCREAMING_SNAKE_CASE					).logits
									lowercase__					:   Dict													=					optax.softmax_cross_entropy(SCREAMING_SNAKE_CASE				,       onehot(SCREAMING_SNAKE_CASE				,       logits.shape[-1]					)					).mean()

									lowercase__					:   Union[str, Any]													=					-(labels.shape[-1] * loss.item())

									lowercase__					:   Union[str, Any]													=					-84.9_127
									self.assertTrue(abs(mtf_score - EXPECTED_SCORE					) < 1E-4					)




