"""simple docstring"""







from dataclasses import dataclass
from typing import Optional, Tuple, Union

import flax
import jax.numpy as jnp

from ..configuration_utils import ConfigMixin, register_to_config
from .scheduling_utils_flax import (
    CommonSchedulerState,
    FlaxKarrasDiffusionSchedulers,
    FlaxSchedulerMixin,
    FlaxSchedulerOutput,
    add_noise_common,
    get_velocity_common,
)


@flax.struct.dataclass
class      _lowercase						:
				lowerCamelCase__									=    4_2
				lowerCamelCase__									=    4_2

				# setable values
				lowerCamelCase__									=    4_2
				lowerCamelCase__									=    4_2
				lowerCamelCase__									=    None

				@classmethod
				def     A	(       cls    :      Optional[Any]    ,      _lowercase    :      CommonSchedulerState    ,      _lowercase    :      jnp.ndarray    ,      _lowercase    :      jnp.ndarray    ,      _lowercase    :      jnp.ndarray    ,      ):


										'''simple docstring'''
										return cls(
										    common=_lowercase    ,      final_alpha_cumprod=_lowercase    ,      init_noise_sigma=_lowercase    ,      timesteps=_lowercase    ,      )


@dataclass
class      _lowercase						(				__lowerCamelCase					):
				lowerCamelCase__									=    4_2


class      _lowercase						(				__lowerCamelCase , __lowerCamelCase					):
				lowerCamelCase__									=    [e.name for e in FlaxKarrasDiffusionSchedulers]

				lowerCamelCase__									=    4_2

				@property
				def     A	(       self    :      int       ):


										'''simple docstring'''
										return True

				@register_to_config
				def __init__(       self    :      Union[str, Any]    ,      _lowercase    :      int = 1_0_0_0    ,      _lowercase    :      float = 0.0001    ,      _lowercase    :      float = 0.02    ,      _lowercase    :      str = "linear"    ,      _lowercase    :      Optional[jnp.ndarray] = None    ,      _lowercase    :      bool = True    ,      _lowercase    :      int = 0    ,      _lowercase    :      str = "epsilon"    ,      _lowercase    :      jnp.dtype = jnp.floataa    ,      ):


										'''simple docstring'''
										UpperCAmelCase__   		=						dtype

				def     A	(       self    :      Tuple    ,      _lowercase    :      Optional[CommonSchedulerState] = None       ):


										'''simple docstring'''
										if common is None:
																UpperCAmelCase__   		=						CommonSchedulerState.create(self       )

										# At every step in ddim, we are looking into the previous alphas_cumprod
										# For the final step, there is no previous alphas_cumprod because we are already at 0
										# `set_alpha_to_one` decides whether we set this parameter simply to one or
										# whether we use the final alpha of the "non-previous" one.
										UpperCAmelCase__   		=						(
										    jnp.array(1.0    ,      dtype=self.dtype       ) if self.config.set_alpha_to_one else common.alphas_cumprod[0]
										)

										# standard deviation of the initial noise distribution
										UpperCAmelCase__   		=						jnp.array(1.0    ,      dtype=self.dtype       )

										UpperCAmelCase__   		=						jnp.arange(0    ,      self.config.num_train_timesteps       ).round()[::-1]

										return DDIMSchedulerState.create(
										    common=_lowercase    ,      final_alpha_cumprod=_lowercase    ,      init_noise_sigma=_lowercase    ,      timesteps=_lowercase    ,      )

				def     A	(       self    :      Optional[Any]    ,      _lowercase    :      DDIMSchedulerState    ,      _lowercase    :      jnp.ndarray    ,      _lowercase    :      Optional[int] = None       ):


										'''simple docstring'''
										return sample

				def     A	(       self    :      int    ,      _lowercase    :      DDIMSchedulerState    ,      _lowercase    :      int    ,      _lowercase    :      Tuple = ()       ):


										'''simple docstring'''
										UpperCAmelCase__   		=						self.config.num_train_timesteps // num_inference_steps
										# creates integer timesteps by multiplying by ratio
										# rounding to avoid issues when num_inference_step is power of 3
										UpperCAmelCase__   		=						(jnp.arange(0    ,      _lowercase       ) * step_ratio).round()[::-1] + self.config.steps_offset

										return state.replace(
										    num_inference_steps=_lowercase    ,      timesteps=_lowercase    ,      )

				def     A	(       self    :      Dict    ,      _lowercase    :      DDIMSchedulerState    ,      _lowercase    :      Any    ,      _lowercase    :      Dict       ):


										'''simple docstring'''
										UpperCAmelCase__   		=						state.common.alphas_cumprod[timestep]
										UpperCAmelCase__   		=						jnp.where(
										    prev_timestep >= 0    ,      state.common.alphas_cumprod[prev_timestep]    ,      state.final_alpha_cumprod       )
										UpperCAmelCase__   		=						1 - alpha_prod_t
										UpperCAmelCase__   		=						1 - alpha_prod_t_prev

										UpperCAmelCase__   		=						(beta_prod_t_prev / beta_prod_t) * (1 - alpha_prod_t / alpha_prod_t_prev)

										return variance

				def     A	(       self    :      List[Any]    ,      _lowercase    :      DDIMSchedulerState    ,      _lowercase    :      jnp.ndarray    ,      _lowercase    :      int    ,      _lowercase    :      jnp.ndarray    ,      _lowercase    :      float = 0.0    ,      _lowercase    :      bool = True    ,      ):


										'''simple docstring'''
										if state.num_inference_steps is None:
																raise ValueError(
																    """Number of inference steps is 'None', you need to run 'set_timesteps' after creating the scheduler"""       )

										# See formulas (12) and (16) of DDIM paper https://arxiv.org/pdf/2010.02502.pdf
										# Ideally, read DDIM paper in-detail understanding

										# Notation (<variable name> -> <name in paper>
										# - pred_noise_t -> e_theta(x_t, t)
										# - pred_original_sample -> f_theta(x_t, t) or x_0
										# - std_dev_t -> sigma_t
										# - eta -> η
										# - pred_sample_direction -> "direction pointing to x_t"
										# - pred_prev_sample -> "x_t-1"

										# 1. get previous step value (=t-1)
										UpperCAmelCase__   		=						timestep - self.config.num_train_timesteps // state.num_inference_steps

										UpperCAmelCase__   		=						state.common.alphas_cumprod
										UpperCAmelCase__   		=						state.final_alpha_cumprod

										# 2. compute alphas, betas
										UpperCAmelCase__   		=						alphas_cumprod[timestep]
										UpperCAmelCase__   		=						jnp.where(prev_timestep >= 0    ,      alphas_cumprod[prev_timestep]    ,      _lowercase       )

										UpperCAmelCase__   		=						1 - alpha_prod_t

										# 3. compute predicted original sample from predicted noise also called
										# "predicted x_0" of formula (12) from https://arxiv.org/pdf/2010.02502.pdf
										if self.config.prediction_type == "epsilon":
																UpperCAmelCase__   		=						(sample - beta_prod_t ** 0.5 * model_output) / alpha_prod_t ** 0.5
																UpperCAmelCase__   		=						model_output
										elif self.config.prediction_type == "sample":
																UpperCAmelCase__   		=						model_output
																UpperCAmelCase__   		=						(sample - alpha_prod_t ** 0.5 * pred_original_sample) / beta_prod_t ** 0.5
										elif self.config.prediction_type == "v_prediction":
																UpperCAmelCase__   		=						(alpha_prod_t**0.5) * sample - (beta_prod_t**0.5) * model_output
																UpperCAmelCase__   		=						(alpha_prod_t**0.5) * model_output + (beta_prod_t**0.5) * sample
										else:
																raise ValueError(
																    f'prediction_type given as {self.config.prediction_type} must be one of `epsilon`, `sample`, or'
																    """ `v_prediction`"""       )

										# 4. compute variance: "sigma_t(η)" -> see formula (16)
										# σ_t = sqrt((1 − α_t−1)/(1 − α_t)) * sqrt(1 − α_t/α_t−1)
										UpperCAmelCase__   		=						self._get_variance(_lowercase    ,      _lowercase    ,      _lowercase       )
										UpperCAmelCase__   		=						eta * variance ** 0.5

										# 5. compute "direction pointing to x_t" of formula (12) from https://arxiv.org/pdf/2010.02502.pdf
										UpperCAmelCase__   		=						(1 - alpha_prod_t_prev - std_dev_t**2) ** 0.5 * pred_epsilon

										# 6. compute x_t without "random noise" of formula (12) from https://arxiv.org/pdf/2010.02502.pdf
										UpperCAmelCase__   		=						alpha_prod_t_prev ** 0.5 * pred_original_sample + pred_sample_direction

										if not return_dict:
																return (prev_sample, state)

										return FlaxDDIMSchedulerOutput(prev_sample=_lowercase    ,      state=_lowercase       )

				def     A	(       self    :      Any    ,      _lowercase    :      DDIMSchedulerState    ,      _lowercase    :      jnp.ndarray    ,      _lowercase    :      jnp.ndarray    ,      _lowercase    :      jnp.ndarray    ,      ):


										'''simple docstring'''
										return add_noise_common(state.common    ,      _lowercase    ,      _lowercase    ,      _lowercase       )

				def     A	(       self    :      Union[str, Any]    ,      _lowercase    :      DDIMSchedulerState    ,      _lowercase    :      jnp.ndarray    ,      _lowercase    :      jnp.ndarray    ,      _lowercase    :      jnp.ndarray    ,      ):


										'''simple docstring'''
										return get_velocity_common(state.common    ,      _lowercase    ,      _lowercase    ,      _lowercase       )

				def __len__(       self    :      int       ):


										'''simple docstring'''
										return self.config.num_train_timesteps


"""simple docstring"""







from __future__ import annotations

from collections import Counter


def 					a__ (						UpperCAmelCase__    :     int     )->		list[int]:
						UpperCAmelCase__   		=						[True] * n
						UpperCAmelCase__   		=						False
						UpperCAmelCase__   		=						False
						UpperCAmelCase__   		=						True

						for i in range(3						,     int(n**0.5 + 1     )						,     2     ):
												UpperCAmelCase__   		=						i * 2
												while index < n:
																		UpperCAmelCase__   		=						False
																		UpperCAmelCase__   		=						index + i

						UpperCAmelCase__   		=						[2]

						for i in range(3						,     UpperCAmelCase__						,     2     ):
												if is_prime[i]:
																		primes.append(UpperCAmelCase__     )

						return primes


def 					a__ (						UpperCAmelCase__    :     int     )->		list[list[int]]:
						UpperCAmelCase__   		=						str(UpperCAmelCase__     )
						UpperCAmelCase__   		=						[]
						UpperCAmelCase__   		=						["""0""", """1""", """2""", """3""", """4""", """5""", """6""", """7""", """8""", """9"""]

						for duplicate in Counter(UpperCAmelCase__     ) - Counter(set(UpperCAmelCase__     )     ):
												UpperCAmelCase__   		=						[int(number_str.replace(UpperCAmelCase__						,     UpperCAmelCase__     )     ) for digit in digits]
												replacements.append(UpperCAmelCase__     )

						return replacements


def 					a__ (						UpperCAmelCase__    :     int = 8     )->		int:
						UpperCAmelCase__   		=						set()

						# Filter primes with less than 3 replaceable digits
						UpperCAmelCase__   		=						{
						    x for x in set(prime_sieve(1_0_0_0_0_0_0     )     ) if len(str(UpperCAmelCase__     )     ) - len(set(str(UpperCAmelCase__     )     )     ) >= 3
						}

						for prime in primes:
												if prime in numbers_checked:
																		continue

												UpperCAmelCase__   		=						digit_replacements(UpperCAmelCase__     )

												for family in replacements:
																		numbers_checked.update(UpperCAmelCase__     )
																		UpperCAmelCase__   		=						primes.intersection(UpperCAmelCase__     )

																		if len(UpperCAmelCase__     ) != family_length:
																								continue

																		return min(UpperCAmelCase__     )

						return -1


if __name__ == "__main__":
						print(solution())

