code: 
  from __future__ import annotations
  
  import random
  from collections.abc import Iterable
  
  
  
  
  
  class    _snake_case      :
  
  
  
  
  
  
  
  			def __init__(       self		,     _a	):
  				__magic_name__	:	dict[str, bool | None]			     =  {literal: None for literal in literals}
  
  
  
  
  
  
  
  			def __str__(       self	):
  				return "{" + " , ".join(self.literals	) + "}"
  
  
  
  
  
  
  
  			def __len__(       self	):
  				return len(self.literals	)
  
  
  
  
  
  
  
  			def     SCREAMING_SNAKE_CASE					(       self		,     _a	):
  				for literal in self.literals:
  					__magic_name__	:	Any			     =  literal[:2]
  					if symbol in model:
  						__magic_name__	:	int			     =  model[symbol]
  					else:
  						continue
  					if value is not None:
  						# Complement assignment if literal is in complemented form
  						if literal.endswith("'"	):
  							__magic_name__	:	Dict			     =  not value
  					__magic_name__	:	Union[str, Any]			     =  value
  
  
  
  
  
  
  
  			def     SCREAMING_SNAKE_CASE					(       self		,     _a	):
  				for literal in self.literals:
  					__magic_name__	:	Union[str, Any]			     =  literal.rstrip("'"	) if literal.endswith("'"	) else literal + "'"
  					if symbol in self.literals:
  						return True
  
  				self.assign(_a	)
  				for value in self.literals.values():
  					if value in (True, None):
  						return value
  				return any(self.literals.values()	)
  
  
  
  
  
  
  
  class    _snake_case      :
  
  
  
  
  
  
  
  			def __init__(       self		,     _a	):
  				__magic_name__	:	Tuple			     =  list(_a	)
  
  
  
  
  
  
  
  			def __str__(       self	):
  				return "{" + " , ".join(str(_a	) for clause in self.clauses	) + "}"
  
  
  
  
  
  
  
  def 						lowerCAmelCase_		(      )       ->	Clause:
  
  
  
  
  
  
  
  	'''simple docstring'''
  
  
  	__magic_name__	:	List[Any]			     =  []
  	__magic_name__	:	int			     =  random.randint(1   ,     5     )
  	__magic_name__	:	Any			     =  "A"
  	__magic_name__	:	Optional[int]			     =  0
  	while i < no_of_literals:
  		__magic_name__	:	Tuple			     =  random.randint(1   ,     5     )
  		__magic_name__	:	List[str]			     =  base_var + str(_snake_case     )
  		__magic_name__	:	Optional[Any]			     =  random.randint(0   ,     1     )
  		if var_complement == 1:
  			var_name += "'"
  		if var_name in literals:
  			i -= 1
  		else:
  			literals.append(_snake_case     )
  		i += 1
  	return Clause(_snake_case     )
  
  
  
  
  
  
  def 						lowerCAmelCase_		(      )       ->	Formula:
  
  
  
  
  
  
  
  	'''simple docstring'''
  
  
  	__magic_name__	:	set[Clause]			     =  set()
  	__magic_name__	:	Any			     =  random.randint(1   ,     10     )
  	while len(_snake_case     ) < no_of_clauses:
  		clauses.add(generate_clause()     )
  	return Formula(_snake_case     )
  
  
  
  
  
  
  def 						lowerCAmelCase_		(      _snake_case   :						Formula     )       ->	tuple[list[Clause], list[str]]:
  
  
  
  
  
  
  
  	'''simple docstring'''
  
  
  	__magic_name__	:	List[Any]			     =  formula.clauses
  	__magic_name__	:	Optional[int]			     =  []
  	for clause in formula.clauses:
  		for literal in clause.literals:
  			__magic_name__	:	Dict			     =  literal[:2]
  			if symbol not in symbols_set:
  				symbols_set.append(_snake_case     )
  	return clauses, symbols_set
  
  
  
  
  
  
  def 						lowerCAmelCase_		(      _snake_case   :						list[Clause]   ,     _snake_case   :						list[str]   ,     _snake_case   :						dict[str, bool | None]     )       ->	tuple[list[str], dict[str, bool | None]]:
  
  
  
  
  
  
  
  	'''simple docstring'''
  
  
  	__magic_name__	:	Optional[int]			     =  []
  	__magic_name__	:	dict[str, bool | None]			     =  {}
  	__magic_name__	:	Any			     =  []
  
  	for clause in clauses:
  		if clause.evaluate(_snake_case     ):
  			continue
  		for literal in clause.literals:
  			literals.append(_snake_case     )
  
  	for s in symbols:
  		__magic_name__	:	Dict			     =  s + "'"
  		if (s in literals and sym not in literals) or (
  		    s not in literals and sym in literals
  		):
  			pure_symbols.append(_snake_case     )
  	for p in pure_symbols:
  		__magic_name__	:	List[str]			     =  None
  	for s in pure_symbols:
  		__magic_name__	:	Any			     =  s + "'"
  		if s in literals:
  			__magic_name__	:	List[str]			     =  True
  		elif sym in literals:
  			__magic_name__	:	Union[str, Any]			     =  False
  	return pure_symbols, assignment
  
  
  
  
  
  
  def 						lowerCAmelCase_		(      _snake_case   :						list[Clause]   ,     _snake_case   :						dict[str, bool | None]     )       ->	tuple[list[str], dict[str, bool | None]]:
  
  
  
  
  
  
  
  	'''simple docstring'''
  
  
  	__magic_name__	:	Optional[Any]			     =  []
  	for clause in clauses:
  		if len(_snake_case     ) == 1:
  			unit_symbols.append(list(clause.literals.keys()     )[0]     )
  		else:
  			__magic_name__	:	int			     =  0, 0
  			for literal, value in clause.literals.items():
  				if value is False:
  					f_count += 1
  				elif value is None:
  					__magic_name__	:	int			     =  literal
  					n_count += 1
  			if f_count == len(_snake_case     ) - 1 and n_count == 1:
  				unit_symbols.append(_snake_case     )
  	__magic_name__	:	dict[str, bool | None]			     =  {}
  	for i in unit_symbols:
  		__magic_name__	:	Tuple			     =  i[:2]
  		__magic_name__	:	Any			     =  len(_snake_case     ) == 2
  	__magic_name__	:	Optional[int]			     =  [i[:2] for i in unit_symbols]
  
  	return unit_symbols, assignment
  
  
  
  
  
  
  def 						lowerCAmelCase_		(      _snake_case   :						list[Clause]   ,     _snake_case   :						list[str]   ,     _snake_case   :						dict[str, bool | None]     )       ->	tuple[bool | None, dict[str, bool | None] | None]:
  
  
  
  
  
  
  
  	'''simple docstring'''
  
  
  	__magic_name__	:	Dict			     =  True
  	for clause in clauses:
  		__magic_name__	:	List[Any]			     =  clause.evaluate(_snake_case     )
  		if clause_check is False:
  			return False, None
  		elif clause_check is None:
  			__magic_name__	:	Union[str, Any]			     =  False
  			continue
  
  	if check_clause_all_true:
  		return True, model
  
  	try:
  		__magic_name__	:	Optional[Any]			     =  find_pure_symbols(_snake_case   ,     _snake_case   ,     _snake_case     )
  	except RecursionError:
  		print("raises a RecursionError and is"     )
  		return None, {}
  	__magic_name__	:	Dict			     =  None
  	if len(_snake_case     ) > 0:
  		__magic_name__	:	List[str]			     =  pure_symbols[0], assignment[pure_symbols[0]]
  
  	if p:
  		__magic_name__	:	Optional[Any]			     =  model
  		__magic_name__	:	Optional[int]			     =  value
  		__magic_name__	:	Dict			     =  list(_snake_case     )
  		if p in tmp_symbols:
  			tmp_symbols.remove(_snake_case     )
  		return dpll_algorithm(_snake_case   ,     _snake_case   ,     _snake_case     )
  
  	__magic_name__	:	Dict			     =  find_unit_clauses(_snake_case   ,     _snake_case     )
  	__magic_name__	:	int			     =  None
  	if len(_snake_case     ) > 0:
  		__magic_name__	:	Any			     =  unit_symbols[0], assignment[unit_symbols[0]]
  	if p:
  		__magic_name__	:	List[Any]			     =  model
  		__magic_name__	:	Tuple			     =  value
  		__magic_name__	:	List[str]			     =  list(_snake_case     )
  		if p in tmp_symbols:
  			tmp_symbols.remove(_snake_case     )
  		return dpll_algorithm(_snake_case   ,     _snake_case   ,     _snake_case     )
  	__magic_name__	:	List[Any]			     =  symbols[0]
  	__magic_name__	:	Union[str, Any]			     =  symbols[1:]
  	__magic_name__	:	Union[str, Any]			     =  model, model
  	__magic_name__	:	Union[str, Any]			     =  True, False
  
  	return dpll_algorithm(_snake_case   ,     _snake_case   ,     _snake_case     ) or dpll_algorithm(_snake_case   ,     _snake_case   ,     _snake_case     )
  
  
  if __name__ == "__main__":
  				import doctest
  
  				doctest.testmod()
  
  				snake_case				:	List[str]							   =						generate_formula()
  				print(F"The formula {formula} is", end=" ")
  
  				snake_case				:	List[str]							   =						generate_parameters(formula)
  				snake_case				:	List[Any]							   =						dpll_algorithm(clauses, symbols, {})
  
  				if solution:
  								print(F"satisfiable with the assignment {model}.")
  				else:
  								print("not satisfiable.")
  
style_context: 
  from sklearn.metrics import roc_auc_score
  
  import datasets
  
  
  snake_case				:	List[Any]							   =						"\nThis metric computes the area under the curve (AUC) for the Receiver Operating Characteristic Curve (ROC). The return values represent how well the model used is predicting the correct classes, based on the input data. A score of `0.5` means that the model is predicting exactly at chance, i.e. the model's predictions are correct at the same rate as if the predictions were being decided by the flip of a fair coin or the roll of a fair die. A score above `0.5` indicates that the model is doing better than chance, while a score below `0.5` indicates that the model is doing worse than chance.\n\nThis metric has three separate use cases:\n    - binary: The case in which there are only two different label classes, and each example gets only one label. This is the default implementation.\n    - multiclass: The case in which there can be more than two different label classes, but each example still gets only one label.\n    - multilabel: The case in which there can be more than two different label classes, and each example can have more than one label.\n"
  
  snake_case				:	Any							   =						"\nArgs:\n- references (array-like of shape (n_samples,) or (n_samples, n_classes)): Ground truth labels. Expects different input based on use case:\n    - binary: expects an array-like of shape (n_samples,)\n    - multiclass: expects an array-like of shape (n_samples,)\n    - multilabel: expects an array-like of shape (n_samples, n_classes)\n- prediction_scores (array-like of shape (n_samples,) or (n_samples, n_classes)): Model predictions. Expects different inputs based on use case:\n    - binary: expects an array-like of shape (n_samples,)\n    - multiclass: expects an array-like of shape (n_samples, n_classes)\n    - multilabel: expects an array-like of shape (n_samples, n_classes)\n- average (`str`): Type of average, and is ignored in the binary use case. Defaults to 'macro'. Options are:\n    - `'micro'`: Calculates metrics globally by considering each element of the label indicator matrix as a label. Only works with the multilabel use case.\n    - `'macro'`: Calculate metrics for each label, and find their unweighted mean.  This does not take label imbalance into account.\n    - `'weighted'`: Calculate metrics for each label, and find their average, weighted by support (i.e. the number of true instances for each label).\n    - `'samples'`: Calculate metrics for each instance, and find their average. Only works with the multilabel use case.\n    - `None`:  No average is calculated, and scores for each class are returned. Only works with the multilabels use case.\n- sample_weight (array-like of shape (n_samples,)): Sample weights. Defaults to None.\n- max_fpr (`float`): If not None, the standardized partial AUC over the range [0, `max_fpr`] is returned. Must be greater than `0` and less than or equal to `1`. Defaults to `None`. Note: For the multiclass use case, `max_fpr` should be either `None` or `1.0` as ROC AUC partial computation is not currently supported for `multiclass`.\n- multi_class (`str`): Only used for multiclass targets, where it is required. Determines the type of configuration to use. Options are:\n    - `'ovr'`: Stands for One-vs-rest. Computes the AUC of each class against the rest. This treats the multiclass case in the same way as the multilabel case. Sensitive to class imbalance even when `average == 'macro'`, because class imbalance affects the composition of each of the 'rest' groupings.\n    - `'ovo'`: Stands for One-vs-one. Computes the average AUC of all possible pairwise combinations of classes. Insensitive to class imbalance when `average == 'macro'`.\n- labels (array-like of shape (n_classes,)): Only used for multiclass targets. List of labels that index the classes in\n    `prediction_scores`. If `None`, the numerical or lexicographical order of the labels in\n    `prediction_scores` is used. Defaults to `None`.\nReturns:\n    roc_auc (`float` or array-like of shape (n_classes,)): Returns array if in multilabel use case and `average='None'`. Otherwise, returns `float`.\nExamples:\n    Example 1:\n        >>> roc_auc_score = datasets.load_metric(\"roc_auc\")\n        >>> refs = [1, 0, 1, 1, 0, 0]\n        >>> pred_scores = [0.5, 0.2, 0.99, 0.3, 0.1, 0.7]\n        >>> results = roc_auc_score.compute(references=refs, prediction_scores=pred_scores)\n        >>> print(round(results['roc_auc'], 2))\n        0.78\n\n    Example 2:\n        >>> roc_auc_score = datasets.load_metric(\"roc_auc\", \"multiclass\")\n        >>> refs = [1, 0, 1, 2, 2, 0]\n        >>> pred_scores = [[0.3, 0.5, 0.2],\n        ...                 [0.7, 0.2, 0.1],\n        ...                 [0.005, 0.99, 0.005],\n        ...                 [0.2, 0.3, 0.5],\n        ...                 [0.1, 0.1, 0.8],\n        ...                 [0.1, 0.7, 0.2]]\n        >>> results = roc_auc_score.compute(references=refs, prediction_scores=pred_scores, multi_class='ovr')\n        >>> print(round(results['roc_auc'], 2))\n        0.85\n\n    Example 3:\n        >>> roc_auc_score = datasets.load_metric(\"roc_auc\", \"multilabel\")\n        >>> refs = [[1, 1, 0],\n        ...         [1, 1, 0],\n        ...         [0, 1, 0],\n        ...         [0, 0, 1],\n        ...         [0, 1, 1],\n        ...         [1, 0, 1]]\n        >>> pred_scores = [[0.3, 0.5, 0.2],\n        ...                 [0.7, 0.2, 0.1],\n        ...                 [0.005, 0.99, 0.005],\n        ...                 [0.2, 0.3, 0.5],\n        ...                 [0.1, 0.1, 0.8],\n        ...                 [0.1, 0.7, 0.2]]\n        >>> results = roc_auc_score.compute(references=refs, prediction_scores=pred_scores, average=None)\n        >>> print([round(res, 2) for res in results['roc_auc']])\n        [0.83, 0.38, 0.94]\n"
  
  snake_case				:	List[Any]							   =						"\\n@article{doi:10.1177/0272989X8900900307,\nauthor = {Donna Katzman McClish},\ntitle ={Analyzing a Portion of the ROC Curve},\njournal = {Medical Decision Making},\nvolume = {9},\nnumber = {3},\npages = {190-195},\nyear = {1989},\ndoi = {10.1177/0272989X8900900307},\n    note ={PMID: 2668680},\nURL = {https://doi.org/10.1177/0272989X8900900307},\neprint = {https://doi.org/10.1177/0272989X8900900307}\n}\n\n\n@article{10.1023/A:1010920819831,\nauthor = {Hand, David J. and Till, Robert J.},\ntitle = {A Simple Generalisation of the Area Under the ROC Curve for Multiple Class Classification Problems},\nyear = {2001},\nissue_date = {November 2001},\npublisher = {Kluwer Academic Publishers},\naddress = {USA},\nvolume = {45},\nnumber = {2},\nissn = {0885-6125},\nurl = {https://doi.org/10.1023/A:1010920819831},\ndoi = {10.1023/A:1010920819831},\njournal = {Mach. Learn.},\nmonth = {oct},\npages = {171–186},\nnumpages = {16},\nkeywords = {Gini index, AUC, error rate, ROC curve, receiver operating characteristic}\n}\n\n\n@article{scikit-learn,\ntitle={Scikit-learn: Machine Learning in {P}ython},\nauthor={Pedregosa, F. and Varoquaux, G. and Gramfort, A. and Michel, V. and Thirion, B. and Grisel, O. and Blondel, M. and Prettenhofer, P. and Weiss, R. and Dubourg, V. and Vanderplas, J. and Passos, A. and Cournapeau, D. and Brucher, M. and Perrot, M. and Duchesnay, E.},\njournal={Journal of Machine Learning Research},\nvolume={12},\npages={2825--2830},\nyear={2011}\n}\n"
  
  
  
  
  
  @datasets.utils.file_utils.add_start_docstrings(_DESCRIPTION       ,		_KWARGS_DESCRIPTION       )
  class    _snake_case      (  datasets.Metric       ):
  
  
  
  
  
  
  
  			def     SCREAMING_SNAKE_CASE					(       self	):
  				return datasets.MetricInfo(
  				    description=_DESCRIPTION		,     citation=_CITATION		,     inputs_description=_KWARGS_DESCRIPTION		,     features=datasets.Features(
  				        {
  				            "prediction_scores": datasets.Sequence(datasets.Value("float"	)	),
  				            "references": datasets.Value("int32"	),
  				        }
  				        if self.config_name == "multiclass"
  				        else {
  				            "references": datasets.Sequence(datasets.Value("int32"	)	),
  				            "prediction_scores": datasets.Sequence(datasets.Value("float"	)	),
  				        }
  				        if self.config_name == "multilabel"
  				        else {
  				            "references": datasets.Value("int32"	),
  				            "prediction_scores": datasets.Value("float"	),
  				        }	)		,     reference_urls=["https://scikit-learn.org/stable/modules/generated/sklearn.metrics.roc_auc_score.html"]		,     )
  
  
  
  
  
  
  
  			def     SCREAMING_SNAKE_CASE					(       self		,     _a		,     _a		,     _a="macro"		,     _a=None		,     _a=None		,     _a="raise"		,     _a=None		,     ):
  				return {
  				    "roc_auc": roc_auc_score(
  				        _a		,     _a		,     average=_a		,     sample_weight=_a		,     max_fpr=_a		,     multi_class=_a		,     labels=_a		,     )
  				}
  