



'''simple docstring'''







from __future__ import annotations

import random
from collections.abc import Iterable




class 				__lowerCAmelCase:






			def __init__(   self  :							int      ,       SCREAMING_SNAKE_CASE  :							list[str]	):




					"""simple docstring"""







					SCREAMING_SNAKE_CASE_						:dict[str, bool | None]										=      {literal: None for literal in literals}






			def __str__(   self  :							Optional[int]	):




					"""simple docstring"""







					return "{" + " , ".join(self.literals	) + "}"






			def __len__(   self  :							Optional[int]	):




					"""simple docstring"""







					return len(self.literals	)






			def 				_lowercase	(   self  :							List[str]      ,       SCREAMING_SNAKE_CASE  :							dict[str, bool | None]	):




					"""simple docstring"""







					for literal in self.literals:
							SCREAMING_SNAKE_CASE_						:Tuple										=      literal[:2]
							if symbol in model:
									SCREAMING_SNAKE_CASE_						:Optional[int]										=      model[symbol]
							else:
									continue
							if value is not None:
									# Complement assignment if literal is in complemented form
									if literal.endswith('\''	):
											SCREAMING_SNAKE_CASE_						:int										=      not value
							SCREAMING_SNAKE_CASE_						:List[str]										=      value






			def 				_lowercase	(   self  :							Any      ,       SCREAMING_SNAKE_CASE  :							dict[str, bool | None]	):




					"""simple docstring"""







					for literal in self.literals:
							SCREAMING_SNAKE_CASE_						:Any										=      literal.rstrip('\''	) if literal.endswith('\''	) else literal + '\''
							if symbol in self.literals:
									return True

					self.assign(SCREAMING_SNAKE_CASE	)
					for value in self.literals.values():
							if value in (True, None):
									return value
					return any(self.literals.values()	)




class 				__lowerCAmelCase:






			def __init__(   self  :							Union[str, Any]      ,       SCREAMING_SNAKE_CASE  :							Iterable[Clause]	):




					"""simple docstring"""







					SCREAMING_SNAKE_CASE_						:Tuple										=      list(SCREAMING_SNAKE_CASE	)






			def __str__(   self  :							Optional[Any]	):




					"""simple docstring"""







					return "{" + " , ".join(str(SCREAMING_SNAKE_CASE	) for clause in self.clauses	) + "}"







def  SCREAMING_SNAKE_CASE__ (				):
		SCREAMING_SNAKE_CASE_						:str										=      []
		SCREAMING_SNAKE_CASE_						:Tuple										=      random.randint(1 ,		5     )
		SCREAMING_SNAKE_CASE_						:str										=      'A'
		SCREAMING_SNAKE_CASE_						:Any										=      0
		while i < no_of_literals:
				SCREAMING_SNAKE_CASE_						:Dict										=      random.randint(1 ,		5     )
				SCREAMING_SNAKE_CASE_						:Optional[Any]										=      base_var + str(SCREAMING_SNAKE_CASE     )
				SCREAMING_SNAKE_CASE_						:Tuple										=      random.randint(0 ,		1     )
				if var_complement == 1:
						var_name += "'"
				if var_name in literals:
						i -= 1
				else:
						literals.append(SCREAMING_SNAKE_CASE     )
				i += 1
		return Clause(SCREAMING_SNAKE_CASE     )







def  SCREAMING_SNAKE_CASE__ (				):
		SCREAMING_SNAKE_CASE_						:set[Clause]										=      set()
		SCREAMING_SNAKE_CASE_						:str										=      random.randint(1 ,		10     )
		while len(SCREAMING_SNAKE_CASE     ) < no_of_clauses:
				clauses.add(generate_clause()     )
		return Formula(SCREAMING_SNAKE_CASE     )







def  SCREAMING_SNAKE_CASE__ (				SCREAMING_SNAKE_CASE     ):
		SCREAMING_SNAKE_CASE_						:Optional[int]										=      formula.clauses
		SCREAMING_SNAKE_CASE_						:int										=      []
		for clause in formula.clauses:
				for literal in clause.literals:
						SCREAMING_SNAKE_CASE_						:Tuple										=      literal[:2]
						if symbol not in symbols_set:
								symbols_set.append(SCREAMING_SNAKE_CASE     )
		return clauses, symbols_set







def  SCREAMING_SNAKE_CASE__ (				SCREAMING_SNAKE_CASE ,		SCREAMING_SNAKE_CASE ,		SCREAMING_SNAKE_CASE     ):
		SCREAMING_SNAKE_CASE_						:List[str]										=      []
		SCREAMING_SNAKE_CASE_						:dict[str, bool | None]										=      {}
		SCREAMING_SNAKE_CASE_						:Tuple										=      []

		for clause in clauses:
				if clause.evaluate(SCREAMING_SNAKE_CASE     ):
						continue
				for literal in clause.literals:
						literals.append(SCREAMING_SNAKE_CASE     )

		for s in symbols:
				SCREAMING_SNAKE_CASE_						:Optional[Any]										=      s + '\''
				if (s in literals and sym not in literals) or (
				    s not in literals and sym in literals
				):
						pure_symbols.append(SCREAMING_SNAKE_CASE     )
		for p in pure_symbols:
				SCREAMING_SNAKE_CASE_						:str										=      None
		for s in pure_symbols:
				SCREAMING_SNAKE_CASE_						:List[str]										=      s + '\''
				if s in literals:
						SCREAMING_SNAKE_CASE_						:Any										=      True
				elif sym in literals:
						SCREAMING_SNAKE_CASE_						:str										=      False
		return pure_symbols, assignment







def  SCREAMING_SNAKE_CASE__ (				SCREAMING_SNAKE_CASE ,		SCREAMING_SNAKE_CASE     ):
		SCREAMING_SNAKE_CASE_						:Union[str, Any]										=      []
		for clause in clauses:
				if len(SCREAMING_SNAKE_CASE     ) == 1:
						unit_symbols.append(list(clause.literals.keys()     )[0]     )
				else:
						SCREAMING_SNAKE_CASE_						,		SCREAMING_SNAKE_CASE_						:Tuple										=      0, 0
						for literal, value in clause.literals.items():
								if value is False:
										f_count += 1
								elif value is None:
										SCREAMING_SNAKE_CASE_						:Tuple										=      literal
										n_count += 1
						if f_count == len(SCREAMING_SNAKE_CASE     ) - 1 and n_count == 1:
								unit_symbols.append(SCREAMING_SNAKE_CASE     )
		SCREAMING_SNAKE_CASE_						:dict[str, bool | None]										=      {}
		for i in unit_symbols:
				SCREAMING_SNAKE_CASE_						:Dict										=      i[:2]
				SCREAMING_SNAKE_CASE_						:str										=      len(SCREAMING_SNAKE_CASE     ) == 2
		SCREAMING_SNAKE_CASE_						:List[str]										=      [i[:2] for i in unit_symbols]

		return unit_symbols, assignment







def  SCREAMING_SNAKE_CASE__ (				SCREAMING_SNAKE_CASE ,		SCREAMING_SNAKE_CASE ,		SCREAMING_SNAKE_CASE     ):
		SCREAMING_SNAKE_CASE_						:Union[str, Any]										=      True
		for clause in clauses:
				SCREAMING_SNAKE_CASE_						:List[str]										=      clause.evaluate(SCREAMING_SNAKE_CASE     )
				if clause_check is False:
						return False, None
				elif clause_check is None:
						SCREAMING_SNAKE_CASE_						:Dict										=      False
						continue

		if check_clause_all_true:
				return True, model

		try:
				SCREAMING_SNAKE_CASE_						,		SCREAMING_SNAKE_CASE_						:List[Any]										=      find_pure_symbols(SCREAMING_SNAKE_CASE ,		SCREAMING_SNAKE_CASE ,		SCREAMING_SNAKE_CASE     )
		except RecursionError:
				print('raises a RecursionError and is'     )
				return None, {}
		SCREAMING_SNAKE_CASE_						:Optional[Any]										=      None
		if len(SCREAMING_SNAKE_CASE     ) > 0:
				SCREAMING_SNAKE_CASE_						,		SCREAMING_SNAKE_CASE_						:Union[str, Any]										=      pure_symbols[0], assignment[pure_symbols[0]]

		if p:
				SCREAMING_SNAKE_CASE_						:Dict										=      model
				SCREAMING_SNAKE_CASE_						:Tuple										=      value
				SCREAMING_SNAKE_CASE_						:List[Any]										=      list(SCREAMING_SNAKE_CASE     )
				if p in tmp_symbols:
						tmp_symbols.remove(SCREAMING_SNAKE_CASE     )
				return dpll_algorithm(SCREAMING_SNAKE_CASE ,		SCREAMING_SNAKE_CASE ,		SCREAMING_SNAKE_CASE     )

		SCREAMING_SNAKE_CASE_						,		SCREAMING_SNAKE_CASE_						:str										=      find_unit_clauses(SCREAMING_SNAKE_CASE ,		SCREAMING_SNAKE_CASE     )
		SCREAMING_SNAKE_CASE_						:Dict										=      None
		if len(SCREAMING_SNAKE_CASE     ) > 0:
				SCREAMING_SNAKE_CASE_						,		SCREAMING_SNAKE_CASE_						:List[str]										=      unit_symbols[0], assignment[unit_symbols[0]]
		if p:
				SCREAMING_SNAKE_CASE_						:Optional[int]										=      model
				SCREAMING_SNAKE_CASE_						:Dict										=      value
				SCREAMING_SNAKE_CASE_						:Dict										=      list(SCREAMING_SNAKE_CASE     )
				if p in tmp_symbols:
						tmp_symbols.remove(SCREAMING_SNAKE_CASE     )
				return dpll_algorithm(SCREAMING_SNAKE_CASE ,		SCREAMING_SNAKE_CASE ,		SCREAMING_SNAKE_CASE     )
		SCREAMING_SNAKE_CASE_						:str										=      symbols[0]
		SCREAMING_SNAKE_CASE_						:List[Any]										=      symbols[1:]
		SCREAMING_SNAKE_CASE_						,		SCREAMING_SNAKE_CASE_						:str										=      model, model
		SCREAMING_SNAKE_CASE_						,		SCREAMING_SNAKE_CASE_						:List[str]										=      True, False

		return dpll_algorithm(SCREAMING_SNAKE_CASE ,		SCREAMING_SNAKE_CASE ,		SCREAMING_SNAKE_CASE     ) or dpll_algorithm(SCREAMING_SNAKE_CASE ,		SCREAMING_SNAKE_CASE ,		SCREAMING_SNAKE_CASE     )


if __name__ == "__main__":
							import doctest

							doctest.testmod()

							SCREAMING_SNAKE_CASE__     : Union[str, Any]				=   generate_formula()
							print(f"""The formula {formula} is""", end=""" """)

							SCREAMING_SNAKE_CASE__	,							SCREAMING_SNAKE_CASE__     : str				=   generate_parameters(formula)
							SCREAMING_SNAKE_CASE__	,							SCREAMING_SNAKE_CASE__     : Union[str, Any]				=   dpll_algorithm(clauses, symbols, {})

							if solution:
														print(f"""satisfiable with the assignment {model}.""")
							else:
														print("""not satisfiable.""")





'''simple docstring'''







from functools import partial
from typing import Any, Callable, Dict, List, Type, TypeVar, Union, overload

import torch
import torch.nn as nn
import torch.types
def 			_lowerCAmelCase(      UpperCAmelCase_      :	torch.Tensor					, UpperCAmelCase_      :	torch.Tensor					, UpperCAmelCase_      :	bool						) ->							torch.Tensor:
							# The first operation in a checkpoint can't be in-place, but it's
							# nice to have in-place addition during inference. Thus...
							if not inplace:
														lowerCAmelCase__  =   ma + ma
							else:
														ma += ma

							return ma
def 			_lowerCAmelCase(      UpperCAmelCase_      :	torch.Tensor					, UpperCAmelCase_      :	List[int]						) ->							torch.Tensor:
							lowerCAmelCase__  =   -1 * len(UpperCAmelCase_						)
							lowerCAmelCase__  =   list(range(len(tensor.shape[:zero_index]						)						)						)
							return tensor.permute(first_inds + [zero_index + i for i in inds]						)
def 			_lowerCAmelCase(      UpperCAmelCase_      :	torch.Tensor					, UpperCAmelCase_      :	int						) ->							torch.Tensor:
							return t.reshape(t.shape[:-no_dims] + (-1,)						)
def 			_lowerCAmelCase(      UpperCAmelCase_      :	torch.Tensor					, UpperCAmelCase_      :	torch.Tensor					, UpperCAmelCase_      :	int					, UpperCAmelCase_      :	float = 1E-4						) ->							torch.Tensor:
							lowerCAmelCase__  =   mask.expand(*value.shape						)
							return torch.sum(mask * value					, dim=UpperCAmelCase_						) / (eps + torch.sum(UpperCAmelCase_					, dim=UpperCAmelCase_						))
def 			_lowerCAmelCase(      UpperCAmelCase_      :	torch.Tensor					, UpperCAmelCase_      :	torch.types.Number = 2.31_25					, UpperCAmelCase_      :	torch.types.Number = 21.68_75					, UpperCAmelCase_      :	int = 64						) ->							torch.Tensor:
							lowerCAmelCase__  =   torch.linspace(UpperCAmelCase_					, UpperCAmelCase_					, no_bins - 1					, device=pts.device						)
							lowerCAmelCase__  =   torch.sqrt(torch.sum((pts.unsqueeze(-2						) - pts.unsqueeze(-3						)) ** 2					, dim=-1						)						)
							return torch.bucketize(UpperCAmelCase_					, UpperCAmelCase_						)
def 			_lowerCAmelCase(      UpperCAmelCase_      :	Callable[[list], Any]					, UpperCAmelCase_      :	List[dict]						) ->							dict:
							lowerCAmelCase__  =   dicts[0]
							lowerCAmelCase__  =   {}
							for k, v in first.items():
														lowerCAmelCase__  =   [d[k] for d in dicts]
														if isinstance(UpperCAmelCase_					, UpperCAmelCase_						):
																					lowerCAmelCase__  =   dict_multimap(UpperCAmelCase_					, UpperCAmelCase_						)
														else:
																					lowerCAmelCase__  =   fn(UpperCAmelCase_						)

							return new_dict
def 			_lowerCAmelCase(      UpperCAmelCase_      :	torch.Tensor					, UpperCAmelCase_      :	torch.Tensor						) ->							torch.Tensor:
							lowerCAmelCase__  =   v_bins.view(((1,) * len(x.shape						)) + (len(UpperCAmelCase_						),)						)
							lowerCAmelCase__  =   x[..., None] - reshaped_bins
							lowerCAmelCase__  =   torch.argmin(torch.abs(UpperCAmelCase_						)					, dim=-1						)
							return nn.functional.one_hot(UpperCAmelCase_					, num_classes=len(UpperCAmelCase_						)						).float()
def 			_lowerCAmelCase(      UpperCAmelCase_      :	torch.Tensor					, UpperCAmelCase_      :	torch.Tensor					, UpperCAmelCase_      :	int = 0					, UpperCAmelCase_      :	int = 0						) ->							torch.Tensor:
							lowerCAmelCase__  =   []
							for i, s in enumerate(data.shape[:no_batch_dims]						):
														lowerCAmelCase__  =   torch.arange(UpperCAmelCase_						)
														lowerCAmelCase__  =   r.view(*(*((1,) * i), -1, *((1,) * (len(inds.shape						) - i - 1)))						)
														ranges.append(UpperCAmelCase_						)

							lowerCAmelCase__  =   [slice(UpperCAmelCase_						) for _ in range(len(data.shape						) - no_batch_dims						)]
							lowerCAmelCase__  =   inds
							ranges.extend(UpperCAmelCase_						)
							# Matt note: Editing this to get around the behaviour of using a list as an array index changing
							# in recent Numpy versions
							return data[tuple(UpperCAmelCase_						)]


_UpperCamelCase  						=			TypeVar("""T""")
def 			_lowerCAmelCase(      UpperCAmelCase_      :	Callable[[T], Any]					, UpperCAmelCase_      :	Dict[Any, Union[dict, list, tuple, T]]					, UpperCAmelCase_      :	Type[T]						) ->							Dict[Any, Union[dict, list, tuple, Any]]:
							lowerCAmelCase__  =   {}
							for k, v in dic.items():
														if isinstance(UpperCAmelCase_					, UpperCAmelCase_						):
																					lowerCAmelCase__  =   dict_map(UpperCAmelCase_					, UpperCAmelCase_					, UpperCAmelCase_						)
														else:
																					lowerCAmelCase__  =   tree_map(UpperCAmelCase_					, UpperCAmelCase_					, UpperCAmelCase_						)

							return new_dict
@overload
def 			_lowerCAmelCase(      UpperCAmelCase_      :	Callable[[T], Any]					, UpperCAmelCase_      :	T					, UpperCAmelCase_      :	Type[T]						) ->							Any:
							...
@overload
def 			_lowerCAmelCase(      UpperCAmelCase_      :	Callable[[T], Any]					, UpperCAmelCase_      :	dict					, UpperCAmelCase_      :	Type[T]						) ->							dict:
							...
@overload
def 			_lowerCAmelCase(      UpperCAmelCase_      :	Callable[[T], Any]					, UpperCAmelCase_      :	list					, UpperCAmelCase_      :	Type[T]						) ->							list:
							...
@overload
def 			_lowerCAmelCase(      UpperCAmelCase_      :	Callable[[T], Any]					, UpperCAmelCase_      :	tuple					, UpperCAmelCase_      :	Type[T]						) ->							tuple:
							...
def 			_lowerCAmelCase(      UpperCAmelCase_      :	str					, UpperCAmelCase_      :	Dict					, UpperCAmelCase_      :	Union[str, Any]						) ->							List[str]:
							if isinstance(UpperCAmelCase_					, UpperCAmelCase_						):
														return dict_map(UpperCAmelCase_					, UpperCAmelCase_					, UpperCAmelCase_						)
							elif isinstance(UpperCAmelCase_					, UpperCAmelCase_						):
														return [tree_map(UpperCAmelCase_					, UpperCAmelCase_					, UpperCAmelCase_						) for x in tree]
							elif isinstance(UpperCAmelCase_					, UpperCAmelCase_						):
														return tuple(tree_map(UpperCAmelCase_					, UpperCAmelCase_					, UpperCAmelCase_						) for x in tree						)
							elif isinstance(UpperCAmelCase_					, UpperCAmelCase_						):
														return fn(UpperCAmelCase_						)
							else:
														print(type(UpperCAmelCase_						)						)
														raise ValueError("""Not supported"""						)






_UpperCamelCase  						=			partial(tree_map, leaf_type=torch.Tensor)





