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__     ,					__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__     ,					__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__     ,					__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__     ,					__magic_name__	:	Dict			     =  find_unit_clauses(_snake_case   ,     _snake_case     )
  	__magic_name__	:	int			     =  None
  	if len(_snake_case     ) > 0:
  		__magic_name__     ,					__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__     ,					__magic_name__	:	Union[str, Any]			     =  model, model
  	__magic_name__     ,					__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   ,snake_case				:	List[str]							   =						generate_parameters(formula)
  				snake_case   ,snake_case				:	List[Any]							   =						dpll_algorithm(clauses, symbols, {})
  
  				if solution:
  								print(F"satisfiable with the assignment {model}.")
  				else:
  								print("not satisfiable.")
  
style_context: 
  from __future__ import annotations
  
  snake_case				:	List[str]							   =						"Alexander Joslin"
  
  
  
  
  
  
  def 						lowerCAmelCase_		(      _snake_case   :						int     )       ->	int:
  
  
  
  
  
  
  
  	'''simple docstring'''
  
  
  	if number <= 0:
  		__magic_name__	:	int			     =  F'''n must be greater than 0. Got n = {number}'''
  		raise ValueError(_snake_case     )
  
  	__magic_name__	:	List[str]			     =  [number + 1] * (number + 1)
  
  	# starting position
  	__magic_name__	:	Any			     =  0
  	for i in range(1   ,     _snake_case     ):
  		__magic_name__	:	Optional[int]			     =  min(table[i + 1]   ,     table[i] + 1     )
  		# check if out of bounds
  		if i * 2 <= number:
  			__magic_name__	:	str			     =  min(table[i * 2]   ,     table[i] + 1     )
  		# check if out of bounds
  		if i * 3 <= number:
  			__magic_name__	:	Any			     =  min(table[i * 3]   ,     table[i] + 1     )
  	return table[number]
  
  
  if __name__ == "__main__":
  				import doctest
  
  				doctest.testmod()
  