



from __future__ import annotations

import random
from collections.abc import Iterable

class 			__snake_case   :







							'''simple docstring'''






							def __init__(				self	,	a_		):
											a__					   =		{literal: None for literal in literals}


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


							def __len__(				self		):
											return len(self.literals		)


							def _a   (				self	,	a_		):
											for literal in self.literals:
															a__					   =		literal[:2]
															if symbol in model:
																			a__					   =		model[symbol]
															else:
																			continue
															if value is not None:
																			# Complement assignment if literal is in complemented form
																			if literal.endswith("""'"""		):
																							a__					   =		not value
															a__					   =		value







							def _a   (				self	,	a_		):
											for literal in self.literals:
															a__					   =		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   :







							'''simple docstring'''






							def __init__(				self	,	a_		):
											a__					   =		list(a_		)







							def __str__(				self		):
											return "{" + " , ".join(str(a_		) for clause in self.clauses		) + "}"




def       A_   (						):

				"""simple docstring"""
				a__					   =		[]
				a__					   =		random.randint(1			,		5		)
				a__					   =		"""A"""
				a__					   =		0
				while i < no_of_literals:
								a__					   =		random.randint(1			,		5		)
								a__					   =		base_var + str(__a		)
								a__					   =		random.randint(0			,		1		)
								if var_complement == 1:
												var_name += "'"
								if var_name in literals:
												i -= 1
								else:
												literals.append(__a		)
								i += 1
				return Clause(__a		)

def       A_   (						):

				"""simple docstring"""
				a__					   =		set()
				a__					   =		random.randint(1			,		10		)
				while len(__a		) < no_of_clauses:
								clauses.add(generate_clause()		)
				return Formula(__a		)

def       A_   (						__a  :					Formula		):

				"""simple docstring"""
				a__					   =		formula.clauses
				a__					   =		[]
				for clause in formula.clauses:
								for literal in clause.literals:
												a__					   =		literal[:2]
												if symbol not in symbols_set:
																symbols_set.append(__a		)
				return clauses, symbols_set

def       A_   (						__a  :					list[Clause]			,		__a  :					list[str]			,		__a  :					dict[str, bool | None]		):

				"""simple docstring"""
				a__					   =		[]
				a__					   =		{}
				a__					   =		[]

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

				for s in symbols:
								a__					   =		s + """'"""
								if (s in literals and sym not in literals) or (
								    s not in literals and sym in literals
								):
												pure_symbols.append(__a		)
				for p in pure_symbols:
								a__					   =		None
				for s in pure_symbols:
								a__					   =		s + """'"""
								if s in literals:
												a__					   =		True
								elif sym in literals:
												a__					   =		False
				return pure_symbols, assignment

def       A_   (						__a  :					list[Clause]			,		__a  :					dict[str, bool | None]		):

				"""simple docstring"""
				a__					   =		[]
				for clause in clauses:
								if len(__a		) == 1:
												unit_symbols.append(list(clause.literals.keys()		)[0]		)
								else:
												a__   ,      a__					   =		0, 0
												for literal, value in clause.literals.items():
																if value is False:
																				f_count += 1
																elif value is None:
																				a__					   =		literal
																				n_count += 1
												if f_count == len(__a		) - 1 and n_count == 1:
																unit_symbols.append(__a		)
				a__					   =		{}
				for i in unit_symbols:
								a__					   =		i[:2]
								a__					   =		len(__a		) == 2
				a__					   =		[i[:2] for i in unit_symbols]

				return unit_symbols, assignment

def       A_   (						__a  :					list[Clause]			,		__a  :					list[str]			,		__a  :					dict[str, bool | None]		):

				"""simple docstring"""
				a__					   =		True
				for clause in clauses:
								a__					   =		clause.evaluate(__a		)
								if clause_check is False:
												return False, None
								elif clause_check is None:
												a__					   =		False
												continue

				if check_clause_all_true:
								return True, model

				try:
								a__   ,      a__					   =		find_pure_symbols(__a			,		__a			,		__a		)
				except RecursionError:
								print("""raises a RecursionError and is"""		)
								return None, {}
				a__					   =		None
				if len(__a		) > 0:
								a__   ,      a__					   =		pure_symbols[0], assignment[pure_symbols[0]]

				if p:
								a__					   =		model
								a__					   =		value
								a__					   =		list(__a		)
								if p in tmp_symbols:
												tmp_symbols.remove(__a		)
								return dpll_algorithm(__a			,		__a			,		__a		)

				a__   ,      a__					   =		find_unit_clauses(__a			,		__a		)
				a__					   =		None
				if len(__a		) > 0:
								a__   ,      a__					   =		unit_symbols[0], assignment[unit_symbols[0]]
				if p:
								a__					   =		model
								a__					   =		value
								a__					   =		list(__a		)
								if p in tmp_symbols:
												tmp_symbols.remove(__a		)
								return dpll_algorithm(__a			,		__a			,		__a		)
				a__					   =		symbols[0]
				a__					   =		symbols[1:]
				a__   ,      a__					   =		model, model
				a__   ,      a__					   =		True, False

				return dpll_algorithm(__a			,		__a			,		__a		) or dpll_algorithm(__a			,		__a			,		__a		)


if __name__ == "__main__":
	import doctest

	doctest.testmod()

	UpperCAmelCase							=       generate_formula()
	print(f"""The formula {formula} is""", end=""" """)

	UpperCAmelCase			,							UpperCAmelCase							=       generate_parameters(formula)
	UpperCAmelCase			,							UpperCAmelCase							=       dpll_algorithm(clauses, symbols, {})

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








from functools import lru_cache







@lru_cache
def      A__				(       SCREAMING_SNAKE_CASE__)  ->       int:
  if not isinstance(SCREAMING_SNAKE_CASE__   ,   SCREAMING_SNAKE_CASE__):
    raise TypeError("""k must be an integer.""")
  if k < 0:
    raise ValueError("""k integer must be greater or equal to zero.""")
  if k == 0:
    return 0
  elif k == 1:
    return 1
  else:
    return fibonacci(k - 1) + fibonacci(k - 2)







def      A__				(       SCREAMING_SNAKE_CASE__   ,   SCREAMING_SNAKE_CASE__)  ->       int:
  __snake_case:		str					    =		len(SCREAMING_SNAKE_CASE__)
  # Find m such that F_m >= n where F_i is the i_th fibonacci number.
  __snake_case:		Optional[int]					    =		0
  while True:
    if fibonacci(SCREAMING_SNAKE_CASE__) >= len_list:
      __snake_case:		str					    =		i
      break
    i += 1
  __snake_case:		Tuple					    =		0
  while fibb_k > 0:
    __snake_case:		str					    =		min(
        offset + fibonacci(fibb_k - 1)   ,   len_list - 1)  # Prevent out of range
    __snake_case:		List[Any]					    =		arr[index_k]
    if item_k_1 == val:
      return index_k
    elif val < item_k_1:
      fibb_k -= 1
    elif val > item_k_1:
      offset += fibonacci(fibb_k - 1)
      fibb_k -= 2
  else:
    return -1


if __name__ == "__main__":
      import doctest

      doctest.testmod()






