
'''simple docstring'''
from __future__ import annotations

import random
from collections.abc import Iterable







class 			__A					:
 def __init__( self      :     List[Any]					, lowerCamelCase      :     Tuple							):



   """simple docstring"""





   __A		:  dict[str, bool | None]       				=		{literal: None for literal in literals}
 def __str__( self      :     str							):



   """simple docstring"""





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



   """simple docstring"""





   return len(self.literals							)
 def 		lowercase_( self      :     str					, lowerCamelCase      :     str							):



   """simple docstring"""





   for literal in self.literals:
     __A		:  Optional[int]       				=		literal[:2]
     if symbol in model:
       __A		:  int       				=		model[symbol]
     else:
       continue
     if value is not None:
       # Complement assignment if literal is in complemented form
       if literal.endswith("""'"""							):
         __A		:  int       				=		not value
     __A		:  Any       				=		value






 def 		lowercase_( self      :     Optional[Any]					, lowerCamelCase      :     List[Any]							):



   """simple docstring"""





   for literal in self.literals:
     __A		:  Optional[Any]       				=		literal.rstrip("""'"""							) if literal.endswith("""'"""							) else literal + "'"
     if symbol in self.literals:
       return True

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







class 			__A					:
 def __init__( self      :     Optional[Any]					, lowerCamelCase      :     List[Any]							):



   """simple docstring"""





   __A		:  str       				=		list(lowerCAmelCase__							)






 def __str__( self      :     Optional[Any]							):



   """simple docstring"""





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







def 							A_  (    )					->							Clause:





  """simple docstring"""

  __A		:  Any       				=		[]
  __A		:  int       				=		random.randint(1      ,  5			)
  __A		:  Any       				=		"A"
  __A		:  int       				=		0
  while i < no_of_literals:
    __A		:  Optional[int]       				=		random.randint(1      ,  5			)
    __A		:  str       				=		base_var + str(__lowerCAmelCase			)
    __A		:  Union[str, Any]       				=		random.randint(0      ,  1			)
    if var_complement == 1:
      var_name += "'"
    if var_name in literals:
      i -= 1
    else:
      literals.append(__lowerCAmelCase			)
    i += 1
  return Clause(__lowerCAmelCase			)







def 							A_  (    )					->							Formula:





  """simple docstring"""

  __A		:  set[Clause]       				=		set()
  __A		:  List[str]       				=		random.randint(1      ,  10			)
  while len(__lowerCAmelCase			) < no_of_clauses:
    clauses.add(generate_clause()			)
  return Formula(__lowerCAmelCase			)







def 							A_  (    __SCREAMING_SNAKE_CASE    :	Formula			)					->							tuple[list[Clause], list[str]]:





  """simple docstring"""

  __A		:  Optional[int]       				=		formula.clauses
  __A		:  str       				=		[]
  for clause in formula.clauses:
    for literal in clause.literals:
      __A		:  Any       				=		literal[:2]
      if symbol not in symbols_set:
        symbols_set.append(__lowerCAmelCase			)
  return clauses, symbols_set







def 							A_  (    __SCREAMING_SNAKE_CASE    :	list[Clause]      ,  __SCREAMING_SNAKE_CASE    :	list[str]      ,  __SCREAMING_SNAKE_CASE    :	dict[str, bool | None]			)					->							tuple[list[str], dict[str, bool | None]]:





  """simple docstring"""

  __A		:  List[Any]       				=		[]
  __A		:  dict[str, bool | None]       				=		{}
  __A		:  Tuple       				=		[]

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

  for s in symbols:
    __A		:  Dict       				=		s + "'"
    if (s in literals and sym not in literals) or (
        s not in literals and sym in literals
    ):
      pure_symbols.append(__lowerCAmelCase			)
  for p in pure_symbols:
    __A		:  Tuple       				=		None
  for s in pure_symbols:
    __A		:  int       				=		s + "'"
    if s in literals:
      __A		:  Tuple       				=		True
    elif sym in literals:
      __A		:  str       				=		False
  return pure_symbols, assignment







def 							A_  (    __SCREAMING_SNAKE_CASE    :	list[Clause]      ,  __SCREAMING_SNAKE_CASE    :	dict[str, bool | None]			)					->							tuple[list[str], dict[str, bool | None]]:





  """simple docstring"""

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

  return unit_symbols, assignment







def 							A_  (    __SCREAMING_SNAKE_CASE    :	list[Clause]      ,  __SCREAMING_SNAKE_CASE    :	list[str]      ,  __SCREAMING_SNAKE_CASE    :	dict[str, bool | None]			)					->							tuple[bool | None, dict[str, bool | None] | None]:





  """simple docstring"""

  __A		:  Union[str, Any]       				=		True
  for clause in clauses:
    __A		:  int       				=		clause.evaluate(__lowerCAmelCase			)
    if clause_check is False:
      return False, None
    elif clause_check is None:
      __A		:  Any       				=		False
      continue

  if check_clause_all_true:
    return True, model

  try:
    __A		:  Optional[Any]       				=		find_pure_symbols(__lowerCAmelCase      ,  __lowerCAmelCase      ,  __lowerCAmelCase			)
  except RecursionError:
    print("""raises a RecursionError and is"""			)
    return None, {}
  __A		:  Optional[int]       				=		None
  if len(__lowerCAmelCase			) > 0:
    __A		:  Union[str, Any]       				=		pure_symbols[0], assignment[pure_symbols[0]]

  if p:
    __A		:  Union[str, Any]       				=		model
    __A		:  List[Any]       				=		value
    __A		:  int       				=		list(__lowerCAmelCase			)
    if p in tmp_symbols:
      tmp_symbols.remove(__lowerCAmelCase			)
    return dpll_algorithm(__lowerCAmelCase      ,  __lowerCAmelCase      ,  __lowerCAmelCase			)

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

  return dpll_algorithm(__lowerCAmelCase      ,  __lowerCAmelCase      ,  __lowerCAmelCase			) or dpll_algorithm(__lowerCAmelCase      ,  __lowerCAmelCase      ,  __lowerCAmelCase			)


if __name__ == "__main__":
     import doctest

     doctest.testmod()

     A__ :		Union[str, Any]					=generate_formula()
     print(F"The formula {formula} is", end=' ')

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

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









"""simple docstring"""
import math






def 				__lowerCAmelCase					(					__lowerCAmelCase			:		Tuple						)    ->       Any:
 _UpperCamelCase      :	str					      =  0
 if n % 2 == 0:
  count += 1
  while n % 2 == 0:
   _UpperCamelCase      :	List[Any]					      =  int(n / 2						)
    # the n input value must be odd so that
    # we can skip one element (ie i += 2)

 _UpperCamelCase      :	Tuple					      =  3

 while i <= int(math.sqrt(__lowerCAmelCase						)						):
  if n % i == 0:
   count += 1
   while n % i == 0:
    _UpperCamelCase      :	Any					      =  int(n / i						)
  _UpperCamelCase      :	Optional[Any]					      =  i + 2

 # this condition checks the prime
 # number n is greater than 2

 if n > 2:
  count += 1
 return count


if __name__ == "__main__":
    _SCREAMING_SNAKE_CASE						 =     5_1_2_4_2_1_8_3
    print(f'The number of distinct prime factors is/are {exact_prime_factor_count(n)}')
    print(f'The value of log(log(n)) is {math.log(math.log(n)):.4f}')



