


from __future__ import annotations

import random
from collections.abc import Iterable


class        __lowerCamelCase      :


				"""simple docstring"""







				def __init__(				self     :    Optional[Any]	,							lowerCamelCase_     :    list[str]					):
								_lowerCAmelCase  						={literal: None for literal in literals}






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






				def __len__(				self     :    List[str]					):
								return len(self.literals					)






				def        lowerCAmelCase__			(				self     :    Tuple	,							lowerCamelCase_     :    dict[str, bool | None]					):
								for literal in self.literals:
												_lowerCAmelCase  						=literal[:2]
												if symbol in model:
																_lowerCAmelCase  						=model[symbol]
												else:
																continue
												if value is not None:
																# Complement assignment if literal is in complemented form
																if literal.endswith("""'"""					):
																				_lowerCAmelCase  						=not value
												_lowerCAmelCase  						=value






				def        lowerCAmelCase__			(				self     :    Any	,							lowerCamelCase_     :    dict[str, bool | None]					):
								for literal in self.literals:
												_lowerCAmelCase  						=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        __lowerCamelCase      :


				"""simple docstring"""







				def __init__(				self     :    Union[str, Any]	,							lowerCamelCase_     :    Iterable[Clause]					):
								_lowerCAmelCase  						=list(lowerCamelCase_					)






				def __str__(				self     :    List[Any]					):
								return "{" + " , ".join(str(lowerCamelCase_					) for clause in self.clauses					) + "}"

def 		snake_case_   (				):






				'''simple docstring'''






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

def 		snake_case_   (				):






				'''simple docstring'''






				_lowerCAmelCase  						=set()
				_lowerCAmelCase  						=random.randint(1       ,     10     )
				while len(lowercase__     ) < no_of_clauses:
								clauses.add(generate_clause()     )
				return Formula(lowercase__     )

def 		snake_case_   (				lowercase__     :    Formula     ):






				'''simple docstring'''






				_lowerCAmelCase  						=formula.clauses
				_lowerCAmelCase  						=[]
				for clause in formula.clauses:
								for literal in clause.literals:
												_lowerCAmelCase  						=literal[:2]
												if symbol not in symbols_set:
																symbols_set.append(lowercase__     )
				return clauses, symbols_set

def 		snake_case_   (				lowercase__     :    list[Clause]       ,     lowercase__     :    list[str]       ,     lowercase__     :    dict[str, bool | None]     ):






				'''simple docstring'''






				_lowerCAmelCase  						=[]
				_lowerCAmelCase  						={}
				_lowerCAmelCase  						=[]

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

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

def 		snake_case_   (				lowercase__     :    list[Clause]       ,     lowercase__     :    dict[str, bool | None]     ):






				'''simple docstring'''






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

				return unit_symbols, assignment

def 		snake_case_   (				lowercase__     :    list[Clause]       ,     lowercase__     :    list[str]       ,     lowercase__     :    dict[str, bool | None]     ):






				'''simple docstring'''






				_lowerCAmelCase  						=True
				for clause in clauses:
								_lowerCAmelCase  						=clause.evaluate(lowercase__     )
								if clause_check is False:
												return False, None
								elif clause_check is None:
												_lowerCAmelCase  						=False
												continue

				if check_clause_all_true:
								return True, model

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

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

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

				return dpll_algorithm(lowercase__       ,     lowercase__       ,     lowercase__     ) or dpll_algorithm(lowercase__       ,     lowercase__       ,     lowercase__     )


if __name__ == "__main__":
		import doctest

		doctest.testmod()

		__SCREAMING_SNAKE_CASE	:							List[Any]           =  generate_formula()
		print(F'The formula {formula} is', end=''' ''')

		__SCREAMING_SNAKE_CASE      , __SCREAMING_SNAKE_CASE	:							List[str]           =  generate_parameters(formula)
		__SCREAMING_SNAKE_CASE      , __SCREAMING_SNAKE_CASE	:							List[str]           =  dpll_algorithm(clauses, symbols, {})

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





from __future__ import annotations

__SCREAMING_SNAKE_CASE	:							Union[str, Any]           =  [2, 1, 4, 5, 6, 0, 8, 9, 1, 2, 0, 6, 4, 2, 0, 6, 5, 3, 2, 7]


class        __lowerCamelCase      :


				"""simple docstring"""







				def __init__(				self     :    Dict	,							lowerCamelCase_     :    int					):
								_lowerCAmelCase  						=-1
								_lowerCAmelCase  						=-1
								_lowerCAmelCase  						=[-1] * length
								_lowerCAmelCase  						=None
								_lowerCAmelCase  						=None






				def __repr__(				self     :    Tuple					):
								return F"Node(min_value={self.minn} max_value={self.maxx})"

def 		snake_case_   (				lowercase__     :    list[int]     ):






				'''simple docstring'''






				_lowerCAmelCase  						=Node(len(lowercase__     )     )
				_lowerCAmelCase						,							_lowerCAmelCase  						=min(lowercase__     ), max(lowercase__     )
				# Leaf node case where the node contains only one unique value
				if root.minn == root.maxx:
								return root
				_lowerCAmelCase  						=(root.minn + root.maxx) // 2

				_lowerCAmelCase  						=[]
				_lowerCAmelCase  						=[]

				for index, num in enumerate(lowercase__     ):
								if num <= pivot:
												left_arr.append(lowercase__     )
								else:
												right_arr.append(lowercase__     )
								_lowerCAmelCase  						=len(lowercase__     )
				_lowerCAmelCase  						=build_tree(lowercase__     )
				_lowerCAmelCase  						=build_tree(lowercase__     )
				return root

def 		snake_case_   (				lowercase__     :    Node | None       ,     lowercase__     :    int       ,     lowercase__     :    int     ):






				'''simple docstring'''






				if index < 0 or node is None:
								return 0
				# Leaf node cases
				if node.minn == node.maxx:
								return index + 1 if node.minn == num else 0
				_lowerCAmelCase  						=(node.minn + node.maxx) // 2
				if num <= pivot:
								# go the left subtree and map index to the left subtree
								return rank_till_index(node.left       ,     lowercase__       ,     node.map_left[index] - 1     )
				else:
								# go to the right subtree and map index to the right subtree
								return rank_till_index(node.right       ,     lowercase__       ,     index - node.map_left[index]     )

def 		snake_case_   (				lowercase__     :    Node | None       ,     lowercase__     :    int       ,     lowercase__     :    int       ,     lowercase__     :    int     ):






				'''simple docstring'''






				if start > end:
								return 0
				_lowerCAmelCase  						=rank_till_index(lowercase__       ,     lowercase__       ,     lowercase__     )
				_lowerCAmelCase  						=rank_till_index(lowercase__       ,     lowercase__       ,     start - 1     )
				return rank_till_end - rank_before_start

def 		snake_case_   (				lowercase__     :    Node | None       ,     lowercase__     :    int       ,     lowercase__     :    int       ,     lowercase__     :    int     ):






				'''simple docstring'''






				if index > (end - start) or start > end or node is None:
								return -1
				# Leaf node case
				if node.minn == node.maxx:
								return node.minn
				# Number of elements in the left subtree in interval [start, end]
				_lowerCAmelCase  						=node.map_left[end] - (
				    node.map_left[start - 1] if start else 0
				)
				if num_elements_in_left_tree > index:
								return quantile(
								    node.left       ,     lowercase__       ,     (node.map_left[start - 1] if start else 0)       ,     node.map_left[end] - 1       ,     )
				else:
								return quantile(
								    node.right       ,     index - num_elements_in_left_tree       ,     start - (node.map_left[start - 1] if start else 0)       ,     end - node.map_left[end]       ,     )

def 		snake_case_   (				lowercase__     :    Node | None       ,     lowercase__     :    int       ,     lowercase__     :    int       ,     lowercase__     :    int       ,     lowercase__     :    int     ):






				'''simple docstring'''






				if (
				    start > end
				    or node is None
				    or start_num > end_num
				    or node.minn > end_num
				    or node.maxx < start_num
				):
								return 0
				if start_num <= node.minn and node.maxx <= end_num:
								return end - start + 1
				_lowerCAmelCase  						=range_counting(
				    node.left       ,     (node.map_left[start - 1] if start else 0)       ,     node.map_left[end] - 1       ,     lowercase__       ,     lowercase__       ,     )
				_lowerCAmelCase  						=range_counting(
				    node.right       ,     start - (node.map_left[start - 1] if start else 0)       ,     end - node.map_left[end]       ,     lowercase__       ,     lowercase__       ,     )
				return left + right


if __name__ == "__main__":
		import doctest

		doctest.testmod()

