code: 
  
  
  
  
  
  '''simple docstring'''
  
  
  
  
  
  
  
  from __future__ import annotations
  
  import random
  from collections.abc import Iterable
  
  
  
  
  
  class 					_UpperCAmelCase							:
    def __init__(						self,__SCREAMING_SNAKE_CASE    ):
  
  
  
       '''simple docstring'''
  
       __lowerCAmelCase      		=       {literal: None for literal in literals}
    def __str__(						self    ):
  
  
  
       '''simple docstring'''
  
       return "{" + " , ".join(self.literals    ) + "}"
    def __len__(						self    ):
  
  
  
       '''simple docstring'''
  
       return len(self.literals    )
    def 						lowerCamelCase__    (						self,__SCREAMING_SNAKE_CASE    ):
  
  
  
       '''simple docstring'''
  
       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,__SCREAMING_SNAKE_CASE    ):
  
  
  
       '''simple docstring'''
  
       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 					_UpperCAmelCase							:
    def __init__(						self,__SCREAMING_SNAKE_CASE    ):
  
  
  
       '''simple docstring'''
  
       __lowerCAmelCase      		=       list(lowerCAmelCase_    )
  
  
  
  
  
  
  
    def __str__(						self    ):
  
  
  
       '''simple docstring'''
  
       return "{" + " , ".join(str(lowerCAmelCase_    ) for clause in self.clauses    ) + "}"
  
  
  
  
  
  def      _lowerCAmelCase		(		)      ->      Clause:
     __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      _lowerCAmelCase		(		)      ->      Formula:
     __lowerCAmelCase      		=       set()
     __lowerCAmelCase      		=       random.randint(1      ,			10			)
     while len(lowercase			) < no_of_clauses:
        clauses.add(generate_clause()			)
     return Formula(lowercase			)
  
  
  
  
  
  def      _lowerCAmelCase		(		lowercase			)      ->      tuple[list[Clause], list[str]]:
     __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      _lowerCAmelCase		(		lowercase      ,			lowercase      ,			lowercase			)      ->      tuple[list[str], dict[str, bool | None]]:
     __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      _lowerCAmelCase		(		lowercase      ,			lowercase			)      ->      tuple[list[str], dict[str, bool | None]]:
     __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      _lowerCAmelCase		(		lowercase      ,			lowercase      ,			lowercase			)      ->      tuple[bool | None, dict[str, bool | None] | None]:
     __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()
  
   _a      :				int											=					generate_formula()
   print(f'The formula {formula} is', end=""" """)
  
   _a					,_a      :				int											=					generate_parameters(formula)
   _a					,_a      :				int											=					dpll_algorithm(clauses, symbols, {})
  
   if solution:
    print(f'satisfiable with the assignment {model}.')
   else:
    print("""not satisfiable.""")
  
  
style_context: 
  
  
  
  
  
  '''simple docstring'''
  
  
  
  
  
  
  
  import math
  import sys
  
  
  
  
  
  class 					_UpperCAmelCase							:
  		def __init__(						self    ):
  
  
  
  					'''simple docstring'''
  
  					__lowerCAmelCase      		=       0
  					__lowerCAmelCase      		=       []
  					__lowerCAmelCase      		=       {}  # To store the pos of node in array
  		def 						lowerCamelCase__    (						self    ):
  
  
  
  					'''simple docstring'''
  
  					return self.cur_size == 0
  		def 						lowerCamelCase__    (						self,__SCREAMING_SNAKE_CASE    ):
  
  
  
  					'''simple docstring'''
  
  					__lowerCAmelCase      		=       self.left(__SCREAMING_SNAKE_CASE    )
  					__lowerCAmelCase      		=       self.right(__SCREAMING_SNAKE_CASE    )
  					if lc < self.cur_size and self.array(__SCREAMING_SNAKE_CASE    )[0] < self.array(__SCREAMING_SNAKE_CASE    )[0]:
  								__lowerCAmelCase      		=       lc
  					else:
  								__lowerCAmelCase      		=       idx
  					if rc < self.cur_size and self.array(__SCREAMING_SNAKE_CASE    )[0] < self.array(__SCREAMING_SNAKE_CASE    )[0]:
  								__lowerCAmelCase      		=       rc
  					if smallest != idx:
  								self.swap(__SCREAMING_SNAKE_CASE,__SCREAMING_SNAKE_CASE    )
  								self.min_heapify(__SCREAMING_SNAKE_CASE    )
  		def 						lowerCamelCase__    (						self,__SCREAMING_SNAKE_CASE    ):
  
  
  
  					'''simple docstring'''
  
  					__lowerCAmelCase      		=       self.cur_size
  					self.cur_size += 1
  					self.array.append((sys.maxsize, tup[1])    )
  					self.decrease_key((sys.maxsize, tup[1]),tup[0]    )
  		def 						lowerCamelCase__    (						self    ):
  
  
  
  					'''simple docstring'''
  
  					__lowerCAmelCase      		=       self.array[0][1]
  					__lowerCAmelCase      		=       self.array[self.cur_size - 1]
  					self.cur_size -= 1
  					self.min_heapify(1    )
  					del self.pos[min_node]
  					return min_node
  		def 						lowerCamelCase__    (						self,__SCREAMING_SNAKE_CASE    ):
  
  
  
  					'''simple docstring'''
  
  					return 2 * i + 1
  		def 						lowerCamelCase__    (						self,__SCREAMING_SNAKE_CASE    ):
  
  
  
  					'''simple docstring'''
  
  					return 2 * i + 2
  		def 						lowerCamelCase__    (						self,__SCREAMING_SNAKE_CASE    ):
  
  
  
  					'''simple docstring'''
  
  					return math.floor(i / 2    )
  		def 						lowerCamelCase__    (						self,__SCREAMING_SNAKE_CASE,__SCREAMING_SNAKE_CASE    ):
  
  
  
  					'''simple docstring'''
  
  					__lowerCAmelCase      		=       j
  					__lowerCAmelCase      		=       i
  					__lowerCAmelCase      		=       self.array[i]
  					__lowerCAmelCase      		=       self.array[j]
  					__lowerCAmelCase      		=       temp
  
  
  
  
  
  
  
  		def 						lowerCamelCase__    (						self,__SCREAMING_SNAKE_CASE,__SCREAMING_SNAKE_CASE    ):
  
  
  
  					'''simple docstring'''
  
  					__lowerCAmelCase      		=       self.pos[tup[1]]
  					# assuming the new_d is atmost old_d
  					__lowerCAmelCase      		=       (new_d, tup[1])
  					while idx > 0 and self.array[self.par(__SCREAMING_SNAKE_CASE    )][0] > self.array[idx][0]:
  								self.swap(__SCREAMING_SNAKE_CASE,self.par(__SCREAMING_SNAKE_CASE    )    )
  								__lowerCAmelCase      		=       self.par(__SCREAMING_SNAKE_CASE    )
  
  
  
  
  
  class 					_UpperCAmelCase							:
  		def __init__(						self,__SCREAMING_SNAKE_CASE    ):
  
  
  
  					'''simple docstring'''
  
  					__lowerCAmelCase      		=       {}  # To store graph: u -> (v,w)
  					__lowerCAmelCase      		=       num  # Number of nodes in graph
  					# To store the distance from source vertex
  					__lowerCAmelCase      		=       [0] * self.num_nodes
  					__lowerCAmelCase      		=       [-1] * self.num_nodes  # To store the path
  		def 						lowerCamelCase__    (						self,__SCREAMING_SNAKE_CASE,__SCREAMING_SNAKE_CASE,__SCREAMING_SNAKE_CASE    ):
  
  
  
  					'''simple docstring'''
  
  					if u in self.adjList:
  								self.adjList[u].append((v, w)    )
  					else:
  								__lowerCAmelCase      		=       [(v, w)]
  
  					# Assuming undirected graph
  					if v in self.adjList:
  								self.adjList[v].append((u, w)    )
  					else:
  								__lowerCAmelCase      		=       [(u, w)]
  		def 						lowerCamelCase__    (						self    ):
  
  
  
  					'''simple docstring'''
  
  					for u in self.adjList:
  								print(__SCREAMING_SNAKE_CASE,"""->""",""" -> """.join(str(f'{v}({w})'    ) for v, w in self.adjList[u]    )    )
  		def 						lowerCamelCase__    (						self,__SCREAMING_SNAKE_CASE    ):
  
  
  
  					'''simple docstring'''
  
  					__lowerCAmelCase      		=       [-1] * self.num_nodes
  					# src is the source node
  					__lowerCAmelCase      		=       0
  					__lowerCAmelCase      		=       PriorityQueue()
  					q.insert((0, src)    )  # (dist from src, node)
  					for u in self.adjList:
  								if u != src:
  											__lowerCAmelCase      		=       sys.maxsize  # Infinity
  											__lowerCAmelCase      		=       -1
  
  					while not q.is_empty():
  								__lowerCAmelCase      		=       q.extract_min()  # Returns node with the min dist from source
  								# Update the distance of all the neighbours of u and
  								# if their prev dist was INFINITY then push them in Q
  								for v, w in self.adjList[u]:
  											__lowerCAmelCase      		=       self.dist[u] + w
  											if self.dist[v] > new_dist:
  														if self.dist[v] == sys.maxsize:
  																	q.insert((new_dist, v)    )
  														else:
  																	q.decrease_key((self.dist[v], v),__SCREAMING_SNAKE_CASE    )
  														__lowerCAmelCase      		=       new_dist
  														__lowerCAmelCase      		=       u
  
          # Show the shortest distances from src
  					self.show_distances(__SCREAMING_SNAKE_CASE    )
  		def 						lowerCamelCase__    (						self,__SCREAMING_SNAKE_CASE    ):
  
  
  
  					'''simple docstring'''
  
  					print(f'Distance from node: {src}'    )
  					for u in range(self.num_nodes    ):
  								print(f'Node {u} has distance: {self.dist[u]}'    )
  
  
  
  
  
  
  
  		def 						lowerCamelCase__    (						self,__SCREAMING_SNAKE_CASE,__SCREAMING_SNAKE_CASE    ):
  
  
  
  					'''simple docstring'''
  
  					__lowerCAmelCase      		=       []
  					__lowerCAmelCase      		=       0
  					__lowerCAmelCase      		=       dest
  					# Backtracking from dest to src
  					while self.par[temp] != -1:
  								path.append(__SCREAMING_SNAKE_CASE    )
  								if temp != src:
  											for v, w in self.adjList[temp]:
  														if v == self.par[temp]:
  																	cost += w
  																	break
  								__lowerCAmelCase      		=       self.par[temp]
  					path.append(__SCREAMING_SNAKE_CASE    )
  					path.reverse()
  
  					print(f'----Path to reach {dest} from {src}----'    )
  					for u in path:
  								print(f'{u}',end=""" """    )
  								if u != dest:
  											print("""-> """,end=""""""    )
  
  					print("""\nTotal cost of path: """,__SCREAMING_SNAKE_CASE    )
  
  
  if __name__ == "__main__":
  	_a      :				str											=					Graph(9)
  	graph.add_edge(0, 1, 4)
  	graph.add_edge(0, 7, 8)
  	graph.add_edge(1, 2, 8)
  	graph.add_edge(1, 7, 1_1)
  	graph.add_edge(2, 3, 7)
  	graph.add_edge(2, 8, 2)
  	graph.add_edge(2, 5, 4)
  	graph.add_edge(3, 4, 9)
  	graph.add_edge(3, 5, 1_4)
  	graph.add_edge(4, 5, 1_0)
  	graph.add_edge(5, 6, 2)
  	graph.add_edge(6, 7, 1)
  	graph.add_edge(6, 8, 6)
  	graph.add_edge(7, 8, 7)
  	graph.show_graph()
  	graph.dijkstra(0)
  	graph.show_path(0, 4)
  
  # OUTPUT
  # 0 -> 1(4) -> 7(8)
  # 1 -> 0(4) -> 2(8) -> 7(11)
  # 7 -> 0(8) -> 1(11) -> 6(1) -> 8(7)
  # 2 -> 1(8) -> 3(7) -> 8(2) -> 5(4)
  # 3 -> 2(7) -> 4(9) -> 5(14)
  # 8 -> 2(2) -> 6(6) -> 7(7)
  # 5 -> 2(4) -> 3(14) -> 4(10) -> 6(2)
  # 4 -> 3(9) -> 5(10)
  # 6 -> 5(2) -> 7(1) -> 8(6)
  # Distance from node: 0
  # Node 0 has distance: 0
  # Node 1 has distance: 4
  # Node 2 has distance: 12
  # Node 3 has distance: 19
  # Node 4 has distance: 21
  # Node 5 has distance: 11
  # Node 6 has distance: 9
  # Node 7 has distance: 8
  # Node 8 has distance: 14
  # ----Path to reach 4 from 0----
  # 0 -> 7 -> 6 -> 5 -> 4
  # Total cost of path:  21
  
  