


from typing import Any



def     __lowerCAmelCase						(							snake_case	:    list						,      snake_case	:    list						,      snake_case	:    dict						,      snake_case	:    dict						,      snake_case	:    dict						,      )       ->							list:
   _validation(
       snake_case						,      snake_case						,      snake_case						,      snake_case						,      snake_case						,      )
   # Creates data structures and fill initial step
   __lowerCamelCase:				dict     =	{}
   __lowerCamelCase:				dict     =	{}
   for state in states_space:
      __lowerCamelCase:				str     =	observations_space[0]
      __lowerCamelCase:				Optional[int]     =	(
          initial_probabilities[state] * emission_probabilities[state][observation]
      )
      __lowerCamelCase:				Optional[Any]     =	None

   # Fills the data structure with the probabilities of
   # different transitions and pointers to previous states
   for o in range(1						,      len(snake_case ) ):
      __lowerCamelCase:				Any     =	observations_space[o]
      __lowerCamelCase:				List[Any]     =	observations_space[o - 1]
      for state in states_space:
         # Calculates the argmax for probability function
         __lowerCamelCase:				Any     =	""""""
         __lowerCamelCase:				int     =	-1
         for k_state in states_space:
            __lowerCamelCase:				Union[str, Any]     =	(
                probabilities[(k_state, prior_observation)]
                * transition_probabilities[k_state][state]
                * emission_probabilities[state][observation]
            )
            if probability > max_probability:
               __lowerCamelCase:				str     =	probability
               __lowerCamelCase:				Optional[Any]     =	k_state

            # Update probabilities and pointers dicts
         __lowerCamelCase:				Optional[Any]     =	(
             probabilities[(arg_max, prior_observation)]
             * transition_probabilities[arg_max][state]
             * emission_probabilities[state][observation]
         )

         __lowerCamelCase:				Optional[int]     =	arg_max

    # The final observation
   __lowerCamelCase:				Tuple     =	observations_space[len(snake_case ) - 1]

   # argmax for given final observation
   __lowerCamelCase:				List[Any]     =	""""""
   __lowerCamelCase:				Dict     =	-1
   for k_state in states_space:
      __lowerCamelCase:				int     =	probabilities[(k_state, final_observation)]
      if probability > max_probability:
         __lowerCamelCase:				int     =	probability
         __lowerCamelCase:				Union[str, Any]     =	k_state
   __lowerCamelCase:				List[str]     =	arg_max

   # Process pointers backwards
   __lowerCamelCase:				str     =	last_state
   __lowerCamelCase:				Union[str, Any]     =	[]
   for o in range(len(snake_case ) - 1						,      -1						,      -1 ):
      result.append(snake_case )
      __lowerCamelCase:				str     =	pointers[previous, observations_space[o]]
   result.reverse()

   return result



def     __lowerCAmelCase						(							snake_case	:    Any						,      snake_case	:    Any						,      snake_case	:    Any						,      snake_case	:    Any						,      snake_case	:    Any						,      )       ->							None:
   _validate_not_empty(
       snake_case						,      snake_case						,      snake_case						,      snake_case						,      snake_case						,      )
   _validate_lists(snake_case						,      snake_case )
   _validate_dicts(
       snake_case						,      snake_case						,      snake_case )



def     __lowerCAmelCase						(							snake_case	:    Any						,      snake_case	:    Any						,      snake_case	:    Any						,      snake_case	:    Any						,      snake_case	:    Any						,      )       ->							None:
   if not all(
       [
           observations_space,
           states_space,
           initial_probabilities,
           transition_probabilities,
           emission_probabilities,
       ] ):
      raise ValueError("""There's an empty parameter""" )



def     __lowerCAmelCase						(							snake_case	:    Any						,      snake_case	:    Any )       ->							None:
   _validate_list(snake_case						,      """observations_space""" )
   _validate_list(snake_case						,      """states_space""" )



def     __lowerCAmelCase						(							snake_case	:    Any						,      snake_case	:    str )       ->							None:
   if not isinstance(_object						,      snake_case ):
      __lowerCamelCase:				Optional[int]     =	f'{var_name} must be a list'
      raise ValueError(snake_case )
   else:
      for x in _object:
         if not isinstance(snake_case						,      snake_case ):
            __lowerCamelCase:				Optional[Any]     =	f'{var_name} must be a list of strings'
            raise ValueError(snake_case )



def     __lowerCAmelCase						(							snake_case	:    Any						,      snake_case	:    Any						,      snake_case	:    Any						,      )       ->							None:
   _validate_dict(snake_case						,      """initial_probabilities"""						,      snake_case )
   _validate_nested_dict(snake_case						,      """transition_probabilities""" )
   _validate_nested_dict(snake_case						,      """emission_probabilities""" )



def     __lowerCAmelCase						(							snake_case	:    Any						,      snake_case	:    str )       ->							None:
   _validate_dict(_object						,      snake_case						,      snake_case )
   for x in _object.values():
      _validate_dict(snake_case						,      snake_case						,      snake_case						,      snake_case )



def     __lowerCAmelCase						(							snake_case	:    Any						,      snake_case	:    str						,      snake_case	:    type						,      snake_case	:    bool = False )       ->							None:
   if not isinstance(_object						,      snake_case ):
      __lowerCamelCase:				Any     =	f'{var_name} must be a dict'
      raise ValueError(snake_case )
   if not all(isinstance(snake_case						,      snake_case ) for x in _object ):
      __lowerCamelCase:				List[str]     =	f'{var_name} all keys must be strings'
      raise ValueError(snake_case )
   if not all(isinstance(snake_case						,      snake_case ) for x in _object.values() ):
      __lowerCamelCase:				Any     =	"""nested dictionary """ if nested else """"""
      __lowerCamelCase:				List[Any]     =	f'{var_name} {nested_text}all values must be {value_type.__name__}'
      raise ValueError(snake_case )


if __name__ == "__main__":
   from doctest import testmod

   testmod()






from __future__ import annotations

from typing import Any



class        a					:





     def __init__(    self					:				str    ,	SCREAMING_SNAKE_CASE_					:				int    ,	SCREAMING_SNAKE_CASE_					:				int    ,	SCREAMING_SNAKE_CASE_					:				float = 0							):
        __lowerCamelCase   ,     __lowerCamelCase:				Dict     =	row, column
        __lowerCamelCase:				Optional[Any]     =	[[default_value for c in range(SCREAMING_SNAKE_CASE_							)] for r in range(SCREAMING_SNAKE_CASE_							)]





     def __str__(    self					:				Union[str, Any]							):
        __lowerCamelCase:				str     =	F'Matrix consist of {self.row} rows and {self.column} columns\n'

        # Make string identifier
        __lowerCamelCase:				Optional[int]     =	0
        for row_vector in self.array:
           for obj in row_vector:
              __lowerCamelCase:				int     =	max(SCREAMING_SNAKE_CASE_    ,	len(str(SCREAMING_SNAKE_CASE_							)							)							)
        __lowerCamelCase:				Union[str, Any]     =	F'%{max_element_length}s'

        # Make string and return
        def single_line(SCREAMING_SNAKE_CASE_					:				list[float]							) -> str:
           nonlocal string_format_identifier
           __lowerCamelCase:				Any     =	"""["""
           line += ", ".join(string_format_identifier % (obj,) for obj in row_vector							)
           line += "]"
           return line

        s += "\n".join(single_line(SCREAMING_SNAKE_CASE_							) for row_vector in self.array							)
        return s





     def __repr__(    self					:				Union[str, Any]							):
        return str(self							)





     def  SCREAMING_SNAKE_CASE__       (    self					:				Any    ,	SCREAMING_SNAKE_CASE_					:				tuple[int, int]							):
        if not (isinstance(SCREAMING_SNAKE_CASE_    ,	(list, tuple)							) and len(SCREAMING_SNAKE_CASE_							) == 2):
           return False
        elif not (0 <= loc[0] < self.row and 0 <= loc[1] < self.column):
           return False
        else:
           return True





     def __getitem__(    self					:				Optional[int]    ,	SCREAMING_SNAKE_CASE_					:				tuple[int, int]							):
        assert self.validate_indicies(SCREAMING_SNAKE_CASE_							)
        return self.array[loc[0]][loc[1]]





     def __setitem__(    self					:				Any    ,	SCREAMING_SNAKE_CASE_					:				tuple[int, int]    ,	SCREAMING_SNAKE_CASE_					:				float							):
        assert self.validate_indicies(SCREAMING_SNAKE_CASE_							)
        __lowerCamelCase:				List[Any]     =	value





     def __add__(    self					:				Union[str, Any]    ,	SCREAMING_SNAKE_CASE_					:				Matrix							):
        assert isinstance(SCREAMING_SNAKE_CASE_    ,	SCREAMING_SNAKE_CASE_							)
        assert self.row == another.row and self.column == another.column

        # Add
        __lowerCamelCase:				Any     =	Matrix(self.row    ,	self.column							)
        for r in range(self.row							):
           for c in range(self.column							):
              __lowerCamelCase:				Tuple     =	self[r, c] + another[r, c]
        return result





     def __neg__(    self					:				Any							):
        __lowerCamelCase:				List[str]     =	Matrix(self.row    ,	self.column							)
        for r in range(self.row							):
           for c in range(self.column							):
              __lowerCamelCase:				int     =	-self[r, c]
        return result





     def __sub__(    self					:				Union[str, Any]    ,	SCREAMING_SNAKE_CASE_					:				Matrix							):
        return self + (-another)





     def __mul__(    self					:				Tuple    ,	SCREAMING_SNAKE_CASE_					:				int | float | Matrix							):

        if isinstance(SCREAMING_SNAKE_CASE_    ,	(int, float)							):  # Scalar multiplication
           __lowerCamelCase:				List[str]     =	Matrix(self.row    ,	self.column							)
           for r in range(self.row							):
              for c in range(self.column							):
                 __lowerCamelCase:				List[Any]     =	self[r, c] * another
           return result
        elif isinstance(SCREAMING_SNAKE_CASE_    ,	SCREAMING_SNAKE_CASE_							):  # Matrix multiplication
           assert self.column == another.row
           __lowerCamelCase:				Optional[int]     =	Matrix(self.row    ,	another.column							)
           for r in range(self.row							):
              for c in range(another.column							):
                 for i in range(self.column							):
                    result[r, c] += self[r, i] * another[i, c]
           return result
        else:
           __lowerCamelCase:				int     =	F'Unsupported type given for another ({type(SCREAMING_SNAKE_CASE_							)})'
           raise TypeError(SCREAMING_SNAKE_CASE_							)





     def  SCREAMING_SNAKE_CASE__       (    self					:				Any							):
        __lowerCamelCase:				int     =	Matrix(self.column    ,	self.row							)
        for r in range(self.row							):
           for c in range(self.column							):
              __lowerCamelCase:				List[str]     =	self[r, c]
        return result







     def  SCREAMING_SNAKE_CASE__       (    self					:				int    ,	SCREAMING_SNAKE_CASE_					:				Matrix    ,	SCREAMING_SNAKE_CASE_					:				Matrix							):
        assert isinstance(SCREAMING_SNAKE_CASE_    ,	SCREAMING_SNAKE_CASE_							) and isinstance(SCREAMING_SNAKE_CASE_    ,	SCREAMING_SNAKE_CASE_							)
        assert self.row == self.column == u.row == v.row  # u, v should be column vector
        assert u.column == v.column == 1  # u, v should be column vector

        # Calculate
        __lowerCamelCase:				Optional[Any]     =	v.transpose()
        __lowerCamelCase:				Any     =	(v_t * self * u)[0, 0] + 1
        if numerator_factor == 0:
           return None  # It's not invertable
        return self - ((self * u) * (v_t * self) * (1.0 / numerator_factor))


# Testing
if __name__ == "__main__":



   def     __lowerCAmelCase						(							)       ->							None:
      # a^(-1)
      __lowerCamelCase:				int     =	Matrix(3						,      3						,      0 )
      for i in range(3 ):
         __lowerCamelCase:				List[Any]     =	1
      print(f'a^(-1) is {ainv}' )
      # u, v
      __lowerCamelCase:				List[str]     =	Matrix(3						,      1						,      0 )
      __lowerCamelCase   ,     __lowerCamelCase   ,     __lowerCamelCase:				Optional[Any]     =	1, 2, -3
      __lowerCamelCase:				Dict     =	Matrix(3						,      1						,      0 )
      __lowerCamelCase   ,     __lowerCamelCase   ,     __lowerCamelCase:				Union[str, Any]     =	4, -2, 5
      print(f'u is {u}' )
      print(f'v is {v}' )
      print(f'uv^T is {u * v.transpose()}' )
      # Sherman Morrison
      print(f'(a + uv^T)^(-1) is {ainv.sherman_morrison(snake_case						,      snake_case )}' )



   def     __lowerCAmelCase						(							)       ->							None:
      import doctest

      doctest.testmod()







   testa()


