


from string import ascii_lowercase, ascii_uppercase

def 			lowerCAmelCase__					(     a__:						str			)    ->						str:


							'''simple docstring'''





							if not sentence:
														return ""
							_UpperCAmelCase											=     dict(zip(a__     ,    a__			)			)
							return lower_to_upper.get(sentence[0]     ,    sentence[0]			) + sentence[1:]


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

		testmod()




from typing import Any

def 			lowerCAmelCase__					(     a__:						list     ,    a__:						list     ,    a__:						dict     ,    a__:						dict     ,    a__:						dict     ,    )    ->						list:


							'''simple docstring'''





							_validation(
							    a__     ,    a__     ,    a__     ,    a__     ,    a__     ,    )
							# Creates data structures and fill initial step
							_UpperCAmelCase											=     {}
							_UpperCAmelCase											=     {}
							for state in states_space:
														_UpperCAmelCase											=     observations_space[0]
														_UpperCAmelCase											=     (
														    initial_probabilities[state] * emission_probabilities[state][observation]
														)
														_UpperCAmelCase											=     None

							# Fills the data structure with the probabilities of
							# different transitions and pointers to previous states
							for o in range(1     ,    len(a__			)			):
														_UpperCAmelCase											=     observations_space[o]
														_UpperCAmelCase											=     observations_space[o - 1]
														for state in states_space:
																					# Calculates the argmax for probability function
																					_UpperCAmelCase											=     ''
																					_UpperCAmelCase											=     -1
																					for k_state in states_space:
																												_UpperCAmelCase											=     (
																												    probabilities[(k_state, prior_observation)]
																												    * transition_probabilities[k_state][state]
																												    * emission_probabilities[state][observation]
																												)
																												if probability > max_probability:
																																			_UpperCAmelCase											=     probability
																																			_UpperCAmelCase											=     k_state

            # Update probabilities and pointers dicts
																					_UpperCAmelCase											=     (
																					    probabilities[(arg_max, prior_observation)]
																					    * transition_probabilities[arg_max][state]
																					    * emission_probabilities[state][observation]
																					)

																					_UpperCAmelCase											=     arg_max

    # The final observation
							_UpperCAmelCase											=     observations_space[len(a__			) - 1]

							# argmax for given final observation
							_UpperCAmelCase											=     ''
							_UpperCAmelCase											=     -1
							for k_state in states_space:
														_UpperCAmelCase											=     probabilities[(k_state, final_observation)]
														if probability > max_probability:
																					_UpperCAmelCase											=     probability
																					_UpperCAmelCase											=     k_state
							_UpperCAmelCase											=     arg_max

							# Process pointers backwards
							_UpperCAmelCase											=     last_state
							_UpperCAmelCase											=     []
							for o in range(len(a__			) - 1     ,    -1     ,    -1			):
														result.append(a__			)
														_UpperCAmelCase											=     pointers[previous, observations_space[o]]
							result.reverse()

							return result

def 			lowerCAmelCase__					(     a__:						Any     ,    a__:						Any     ,    a__:						Any     ,    a__:						Any     ,    a__:						Any     ,    )    ->						None:


							'''simple docstring'''





							_validate_not_empty(
							    a__     ,    a__     ,    a__     ,    a__     ,    a__     ,    )
							_validate_lists(a__     ,    a__			)
							_validate_dicts(
							    a__     ,    a__     ,    a__			)

def 			lowerCAmelCase__					(     a__:						Any     ,    a__:						Any     ,    a__:						Any     ,    a__:						Any     ,    a__:						Any     ,    )    ->						None:


							'''simple docstring'''





							if not all(
							    [
							        observations_space,
							        states_space,
							        initial_probabilities,
							        transition_probabilities,
							        emission_probabilities,
							    ]			):
														raise ValueError('There\'s an empty parameter'			)

def 			lowerCAmelCase__					(     a__:						Any     ,    a__:						Any			)    ->						None:


							'''simple docstring'''





							_validate_list(a__     ,    'observations_space'			)
							_validate_list(a__     ,    'states_space'			)

def 			lowerCAmelCase__					(     a__:						Any     ,    a__:						str			)    ->						None:


							'''simple docstring'''





							if not isinstance(_object     ,    a__			):
														_UpperCAmelCase											=     F'''{var_name} must be a list'''
														raise ValueError(a__			)
							else:
														for x in _object:
																					if not isinstance(a__     ,    a__			):
																												_UpperCAmelCase											=     F'''{var_name} must be a list of strings'''
																												raise ValueError(a__			)

def 			lowerCAmelCase__					(     a__:						Any     ,    a__:						Any     ,    a__:						Any     ,    )    ->						None:


							'''simple docstring'''





							_validate_dict(a__     ,    'initial_probabilities'     ,    a__			)
							_validate_nested_dict(a__     ,    'transition_probabilities'			)
							_validate_nested_dict(a__     ,    'emission_probabilities'			)

def 			lowerCAmelCase__					(     a__:						Any     ,    a__:						str			)    ->						None:


							'''simple docstring'''





							_validate_dict(_object     ,    a__     ,    a__			)
							for x in _object.values():
														_validate_dict(a__     ,    a__     ,    a__     ,    a__			)

def 			lowerCAmelCase__					(     a__:						Any     ,    a__:						str     ,    a__:						type     ,    a__:						bool = False			)    ->						None:


							'''simple docstring'''





							if not isinstance(_object     ,    a__			):
														_UpperCAmelCase											=     F'''{var_name} must be a dict'''
														raise ValueError(a__			)
							if not all(isinstance(a__     ,    a__			) for x in _object			):
														_UpperCAmelCase											=     F'''{var_name} all keys must be strings'''
														raise ValueError(a__			)
							if not all(isinstance(a__     ,    a__			) for x in _object.values()			):
														_UpperCAmelCase											=     'nested dictionary ' if nested else ''
														_UpperCAmelCase											=     F'''{var_name} {nested_text}all values must be {value_type.__name__}'''
														raise ValueError(a__			)


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

		testmod()
