




from typing import Any


def 		lowerCamelCase__       (						_A	,      _A	,      _A	,      _A	,      _A	,      ):







	'''simple docstring'''





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

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

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

			snake_case_						       =	arg_max

		# The final observation
	snake_case_						       =	observations_space[len(_A				) - 1]

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

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

	return result


def 		lowerCamelCase__       (						_A	,      _A	,      _A	,      _A	,      _A	,      ):







	'''simple docstring'''





	_validate_not_empty(
	    _A	,      _A	,      _A	,      _A	,      _A	,      )
	_validate_lists(_A	,      _A				)
	_validate_dicts(
	    _A	,      _A	,      _A				)


def 		lowerCamelCase__       (						_A	,      _A	,      _A	,      _A	,      _A	,      ):







	'''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	,      _A				):







	'''simple docstring'''





	_validate_list(_A	,      "observations_space"				)
	_validate_list(_A	,      "states_space"				)


def 		lowerCamelCase__       (						_A	,      _A				):







	'''simple docstring'''





	if not isinstance(_object	,      _A				):
		snake_case_						       =	f"{var_name} must be a list"
		raise ValueError(_A				)
	else:
		for x in _object:
			if not isinstance(_A	,      _A				):
				snake_case_						       =	f"{var_name} must be a list of strings"
				raise ValueError(_A				)


def 		lowerCamelCase__       (						_A	,      _A	,      _A	,      ):







	'''simple docstring'''





	_validate_dict(_A	,      "initial_probabilities"	,      _A				)
	_validate_nested_dict(_A	,      "transition_probabilities"				)
	_validate_nested_dict(_A	,      "emission_probabilities"				)


def 		lowerCamelCase__       (						_A	,      _A				):







	'''simple docstring'''





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


def 		lowerCamelCase__       (						_A	,      _A	,      _A	,      _A = False				):







	'''simple docstring'''





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


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

		testmod()

"""simple docstring"""


from __future__ import annotations

from numpy import array, cos, cross, floataa, radians, sin
from numpy.typing import NDArray






def      __UpperCAmelCase       (      lowercase ,lowercase ,lowercase = False		):







  """simple docstring"""
  if radian_mode:
    return [magnitude * cos(lowercase		), magnitude * sin(lowercase		)]
  return [magnitude * cos(radians(lowercase		)		), magnitude * sin(radians(lowercase		)		)]






def      __UpperCAmelCase       (      lowercase ,lowercase ,lowercase = 10**-1		):







  """simple docstring"""
  _UpperCAmelCase    			=   cross(lowercase ,lowercase		)
  _UpperCAmelCase    			=   sum(lowercase		)
  return abs(lowercase		) < eps


if __name__ == "__main__":
    # Test to check if it works
    UpperCAmelCase__			=    array(
        [
            polar_force(718.4, 1_8_0 - 3_0),
            polar_force(879.54, 4_5),
            polar_force(1_0_0, -9_0),
        ]
    )

    UpperCAmelCase__			=    array([[0, 0], [0, 0], [0, 0]])

    assert in_static_equilibrium(forces, location)

    # Problem 1 in image_data/2D_problems.jpg
    UpperCAmelCase__			=    array(
        [
            polar_force(3_0 * 9.81, 1_5),
            polar_force(2_1_5, 1_8_0 - 4_5),
            polar_force(2_6_4, 9_0 - 3_0),
        ]
    )

    UpperCAmelCase__			=    array([[0, 0], [0, 0], [0, 0]])

    assert in_static_equilibrium(forces, location)

    # Problem in image_data/2D_problems_1.jpg
    UpperCAmelCase__			=    array([[0, -2_0_0_0], [0, -1_2_0_0], [0, 1_5_6_0_0], [0, -1_2_4_0_0]])

    UpperCAmelCase__			=    array([[0, 0], [6, 0], [1_0, 0], [1_2, 0]])

    assert in_static_equilibrium(forces, location)

    import doctest

    doctest.testmod()
