



'''simple docstring'''




from __future__ import annotations

from typing import Any



def  __lowerCamelCase    (   *_lowercase						) ->     list[list[int]]:
							if all(_check_not_integer(_lowercase						) for m in matrix_s						):
														for i in matrix_s[1:]:
																					_verify_matrix_sizes(matrix_s[0]      ,							_lowercase						)
														return [[sum(_lowercase						) for t in zip(*_lowercase						)] for m in zip(*_lowercase						)]
							raise TypeError("""Expected a matrix, got int/list instead"""						)



def  __lowerCamelCase    (   _lowercase      ,							_lowercase						) ->     list[list[int]]:
							if (
							    _check_not_integer(_lowercase						)
							    and _check_not_integer(_lowercase						)
							    and _verify_matrix_sizes(_lowercase      ,							_lowercase						)
							):
														return [[i - j for i, j in zip(*_lowercase						)] for m in zip(_lowercase      ,							_lowercase						)]
							raise TypeError("""Expected a matrix, got int/list instead"""						)



def  __lowerCamelCase    (   _lowercase      ,							_lowercase						) ->     list[list[float]]:
							return [[x * n for x in row] for row in matrix]



def  __lowerCamelCase    (   _lowercase      ,							_lowercase						) ->     list[list[int]]:
							if _check_not_integer(_lowercase						) and _check_not_integer(_lowercase						):
														UpperCAmelCase	,	UpperCAmelCase	:						int					=						_verify_matrix_sizes(_lowercase      ,							_lowercase						)

							if cols[0] != rows[1]:
														UpperCAmelCase	:						Optional[int]					=						(
														    """Cannot multiply matrix of dimensions """
														    F'''({rows[0]},{cols[0]}) and ({rows[1]},{cols[1]})'''
														)
														raise ValueError(_lowercase						)
							return [
							    [sum(m * n for m, n in zip(_lowercase      ,							_lowercase						)						) for j in zip(*_lowercase						)] for i in matrix_a
							]



def  __lowerCamelCase    (   _lowercase						) ->     list[list[int]]:
							UpperCAmelCase	:						Tuple					=						int(_lowercase						)
							return [[int(row == column						) for column in range(_lowercase						)] for row in range(_lowercase						)]



def  __lowerCamelCase    (   _lowercase      ,							_lowercase = True						) ->     list[list[int]] | map[list[int]]:
							if _check_not_integer(_lowercase						):
														if return_map:
																					return map(_lowercase      ,							zip(*_lowercase						)						)
														else:
																					return list(map(_lowercase      ,							zip(*_lowercase						)						)						)
							raise TypeError("""Expected a matrix, got int/list instead"""						)



def  __lowerCamelCase    (   _lowercase      ,							_lowercase      ,							_lowercase						) ->     list[list[int]]:
							UpperCAmelCase	:						Optional[Any]					=						matrix[:row] + matrix[row + 1 :]
							return [row[:column] + row[column + 1 :] for row in minor]



def  __lowerCamelCase    (   _lowercase						) ->     Any:
							if len(_lowercase						) == 1:
														return matrix[0][0]

							return sum(
							    x * determinant(minor(_lowercase      ,							0      ,							_lowercase						)						) * (-1) ** i
							    for i, x in enumerate(matrix[0]						)						)



def  __lowerCamelCase    (   _lowercase						) ->     list[list[float]] | None:
							UpperCAmelCase	:						Dict					=						determinant(_lowercase						)
							if det == 0:
														return None

							UpperCAmelCase	:						str					=						[
							    [determinant(minor(_lowercase      ,							_lowercase      ,							_lowercase						)						) for j in range(len(_lowercase						)						)]
							    for i in range(len(_lowercase						)						)
							]

							UpperCAmelCase	:						List[str]					=						[
							    [x * (-1) ** (row + col) for col, x in enumerate(matrix_minor[row]						)]
							    for row in range(len(_lowercase						)						)
							]
							UpperCAmelCase	:						int					=						list(transpose(_lowercase						)						)
							return scalar_multiply(_lowercase      ,							1 / det						)



def  __lowerCamelCase    (   _lowercase						) ->     bool:
							return not isinstance(_lowercase      ,							_lowercase						) and not isinstance(matrix[0]      ,							_lowercase						)



def  __lowerCamelCase    (   _lowercase						) ->     tuple[int, int]:
							return len(_lowercase						), len(matrix[0]						)



def  __lowerCamelCase    (   _lowercase      ,							_lowercase						) ->     tuple[tuple[int, int], tuple[int, int]]:
							UpperCAmelCase	:						Any					=						_shape(_lowercase						) + _shape(_lowercase						)
							if shape[0] != shape[3] or shape[1] != shape[2]:
														UpperCAmelCase	:						Union[str, Any]					=						(
														    """operands could not be broadcast together with shape """
														    F'''({shape[0], shape[1]}), ({shape[2], shape[3]})'''
														)
														raise ValueError(_lowercase						)
							return (shape[0], shape[2]), (shape[1], shape[3])



def  __lowerCamelCase    (   ) ->     None:
							UpperCAmelCase	:						Tuple					=						[[1_2, 1_0], [3, 9]]
							UpperCAmelCase	:						Dict					=						[[3, 4], [7, 4]]
							UpperCAmelCase	:						str					=						[[1_1, 1_2, 1_3, 1_4], [2_1, 2_2, 2_3, 2_4], [3_1, 3_2, 3_3, 3_4], [4_1, 4_2, 4_3, 4_4]]
							UpperCAmelCase	:						List[str]					=						[[3, 0, 2], [2, 0, -2], [0, 1, 1]]
							print(F'''Add Operation, {add(_lowercase      ,							_lowercase						) = } \n'''						)
							print(
							    F'''Multiply Operation, {multiply(_lowercase      ,							_lowercase						) = } \n'''      ,							)
							print(F'''Identity: {identity(5						)}\n'''						)
							print(F'''Minor of {matrix_c} = {minor(_lowercase      ,							1      ,							2						)} \n'''						)
							print(F'''Determinant of {matrix_b} = {determinant(_lowercase						)} \n'''						)
							print(F'''Inverse of {matrix_d} = {inverse(_lowercase						)}\n'''						)


if __name__ == "__main__":
							import doctest

							doctest.testmod()
							main()






'''simple docstring'''




from __future__ import annotations

from collections.abc import Iterable

class      UpperCamelCase_					:





	def __init__(    self						,						A						,						A       )     ->     int:
								UpperCAmelCase	,	UpperCAmelCase	:						Any					=						float(A       ), float(A       )





	def __eq__(    self						,						A       )     ->     str:
								return self.x == other.x and self.y == other.y





	def __ne__(    self						,						A       )     ->     Dict:
								return not self == other





	def __gt__(    self						,						A       )     ->     Dict:
								if self.x > other.x:
															return True
								elif self.x == other.x:
															return self.y > other.y
								return False





	def __lt__(    self						,						A       )     ->     str:
								return not self > other





	def __ge__(    self						,						A       )     ->     Union[str, Any]:
								if self.x > other.x:
															return True
								elif self.x == other.x:
															return self.y >= other.y
								return False





	def __le__(    self						,						A       )     ->     List[str]:
								if self.x < other.x:
															return True
								elif self.x == other.x:
															return self.y <= other.y
								return False





	def __repr__(    self       )     ->     Tuple:
								return f'''({self.x}, {self.y})'''






	def __hash__(    self       )     ->     List[str]:
								return hash(self.x       )




def  __lowerCamelCase    (   _lowercase      ,							) ->     list[Point]:
							UpperCAmelCase	:						list[Point]					=						[]
							if list_of_tuples:
														for p in list_of_tuples:
																					if isinstance(_lowercase      ,							_lowercase						):
																												points.append(_lowercase						)
																					else:
																												try:
																																			points.append(Point(p[0]      ,							p[1]						)						)
																												except (IndexError, TypeError):
																																			print(
																																			    F'''Ignoring deformed point {p}. All points'''
																																			    """ must have at least 2 coordinates."""						)
							return points



def  __lowerCamelCase    (   _lowercase						) ->     list[Point]:

							if not hasattr(_lowercase      ,							"""__iter__"""						):
														UpperCAmelCase	:						List[str]					=						F'''Expecting an iterable object but got an non-iterable type {points}'''
														raise ValueError(_lowercase						)

							if not points:
														UpperCAmelCase	:						Optional[int]					=						F'''Expecting a list of points but got {points}'''
														raise ValueError(_lowercase						)

							return _construct_points(_lowercase						)



def  __lowerCamelCase    (   _lowercase      ,							_lowercase      ,							_lowercase						) ->     float:
							UpperCAmelCase	:						Dict					=						(a.x * b.y + b.x * c.y + c.x * a.y) - (a.y * b.x + b.y * c.x + c.y * a.x)
							return det



def  __lowerCamelCase    (   _lowercase						) ->     list[Point]:
							UpperCAmelCase	:						Optional[int]					=						sorted(_validate_input(_lowercase						)						)
							UpperCAmelCase	:						int					=						len(_lowercase						)
							UpperCAmelCase	:						Any					=						set()

							for i in range(n - 1						):
														for j in range(i + 1      ,							_lowercase						):
																					UpperCAmelCase	:						Optional[Any]					=						False
																					UpperCAmelCase	:						Optional[int]					=						True
																					for k in range(_lowercase						):
																												if k != i and k != j:
																																			UpperCAmelCase	:						Union[str, Any]					=						_det(points[i]      ,							points[j]      ,							points[k]						)

																																			if det_k > 0:
																																										UpperCAmelCase	:						Optional[int]					=						True
																																			elif det_k < 0:
																																										UpperCAmelCase	:						int					=						True
																																			else:
																																										# point[i], point[j], point[k] all lie on a straight line
																																										# if point[k] is to the left of point[i] or it's to the
																																										# right of point[j], then point[i], point[j] cannot be
																																										# part of the convex hull of A
																																										if points[k] < points[i] or points[k] > points[j]:
																																																	UpperCAmelCase	:						List[Any]					=						False
																																																	break

																												if points_left_of_ij and points_right_of_ij:
																																			UpperCAmelCase	:						Union[str, Any]					=						False
																																			break

																					if ij_part_of_convex_hull:
																												convex_set.update([points[i], points[j]]						)

							return sorted(_lowercase						)



def  __lowerCamelCase    (   _lowercase						) ->     list[Point]:
							UpperCAmelCase	:						List[str]					=						sorted(_validate_input(_lowercase						)						)
							UpperCAmelCase	:						Optional[Any]					=						len(_lowercase						)

							# divide all the points into an upper hull and a lower hull
							# the left most point and the right most point are definitely
							# members of the convex hull by definition.
							# use these two anchors to divide all the points into two hulls,
							# an upper hull and a lower hull.

							# all points to the left (above) the line joining the extreme points belong to the
							# upper hull
							# all points to the right (below) the line joining the extreme points below to the
							# lower hull
							# ignore all points on the line joining the extreme points since they cannot be
							# part of the convex hull

							UpperCAmelCase	:						List[str]					=						points[0]
							UpperCAmelCase	:						Union[str, Any]					=						points[n - 1]

							UpperCAmelCase	:						int					=						{left_most_point, right_most_point}
							UpperCAmelCase	:						Dict					=						[]
							UpperCAmelCase	:						Any					=						[]

							for i in range(1      ,							n - 1						):
														UpperCAmelCase	:						Any					=						_det(_lowercase      ,							_lowercase      ,							points[i]						)

														if det > 0:
																					upper_hull.append(points[i]						)
														elif det < 0:
																					lower_hull.append(points[i]						)

							_construct_hull(_lowercase      ,							_lowercase      ,							_lowercase      ,							_lowercase						)
							_construct_hull(_lowercase      ,							_lowercase      ,							_lowercase      ,							_lowercase						)

							return sorted(_lowercase						)



def  __lowerCamelCase    (   _lowercase      ,							_lowercase      ,							_lowercase      ,							_lowercase						) ->     None:
							if points:
														UpperCAmelCase	:						Union[str, Any]					=						None
														UpperCAmelCase	:						int					=						float("""-inf"""						)
														UpperCAmelCase	:						Tuple					=						[]

														for p in points:
																					UpperCAmelCase	:						List[Any]					=						_det(_lowercase      ,							_lowercase      ,							_lowercase						)

																					if det > 0:
																												candidate_points.append(_lowercase						)

																												if det > extreme_point_distance:
																																			UpperCAmelCase	:						List[str]					=						det
																																			UpperCAmelCase	:						Optional[Any]					=						p

														if extreme_point:
																					_construct_hull(_lowercase      ,							_lowercase      ,							_lowercase      ,							_lowercase						)
																					convex_set.add(_lowercase						)
																					_construct_hull(_lowercase      ,							_lowercase      ,							_lowercase      ,							_lowercase						)



def  __lowerCamelCase    (   _lowercase						) ->     list[Point]:
							UpperCAmelCase	:						Any					=						sorted(_validate_input(_lowercase						)						)
							UpperCAmelCase	:						Union[str, Any]					=						len(_lowercase						)

							UpperCAmelCase	:						Dict					=						points[:2]
							for i in range(2      ,							_lowercase						):
														UpperCAmelCase	:						Dict					=						_det(convex_hull[1]      ,							convex_hull[0]      ,							points[i]						)
														if det > 0:
																					convex_hull.insert(0      ,							points[i]						)
																					break
														elif det < 0:
																					convex_hull.append(points[i]						)
																					break
														else:
																					UpperCAmelCase	:						Dict					=						points[i]
							i += 1

							for j in range(_lowercase      ,							_lowercase						):
														if (
														    _det(convex_hull[0]      ,							convex_hull[-1]      ,							points[j]						) > 0
														    and _det(convex_hull[-1]      ,							convex_hull[0]      ,							points[1]						) < 0
														):
																					# The point lies within the convex hull
																					continue

														convex_hull.insert(0      ,							points[j]						)
														convex_hull.append(points[j]						)
														while _det(convex_hull[0]      ,							convex_hull[1]      ,							convex_hull[2]						) >= 0:
																					del convex_hull[1]
														while _det(convex_hull[-1]      ,							convex_hull[-2]      ,							convex_hull[-3]						) <= 0:
																					del convex_hull[-2]

    # `convex_hull` is contains the convex hull in circular order
							return sorted(convex_hull[1:] if len(_lowercase						) > 3 else convex_hull						)



def  __lowerCamelCase    (   ) ->     Tuple:
							UpperCAmelCase	:						Tuple					=						[
							    (0, 3),
							    (2, 2),
							    (1, 1),
							    (2, 1),
							    (3, 0),
							    (0, 0),
							    (3, 3),
							    (2, -1),
							    (2, -4),
							    (1, -3),
							]
							# the convex set of points is
							# [(0, 0), (0, 3), (1, -3), (2, -4), (3, 0), (3, 3)]
							UpperCAmelCase	:						Tuple					=						convex_hull_bf(_lowercase						)

							UpperCAmelCase	:						List[Any]					=						convex_hull_recursive(_lowercase						)
							assert results_bf == results_recursive

							UpperCAmelCase	:						Union[str, Any]					=						convex_hull_melkman(_lowercase						)
							assert results_bf == results_melkman

							print(_lowercase						)


if __name__ == "__main__":
							main()

