

def       a__					(		UpperCAmelCase							:      int      )  ->						List[Any]:
		UpperCAmelCase	:							Union[str, Any]						=			0
		UpperCAmelCase	:							Dict						=			len(a__      )

		for i in range(n - 1      ):
				for j in range(i + 1    ,			a__      ):
						if arr[i] > arr[j]:
								num_inversions += 1

		return num_inversions

def       a__					(		UpperCAmelCase							:      Dict      )  ->						str:
		if len(a__      ) <= 1:
				return arr, 0
		UpperCAmelCase	:							Any						=			len(a__      ) // 2
		UpperCAmelCase	:							Any						=			arr[0:mid]
		UpperCAmelCase	:							Dict						=			arr[mid:]

		UpperCAmelCase			,						UpperCAmelCase	:							Dict						=			count_inversions_recursive(a__      )
		UpperCAmelCase			,						UpperCAmelCase	:							List[str]						=			count_inversions_recursive(a__      )
		UpperCAmelCase			,						UpperCAmelCase	:							Optional[Any]						=			_count_cross_inversions(a__    ,			a__      )

		UpperCAmelCase	:							Union[str, Any]						=			inversion_p + inversions_q + cross_inversions
		return c, num_inversions

def       a__					(		UpperCAmelCase							:      Optional[Any]    ,			UpperCAmelCase							:      Any      )  ->						int:
		UpperCAmelCase	:							Dict						=			[]
		UpperCAmelCase	:							Dict						=			0
		while i < len(a__      ) and j < len(a__      ):
				if p[i] > q[j]:
						# if P[1] > Q[j], then P[k] > Q[k] for all  i < k <= len(P)
						# These are all inversions. The claim emerges from the
						# property that P is sorted.
						num_inversion += len(a__      ) - i
						r.append(q[j]      )
						j += 1
				else:
						r.append(p[i]      )
						i += 1

		if i < len(a__      ):
				r.extend(p[i:]      )
		else:
				r.extend(q[j:]      )

		return r, num_inversion

def       a__					(		)  ->						List[Any]:
		UpperCAmelCase	:							Any						=			[10, 2, 1, 5, 5, 2, 11]

		# this arr has 8 inversions:
		# (10, 2), (10, 1), (10, 5), (10, 5), (10, 2), (2, 1), (5, 2), (5, 2)

		UpperCAmelCase	:							int						=			count_inversions_bf(a__      )
		UpperCAmelCase			,						UpperCAmelCase	:							Dict						=			count_inversions_recursive(a__      )

		assert num_inversions_bf == num_inversions_recursive == 8

		print('''number of inversions = '''    ,			a__      )

		# testing an array with zero inversion (a sorted arr_1)

		arr_a.sort()
		UpperCAmelCase	:							List[str]						=			count_inversions_bf(a__      )
		UpperCAmelCase			,						UpperCAmelCase	:							Tuple						=			count_inversions_recursive(a__      )

		assert num_inversions_bf == num_inversions_recursive == 0
		print('''number of inversions = '''    ,			a__      )

		# an empty list should also have zero inversions
		UpperCAmelCase	:							List[Any]						=			[]
		UpperCAmelCase	:							Union[str, Any]						=			count_inversions_bf(a__      )
		UpperCAmelCase			,						UpperCAmelCase	:							Optional[Any]						=			count_inversions_recursive(a__      )

		assert num_inversions_bf == num_inversions_recursive == 0
		print('''number of inversions = '''    ,			a__      )


if __name__ == "__main__":
					main()








def       a__					(		UpperCAmelCase							:      int      )  ->						bool:
  if not isinstance(UpperCAmelCase    ,			UpperCAmelCase      ):
    UpperCAmelCase	:							List[str]						=			f'''Input value of [number={number}] must be an integer'''
    raise TypeError(UpperCAmelCase      )
  if number < 0:
    return False
  UpperCAmelCase	:							List[str]						=			number * number
  while number > 0:
    if number % 10 != number_square % 10:
      return False
    number //= 10
    number_square //= 10
  return True


if __name__ == "__main__":
     import doctest

     doctest.testmod()





