code: from sys import getsizeof
  from typing import (
      TYPE_CHECKING,
      Any,
      Hashable,
      Iterable,
      List,
      Optional,
      Sequence,
      Tuple,
      Union,
  )
  import warnings
  
  import numpy as np
  
  from pandas._config import get_option
  
  from pandas._libs import algos as libalgos, index as libindex, lib
  from pandas._libs.hashtable import duplicated_int64
  from pandas._typing import AnyArrayLike, ArrayLike, Scalar
  from pandas.compat.numpy import function as nv
  from pandas.errors import PerformanceWarning, UnsortedIndexError
  from pandas.util._decorators import Appender, cache_readonly
  
  from pandas.core.dtypes.cast import coerce_indexer_dtype
  from pandas.core.dtypes.common import (
      ensure_int64,
      ensure_platform_int,
      is_categorical_dtype,
      is_hashable,
      is_integer,
      is_iterator,
      is_list_like,
      is_object_dtype,
      is_scalar,
      pandas_dtype,
  )
  from pandas.core.dtypes.dtypes import ExtensionDtype
  from pandas.core.dtypes.generic import ABCDataFrame
  from pandas.core.dtypes.missing import array_equivalent, isna
  
  import pandas.core.algorithms as algos
  from pandas.core.arrays import Categorical
  from pandas.core.arrays.categorical import factorize_from_iterables
  import pandas.core.common as com
  import pandas.core.indexes.base as ibase
  from pandas.core.indexes.base import (
      Index,
      InvalidIndexError,
      _index_shared_docs,
      ensure_index,
  )
  from pandas.core.indexes.frozen import FrozenList
  import pandas.core.missing as missing
  from pandas.core.sorting import (
      get_group_index,
      indexer_from_factorized,
      lexsort_indexer,
  )
  
  from pandas.io.formats.printing import (
      format_object_attrs,
      format_object_summary,
      pprint_thing,
  )
  
  if TYPE_CHECKING:
      from pandas import Series  # noqa:F401
  
  _index_doc_kwargs = dict(ibase._index_doc_kwargs)
  _index_doc_kwargs.update(
      dict(klass="MultiIndex", target_klass="MultiIndex or list of tuples")
  )
  
  
  class MultiIndexUIntEngine(libindex.BaseMultiIndexCodesEngine, libindex.UInt64Engine):
      """
      This class manages a MultiIndex by mapping label combinations to positive
      integers.
      """
  
      _base = libindex.UInt64Engine
  
      def _codes_to_ints(self, codes):
          """
          Transform combination(s) of uint64 in one uint64 (each), in a strictly
          monotonic way (i.e. respecting the lexicographic order of integer
          combinations): see BaseMultiIndexCodesEngine documentation.
  
          Parameters
          ----------
          codes : 1- or 2-dimensional array of dtype uint64
              Combinations of integers (one per row)
  
          Returns
          -------
          scalar or 1-dimensional array, of dtype uint64
              Integer(s) representing one combination (each).
          """
          # Shift the representation of each level by the pre-calculated number
          # of bits:
          codes <<= self.offsets
  
          # Now sum and OR are in fact interchangeable. This is a simple
          # composition of the (disjunct) significant bits of each level (i.e.
          # each column in "codes") in a single positive integer:
          if codes.ndim == 1:
              # Single key
              return np.bitwise_or.reduce(codes)
  
          # Multiple keys
          return np.bitwise_or.reduce(codes, axis=1)
  
  
  class MultiIndexPyIntEngine(libindex.BaseMultiIndexCodesEngine, libindex.ObjectEngine):
      """
      This class manages those (extreme) cases in which the number of possible
      label combinations overflows the 64 bits integers, and uses an ObjectEngine
      containing Python integers.
      """
  
      _base = libindex.ObjectEngine
  
      def _codes_to_ints(self, codes):
          """
          Transform combination(s) of uint64 in one Python integer (each), in a
          strictly monotonic way (i.e. respecting the lexicographic order of
          integer combinations): see BaseMultiIndexCodesEngine documentation.
  
          Parameters
          ----------
          codes : 1- or 2-dimensional array of dtype uint64
              Combinations of integers (one per row)
  
          Returns
          -------
          int, or 1-dimensional array of dtype object
              Integer(s) representing one combination (each).
          """
          # Shift the representation of each level by the pre-calculated number
          # of bits. Since this can overflow uint64, first make sure we are
          # working with Python integers:
          codes = codes.astype("object") << self.offsets
  
          # Now sum and OR are in fact interchangeable. This is a simple
          # composition of the (disjunct) significant bits of each level (i.e.
          # each column in "codes") in a single positive integer (per row):
          if codes.ndim == 1:
              # Single key
              return np.bitwise_or.reduce(codes)
  
          # Multiple keys
          return np.bitwise_or.reduce(codes, axis=1)
  
  
  class MultiIndex(Index):
      """
      A multi-level, or hierarchical, index object for pandas objects.
  
      Parameters
      ----------
      levels : sequence of arrays
          The unique labels for each level.
      codes : sequence of arrays
          Integers for each level designating which label at each location.
  
          .. versionadded:: 0.24.0
      sortorder : optional int
          Level of sortedness (must be lexicographically sorted by that
          level).
      names : optional sequence of objects
          Names for each of the index levels. (name is accepted for compat).
      copy : bool, default False
          Copy the meta-data.
      verify_integrity : bool, default True
          Check that the levels/codes are consistent and valid.
  
      Attributes
      ----------
      names
      levels
      codes
      nlevels
      levshape
  
      Methods
      -------
      from_arrays
      from_tuples
      from_product
      from_frame
      set_levels
      set_codes
      to_frame
      to_flat_index
      is_lexsorted
      sortlevel
      droplevel
      swaplevel
      reorder_levels
      remove_unused_levels
      get_locs
  
      See Also
      --------
      MultiIndex.from_arrays  : Convert list of arrays to MultiIndex.
      MultiIndex.from_product : Create a MultiIndex from the cartesian product
                                of iterables.
      MultiIndex.from_tuples  : Convert list of tuples to a MultiIndex.
      MultiIndex.from_frame   : Make a MultiIndex from a DataFrame.
      Index : The base pandas Index type.
  
      Notes
      -----
      See the `user guide
      <https://pandas.pydata.org/pandas-docs/stable/user_guide/advanced.html>`_
      for more.
  
      Examples
      --------
      A new ``MultiIndex`` is typically constructed using one of the helper
      methods :meth:`MultiIndex.from_arrays`, :meth:`MultiIndex.from_product`
      and :meth:`MultiIndex.from_tuples`. For example (using ``.from_arrays``):
  
      >>> arrays = [[1, 1, 2, 2], ['red', 'blue', 'red', 'blue']]
      >>> pd.MultiIndex.from_arrays(arrays, names=('number', 'color'))
      MultiIndex([(1,  'red'),
                  (1, 'blue'),
                  (2,  'red'),
                  (2, 'blue')],
                 names=['number', 'color'])
  
      See further examples for how to construct a MultiIndex in the doc strings
      of the mentioned helper methods.
      """
  
      _deprecations = Index._deprecations | frozenset()
  
      # initialize to zero-length tuples to make everything work
      _typ = "multiindex"
      _names = FrozenList()
      _levels = FrozenList()
      _codes = FrozenList()
      _comparables = ["names"]
      rename = Index.set_names
  
      _tuples = None
      sortorder: Optional[int]
  
      # --------------------------------------------------------------------
      # Constructors
  
      def __new__(
          cls,
          levels=None,
          codes=None,
          sortorder=None,
          names=None,
          dtype=None,
          copy=False,
          name=None,
          verify_integrity: bool = True,
          _set_identity: bool = True,
      ):
  
          # compat with Index
          if name is not None:
              names = name
          if levels is None or codes is None:
              raise TypeError("Must pass both levels and codes")
          if len(levels) != len(codes):
              raise ValueError("Length of levels and codes must be the same.")
          if len(levels) == 0:
              raise ValueError("Must pass non-zero number of levels/codes")
  
          result = object.__new__(MultiIndex)
  
          # we've already validated levels and codes, so shortcut here
          result._set_levels(levels, copy=copy, validate=False)
          result._set_codes(codes, copy=copy, validate=False)
  
          result._names = [None] * len(levels)
          if names is not None:
              # handles name validation
              result._set_names(names)
  
          if sortorder is not None:
              result.sortorder = int(sortorder)
          else:
              result.sortorder = sortorder
  
          if verify_integrity:
              new_codes = result._verify_integrity()
              result._codes = new_codes
  
          if _set_identity:
              result._reset_identity()
  
          return result
  
      def _validate_codes(self, level: List, code: List):
          """
          Reassign code values as -1 if their corresponding levels are NaN.
  
          Parameters
          ----------
          code : list
              Code to reassign.
          level : list
              Level to check for missing values (NaN, NaT, None).
  
          Returns
          -------
          new code where code value = -1 if it corresponds
          to a level with missing values (NaN, NaT, None).
          """
          null_mask = isna(level)
          if np.any(null_mask):
              code = np.where(null_mask[code], -1, code)
          return code
  
      def _verify_integrity(
          self, codes: Optional[List] = None, levels: Optional[List] = None
      ):
          """
          Parameters
          ----------
          codes : optional list
              Codes to check for validity. Defaults to current codes.
          levels : optional list
              Levels to check for validity. Defaults to current levels.
  
          Raises
          ------
          ValueError
              If length of levels and codes don't match, if the codes for any
              level would exceed level bounds, or there are any duplicate levels.
  
          Returns
          -------
          new codes where code value = -1 if it corresponds to a
          NaN level.
          """
          # NOTE: Currently does not check, among other things, that cached
          # nlevels matches nor that sortorder matches actually sortorder.
          codes = codes or self.codes
          levels = levels or self.levels
  
          if len(levels) != len(codes):
              raise ValueError(
                  "Length of levels and codes must match. NOTE: "
                  "this index is in an inconsistent state."
              )
          codes_length = len(codes[0])
          for i, (level, level_codes) in enumerate(zip(levels, codes)):
              if len(level_codes) != codes_length:
                  raise ValueError(
                      f"Unequal code lengths: {[len(code_) for code_ in codes]}"
                  )
              if len(level_codes) and level_codes.max() >= len(level):
                  raise ValueError(
                      f"On level {i}, code max ({level_codes.max()}) >= length of "
                      f"level ({len(level)}). NOTE: this index is in an "
                      "inconsistent state"
                  )
              if len(level_codes) and level_codes.min() < -1:
                  raise ValueError(f"On level {i}, code value ({level_codes.min()}) < -1")
              if not level.is_unique:
                  raise ValueError(
                      f"Level values must be unique: {list(level)} on level {i}"
                  )
          if self.sortorder is not None:
              if self.sortorder > self._lexsort_depth():
                  raise ValueError(
                      "Value for sortorder must be inferior or equal to actual "
                      f"lexsort_depth: sortorder {self.sortorder} "
                      f"with lexsort_depth {self._lexsort_depth()}"
                  )
  
          codes = [
              self._validate_codes(level, code) for level, code in zip(levels, codes)
          ]
          new_codes = FrozenList(codes)
          return new_codes
  
      @classmethod
      def from_arrays(cls, arrays, sortorder=None, names=lib.no_default):
          """
          Convert arrays to MultiIndex.
  
          Parameters
          ----------
          arrays : list / sequence of array-likes
              Each array-like gives one level's value for each data point.
              len(arrays) is the number of levels.
          sortorder : int or None
              Level of sortedness (must be lexicographically sorted by that
              level).
          names : list / sequence of str, optional
              Names for the levels in the index.
  
          Returns
          -------
          MultiIndex
  
          See Also
          --------
          MultiIndex.from_tuples : Convert list of tuples to MultiIndex.
          MultiIndex.from_product : Make a MultiIndex from cartesian product
                                    of iterables.
          MultiIndex.from_frame : Make a MultiIndex from a DataFrame.
  
          Examples
          --------
          >>> arrays = [[1, 1, 2, 2], ['red', 'blue', 'red', 'blue']]
          >>> pd.MultiIndex.from_arrays(arrays, names=('number', 'color'))
          MultiIndex([(1,  'red'),
                      (1, 'blue'),
                      (2,  'red'),
                      (2, 'blue')],
                     names=['number', 'color'])
          """
          error_msg = "Input must be a list / sequence of array-likes."
          if not is_list_like(arrays):
              raise TypeError(error_msg)
          elif is_iterator(arrays):
              arrays = list(arrays)
  
          # Check if elements of array are list-like
          for array in arrays:
              if not is_list_like(array):
                  raise TypeError(error_msg)
  
          # Check if lengths of all arrays are equal or not,
          # raise ValueError, if not
          for i in range(1, len(arrays)):
              if len(arrays[i]) != len(arrays[i - 1]):
                  raise ValueError("all arrays must be same length")
  
          codes, levels = factorize_from_iterables(arrays)
          if names is lib.no_default:
              names = [getattr(arr, "name", None) for arr in arrays]
  
          return MultiIndex(
              levels=levels,
              codes=codes,
              sortorder=sortorder,
              names=names,
              verify_integrity=False,
          )
  
      @classmethod
      def from_tuples(cls, tuples, sortorder=None, names=None):
          """
          Convert list of tuples to MultiIndex.
  
          Parameters
          ----------
          tuples : list / sequence of tuple-likes
              Each tuple is the index of one row/column.
          sortorder : int or None
              Level of sortedness (must be lexicographically sorted by that
              level).
          names : list / sequence of str, optional
              Names for the levels in the index.
  
          Returns
          -------
          MultiIndex
  
          See Also
          --------
          MultiIndex.from_arrays : Convert list of arrays to MultiIndex.
          MultiIndex.from_product : Make a MultiIndex from cartesian product
                                    of iterables.
          MultiIndex.from_frame : Make a MultiIndex from a DataFrame.
  
          Examples
          --------
          >>> tuples = [(1, 'red'), (1, 'blue'),
          ...           (2, 'red'), (2, 'blue')]
          >>> pd.MultiIndex.from_tuples(tuples, names=('number', 'color'))
          MultiIndex([(1,  'red'),
                      (1, 'blue'),
                      (2,  'red'),
                      (2, 'blue')],
                     names=['number', 'color'])
          """
          if not is_list_like(tuples):
              raise TypeError("Input must be a list / sequence of tuple-likes.")
          elif is_iterator(tuples):
              tuples = list(tuples)
  
          if len(tuples) == 0:
              if names is None:
                  raise TypeError("Cannot infer number of levels from empty list")
              arrays = [[]] * len(names)
          elif isinstance(tuples, (np.ndarray, Index)):
              if isinstance(tuples, Index):
                  tuples = tuples._values
  
              arrays = list(lib.tuples_to_object_array(tuples).T)
          elif isinstance(tuples, list):
              arrays = list(lib.to_object_array_tuples(tuples).T)
          else:
              arrays = zip(*tuples)
  
          return MultiIndex.from_arrays(arrays, sortorder=sortorder, names=names)
  
      @classmethod
      def from_product(cls, iterables, sortorder=None, names=lib.no_default):
          """
          Make a MultiIndex from the cartesian product of multiple iterables.
  
          Parameters
          ----------
          iterables : list / sequence of iterables
              Each iterable has unique labels for each level of the index.
          sortorder : int or None
              Level of sortedness (must be lexicographically sorted by that
              level).
          names : list / sequence of str, optional
              Names for the levels in the index.
  
              .. versionchanged:: 1.0.0
  
                 If not explicitly provided, names will be inferred from the
                 elements of iterables if an element has a name attribute
  
          Returns
          -------
          MultiIndex
  
          See Also
          --------
          MultiIndex.from_arrays : Convert list of arrays to MultiIndex.
          MultiIndex.from_tuples : Convert list of tuples to MultiIndex.
          MultiIndex.from_frame : Make a MultiIndex from a DataFrame.
  
          Examples
          --------
          >>> numbers = [0, 1, 2]
          >>> colors = ['green', 'purple']
          >>> pd.MultiIndex.from_product([numbers, colors],
          ...                            names=['number', 'color'])
          MultiIndex([(0,  'green'),
                      (0, 'purple'),
                      (1,  'green'),
                      (1, 'purple'),
                      (2,  'green'),
                      (2, 'purple')],
                     names=['number', 'color'])
          """
          from pandas.core.reshape.util import cartesian_product
  
          if not is_list_like(iterables):
              raise TypeError("Input must be a list / sequence of iterables.")
          elif is_iterator(iterables):
              iterables = list(iterables)
  
          codes, levels = factorize_from_iterables(iterables)
          if names is lib.no_default:
              names = [getattr(it, "name", None) for it in iterables]
  
          codes = cartesian_product(codes)
          return MultiIndex(levels, codes, sortorder=sortorder, names=names)
  
      @classmethod
      def from_frame(cls, df, sortorder=None, names=None):
          """
          Make a MultiIndex from a DataFrame.
  
          .. versionadded:: 0.24.0
  
          Parameters
          ----------
          df : DataFrame
              DataFrame to be converted to MultiIndex.
          sortorder : int, optional
              Level of sortedness (must be lexicographically sorted by that
              level).
          names : list-like, optional
              If no names are provided, use the column names, or tuple of column
              names if the columns is a MultiIndex. If a sequence, overwrite
              names with the given sequence.
  
          Returns
          -------
          MultiIndex
              The MultiIndex representation of the given DataFrame.
  
          See Also
          --------
          MultiIndex.from_arrays : Convert list of arrays to MultiIndex.
          MultiIndex.from_tuples : Convert list of tuples to MultiIndex.
          MultiIndex.from_product : Make a MultiIndex from cartesian product
                                    of iterables.
  
          Examples
          --------
          >>> df = pd.DataFrame([['HI', 'Temp'], ['HI', 'Precip'],
          ...                    ['NJ', 'Temp'], ['NJ', 'Precip']],
          ...                   columns=['a', 'b'])
          >>> df
                a       b
          0    HI    Temp
          1    HI  Precip
          2    NJ    Temp
          3    NJ  Precip
  
          >>> pd.MultiIndex.from_frame(df)
          MultiIndex([('HI',   'Temp'),
                      ('HI', 'Precip'),
                      ('NJ',   'Temp'),
                      ('NJ', 'Precip')],
                     names=['a', 'b'])
  
          Using explicit names, instead of the column names
  
          >>> pd.MultiIndex.from_frame(df, names=['state', 'observation'])
          MultiIndex([('HI',   'Temp'),
                      ('HI', 'Precip'),
                      ('NJ',   'Temp'),
                      ('NJ', 'Precip')],
                     names=['state', 'observation'])
          """
          if not isinstance(df, ABCDataFrame):
              raise TypeError("Input must be a DataFrame")
  
          column_names, columns = zip(*df.items())
          names = column_names if names is None else names
          return cls.from_arrays(columns, sortorder=sortorder, names=names)
  
      # --------------------------------------------------------------------
  
      @property
      def _values(self):
          # We override here, since our parent uses _data, which we don't use.
          return self.values
  
      @property
      def values(self):
          if self._tuples is not None:
              return self._tuples
  
          values = []
  
          for i in range(self.nlevels):
              vals = self._get_level_values(i)
              if is_categorical_dtype(vals):
                  vals = vals._internal_get_values()
              if isinstance(vals.dtype, ExtensionDtype) or hasattr(vals, "_box_values"):
                  vals = vals.astype(object)
              vals = np.array(vals, copy=False)
              values.append(vals)
  
          self._tuples = lib.fast_zip(values)
          return self._tuples
  
      @property
      def array(self):
          """
          Raises a ValueError for `MultiIndex` because there's no single
          array backing a MultiIndex.
  
          Raises
          ------
          ValueError
          """
          raise ValueError(
              "MultiIndex has no single backing array. Use "
              "'MultiIndex.to_numpy()' to get a NumPy array of tuples."
          )
  
      @property
      def shape(self):
          """
          Return a tuple of the shape of the underlying data.
          """
          # overriding the base Index.shape definition to avoid materializing
          # the values (GH-27384, GH-27775)
          return (len(self),)
  
      def __len__(self) -> int:
          return len(self.codes[0])
  
      # --------------------------------------------------------------------
      # Levels Methods
  
      @cache_readonly
      def levels(self):
          # Use cache_readonly to ensure that self.get_locs doesn't repeatedly
          # create new IndexEngine
          # https://github.com/pandas-dev/pandas/issues/31648
          result = [
              x._shallow_copy(name=name) for x, name in zip(self._levels, self._names)
          ]
          for level in result:
              # disallow midx.levels[0].name = "foo"
              level._no_setting_name = True
          return FrozenList(result)
  
      def _set_levels(
          self, levels, level=None, copy=False, validate=True, verify_integrity=False
      ):
          # This is NOT part of the levels property because it should be
          # externally not allowed to set levels. User beware if you change
          # _levels directly
          if validate:
              if len(levels) == 0:
                  raise ValueError("Must set non-zero number of levels.")
              if level is None and len(levels) != self.nlevels:
                  raise ValueError("Length of levels must match number of levels.")
              if level is not None and len(levels) != len(level):
                  raise ValueError("Length of levels must match length of level.")
  
          if level is None:
              new_levels = FrozenList(
                  ensure_index(lev, copy=copy)._shallow_copy() for lev in levels
              )
          else:
              level_numbers = [self._get_level_number(lev) for lev in level]
              new_levels = list(self._levels)
              for lev_num, lev in zip(level_numbers, levels):
                  new_levels[lev_num] = ensure_index(lev, copy=copy)._shallow_copy()
              new_levels = FrozenList(new_levels)
  
          if verify_integrity:
              new_codes = self._verify_integrity(levels=new_levels)
              self._codes = new_codes
  
          names = self.names
          self._levels = new_levels
          if any(names):
              self._set_names(names)
  
          self._tuples = None
          self._reset_cache()
  
      def set_levels(self, levels, level=None, inplace=False, verify_integrity=True):
          """
          Set new levels on MultiIndex. Defaults to returning new index.
  
          Parameters
          ----------
          levels : sequence or list of sequence
              New level(s) to apply.
          level : int, level name, or sequence of int/level names (default None)
              Level(s) to set (None for all levels).
          inplace : bool
              If True, mutates in place.
          verify_integrity : bool, default True
              If True, checks that levels and codes are compatible.
  
          Returns
          -------
          new index (of same type and class...etc)
  
          Examples
          --------
          >>> idx = pd.MultiIndex.from_tuples([(1, 'one'), (1, 'two'),
                                              (2, 'one'), (2, 'two'),
                                              (3, 'one'), (3, 'two')],
                                              names=['foo', 'bar'])
          >>> idx.set_levels([['a', 'b', 'c'], [1, 2]])
          MultiIndex([('a', 1),
                      ('a', 2),
                      ('b', 1),
                      ('b', 2),
                      ('c', 1),
                      ('c', 2)],
                     names=['foo', 'bar'])
          >>> idx.set_levels(['a', 'b', 'c'], level=0)
          MultiIndex([('a', 'one'),
                      ('a', 'two'),
                      ('b', 'one'),
                      ('b', 'two'),
                      ('c', 'one'),
                      ('c', 'two')],
                     names=['foo', 'bar'])
          >>> idx.set_levels(['a', 'b'], level='bar')
          MultiIndex([(1, 'a'),
                      (1, 'b'),
                      (2, 'a'),
                      (2, 'b'),
                      (3, 'a'),
                      (3, 'b')],
                     names=['foo', 'bar'])
  
          If any of the levels passed to ``set_levels()`` exceeds the
          existing length, all of the values from that argument will
          be stored in the MultiIndex levels, though the values will
          be truncated in the MultiIndex output.
  
          >>> idx.set_levels([['a', 'b', 'c'], [1, 2, 3, 4]], level=[0, 1])
          MultiIndex([('a', 1),
                      ('a', 2),
                      ('b', 1),
                      ('b', 2)],
                     names=['foo', 'bar'])
          >>> idx.set_levels([['a', 'b', 'c'], [1, 2, 3, 4]], level=[0, 1]).levels
          FrozenList([['a', 'b', 'c'], [1, 2, 3, 4]])
          """
          if is_list_like(levels) and not isinstance(levels, Index):
              levels = list(levels)
  
          if level is not None and not is_list_like(level):
              if not is_list_like(levels):
                  raise TypeError("Levels must be list-like")
              if is_list_like(levels[0]):
                  raise TypeError("Levels must be list-like")
              level = [level]
              levels = [levels]
          elif level is None or is_list_like(level):
              if not is_list_like(levels) or not is_list_like(levels[0]):
                  raise TypeError("Levels must be list of lists-like")
  
          if inplace:
              idx = self
          else:
              idx = self._shallow_copy()
          idx._reset_identity()
          idx._set_levels(
              levels, level=level, validate=True, verify_integrity=verify_integrity
          )
          if not inplace:
              return idx
  
      @property
      def nlevels(self) -> int:
          """
          Integer number of levels in this MultiIndex.
          """
          return len(self._levels)
  
      @property
      def levshape(self):
          """
          A tuple with the length of each level.
          """
          return tuple(len(x) for x in self.levels)
  
      # --------------------------------------------------------------------
      # Codes Methods
  
      @property
      def codes(self):
          return self._codes
  
      def _set_codes(
          self, codes, level=None, copy=False, validate=True, verify_integrity=False
      ):
          if validate:
              if level is None and len(codes) != self.nlevels:
                  raise ValueError("Length of codes must match number of levels")
              if level is not None and len(codes) != len(level):
                  raise ValueError("Length of codes must match length of levels.")
  
          if level is None:
              new_codes = FrozenList(
                  _coerce_indexer_frozen(level_codes, lev, copy=copy).view()
                  for lev, level_codes in zip(self._levels, codes)
              )
          else:
              level_numbers = [self._get_level_number(lev) for lev in level]
              new_codes = list(self._codes)
              for lev_num, level_codes in zip(level_numbers, codes):
                  lev = self.levels[lev_num]
                  new_codes[lev_num] = _coerce_indexer_frozen(level_codes, lev, copy=copy)
              new_codes = FrozenList(new_codes)
  
          if verify_integrity:
              new_codes = self._verify_integrity(codes=new_codes)
  
          self._codes = new_codes
  
          self._tuples = None
          self._reset_cache()
  
      def set_codes(self, codes, level=None, inplace=False, verify_integrity=True):
          """
          Set new codes on MultiIndex. Defaults to returning
          new index.
  
          .. versionadded:: 0.24.0
  
             New name for deprecated method `set_labels`.
  
          Parameters
          ----------
          codes : sequence or list of sequence
              New codes to apply.
          level : int, level name, or sequence of int/level names (default None)
              Level(s) to set (None for all levels).
          inplace : bool
              If True, mutates in place.
          verify_integrity : bool (default True)
              If True, checks that levels and codes are compatible.
  
          Returns
          -------
          new index (of same type and class...etc)
  
          Examples
          --------
          >>> idx = pd.MultiIndex.from_tuples([(1, 'one'),
                                               (1, 'two'),
                                               (2, 'one'),
                                               (2, 'two')],
                                              names=['foo', 'bar'])
          >>> idx.set_codes([[1, 0, 1, 0], [0, 0, 1, 1]])
          MultiIndex([(2, 'one'),
                      (1, 'one'),
                      (2, 'two'),
                      (1, 'two')],
                     names=['foo', 'bar'])
          >>> idx.set_codes([1, 0, 1, 0], level=0)
          MultiIndex([(2, 'one'),
                      (1, 'two'),
                      (2, 'one'),
                      (1, 'two')],
                     names=['foo', 'bar'])
          >>> idx.set_codes([0, 0, 1, 1], level='bar')
          MultiIndex([(1, 'one'),
                      (1, 'one'),
                      (2, 'two'),
                      (2, 'two')],
                     names=['foo', 'bar'])
          >>> idx.set_codes([[1, 0, 1, 0], [0, 0, 1, 1]], level=[0, 1])
          MultiIndex([(2, 'one'),
                      (1, 'one'),
                      (2, 'two'),
                      (1, 'two')],
                     names=['foo', 'bar'])
          """
          if level is not None and not is_list_like(level):
              if not is_list_like(codes):
                  raise TypeError("Codes must be list-like")
              if is_list_like(codes[0]):
                  raise TypeError("Codes must be list-like")
              level = [level]
              codes = [codes]
          elif level is None or is_list_like(level):
              if not is_list_like(codes) or not is_list_like(codes[0]):
                  raise TypeError("Codes must be list of lists-like")
  
          if inplace:
              idx = self
          else:
              idx = self._shallow_copy()
          idx._reset_identity()
          idx._set_codes(codes, level=level, verify_integrity=verify_integrity)
          if not inplace:
              return idx
  
      # --------------------------------------------------------------------
      # Index Internals
  
      @cache_readonly
      def _engine(self):
          # Calculate the number of bits needed to represent labels in each
          # level, as log2 of their sizes (including -1 for NaN):
          sizes = np.ceil(np.log2([len(l) + 1 for l in self.levels]))
  
          # Sum bit counts, starting from the _right_....
          lev_bits = np.cumsum(sizes[::-1])[::-1]
  
          # ... in order to obtain offsets such that sorting the combination of
          # shifted codes (one for each level, resulting in a unique integer) is
          # equivalent to sorting lexicographically the codes themselves. Notice
          # that each level needs to be shifted by the number of bits needed to
          # represent the _previous_ ones:
          offsets = np.concatenate([lev_bits[1:], [0]]).astype("uint64")
  
          # Check the total number of bits needed for our representation:
          if lev_bits[0] > 64:
              # The levels would overflow a 64 bit uint - use Python integers:
              return MultiIndexPyIntEngine(self.levels, self.codes, offsets)
          return MultiIndexUIntEngine(self.levels, self.codes, offsets)
  
      @property
      def _constructor(self):
          return MultiIndex.from_tuples
  
      @Appender(Index._shallow_copy.__doc__)
      def _shallow_copy(self, values=None, **kwargs):
          if values is not None:
              names = kwargs.pop("names", kwargs.pop("name", self.names))
              # discards freq
              kwargs.pop("freq", None)
              return MultiIndex.from_tuples(values, names=names, **kwargs)
          return self.copy(**kwargs)
  
      def _shallow_copy_with_infer(self, values, **kwargs):
          # On equal MultiIndexes the difference is empty.
          # Therefore, an empty MultiIndex is returned GH13490
          if len(values) == 0:
              return MultiIndex(
                  levels=[[] for _ in range(self.nlevels)],
                  codes=[[] for _ in range(self.nlevels)],
                  **kwargs,
              )
          return self._shallow_copy(values, **kwargs)
  
      # --------------------------------------------------------------------
  
      def copy(
          self,
          names=None,
          dtype=None,
          levels=None,
          codes=None,
          deep=False,
          name=None,
          _set_identity=False,
      ):
          """
          Make a copy of this object. Names, dtype, levels and codes can be
          passed and will be set on new copy.
  
          Parameters
          ----------
          names : sequence, optional
          dtype : numpy dtype or pandas type, optional
          levels : sequence, optional
          codes : sequence, optional
          deep : bool, default False
          name : Label
              Kept for compatibility with 1-dimensional Index. Should not be used.
  
          Returns
          -------
          MultiIndex
  
          Notes
          -----
          In most cases, there should be no functional difference from using
          ``deep``, but if ``deep`` is passed it will attempt to deepcopy.
          This could be potentially expensive on large MultiIndex objects.
          """
          names = self._validate_names(name=name, names=names, deep=deep)
          if deep:
              from copy import deepcopy
  
              if levels is None:
                  levels = deepcopy(self.levels)
              if codes is None:
                  codes = deepcopy(self.codes)
          else:
              if levels is None:
                  levels = self.levels
              if codes is None:
                  codes = self.codes
          return MultiIndex(
              levels=levels,
              codes=codes,
              names=names,
              sortorder=self.sortorder,
              verify_integrity=False,
              _set_identity=_set_identity,
          )
  
      def __array__(self, dtype=None) -> np.ndarray:
          """ the array interface, return my values """
          return self.values
  
      def view(self, cls=None):
          """ this is defined as a copy with the same identity """
          result = self.copy()
          result._id = self._id
          return result
  
      @Appender(Index.__contains__.__doc__)
      def __contains__(self, key: Any) -> bool:
          hash(key)
          try:
              self.get_loc(key)
              return True
          except (LookupError, TypeError, ValueError):
              return False
  
      @cache_readonly
      def dtype(self) -> np.dtype:
          return np.dtype("O")
  
      def _is_memory_usage_qualified(self) -> bool:
          """ return a boolean if we need a qualified .info display """
  
          def f(l):
              return "mixed" in l or "string" in l or "unicode" in l
  
          return any(f(l) for l in self._inferred_type_levels)
  
      @Appender(Index.memory_usage.__doc__)
      def memory_usage(self, deep: bool = False) -> int:
          # we are overwriting our base class to avoid
          # computing .values here which could materialize
          # a tuple representation unnecessarily
          return self._nbytes(deep)
  
      @cache_readonly
      def nbytes(self) -> int:
          """ return the number of bytes in the underlying data """
          return self._nbytes(False)
  
      def _nbytes(self, deep: bool = False) -> int:
          """
          return the number of bytes in the underlying data
          deeply introspect the level data if deep=True
  
          include the engine hashtable
  
          *this is in internal routine*
  
          """
          # for implementations with no useful getsizeof (PyPy)
          objsize = 24
  
          level_nbytes = sum(i.memory_usage(deep=deep) for i in self.levels)
          label_nbytes = sum(i.nbytes for i in self.codes)
          names_nbytes = sum(getsizeof(i, objsize) for i in self.names)
          result = level_nbytes + label_nbytes + names_nbytes
  
          # include our engine hashtable
          result += self._engine.sizeof(deep=deep)
          return result
  
      # --------------------------------------------------------------------
      # Rendering Methods
  
      def _formatter_func(self, tup):
          """
          Formats each item in tup according to its level's formatter function.
          """
          formatter_funcs = [level._formatter_func for level in self.levels]
          return tuple(func(val) for func, val in zip(formatter_funcs, tup))
  
      def _format_data(self, name=None):
          """
          Return the formatted data as a unicode string
          """
          return format_object_summary(
              self, self._formatter_func, name=name, line_break_each_value=True
          )
  
      def _format_attrs(self):
          """
          Return a list of tuples of the (attr,formatted_value).
          """
          return format_object_attrs(self, include_dtype=False)
  
      def _format_native_types(self, na_rep="nan", **kwargs):
          new_levels = []
          new_codes = []
  
          # go through the levels and format them
          for level, level_codes in zip(self.levels, self.codes):
              level = level._format_native_types(na_rep=na_rep, **kwargs)
              # add nan values, if there are any
              mask = level_codes == -1
              if mask.any():
                  nan_index = len(level)
                  level = np.append(level, na_rep)
                  assert not level_codes.flags.writeable  # i.e. copy is needed
                  level_codes = level_codes.copy()  # make writeable
                  level_codes[mask] = nan_index
              new_levels.append(level)
              new_codes.append(level_codes)
  
          if len(new_levels) == 1:
              # a single-level multi-index
              return Index(new_levels[0].take(new_codes[0]))._format_native_types()
          else:
              # reconstruct the multi-index
              mi = MultiIndex(
                  levels=new_levels,
                  codes=new_codes,
                  names=self.names,
                  sortorder=self.sortorder,
                  verify_integrity=False,
              )
              return mi.values
  
      def format(
          self,
          space=2,
          sparsify=None,
          adjoin=True,
          names=False,
          na_rep=None,
          formatter=None,
      ):
          if len(self) == 0:
              return []
  
          stringified_levels = []
          for lev, level_codes in zip(self.levels, self.codes):
              na = na_rep if na_rep is not None else _get_na_rep(lev.dtype.type)
  
              if len(lev) > 0:
  
                  formatted = lev.take(level_codes).format(formatter=formatter)
  
                  # we have some NA
                  mask = level_codes == -1
                  if mask.any():
                      formatted = np.array(formatted, dtype=object)
                      formatted[mask] = na
                      formatted = formatted.tolist()
  
              else:
                  # weird all NA case
                  formatted = [
                      pprint_thing(na if isna(x) else x, escape_chars=("\t", "\r", "\n"))
                      for x in algos.take_1d(lev._values, level_codes)
                  ]
              stringified_levels.append(formatted)
  
          result_levels = []
          for lev, name in zip(stringified_levels, self.names):
              level = []
  
              if names:
                  level.append(
                      pprint_thing(name, escape_chars=("\t", "\r", "\n"))
                      if name is not None
                      else ""
                  )
  
              level.extend(np.array(lev, dtype=object))
              result_levels.append(level)
  
          if sparsify is None:
              sparsify = get_option("display.multi_sparse")
  
          if sparsify:
              sentinel = ""
              # GH3547
              # use value of sparsify as sentinel,  unless it's an obvious
              # "Truthy" value
              if sparsify not in [True, 1]:
                  sentinel = sparsify
              # little bit of a kludge job for #1217
              result_levels = _sparsify(
                  result_levels, start=int(names), sentinel=sentinel
              )
  
          if adjoin:
              from pandas.io.formats.format import _get_adjustment
  
              adj = _get_adjustment()
              return adj.adjoin(space, *result_levels).split("\n")
          else:
              return result_levels
  
      # --------------------------------------------------------------------
      # Names Methods
  
      def _get_names(self):
          return FrozenList(self._names)
  
      def _set_names(self, names, level=None, validate=True):
          """
          Set new names on index. Each name has to be a hashable type.
  
          Parameters
          ----------
          values : str or sequence
              name(s) to set
          level : int, level name, or sequence of int/level names (default None)
              If the index is a MultiIndex (hierarchical), level(s) to set (None
              for all levels).  Otherwise level must be None
          validate : boolean, default True
              validate that the names match level lengths
  
          Raises
          ------
          TypeError if each name is not hashable.
  
          Notes
          -----
          sets names on levels. WARNING: mutates!
  
          Note that you generally want to set this *after* changing levels, so
          that it only acts on copies
          """
          # GH 15110
          # Don't allow a single string for names in a MultiIndex
          if names is not None and not is_list_like(names):
              raise ValueError("Names should be list-like for a MultiIndex")
          names = list(names)
  
          if validate:
              if level is not None and len(names) != len(level):
                  raise ValueError("Length of names must match length of level.")
              if level is None and len(names) != self.nlevels:
                  raise ValueError(
                      "Length of names must match number of levels in MultiIndex."
                  )
  
          if level is None:
              level = range(self.nlevels)
          else:
              level = [self._get_level_number(lev) for lev in level]
  
          # set the name
          for lev, name in zip(level, names):
              if name is not None:
                  # GH 20527
                  # All items in 'names' need to be hashable:
                  if not is_hashable(name):
                      raise TypeError(
                          f"{type(self).__name__}.name must be a hashable type"
                      )
              self._names[lev] = name
  
          # If .levels has been accessed, the names in our cache will be stale.
          self._reset_cache()
  
      names = property(
          fset=_set_names, fget=_get_names, doc="""\nNames of levels in MultiIndex.\n"""
      )
  
      # --------------------------------------------------------------------
  
      @Appender(Index._get_grouper_for_level.__doc__)
      def _get_grouper_for_level(self, mapper, level):
          indexer = self.codes[level]
          level_index = self.levels[level]
  
          if mapper is not None:
              # Handle group mapping function and return
              level_values = self.levels[level].take(indexer)
              grouper = level_values.map(mapper)
              return grouper, None, None
  
          codes, uniques = algos.factorize(indexer, sort=True)
  
          if len(uniques) > 0 and uniques[0] == -1:
              # Handle NAs
              mask = indexer != -1
              ok_codes, uniques = algos.factorize(indexer[mask], sort=True)
  
              codes = np.empty(len(indexer), dtype=indexer.dtype)
              codes[mask] = ok_codes
              codes[~mask] = -1
  
          if len(uniques) < len(level_index):
              # Remove unobserved levels from level_index
              level_index = level_index.take(uniques)
          else:
              # break references back to us so that setting the name
              # on the output of a groupby doesn't reflect back here.
              level_index = level_index.copy()
  
          if level_index._can_hold_na:
              grouper = level_index.take(codes, fill_value=True)
          else:
              grouper = level_index.take(codes)
  
          return grouper, codes, level_index
  
      @cache_readonly
      def inferred_type(self) -> str:
          return "mixed"
  
      def _get_level_number(self, level) -> int:
          count = self.names.count(level)
          if (count > 1) and not is_integer(level):
              raise ValueError(
                  f"The name {level} occurs multiple times, use a level number"
              )
          try:
              level = self.names.index(level)
          except ValueError as err:
              if not is_integer(level):
                  raise KeyError(f"Level {level} not found") from err
              elif level < 0:
                  level += self.nlevels
                  if level < 0:
                      orig_level = level - self.nlevels
                      raise IndexError(
                          f"Too many levels: Index has only {self.nlevels} levels, "
                          f"{orig_level} is not a valid level number"
                      ) from err
              # Note: levels are zero-based
              elif level >= self.nlevels:
                  raise IndexError(
                      f"Too many levels: Index has only {self.nlevels} levels, "
                      f"not {level + 1}"
                  ) from err
          return level
  
      @property
      def _has_complex_internals(self) -> bool:
          # used to avoid libreduction code paths, which raise or require conversion
          return True
  
      @cache_readonly
      def is_monotonic_increasing(self) -> bool:
          """
          return if the index is monotonic increasing (only equal or
          increasing) values.
          """
          if all(x.is_monotonic for x in self.levels):
              # If each level is sorted, we can operate on the codes directly. GH27495
              return libalgos.is_lexsorted(
                  [x.astype("int64", copy=False) for x in self.codes]
              )
  
          # reversed() because lexsort() wants the most significant key last.
          values = [
              self._get_level_values(i).values for i in reversed(range(len(self.levels)))
          ]
          try:
              sort_order = np.lexsort(values)
              return Index(sort_order).is_monotonic
          except TypeError:
  
              # we have mixed types and np.lexsort is not happy
              return Index(self.values).is_monotonic
  
      @cache_readonly
      def is_monotonic_decreasing(self) -> bool:
          """
          return if the index is monotonic decreasing (only equal or
          decreasing) values.
          """
          # monotonic decreasing if and only if reverse is monotonic increasing
          return self[::-1].is_monotonic_increasing
  
      @cache_readonly
      def _inferred_type_levels(self):
          """ return a list of the inferred types, one for each level """
          return [i.inferred_type for i in self.levels]
  
      @Appender(Index.duplicated.__doc__)
      def duplicated(self, keep="first"):
          shape = map(len, self.levels)
          ids = get_group_index(self.codes, shape, sort=False, xnull=False)
  
          return duplicated_int64(ids, keep)
  
      def fillna(self, value=None, downcast=None):
          """
          fillna is not implemented for MultiIndex
          """
          raise NotImplementedError("isna is not defined for MultiIndex")
  
      @Appender(Index.dropna.__doc__)
      def dropna(self, how="any"):
          nans = [level_codes == -1 for level_codes in self.codes]
          if how == "any":
              indexer = np.any(nans, axis=0)
          elif how == "all":
              indexer = np.all(nans, axis=0)
          else:
              raise ValueError(f"invalid how option: {how}")
  
          new_codes = [level_codes[~indexer] for level_codes in self.codes]
          return self.copy(codes=new_codes, deep=True)
  
      def _get_level_values(self, level, unique=False):
          """
          Return vector of label values for requested level,
          equal to the length of the index
  
          **this is an internal method**
  
          Parameters
          ----------
          level : int level
          unique : bool, default False
              if True, drop duplicated values
  
          Returns
          -------
          values : ndarray
          """
          lev = self.levels[level]
          level_codes = self.codes[level]
          name = self._names[level]
          if unique:
              level_codes = algos.unique(level_codes)
          filled = algos.take_1d(lev._values, level_codes, fill_value=lev._na_value)
          return lev._shallow_copy(filled, name=name)
  
      def get_level_values(self, level):
          """
          Return vector of label values for requested level,
          equal to the length of the index.
  
          Parameters
          ----------
          level : int or str
              ``level`` is either the integer position of the level in the
              MultiIndex, or the name of the level.
  
          Returns
          -------
          values : Index
              Values is a level of this MultiIndex converted to
              a single :class:`Index` (or subclass thereof).
  
          Examples
          --------
          Create a MultiIndex:
  
          >>> mi = pd.MultiIndex.from_arrays((list('abc'), list('def')))
          >>> mi.names = ['level_1', 'level_2']
  
          Get level values by supplying level as either integer or name:
  
          >>> mi.get_level_values(0)
          Index(['a', 'b', 'c'], dtype='object', name='level_1')
          >>> mi.get_level_values('level_2')
          Index(['d', 'e', 'f'], dtype='object', name='level_2')
          """
          level = self._get_level_number(level)
          values = self._get_level_values(level)
          return values
  
      @Appender(Index.unique.__doc__)
      def unique(self, level=None):
  
          if level is None:
              return super().unique()
          else:
              level = self._get_level_number(level)
              return self._get_level_values(level=level, unique=True)
  
      def _to_safe_for_reshape(self):
          """ convert to object if we are a categorical """
          return self.set_levels([i._to_safe_for_reshape() for i in self.levels])
  
      def to_frame(self, index=True, name=None):
          """
          Create a DataFrame with the levels of the MultiIndex as columns.
  
          Column ordering is determined by the DataFrame constructor with data as
          a dict.
  
          .. versionadded:: 0.24.0
  
          Parameters
          ----------
          index : bool, default True
              Set the index of the returned DataFrame as the original MultiIndex.
  
          name : list / sequence of str, optional
              The passed names should substitute index level names.
  
          Returns
          -------
          DataFrame : a DataFrame containing the original MultiIndex data.
  
          See Also
          --------
          DataFrame
          """
          from pandas import DataFrame
  
          if name is not None:
              if not is_list_like(name):
                  raise TypeError("'name' must be a list / sequence of column names.")
  
              if len(name) != len(self.levels):
                  raise ValueError(
                      "'name' should have same length as number of levels on index."
                  )
              idx_names = name
          else:
              idx_names = self.names
  
          # Guarantee resulting column order - PY36+ dict maintains insertion order
          result = DataFrame(
              {
                  (level if lvlname is None else lvlname): self._get_level_values(level)
                  for lvlname, level in zip(idx_names, range(len(self.levels)))
              },
              copy=False,
          )
  
          if index:
              result.index = self
          return result
  
      def to_flat_index(self):
          """
          Convert a MultiIndex to an Index of Tuples containing the level values.
  
          .. versionadded:: 0.24.0
  
          Returns
          -------
          pd.Index
              Index with the MultiIndex data represented in Tuples.
  
          Notes
          -----
          This method will simply return the caller if called by anything other
          than a MultiIndex.
  
          Examples
          --------
          >>> index = pd.MultiIndex.from_product(
          ...     [['foo', 'bar'], ['baz', 'qux']],
          ...     names=['a', 'b'])
          >>> index.to_flat_index()
          Index([('foo', 'baz'), ('foo', 'qux'),
                 ('bar', 'baz'), ('bar', 'qux')],
                dtype='object')
          """
          return Index(self.values, tupleize_cols=False)
  
      @property
      def is_all_dates(self) -> bool:
          return False
  
      def is_lexsorted(self) -> bool:
          """
          Return True if the codes are lexicographically sorted.
  
          Returns
          -------
          bool
          """
          return self.lexsort_depth == self.nlevels
  
      @cache_readonly
      def lexsort_depth(self):
          if self.sortorder is not None:
              return self.sortorder
  
          return self._lexsort_depth()
  
      def _lexsort_depth(self) -> int:
          """
          Compute and return the lexsort_depth, the number of levels of the
          MultiIndex that are sorted lexically
  
          Returns
          -------
          int
          """
          int64_codes = [ensure_int64(level_codes) for level_codes in self.codes]
          for k in range(self.nlevels, 0, -1):
              if libalgos.is_lexsorted(int64_codes[:k]):
                  return k
          return 0
  
      def _sort_levels_monotonic(self):
          """
          This is an *internal* function.
  
          Create a new MultiIndex from the current to monotonically sorted
          items IN the levels. This does not actually make the entire MultiIndex
          monotonic, JUST the levels.
  
          The resulting MultiIndex will have the same outward
          appearance, meaning the same .values and ordering. It will also
          be .equals() to the original.
  
          Returns
          -------
          MultiIndex
  
          Examples
          --------
          >>> mi = pd.MultiIndex(levels=[['a', 'b'], ['bb', 'aa']],
          ...                    codes=[[0, 0, 1, 1], [0, 1, 0, 1]])
          >>> mi
          MultiIndex([('a', 'bb'),
                      ('a', 'aa'),
                      ('b', 'bb'),
                      ('b', 'aa')],
                     )
  
          >>> mi.sort_values()
          MultiIndex([('a', 'aa'),
                      ('a', 'bb'),
                      ('b', 'aa'),
                      ('b', 'bb')],
                     )
          """
          if self.is_lexsorted() and self.is_monotonic:
              return self
  
          new_levels = []
          new_codes = []
  
          for lev, level_codes in zip(self.levels, self.codes):
  
              if not lev.is_monotonic:
                  try:
                      # indexer to reorder the levels
                      indexer = lev.argsort()
                  except TypeError:
                      pass
                  else:
                      lev = lev.take(indexer)
  
                      # indexer to reorder the level codes
                      indexer = ensure_int64(indexer)
                      ri = lib.get_reverse_indexer(indexer, len(indexer))
                      level_codes = algos.take_1d(ri, level_codes)
  
              new_levels.append(lev)
              new_codes.append(level_codes)
  
          return MultiIndex(
              new_levels,
              new_codes,
              names=self.names,
              sortorder=self.sortorder,
              verify_integrity=False,
          )
  
      def remove_unused_levels(self):
          """
          Create a new MultiIndex from the current that removes
          unused levels, meaning that they are not expressed in the labels.
  
          The resulting MultiIndex will have the same outward
          appearance, meaning the same .values and ordering. It will also
          be .equals() to the original.
  
          Returns
          -------
          MultiIndex
  
          Examples
          --------
          >>> mi = pd.MultiIndex.from_product([range(2), list('ab')])
          >>> mi
          MultiIndex([(0, 'a'),
                      (0, 'b'),
                      (1, 'a'),
                      (1, 'b')],
                     )
  
          >>> mi[2:]
          MultiIndex([(1, 'a'),
                      (1, 'b')],
                     )
  
          The 0 from the first level is not represented
          and can be removed
  
          >>> mi2 = mi[2:].remove_unused_levels()
          >>> mi2.levels
          FrozenList([[1], ['a', 'b']])
          """
          new_levels = []
          new_codes = []
  
          changed = False
          for lev, level_codes in zip(self.levels, self.codes):
  
              # Since few levels are typically unused, bincount() is more
              # efficient than unique() - however it only accepts positive values
              # (and drops order):
              uniques = np.where(np.bincount(level_codes + 1) > 0)[0] - 1
              has_na = int(len(uniques) and (uniques[0] == -1))
  
              if len(uniques) != len(lev) + has_na:
                  # We have unused levels
                  changed = True
  
                  # Recalculate uniques, now preserving order.
                  # Can easily be cythonized by exploiting the already existing
                  # "uniques" and stop parsing "level_codes" when all items
                  # are found:
                  uniques = algos.unique(level_codes)
                  if has_na:
                      na_idx = np.where(uniques == -1)[0]
                      # Just ensure that -1 is in first position:
                      uniques[[0, na_idx[0]]] = uniques[[na_idx[0], 0]]
  
                  # codes get mapped from uniques to 0:len(uniques)
                  # -1 (if present) is mapped to last position
                  code_mapping = np.zeros(len(lev) + has_na)
                  # ... and reassigned value -1:
                  code_mapping[uniques] = np.arange(len(uniques)) - has_na
  
                  level_codes = code_mapping[level_codes]
  
                  # new levels are simple
                  lev = lev.take(uniques[has_na:])
  
              new_levels.append(lev)
              new_codes.append(level_codes)
  
          result = self.view()
  
          if changed:
              result._reset_identity()
              result._set_levels(new_levels, validate=False)
              result._set_codes(new_codes, validate=False)
  
          return result
  
      # --------------------------------------------------------------------
      # Pickling Methods
  
      def __reduce__(self):
          """Necessary for making this object picklable"""
          d = dict(
              levels=list(self.levels),
              codes=list(self.codes),
              sortorder=self.sortorder,
              names=list(self.names),
          )
          return ibase._new_Index, (type(self), d), None
  
      # --------------------------------------------------------------------
  
      def __getitem__(self, key):
          if is_scalar(key):
              key = com.cast_scalar_indexer(key)
  
              retval = []
              for lev, level_codes in zip(self.levels, self.codes):
                  if level_codes[key] == -1:
                      retval.append(np.nan)
                  else:
                      retval.append(lev[level_codes[key]])
  
              return tuple(retval)
          else:
              if com.is_bool_indexer(key):
                  key = np.asarray(key, dtype=bool)
                  sortorder = self.sortorder
              else:
                  # cannot be sure whether the result will be sorted
                  sortorder = None
  
                  if isinstance(key, Index):
                      key = np.asarray(key)
  
              new_codes = [level_codes[key] for level_codes in self.codes]
  
              return MultiIndex(
                  levels=self.levels,
                  codes=new_codes,
                  names=self.names,
                  sortorder=sortorder,
                  verify_integrity=False,
              )
  
      @Appender(_index_shared_docs["take"] % _index_doc_kwargs)
      def take(self, indices, axis=0, allow_fill=True, fill_value=None, **kwargs):
          nv.validate_take(tuple(), kwargs)
          indices = ensure_platform_int(indices)
          taken = self._assert_take_fillable(
              self.codes,
              indices,
              allow_fill=allow_fill,
              fill_value=fill_value,
              na_value=-1,
          )
          return MultiIndex(
              levels=self.levels, codes=taken, names=self.names, verify_integrity=False
          )
  
      def _assert_take_fillable(
          self, values, indices, allow_fill=True, fill_value=None, na_value=None
      ):
          """ Internal method to handle NA filling of take """
          # only fill if we are passing a non-None fill_value
          if allow_fill and fill_value is not None:
              if (indices < -1).any():
                  msg = (
                      "When allow_fill=True and fill_value is not None, "
                      "all indices must be >= -1"
                  )
                  raise ValueError(msg)
              taken = [lab.take(indices) for lab in self.codes]
              mask = indices == -1
              if mask.any():
                  masked = []
                  for new_label in taken:
                      label_values = new_label
                      label_values[mask] = na_value
                      masked.append(np.asarray(label_values))
                  taken = masked
          else:
              taken = [lab.take(indices) for lab in self.codes]
          return taken
  
      def append(self, other):
          """
          Append a collection of Index options together
  
          Parameters
          ----------
          other : Index or list/tuple of indices
  
          Returns
          -------
          appended : Index
          """
          if not isinstance(other, (list, tuple)):
              other = [other]
  
          if all(
              (isinstance(o, MultiIndex) and o.nlevels >= self.nlevels) for o in other
          ):
              arrays = []
              for i in range(self.nlevels):
                  label = self._get_level_values(i)
                  appended = [o._get_level_values(i) for o in other]
                  arrays.append(label.append(appended))
              return MultiIndex.from_arrays(arrays, names=self.names)
  
          to_concat = (self.values,) + tuple(k._values for k in other)
          new_tuples = np.concatenate(to_concat)
  
          # if all(isinstance(x, MultiIndex) for x in other):
          try:
              return MultiIndex.from_tuples(new_tuples, names=self.names)
          except (TypeError, IndexError):
              return Index(new_tuples)
  
      def argsort(self, *args, **kwargs) -> np.ndarray:
          return self.values.argsort(*args, **kwargs)
  
      @Appender(_index_shared_docs["repeat"] % _index_doc_kwargs)
      def repeat(self, repeats, axis=None):
          nv.validate_repeat(tuple(), dict(axis=axis))
          repeats = ensure_platform_int(repeats)
          return MultiIndex(
              levels=self.levels,
              codes=[
                  level_codes.view(np.ndarray).astype(np.intp).repeat(repeats)
                  for level_codes in self.codes
              ],
              names=self.names,
              sortorder=self.sortorder,
              verify_integrity=False,
          )
  
      def where(self, cond, other=None):
          raise NotImplementedError(".where is not supported for MultiIndex operations")
  
      def drop(self, codes, level=None, errors="raise"):
          """
          Make new MultiIndex with passed list of codes deleted
  
          Parameters
          ----------
          codes : array-like
              Must be a list of tuples
          level : int or level name, default None
          errors : str, default 'raise'
  
          Returns
          -------
          dropped : MultiIndex
          """
          if level is not None:
              return self._drop_from_level(codes, level, errors)
  
          if not isinstance(codes, (np.ndarray, Index)):
              try:
                  codes = com.index_labels_to_array(codes, dtype=object)
              except ValueError:
                  pass
  
          inds = []
          for level_codes in codes:
              try:
                  loc = self.get_loc(level_codes)
                  # get_loc returns either an integer, a slice, or a boolean
                  # mask
                  if isinstance(loc, int):
                      inds.append(loc)
                  elif isinstance(loc, slice):
                      inds.extend(range(loc.start, loc.stop))
                  elif com.is_bool_indexer(loc):
                      if self.lexsort_depth == 0:
                          warnings.warn(
                              "dropping on a non-lexsorted multi-index "
                              "without a level parameter may impact performance.",
                              PerformanceWarning,
                              stacklevel=3,
                          )
                      loc = loc.nonzero()[0]
                      inds.extend(loc)
                  else:
                      msg = f"unsupported indexer of type {type(loc)}"
                      raise AssertionError(msg)
              except KeyError:
                  if errors != "ignore":
                      raise
  
          return self.delete(inds)
  
      def _drop_from_level(self, codes, level, errors="raise"):
          codes = com.index_labels_to_array(codes)
          i = self._get_level_number(level)
          index = self.levels[i]
          values = index.get_indexer(codes)
  
          mask = ~algos.isin(self.codes[i], values)
          if mask.all() and errors != "ignore":
              raise KeyError(f"labels {codes} not found in level")
  
          return self[mask]
  
      def swaplevel(self, i=-2, j=-1):
          """
          Swap level i with level j.
  
          Calling this method does not change the ordering of the values.
  
          Parameters
          ----------
          i : int, str, default -2
              First level of index to be swapped. Can pass level name as string.
              Type of parameters can be mixed.
          j : int, str, default -1
              Second level of index to be swapped. Can pass level name as string.
              Type of parameters can be mixed.
  
          Returns
          -------
          MultiIndex
              A new MultiIndex.
  
          See Also
          --------
          Series.swaplevel : Swap levels i and j in a MultiIndex.
          Dataframe.swaplevel : Swap levels i and j in a MultiIndex on a
              particular axis.
  
          Examples
          --------
          >>> mi = pd.MultiIndex(levels=[['a', 'b'], ['bb', 'aa']],
          ...                    codes=[[0, 0, 1, 1], [0, 1, 0, 1]])
          >>> mi
          MultiIndex([('a', 'bb'),
                      ('a', 'aa'),
                      ('b', 'bb'),
                      ('b', 'aa')],
                     )
          >>> mi.swaplevel(0, 1)
          MultiIndex([('bb', 'a'),
                      ('aa', 'a'),
                      ('bb', 'b'),
                      ('aa', 'b')],
                     )
          """
          new_levels = list(self.levels)
          new_codes = list(self.codes)
          new_names = list(self.names)
  
          i = self._get_level_number(i)
          j = self._get_level_number(j)
  
          new_levels[i], new_levels[j] = new_levels[j], new_levels[i]
          new_codes[i], new_codes[j] = new_codes[j], new_codes[i]
          new_names[i], new_names[j] = new_names[j], new_names[i]
  
          return MultiIndex(
              levels=new_levels, codes=new_codes, names=new_names, verify_integrity=False
          )
  
      def reorder_levels(self, order):
          """
          Rearrange levels using input order. May not drop or duplicate levels.
  
          Parameters
          ----------
          order : list of int or list of str
              List representing new level order. Reference level by number
              (position) or by key (label).
  
          Returns
          -------
          MultiIndex
          """
          order = [self._get_level_number(i) for i in order]
          if len(order) != self.nlevels:
              raise AssertionError(
                  f"Length of order must be same as number of levels ({self.nlevels}), "
                  f"got {len(order)}"
              )
          new_levels = [self.levels[i] for i in order]
          new_codes = [self.codes[i] for i in order]
          new_names = [self.names[i] for i in order]
  
          return MultiIndex(
              levels=new_levels, codes=new_codes, names=new_names, verify_integrity=False
          )
  
      def _get_codes_for_sorting(self):
          """
          we categorizing our codes by using the
          available categories (all, not just observed)
          excluding any missing ones (-1); this is in preparation
          for sorting, where we need to disambiguate that -1 is not
          a valid valid
          """
  
          def cats(level_codes):
              return np.arange(
                  np.array(level_codes).max() + 1 if len(level_codes) else 0,
                  dtype=level_codes.dtype,
              )
  
          return [
              Categorical.from_codes(level_codes, cats(level_codes), ordered=True)
              for level_codes in self.codes
          ]
  
      def sortlevel(self, level=0, ascending=True, sort_remaining=True):
          """
          Sort MultiIndex at the requested level. The result will respect the
          original ordering of the associated factor at that level.
  
          Parameters
          ----------
          level : list-like, int or str, default 0
              If a string is given, must be a name of the level.
              If list-like must be names or ints of levels.
          ascending : bool, default True
              False to sort in descending order.
              Can also be a list to specify a directed ordering.
          sort_remaining : sort by the remaining levels after level
  
          Returns
          -------
          sorted_index : pd.MultiIndex
              Resulting index.
          indexer : np.ndarray
              Indices of output values in original index.
          """
          if isinstance(level, (str, int)):
              level = [level]
          level = [self._get_level_number(lev) for lev in level]
          sortorder = None
  
          # we have a directed ordering via ascending
          if isinstance(ascending, list):
              if not len(level) == len(ascending):
                  raise ValueError("level must have same length as ascending")
  
              indexer = lexsort_indexer(
                  [self.codes[lev] for lev in level], orders=ascending
              )
  
          # level ordering
          else:
  
              codes = list(self.codes)
              shape = list(self.levshape)
  
              # partition codes and shape
              primary = tuple(codes[lev] for lev in level)
              primshp = tuple(shape[lev] for lev in level)
  
              # Reverse sorted to retain the order of
              # smaller indices that needs to be removed
              for lev in sorted(level, reverse=True):
                  codes.pop(lev)
                  shape.pop(lev)
  
              if sort_remaining:
                  primary += primary + tuple(codes)
                  primshp += primshp + tuple(shape)
              else:
                  sortorder = level[0]
  
              indexer = indexer_from_factorized(primary, primshp, compress=False)
  
              if not ascending:
                  indexer = indexer[::-1]
  
          indexer = ensure_platform_int(indexer)
          new_codes = [level_codes.take(indexer) for level_codes in self.codes]
  
          new_index = MultiIndex(
              codes=new_codes,
              levels=self.levels,
              names=self.names,
              sortorder=sortorder,
              verify_integrity=False,
          )
  
          return new_index, indexer
  
      def reindex(self, target, method=None, level=None, limit=None, tolerance=None):
          """
          Create index with target's values (move/add/delete values as necessary)
  
          Returns
          -------
          new_index : pd.MultiIndex
              Resulting index
          indexer : np.ndarray or None
              Indices of output values in original index.
  
          """
          # GH6552: preserve names when reindexing to non-named target
          # (i.e. neither Index nor Series).
          preserve_names = not hasattr(target, "names")
  
          if level is not None:
              if method is not None:
                  raise TypeError("Fill method not supported if level passed")
  
              # GH7774: preserve dtype/tz if target is empty and not an Index.
              # target may be an iterator
              target = ibase._ensure_has_len(target)
              if len(target) == 0 and not isinstance(target, Index):
                  idx = self.levels[level]
                  attrs = idx._get_attributes_dict()
                  attrs.pop("freq", None)  # don't preserve freq
                  target = type(idx)._simple_new(np.empty(0, dtype=idx.dtype), **attrs)
              else:
                  target = ensure_index(target)
              target, indexer, _ = self._join_level(
                  target, level, how="right", return_indexers=True, keep_order=False
              )
          else:
              target = ensure_index(target)
              if self.equals(target):
                  indexer = None
              else:
                  if self.is_unique:
                      indexer = self.get_indexer(
                          target, method=method, limit=limit, tolerance=tolerance
                      )
                  else:
                      raise ValueError("cannot handle a non-unique multi-index!")
  
          if not isinstance(target, MultiIndex):
              if indexer is None:
                  target = self
              elif (indexer >= 0).all():
                  target = self.take(indexer)
              else:
                  # hopefully?
                  target = MultiIndex.from_tuples(target)
  
          if (
              preserve_names
              and target.nlevels == self.nlevels
              and target.names != self.names
          ):
              target = target.copy(deep=False)
              target.names = self.names
  
          return target, indexer
  
      # --------------------------------------------------------------------
      # Indexing Methods
  
      def get_value(self, series, key):
          # Label-based
          if not is_hashable(key) or is_iterator(key):
              # We allow tuples if they are hashable, whereas other Index
              #  subclasses require scalar.
              # We have to explicitly exclude generators, as these are hashable.
              raise InvalidIndexError(key)
  
          try:
              loc = self.get_loc(key)
          except KeyError:
              if is_integer(key):
                  loc = key
              else:
                  raise
  
          return self._get_values_for_loc(series, loc, key)
  
      def _get_values_for_loc(self, series: "Series", loc, key):
          """
          Do a positional lookup on the given Series, returning either a scalar
          or a Series.
  
          Assumes that `series.index is self`
          """
          new_values = series._values[loc]
          if is_scalar(loc):
              return new_values
  
          new_index = self[loc]
          new_index = maybe_droplevels(new_index, key)
          new_ser = series._constructor(new_values, index=new_index, name=series.name)
          return new_ser.__finalize__(series)
  
      def _convert_listlike_indexer(self, keyarr):
          """
          Parameters
          ----------
          keyarr : list-like
              Indexer to convert.
  
          Returns
          -------
          tuple (indexer, keyarr)
              indexer is an ndarray or None if cannot convert
              keyarr are tuple-safe keys
          """
          indexer, keyarr = super()._convert_listlike_indexer(keyarr)
  
          # are we indexing a specific level
          if indexer is None and len(keyarr) and not isinstance(keyarr[0], tuple):
              level = 0
              _, indexer = self.reindex(keyarr, level=level)
  
              # take all
              if indexer is None:
                  indexer = np.arange(len(self))
  
              check = self.levels[0].get_indexer(keyarr)
              mask = check == -1
              if mask.any():
                  raise KeyError(f"{keyarr[mask]} not in index")
  
          return indexer, keyarr
  
      def _get_partial_string_timestamp_match_key(self, key):
          """
          Translate any partial string timestamp matches in key, returning the
          new key.
  
          Only relevant for MultiIndex.
          """
          # GH#10331
          if isinstance(key, str) and self.levels[0]._supports_partial_string_indexing:
              # Convert key '2016-01-01' to
              # ('2016-01-01'[, slice(None, None, None)]+)
              key = tuple([key] + [slice(None)] * (len(self.levels) - 1))
  
          if isinstance(key, tuple):
              # Convert (..., '2016-01-01', ...) in tuple to
              # (..., slice('2016-01-01', '2016-01-01', None), ...)
              new_key = []
              for i, component in enumerate(key):
                  if (
                      isinstance(component, str)
                      and self.levels[i]._supports_partial_string_indexing
                  ):
                      new_key.append(slice(component, component, None))
                  else:
                      new_key.append(component)
              key = tuple(new_key)
  
          return key
  
      @Appender(_index_shared_docs["get_indexer"] % _index_doc_kwargs)
      def get_indexer(self, target, method=None, limit=None, tolerance=None):
          method = missing.clean_reindex_fill_method(method)
          target = ensure_index(target)
  
          # empty indexer
          if is_list_like(target) and not len(target):
              return ensure_platform_int(np.array([]))
  
          if not isinstance(target, MultiIndex):
              try:
                  target = MultiIndex.from_tuples(target)
              except (TypeError, ValueError):
  
                  # let's instead try with a straight Index
                  if method is None:
                      return Index(self.values).get_indexer(
                          target, method=method, limit=limit, tolerance=tolerance
                      )
  
          if not self.is_unique:
              raise ValueError("Reindexing only valid with uniquely valued Index objects")
  
          if method == "pad" or method == "backfill":
              if tolerance is not None:
                  raise NotImplementedError(
                      "tolerance not implemented yet for MultiIndex"
                  )
              indexer = self._engine.get_indexer(target, method, limit)
          elif method == "nearest":
              raise NotImplementedError(
                  "method='nearest' not implemented yet "
                  "for MultiIndex; see GitHub issue 9365"
              )
          else:
              indexer = self._engine.get_indexer(target)
  
          return ensure_platform_int(indexer)
  
      @Appender(_index_shared_docs["get_indexer_non_unique"] % _index_doc_kwargs)
      def get_indexer_non_unique(self, target):
          return super().get_indexer_non_unique(target)
  
      def get_slice_bound(
          self, label: Union[Hashable, Sequence[Hashable]], side: str, kind: str
      ) -> int:
          """
          For an ordered MultiIndex, compute slice bound
          that corresponds to given label.
  
          Returns leftmost (one-past-the-rightmost if `side=='right') position
          of given label.
  
          Parameters
          ----------
          label : object or tuple of objects
          side : {'left', 'right'}
          kind : {'loc', 'getitem'}
  
          Returns
          -------
          int
              Index of label.
  
          Notes
          -----
          This method only works if level 0 index of the MultiIndex is lexsorted.
  
          Examples
          --------
          >>> mi = pd.MultiIndex.from_arrays([list('abbc'), list('gefd')])
  
          Get the locations from the leftmost 'b' in the first level
          until the end of the multiindex:
  
          >>> mi.get_slice_bound('b', side="left", kind="loc")
          1
  
          Like above, but if you get the locations from the rightmost
          'b' in the first level and 'f' in the second level:
  
          >>> mi.get_slice_bound(('b','f'), side="right", kind="loc")
          3
  
          See Also
          --------
          MultiIndex.get_loc : Get location for a label or a tuple of labels.
          MultiIndex.get_locs : Get location for a label/slice/list/mask or a
                                sequence of such.
          """
          if not isinstance(label, tuple):
              label = (label,)
          return self._partial_tup_index(label, side=side)
  
      def slice_locs(self, start=None, end=None, step=None, kind=None):
          """
          For an ordered MultiIndex, compute the slice locations for input
          labels.
  
          The input labels can be tuples representing partial levels, e.g. for a
          MultiIndex with 3 levels, you can pass a single value (corresponding to
          the first level), or a 1-, 2-, or 3-tuple.
  
          Parameters
          ----------
          start : label or tuple, default None
              If None, defaults to the beginning
          end : label or tuple
              If None, defaults to the end
          step : int or None
              Slice step
          kind : string, optional, defaults None
  
          Returns
          -------
          (start, end) : (int, int)
  
          Notes
          -----
          This method only works if the MultiIndex is properly lexsorted. So,
          if only the first 2 levels of a 3-level MultiIndex are lexsorted,
          you can only pass two levels to ``.slice_locs``.
  
          Examples
          --------
          >>> mi = pd.MultiIndex.from_arrays([list('abbd'), list('deff')],
          ...                                names=['A', 'B'])
  
          Get the slice locations from the beginning of 'b' in the first level
          until the end of the multiindex:
  
          >>> mi.slice_locs(start='b')
          (1, 4)
  
          Like above, but stop at the end of 'b' in the first level and 'f' in
          the second level:
  
          >>> mi.slice_locs(start='b', end=('b', 'f'))
          (1, 3)
  
          See Also
          --------
          MultiIndex.get_loc : Get location for a label or a tuple of labels.
          MultiIndex.get_locs : Get location for a label/slice/list/mask or a
                                sequence of such.
          """
          # This function adds nothing to its parent implementation (the magic
          # happens in get_slice_bound method), but it adds meaningful doc.
          return super().slice_locs(start, end, step, kind=kind)
  
      def _partial_tup_index(self, tup, side="left"):
          if len(tup) > self.lexsort_depth:
              raise UnsortedIndexError(
                  f"Key length ({len(tup)}) was greater than MultiIndex lexsort depth "
                  f"({self.lexsort_depth})"
              )
  
          n = len(tup)
          start, end = 0, len(self)
          zipped = zip(tup, self.levels, self.codes)
          for k, (lab, lev, labs) in enumerate(zipped):
              section = labs[start:end]
  
              if lab not in lev and not isna(lab):
                  if not lev.is_type_compatible(lib.infer_dtype([lab], skipna=False)):
                      raise TypeError(f"Level type mismatch: {lab}")
  
                  # short circuit
                  loc = lev.searchsorted(lab, side=side)
                  if side == "right" and loc >= 0:
                      loc -= 1
                  return start + section.searchsorted(loc, side=side)
  
              idx = self._get_loc_single_level_index(lev, lab)
              if k < n - 1:
                  end = start + section.searchsorted(idx, side="right")
                  start = start + section.searchsorted(idx, side="left")
              else:
                  return start + section.searchsorted(idx, side=side)
  
      def _get_loc_single_level_index(self, level_index: Index, key: Hashable) -> int:
          """
          If key is NA value, location of index unify as -1.
  
          Parameters
          ----------
          level_index: Index
          key : label
  
          Returns
          -------
          loc : int
              If key is NA value, loc is -1
              Else, location of key in index.
  
          See Also
          --------
          Index.get_loc : The get_loc method for (single-level) index.
          """
          if is_scalar(key) and isna(key):
              return -1
          else:
              return level_index.get_loc(key)
  
      def get_loc(self, key, method=None):
          """
          Get location for a label or a tuple of labels as an integer, slice or
          boolean mask.
  
          Parameters
          ----------
          key : label or tuple of labels (one for each level)
          method : None
  
          Returns
          -------
          loc : int, slice object or boolean mask
              If the key is past the lexsort depth, the return may be a
              boolean mask array, otherwise it is always a slice or int.
  
          See Also
          --------
          Index.get_loc : The get_loc method for (single-level) index.
          MultiIndex.slice_locs : Get slice location given start label(s) and
                                  end label(s).
          MultiIndex.get_locs : Get location for a label/slice/list/mask or a
                                sequence of such.
  
          Notes
          -----
          The key cannot be a slice, list of same-level labels, a boolean mask,
          or a sequence of such. If you want to use those, use
          :meth:`MultiIndex.get_locs` instead.
  
          Examples
          --------
          >>> mi = pd.MultiIndex.from_arrays([list('abb'), list('def')])
  
          >>> mi.get_loc('b')
          slice(1, 3, None)
  
          >>> mi.get_loc(('b', 'e'))
          1
          """
          if method is not None:
              raise NotImplementedError(
                  "only the default get_loc method is "
                  "currently supported for MultiIndex"
              )
  
          def _maybe_to_slice(loc):
              """convert integer indexer to boolean mask or slice if possible"""
              if not isinstance(loc, np.ndarray) or loc.dtype != "int64":
                  return loc
  
              loc = lib.maybe_indices_to_slice(loc, len(self))
              if isinstance(loc, slice):
                  return loc
  
              mask = np.empty(len(self), dtype="bool")
              mask.fill(False)
              mask[loc] = True
              return mask
  
          if not isinstance(key, (tuple, list)):
              # not including list here breaks some indexing, xref #30892
              loc = self._get_level_indexer(key, level=0)
              return _maybe_to_slice(loc)
  
          keylen = len(key)
          if self.nlevels < keylen:
              raise KeyError(
                  f"Key length ({keylen}) exceeds index depth ({self.nlevels})"
              )
  
          if keylen == self.nlevels and self.is_unique:
              return self._engine.get_loc(key)
  
          # -- partial selection or non-unique index
          # break the key into 2 parts based on the lexsort_depth of the index;
          # the first part returns a continuous slice of the index; the 2nd part
          # needs linear search within the slice
          i = self.lexsort_depth
          lead_key, follow_key = key[:i], key[i:]
          start, stop = (
              self.slice_locs(lead_key, lead_key) if lead_key else (0, len(self))
          )
  
          if start == stop:
              raise KeyError(key)
  
          if not follow_key:
              return slice(start, stop)
  
          warnings.warn(
              "indexing past lexsort depth may impact performance.",
              PerformanceWarning,
              stacklevel=10,
          )
  
          loc = np.arange(start, stop, dtype="int64")
  
          for i, k in enumerate(follow_key, len(lead_key)):
              mask = self.codes[i][loc] == self._get_loc_single_level_index(
                  self.levels[i], k
              )
              if not mask.all():
                  loc = loc[mask]
              if not len(loc):
                  raise KeyError(key)
  
          return _maybe_to_slice(loc) if len(loc) != stop - start else slice(start, stop)
  
      def get_loc_level(self, key, level=0, drop_level: bool = True):
          """
          Get both the location for the requested label(s) and the
          resulting sliced index.
  
          Parameters
          ----------
          key : label or sequence of labels
          level : int/level name or list thereof, optional
          drop_level : bool, default True
              If ``False``, the resulting index will not drop any level.
  
          Returns
          -------
          loc : A 2-tuple where the elements are:
                Element 0: int, slice object or boolean array
                Element 1: The resulting sliced multiindex/index. If the key
                contains all levels, this will be ``None``.
  
          See Also
          --------
          MultiIndex.get_loc  : Get location for a label or a tuple of labels.
          MultiIndex.get_locs : Get location for a label/slice/list/mask or a
                                sequence of such.
  
          Examples
          --------
          >>> mi = pd.MultiIndex.from_arrays([list('abb'), list('def')],
          ...                                names=['A', 'B'])
  
          >>> mi.get_loc_level('b')
          (slice(1, 3, None), Index(['e', 'f'], dtype='object', name='B'))
  
          >>> mi.get_loc_level('e', level='B')
          (array([False,  True, False], dtype=bool),
          Index(['b'], dtype='object', name='A'))
  
          >>> mi.get_loc_level(['b', 'e'])
          (1, None)
          """
          # different name to distinguish from maybe_droplevels
          def maybe_mi_droplevels(indexer, levels, drop_level: bool):
              if not drop_level:
                  return self[indexer]
              # kludgearound
              orig_index = new_index = self[indexer]
              levels = [self._get_level_number(i) for i in levels]
              for i in sorted(levels, reverse=True):
                  try:
                      new_index = new_index.droplevel(i)
                  except ValueError:
  
                      # no dropping here
                      return orig_index
              return new_index
  
          if isinstance(level, (tuple, list)):
              if len(key) != len(level):
                  raise AssertionError(
                      "Key for location must have same length as number of levels"
                  )
              result = None
              for lev, k in zip(level, key):
                  loc, new_index = self.get_loc_level(k, level=lev)
                  if isinstance(loc, slice):
                      mask = np.zeros(len(self), dtype=bool)
                      mask[loc] = True
                      loc = mask
  
                  result = loc if result is None else result & loc
  
              return result, maybe_mi_droplevels(result, level, drop_level)
  
          level = self._get_level_number(level)
  
          # kludge for #1796
          if isinstance(key, list):
              key = tuple(key)
  
          if isinstance(key, tuple) and level == 0:
  
              try:
                  if key in self.levels[0]:
                      indexer = self._get_level_indexer(key, level=level)
                      new_index = maybe_mi_droplevels(indexer, [0], drop_level)
                      return indexer, new_index
              except (TypeError, InvalidIndexError):
                  pass
  
              if not any(isinstance(k, slice) for k in key):
  
                  # partial selection
                  # optionally get indexer to avoid re-calculation
                  def partial_selection(key, indexer=None):
                      if indexer is None:
                          indexer = self.get_loc(key)
                      ilevels = [
                          i for i in range(len(key)) if key[i] != slice(None, None)
                      ]
                      return indexer, maybe_mi_droplevels(indexer, ilevels, drop_level)
  
                  if len(key) == self.nlevels and self.is_unique:
                      # Complete key in unique index -> standard get_loc
                      try:
                          return (self._engine.get_loc(key), None)
                      except KeyError as e:
                          raise KeyError(key) from e
                  else:
                      return partial_selection(key)
              else:
                  indexer = None
                  for i, k in enumerate(key):
                      if not isinstance(k, slice):
                          k = self._get_level_indexer(k, level=i)
                          if isinstance(k, slice):
                              # everything
                              if k.start == 0 and k.stop == len(self):
                                  k = slice(None, None)
                          else:
                              k_index = k
  
                      if isinstance(k, slice):
                          if k == slice(None, None):
                              continue
                          else:
                              raise TypeError(key)
  
                      if indexer is None:
                          indexer = k_index
                      else:  # pragma: no cover
                          indexer &= k_index
                  if indexer is None:
                      indexer = slice(None, None)
                  ilevels = [i for i in range(len(key)) if key[i] != slice(None, None)]
                  return indexer, maybe_mi_droplevels(indexer, ilevels, drop_level)
          else:
              indexer = self._get_level_indexer(key, level=level)
              return indexer, maybe_mi_droplevels(indexer, [level], drop_level)
  
      def _get_level_indexer(self, key, level=0, indexer=None):
          # return an indexer, boolean array or a slice showing where the key is
          # in the totality of values
          # if the indexer is provided, then use this
  
          level_index = self.levels[level]
          level_codes = self.codes[level]
  
          def convert_indexer(start, stop, step, indexer=indexer, codes=level_codes):
              # given the inputs and the codes/indexer, compute an indexer set
              # if we have a provided indexer, then this need not consider
              # the entire labels set
  
              r = np.arange(start, stop, step)
              if indexer is not None and len(indexer) != len(codes):
  
                  # we have an indexer which maps the locations in the labels
                  # that we have already selected (and is not an indexer for the
                  # entire set) otherwise this is wasteful so we only need to
                  # examine locations that are in this set the only magic here is
                  # that the result are the mappings to the set that we have
                  # selected
                  from pandas import Series
  
                  mapper = Series(indexer)
                  indexer = codes.take(ensure_platform_int(indexer))
                  result = Series(Index(indexer).isin(r).nonzero()[0])
                  m = result.map(mapper)._ndarray_values
  
              else:
                  m = np.zeros(len(codes), dtype=bool)
                  m[np.in1d(codes, r, assume_unique=Index(codes).is_unique)] = True
  
              return m
  
          if isinstance(key, slice):
              # handle a slice, returning a slice if we can
              # otherwise a boolean indexer
  
              try:
                  if key.start is not None:
                      start = level_index.get_loc(key.start)
                  else:
                      start = 0
                  if key.stop is not None:
                      stop = level_index.get_loc(key.stop)
                  else:
                      stop = len(level_index) - 1
                  step = key.step
              except KeyError:
  
                  # we have a partial slice (like looking up a partial date
                  # string)
                  start = stop = level_index.slice_indexer(
                      key.start, key.stop, key.step, kind="loc"
                  )
                  step = start.step
  
              if isinstance(start, slice) or isinstance(stop, slice):
                  # we have a slice for start and/or stop
                  # a partial date slicer on a DatetimeIndex generates a slice
                  # note that the stop ALREADY includes the stopped point (if
                  # it was a string sliced)
                  start = getattr(start, "start", start)
                  stop = getattr(stop, "stop", stop)
                  return convert_indexer(start, stop, step)
  
              elif level > 0 or self.lexsort_depth == 0 or step is not None:
                  # need to have like semantics here to right
                  # searching as when we are using a slice
                  # so include the stop+1 (so we include stop)
                  return convert_indexer(start, stop + 1, step)
              else:
                  # sorted, so can return slice object -> view
                  i = level_codes.searchsorted(start, side="left")
                  j = level_codes.searchsorted(stop, side="right")
                  return slice(i, j, step)
  
          else:
  
              code = self._get_loc_single_level_index(level_index, key)
  
              if level > 0 or self.lexsort_depth == 0:
                  # Desired level is not sorted
                  locs = np.array(level_codes == code, dtype=bool, copy=False)
                  if not locs.any():
                      # The label is present in self.levels[level] but unused:
                      raise KeyError(key)
                  return locs
  
              i = level_codes.searchsorted(code, side="left")
              j = level_codes.searchsorted(code, side="right")
              if i == j:
                  # The label is present in self.levels[level] but unused:
                  raise KeyError(key)
              return slice(i, j)
  
      def get_locs(self, seq):
          """
          Get location for a sequence of labels.
  
          Parameters
          ----------
          seq : label, slice, list, mask or a sequence of such
             You should use one of the above for each level.
             If a level should not be used, set it to ``slice(None)``.
  
          Returns
          -------
          numpy.ndarray
              NumPy array of integers suitable for passing to iloc.
  
          See Also
          --------
          MultiIndex.get_loc : Get location for a label or a tuple of labels.
          MultiIndex.slice_locs : Get slice location given start label(s) and
                                  end label(s).
  
          Examples
          --------
          >>> mi = pd.MultiIndex.from_arrays([list('abb'), list('def')])
  
          >>> mi.get_locs('b')  # doctest: +SKIP
          array([1, 2], dtype=int64)
  
          >>> mi.get_locs([slice(None), ['e', 'f']])  # doctest: +SKIP
          array([1, 2], dtype=int64)
  
          >>> mi.get_locs([[True, False, True], slice('e', 'f')])  # doctest: +SKIP
          array([2], dtype=int64)
          """
          from pandas.core.indexes.numeric import Int64Index
  
          # must be lexsorted to at least as many levels
          true_slices = [i for (i, s) in enumerate(com.is_true_slices(seq)) if s]
          if true_slices and true_slices[-1] >= self.lexsort_depth:
              raise UnsortedIndexError(
                  "MultiIndex slicing requires the index to be lexsorted: slicing "
                  f"on levels {true_slices}, lexsort depth {self.lexsort_depth}"
              )
          # indexer
          # this is the list of all values that we want to select
          n = len(self)
          indexer = None
  
          def _convert_to_indexer(r):
              # return an indexer
              if isinstance(r, slice):
                  m = np.zeros(n, dtype=bool)
                  m[r] = True
                  r = m.nonzero()[0]
              elif com.is_bool_indexer(r):
                  if len(r) != n:
                      raise ValueError(
                          "cannot index with a boolean indexer "
                          "that is not the same length as the "
                          "index"
                      )
                  r = r.nonzero()[0]
              return Int64Index(r)
  
          def _update_indexer(idxr, indexer=indexer):
              if indexer is None:
                  indexer = Index(np.arange(n))
              if idxr is None:
                  return indexer
              return indexer & idxr
  
          for i, k in enumerate(seq):
  
              if com.is_bool_indexer(k):
                  # a boolean indexer, must be the same length!
                  k = np.asarray(k)
                  indexer = _update_indexer(_convert_to_indexer(k), indexer=indexer)
  
              elif is_list_like(k):
                  # a collection of labels to include from this level (these
                  # are or'd)
                  indexers = None
                  for x in k:
                      try:
                          idxrs = _convert_to_indexer(
                              self._get_level_indexer(x, level=i, indexer=indexer)
                          )
                          indexers = idxrs if indexers is None else indexers | idxrs
                      except KeyError:
  
                          # ignore not founds
                          continue
  
                  if indexers is not None:
                      indexer = _update_indexer(indexers, indexer=indexer)
                  else:
                      # no matches we are done
                      return Int64Index([])._ndarray_values
  
              elif com.is_null_slice(k):
                  # empty slice
                  indexer = _update_indexer(None, indexer=indexer)
  
              elif isinstance(k, slice):
  
                  # a slice, include BOTH of the labels
                  indexer = _update_indexer(
                      _convert_to_indexer(
                          self._get_level_indexer(k, level=i, indexer=indexer)
                      ),
                      indexer=indexer,
                  )
              else:
                  # a single label
                  indexer = _update_indexer(
                      _convert_to_indexer(
                          self.get_loc_level(k, level=i, drop_level=False)[0]
                      ),
                      indexer=indexer,
                  )
  
          # empty indexer
          if indexer is None:
              return Int64Index([])._ndarray_values
  
          indexer = self._reorder_indexer(seq, indexer)
  
          return indexer._ndarray_values
  
      def _reorder_indexer(
          self, seq: Tuple[Union[Scalar, Iterable, AnyArrayLike], ...], indexer: ArrayLike
      ) -> ArrayLike:
          """
          Reorder an indexer of a MultiIndex (self) so that the label are in the
          same order as given in seq
  
          Parameters
          ----------
          seq : label/slice/list/mask or a sequence of such
          indexer: an Int64Index indexer of self
  
          Returns
          -------
          indexer : a sorted Int64Index indexer of self ordered as seq
          """
          # If the index is lexsorted and the list_like label in seq are sorted
          # then we do not need to sort
          if self.is_lexsorted():
              need_sort = False
              for i, k in enumerate(seq):
                  if is_list_like(k):
                      if not need_sort:
                          k_codes = self.levels[i].get_indexer(k)
                          k_codes = k_codes[k_codes >= 0]  # Filter absent keys
                          # True if the given codes are not ordered
                          need_sort = (k_codes[:-1] > k_codes[1:]).any()
              # Bail out if both index and seq are sorted
              if not need_sort:
                  return indexer
  
          n = len(self)
          keys: Tuple[np.ndarray, ...] = tuple()
          # For each level of the sequence in seq, map the level codes with the
          # order they appears in a list-like sequence
          # This mapping is then use to reorder the indexer
          for i, k in enumerate(seq):
              if com.is_bool_indexer(k):
                  new_order = np.arange(n)[indexer]
              elif is_list_like(k):
                  # Generate a map with all level codes as sorted initially
                  key_order_map = np.ones(len(self.levels[i]), dtype=np.uint64) * len(
                      self.levels[i]
                  )
                  # Set order as given in the indexer list
                  level_indexer = self.levels[i].get_indexer(k)
                  level_indexer = level_indexer[level_indexer >= 0]  # Filter absent keys
                  key_order_map[level_indexer] = np.arange(len(level_indexer))
  
                  new_order = key_order_map[self.codes[i][indexer]]
              else:
                  # For all other case, use the same order as the level
                  new_order = np.arange(n)[indexer]
              keys = (new_order,) + keys
  
          # Find the reordering using lexsort on the keys mapping
          ind = np.lexsort(keys)
          return indexer[ind]
  
      def truncate(self, before=None, after=None):
          """
          Slice index between two labels / tuples, return new MultiIndex
  
          Parameters
          ----------
          before : label or tuple, can be partial. Default None
              None defaults to start
          after : label or tuple, can be partial. Default None
              None defaults to end
  
          Returns
          -------
          truncated : MultiIndex
          """
          if after and before and after < before:
              raise ValueError("after < before")
  
          i, j = self.levels[0].slice_locs(before, after)
          left, right = self.slice_locs(before, after)
  
          new_levels = list(self.levels)
          new_levels[0] = new_levels[0][i:j]
  
          new_codes = [level_codes[left:right] for level_codes in self.codes]
          new_codes[0] = new_codes[0] - i
  
          return MultiIndex(levels=new_levels, codes=new_codes, verify_integrity=False)
  
      def equals(self, other) -> bool:
          """
          Determines if two MultiIndex objects have the same labeling information
          (the levels themselves do not necessarily have to be the same)
  
          See Also
          --------
          equal_levels
          """
          if self.is_(other):
              return True
  
          if not isinstance(other, Index):
              return False
  
          if not isinstance(other, MultiIndex):
              # d-level MultiIndex can equal d-tuple Index
              if not is_object_dtype(other.dtype):
                  if self.nlevels != other.nlevels:
                      return False
  
              other_vals = com.values_from_object(ensure_index(other))
              return array_equivalent(self._ndarray_values, other_vals)
  
          if self.nlevels != other.nlevels:
              return False
  
          if len(self) != len(other):
              return False
  
          for i in range(self.nlevels):
              self_codes = self.codes[i]
              self_codes = self_codes[self_codes != -1]
              self_values = algos.take_nd(
                  np.asarray(self.levels[i]._values), self_codes, allow_fill=False
              )
  
              other_codes = other.codes[i]
              other_codes = other_codes[other_codes != -1]
              other_values = algos.take_nd(
                  np.asarray(other.levels[i]._values), other_codes, allow_fill=False
              )
  
              # since we use NaT both datetime64 and timedelta64
              # we can have a situation where a level is typed say
              # timedelta64 in self (IOW it has other values than NaT)
              # but types datetime64 in other (where its all NaT)
              # but these are equivalent
              if len(self_values) == 0 and len(other_values) == 0:
                  continue
  
              if not array_equivalent(self_values, other_values):
                  return False
  
          return True
  
      def equal_levels(self, other) -> bool:
          """
          Return True if the levels of both MultiIndex objects are the same
  
          """
          if self.nlevels != other.nlevels:
              return False
  
          for i in range(self.nlevels):
              if not self.levels[i].equals(other.levels[i]):
                  return False
          return True
  
      # --------------------------------------------------------------------
      # Set Methods
  
      def union(self, other, sort=None):
          """
          Form the union of two MultiIndex objects
  
          Parameters
          ----------
          other : MultiIndex or array / Index of tuples
          sort : False or None, default None
              Whether to sort the resulting Index.
  
              * None : Sort the result, except when
  
                1. `self` and `other` are equal.
                2. `self` has length 0.
                3. Some values in `self` or `other` cannot be compared.
                   A RuntimeWarning is issued in this case.
  
              * False : do not sort the result.
  
              .. versionadded:: 0.24.0
  
              .. versionchanged:: 0.24.1
  
                 Changed the default value from ``True`` to ``None``
                 (without change in behaviour).
  
          Returns
          -------
          Index
  
          >>> index.union(index2)
          """
          self._validate_sort_keyword(sort)
          self._assert_can_do_setop(other)
          other, result_names = self._convert_can_do_setop(other)
  
          if len(other) == 0 or self.equals(other):
              return self
  
          # TODO: Index.union returns other when `len(self)` is 0.
  
          uniq_tuples = lib.fast_unique_multiple(
              [self._ndarray_values, other._ndarray_values], sort=sort
          )
  
          return MultiIndex.from_arrays(
              zip(*uniq_tuples), sortorder=0, names=result_names
          )
  
      def intersection(self, other, sort=False):
          """
          Form the intersection of two MultiIndex objects.
  
          Parameters
          ----------
          other : MultiIndex or array / Index of tuples
          sort : False or None, default False
              Sort the resulting MultiIndex if possible
  
              .. versionadded:: 0.24.0
  
              .. versionchanged:: 0.24.1
  
                 Changed the default from ``True`` to ``False``, to match
                 behaviour from before 0.24.0
  
          Returns
          -------
          Index
          """
          self._validate_sort_keyword(sort)
          self._assert_can_do_setop(other)
          other, result_names = self._convert_can_do_setop(other)
  
          if self.equals(other):
              return self
  
          lvals = self._ndarray_values
          rvals = other._ndarray_values
  
          uniq_tuples = None  # flag whether _inner_indexer was succesful
          if self.is_monotonic and other.is_monotonic:
              try:
                  uniq_tuples = self._inner_indexer(lvals, rvals)[0]
                  sort = False  # uniq_tuples is already sorted
              except TypeError:
                  pass
  
          if uniq_tuples is None:
              other_uniq = set(rvals)
              seen = set()
              uniq_tuples = [
                  x for x in lvals if x in other_uniq and not (x in seen or seen.add(x))
              ]
  
          if sort is None:
              uniq_tuples = sorted(uniq_tuples)
  
          if len(uniq_tuples) == 0:
              return MultiIndex(
                  levels=self.levels,
                  codes=[[]] * self.nlevels,
                  names=result_names,
                  verify_integrity=False,
              )
          else:
              return MultiIndex.from_arrays(
                  zip(*uniq_tuples), sortorder=0, names=result_names
              )
  
      def difference(self, other, sort=None):
          """
          Compute set difference of two MultiIndex objects
  
          Parameters
          ----------
          other : MultiIndex
          sort : False or None, default None
              Sort the resulting MultiIndex if possible
  
              .. versionadded:: 0.24.0
  
              .. versionchanged:: 0.24.1
  
                 Changed the default value from ``True`` to ``None``
                 (without change in behaviour).
  
          Returns
          -------
          diff : MultiIndex
          """
          self._validate_sort_keyword(sort)
          self._assert_can_do_setop(other)
          other, result_names = self._convert_can_do_setop(other)
  
          if len(other) == 0:
              return self
  
          if self.equals(other):
              return MultiIndex(
                  levels=self.levels,
                  codes=[[]] * self.nlevels,
                  names=result_names,
                  verify_integrity=False,
              )
  
          this = self._get_unique_index()
  
          indexer = this.get_indexer(other)
          indexer = indexer.take((indexer != -1).nonzero()[0])
  
          label_diff = np.setdiff1d(np.arange(this.size), indexer, assume_unique=True)
          difference = this.values.take(label_diff)
          if sort is None:
              difference = sorted(difference)
  
          if len(difference) == 0:
              return MultiIndex(
                  levels=[[]] * self.nlevels,
                  codes=[[]] * self.nlevels,
                  names=result_names,
                  verify_integrity=False,
              )
          else:
              return MultiIndex.from_tuples(difference, sortorder=0, names=result_names)
  
      def _convert_can_do_setop(self, other):
          result_names = self.names
  
          if not hasattr(other, "names"):
              if len(other) == 0:
                  other = MultiIndex(
                      levels=[[]] * self.nlevels,
                      codes=[[]] * self.nlevels,
                      verify_integrity=False,
                  )
              else:
                  msg = "other must be a MultiIndex or a list of tuples"
                  try:
                      other = MultiIndex.from_tuples(other)
                  except TypeError as err:
                      raise TypeError(msg) from err
          else:
              result_names = self.names if self.names == other.names else None
          return other, result_names
  
      # --------------------------------------------------------------------
  
      @Appender(Index.astype.__doc__)
      def astype(self, dtype, copy=True):
          dtype = pandas_dtype(dtype)
          if is_categorical_dtype(dtype):
              msg = "> 1 ndim Categorical are not supported at this time"
              raise NotImplementedError(msg)
          elif not is_object_dtype(dtype):
              raise TypeError(
                  f"Setting {type(self)} dtype to anything other "
                  "than object is not supported"
              )
          elif copy is True:
              return self._shallow_copy()
          return self
  
      def insert(self, loc: int, item):
          """
          Make new MultiIndex inserting new item at location
  
          Parameters
          ----------
          loc : int
          item : tuple
              Must be same length as number of levels in the MultiIndex
  
          Returns
          -------
          new_index : Index
          """
          # Pad the key with empty strings if lower levels of the key
          # aren't specified:
          if not isinstance(item, tuple):
              item = (item,) + ("",) * (self.nlevels - 1)
          elif len(item) != self.nlevels:
              raise ValueError("Item must have length equal to number of levels.")
  
          new_levels = []
          new_codes = []
          for k, level, level_codes in zip(item, self.levels, self.codes):
              if k not in level:
                  # have to insert into level
                  # must insert at end otherwise you have to recompute all the
                  # other codes
                  lev_loc = len(level)
                  level = level.insert(lev_loc, k)
              else:
                  lev_loc = level.get_loc(k)
  
              new_levels.append(level)
              new_codes.append(np.insert(ensure_int64(level_codes), loc, lev_loc))
  
          return MultiIndex(
              levels=new_levels, codes=new_codes, names=self.names, verify_integrity=False
          )
  
      def delete(self, loc):
          """
          Make new index with passed location deleted
  
          Returns
          -------
          new_index : MultiIndex
          """
          new_codes = [np.delete(level_codes, loc) for level_codes in self.codes]
          return MultiIndex(
              levels=self.levels,
              codes=new_codes,
              names=self.names,
              verify_integrity=False,
          )
  
      def _wrap_joined_index(self, joined, other):
          names = self.names if self.names == other.names else None
          return MultiIndex.from_tuples(joined, names=names)
  
      @Appender(Index.isin.__doc__)
      def isin(self, values, level=None):
          if level is None:
              values = MultiIndex.from_tuples(values, names=self.names).values
              return algos.isin(self.values, values)
          else:
              num = self._get_level_number(level)
              levs = self.get_level_values(num)
  
              if levs.size == 0:
                  return np.zeros(len(levs), dtype=np.bool_)
              return levs.isin(values)
  
  
  MultiIndex._add_numeric_methods_disabled()
  MultiIndex._add_numeric_methods_add_sub_disabled()
  MultiIndex._add_logical_methods_disabled()
  
  
  def _sparsify(label_list, start: int = 0, sentinel=""):
      pivoted = list(zip(*label_list))
      k = len(label_list)
  
      result = pivoted[: start + 1]
      prev = pivoted[start]
  
      for cur in pivoted[start + 1 :]:
          sparse_cur = []
  
          for i, (p, t) in enumerate(zip(prev, cur)):
              if i == k - 1:
                  sparse_cur.append(t)
                  result.append(sparse_cur)
                  break
  
              if p == t:
                  sparse_cur.append(sentinel)
              else:
                  sparse_cur.extend(cur[i:])
                  result.append(sparse_cur)
                  break
  
          prev = cur
  
      return list(zip(*result))
  
  
  def _get_na_rep(dtype) -> str:
      return {np.datetime64: "NaT", np.timedelta64: "NaT"}.get(dtype, "NaN")
  
  
  def maybe_droplevels(index, key):
      """
      Attempt to drop level or levels from the given index.
  
      Parameters
      ----------
      index: Index
      key : scalar or tuple
  
      Returns
      -------
      Index
      """
      # drop levels
      original_index = index
      if isinstance(key, tuple):
          for _ in key:
              try:
                  index = index.droplevel(0)
              except ValueError:
                  # we have dropped too much, so back out
                  return original_index
      else:
          try:
              index = index.droplevel(0)
          except ValueError:
              pass
  
      return index
  
  
  def _coerce_indexer_frozen(array_like, categories, copy: bool = False) -> np.ndarray:
      """
      Coerce the array_like indexer to the smallest integer dtype that can encode all
      of the given categories.
  
      Parameters
      ----------
      array_like : array-like
      categories : array-like
      copy : bool
  
      Returns
      -------
      np.ndarray
          Non-writeable.
      """
      array_like = coerce_indexer_dtype(array_like, categories)
      if copy:
          array_like = array_like.copy()
      array_like.flags.writeable = False
      return array_like
  
apis: pandas.core.common.cast_scalar_indexer pandas.core.indexes.frozen.FrozenList pandas.io.formats.format._get_adjustment pandas.core.algorithms.unique pandas.io.formats.printing.format_object_summary pandas._libs.lib.fast_zip pandas.Series numpy.dtype numpy.any numpy.asarray pandas.core.common.is_true_slices pandas.core.dtypes.common.ensure_int64 pandas.core.sorting.lexsort_indexer pandas.core.dtypes.common.is_integer pandas.core.dtypes.common.pandas_dtype pandas.core.dtypes.common.is_list_like pandas.core.common.is_null_slice pandas.core.dtypes.cast.coerce_indexer_dtype pandas.core.indexes.base.InvalidIndexError pandas._libs.algos.is_lexsorted numpy.append pandas.core.dtypes.common.is_categorical_dtype pandas._libs.lib.fast_unique_multiple pandas.core.indexes.base._ensure_has_len pandas.core.dtypes.common.is_hashable pandas.io.formats.printing.pprint_thing pandas.core.algorithms.take_1d pandas.core.dtypes.common.is_iterator pandas.core.indexes.base.ensure_index numpy.delete pandas.core.common.index_labels_to_array numpy.where pandas.io.formats.printing.format_object_attrs pandas._libs.lib.infer_dtype pandas.core.indexes.base.Index numpy.bincount numpy.zeros pandas._libs.lib.to_object_array_tuples pandas.core.reshape.util.cartesian_product pandas._libs.hashtable.duplicated_int64 numpy.arange numpy.lexsort pandas._config.get_option pandas.core.common.is_bool_indexer pandas.core.algorithms.isin numpy.all pandas.core.algorithms.factorize pandas._libs.lib.tuples_to_object_array pandas.errors.UnsortedIndexError pandas.core.dtypes.common.ensure_platform_int pandas.core.dtypes.common.is_scalar pandas.core.dtypes.missing.isna pandas.core.sorting.indexer_from_factorized pandas.util._decorators.Appender pandas.core.missing.clean_reindex_fill_method numpy.cumsum numpy.empty pandas.core.arrays.categorical.factorize_from_iterables pandas.core.indexes.numeric.Int64Index pandas.core.dtypes.common.is_object_dtype numpy.bitwise_or.reduce pandas.core.sorting.get_group_index numpy.array numpy.concatenate pandas.core.dtypes.missing.array_equivalent