base.py

from __future__ import annotations

from copy import copy as copy_func
from datetime import datetime
from itertools import zip_longest
import operator
from typing import (
    TYPE_CHECKING,
    Any,
    Callable,
    Dict,
    FrozenSet,
    Hashable,
    List,
    NewType,
    Optional,
    Sequence,
    Set,
    Tuple,
    Type,
    TypeVar,
    Union,
    cast,
)
import warnings

import numpy as np

from pandas._libs import (
    algos as libalgos,
    index as libindex,
    lib,
)
import pandas._libs.join as libjoin
from pandas._libs.lib import (
    is_datetime_array,
    no_default,
)
from pandas._libs.tslibs import (
    IncompatibleFrequency,
    OutOfBoundsDatetime,
    Timestamp,
    tz_compare,
)
from pandas._typing import (
    AnyArrayLike,
    ArrayLike,
    Dtype,
    DtypeObj,
    Shape,
    T,
    final,
)
from pandas.compat.numpy import function as nv
from pandas.errors import (
    DuplicateLabelError,
    InvalidIndexError,
)
from pandas.util._decorators import (
    Appender,
    cache_readonly,
    doc,
)

from pandas.core.dtypes.cast import (
    can_hold_element,
    find_common_type,
    infer_dtype_from,
    maybe_cast_to_integer_array,
    validate_numeric_casting,
)
from pandas.core.dtypes.common import (
    ensure_int64,
    ensure_object,
    ensure_platform_int,
    is_bool_dtype,
    is_categorical_dtype,
    is_dtype_equal,
    is_ea_or_datetimelike_dtype,
    is_extension_array_dtype,
    is_float,
    is_float_dtype,
    is_hashable,
    is_integer,
    is_integer_dtype,
    is_interval_dtype,
    is_iterator,
    is_list_like,
    is_object_dtype,
    is_scalar,
    is_signed_integer_dtype,
    is_unsigned_integer_dtype,
    needs_i8_conversion,
    pandas_dtype,
    validate_all_hashable,
)
from pandas.core.dtypes.concat import concat_compat
from pandas.core.dtypes.dtypes import (
    CategoricalDtype,
    DatetimeTZDtype,
    ExtensionDtype,
    IntervalDtype,
    PandasDtype,
    PeriodDtype,
)
from pandas.core.dtypes.generic import (
    ABCDatetimeIndex,
    ABCMultiIndex,
    ABCPeriodIndex,
    ABCSeries,
    ABCTimedeltaIndex,
)
from pandas.core.dtypes.inference import is_dict_like
from pandas.core.dtypes.missing import (
    array_equivalent,
    is_valid_na_for_dtype,
    isna,
)

from pandas.core import (
    missing,
    ops,
)
from pandas.core.accessor import CachedAccessor
import pandas.core.algorithms as algos
from pandas.core.array_algos.putmask import (
    setitem_datetimelike_compat,
    validate_putmask,
)
from pandas.core.arrays import (
    Categorical,
    ExtensionArray,
)
from pandas.core.arrays.datetimes import (
    tz_to_dtype,
    validate_tz_from_dtype,
)
from pandas.core.arrays.sparse import SparseDtype
from pandas.core.base import (
    IndexOpsMixin,
    PandasObject,
)
import pandas.core.common as com
from pandas.core.construction import (
    ensure_wrapped_if_datetimelike,
    extract_array,
)
from pandas.core.indexers import deprecate_ndim_indexing
from pandas.core.indexes.frozen import FrozenList
from pandas.core.ops import get_op_result_name
from pandas.core.ops.invalid import make_invalid_op
from pandas.core.sorting import (
    ensure_key_mapped,
    get_group_index_sorter,
    nargsort,
)
from pandas.core.strings import StringMethods

from pandas.io.formats.printing import (
    PrettyDict,
    default_pprint,
    format_object_attrs,
    format_object_summary,
    pprint_thing,
)

if TYPE_CHECKING:
    from pandas import (
        CategoricalIndex,
        DataFrame,
        IntervalIndex,
        MultiIndex,
        RangeIndex,
        Series,
    )
    from pandas.core.indexes.datetimelike import DatetimeIndexOpsMixin


__all__ = ["Index"]

_unsortable_types = frozenset(("mixed", "mixed-integer"))

_index_doc_kwargs = {
    "klass": "Index",
    "inplace": "",
    "target_klass": "Index",
    "raises_section": "",
    "unique": "Index",
    "duplicated": "np.ndarray",
}
_index_shared_docs = {}
str_t = str


_o_dtype = np.dtype(object)


_Identity = NewType("_Identity", object)


def disallow_kwargs(kwargs: Dict[str, Any]):
    if kwargs:
        raise TypeError(f"Unexpected keyword arguments {repr(set(kwargs))}")


def _new_Index(cls, d):
    """
    This is called upon unpickling, rather than the default which doesn't
    have arguments and breaks __new__.
    """
    # required for backward compat, because PI can't be instantiated with
    # ordinals through __new__ GH #13277
    if issubclass(cls, ABCPeriodIndex):
        from pandas.core.indexes.period import _new_PeriodIndex

        return _new_PeriodIndex(cls, **d)

    if issubclass(cls, ABCMultiIndex):
        if "labels" in d and "codes" not in d:
            # GH#23752 "labels" kwarg has been replaced with "codes"
            d["codes"] = d.pop("labels")

    return cls.__new__(cls, **d)


_IndexT = TypeVar("_IndexT", bound="Index")


class Index(IndexOpsMixin, PandasObject):
    """
    Immutable sequence used for indexing and alignment. The basic object
    storing axis labels for all pandas objects.

    Parameters
    ----------
    data : array-like (1-dimensional)
    dtype : NumPy dtype (default: object)
        If dtype is None, we find the dtype that best fits the data.
        If an actual dtype is provided, we coerce to that dtype if it's safe.
        Otherwise, an error will be raised.
    copy : bool
        Make a copy of input ndarray.
    name : object
        Name to be stored in the index.
    tupleize_cols : bool (default: True)
        When True, attempt to create a MultiIndex if possible.

    See Also
    --------
    RangeIndex : Index implementing a monotonic integer range.
    CategoricalIndex : Index of :class:`Categorical` s.
    MultiIndex : A multi-level, or hierarchical Index.
    IntervalIndex : An Index of :class:`Interval` s.
    DatetimeIndex : Index of datetime64 data.
    TimedeltaIndex : Index of timedelta64 data.
    PeriodIndex : Index of Period data.
    Int64Index : A special case of :class:`Index` with purely integer labels.
    UInt64Index : A special case of :class:`Index` with purely unsigned integer labels.
    Float64Index : A special case of :class:`Index` with purely float labels.

    Notes
    -----
    An Index instance can **only** contain hashable objects

    Examples
    --------
    >>> pd.Index([1, 2, 3])
    Int64Index([1, 2, 3], dtype='int64')

    >>> pd.Index(list('abc'))
    Index(['a', 'b', 'c'], dtype='object')
    """

    # tolist is not actually deprecated, just suppressed in the __dir__
    _hidden_attrs: FrozenSet[str] = (
        PandasObject._hidden_attrs
        | IndexOpsMixin._hidden_attrs
        | frozenset(["contains", "set_value"])
    )

    # To hand over control to subclasses
    _join_precedence = 1

    # Cython methods; see github.com/cython/cython/issues/2647
    #  for why we need to wrap these instead of making them class attributes
    # Moreover, cython will choose the appropriate-dtyped sub-function
    #  given the dtypes of the passed arguments
    def _left_indexer_unique(self, left: np.ndarray, right: np.ndarray) -> np.ndarray:
        return libjoin.left_join_indexer_unique(left, right)

    def _left_indexer(
        self, left: np.ndarray, right: np.ndarray
    ) -> Tuple[np.ndarray, np.ndarray, np.ndarray]:
        return libjoin.left_join_indexer(left, right)

    def _inner_indexer(
        self, left: np.ndarray, right: np.ndarray
    ) -> Tuple[np.ndarray, np.ndarray, np.ndarray]:
        return libjoin.inner_join_indexer(left, right)

    def _outer_indexer(
        self, left: np.ndarray, right: np.ndarray
    ) -> Tuple[np.ndarray, np.ndarray, np.ndarray]:
        return libjoin.outer_join_indexer(left, right)

    _typ = "index"
    _data: Union[ExtensionArray, np.ndarray]
    _id: Optional[_Identity] = None
    _name: Hashable = None
    # MultiIndex.levels previously allowed setting the index name. We
    # don't allow this anymore, and raise if it happens rather than
    # failing silently.
    _no_setting_name: bool = False
    _comparables = ["name"]
    _attributes = ["name"]
    _is_numeric_dtype = False
    _can_hold_na = True
    _can_hold_strings = True

    # would we like our indexing holder to defer to us
    _defer_to_indexing = False

    _engine_type = libindex.ObjectEngine
    # whether we support partial string indexing. Overridden
    # in DatetimeIndex and PeriodIndex
    _supports_partial_string_indexing = False

    _accessors = {"str"}

    str = CachedAccessor("str", StringMethods)

    # --------------------------------------------------------------------
    # Constructors

    def __new__(
        cls, data=None, dtype=None, copy=False, name=None, tupleize_cols=True, **kwargs
    ) -> Index:

        if kwargs:
            warnings.warn(
                "Passing keywords other than 'data', 'dtype', 'copy', 'name', "
                "'tupleize_cols' is deprecated and will raise TypeError in a "
                "future version.  Use the specific Index subclass directly instead",
                FutureWarning,
                stacklevel=2,
            )

        from pandas.core.arrays import PandasArray
        from pandas.core.indexes.range import RangeIndex

        name = maybe_extract_name(name, data, cls)

        if dtype is not None:
            dtype = pandas_dtype(dtype)
        if "tz" in kwargs:
            tz = kwargs.pop("tz")
            validate_tz_from_dtype(dtype, tz)
            dtype = tz_to_dtype(tz)

        if isinstance(data, PandasArray):
            # ensure users don't accidentally put a PandasArray in an index.
            data = data.to_numpy()
        if isinstance(dtype, PandasDtype):
            dtype = dtype.numpy_dtype

        data_dtype = getattr(data, "dtype", None)

        # range
        if isinstance(data, RangeIndex):
            result = RangeIndex(start=data, copy=copy, name=name)
            if dtype is not None:
                return result.astype(dtype, copy=False)
            return result
        elif isinstance(data, range):
            result = RangeIndex.from_range(data, name=name)
            if dtype is not None:
                return result.astype(dtype, copy=False)
            return result

        elif is_ea_or_datetimelike_dtype(dtype):
            # non-EA dtype indexes have special casting logic, so we punt here
            klass = cls._dtype_to_subclass(dtype)
            if klass is not Index:
                return klass(data, dtype=dtype, copy=copy, name=name, **kwargs)

            ea_cls = dtype.construct_array_type()
            data = ea_cls._from_sequence(data, dtype=dtype, copy=copy)
            data = np.asarray(data, dtype=object)
            disallow_kwargs(kwargs)
            return Index._simple_new(data, name=name)

        elif is_ea_or_datetimelike_dtype(data_dtype):
            klass = cls._dtype_to_subclass(data_dtype)
            if klass is not Index:
                result = klass(data, copy=copy, name=name, **kwargs)
                if dtype is not None:
                    return result.astype(dtype, copy=False)
                return result

            data = np.array(data, dtype=object, copy=copy)
            disallow_kwargs(kwargs)
            return Index._simple_new(data, name=name)

        # index-like
        elif isinstance(data, (np.ndarray, Index, ABCSeries)):

            if dtype is not None:
                # we need to avoid having numpy coerce
                # things that look like ints/floats to ints unless
                # they are actually ints, e.g. '0' and 0.0
                # should not be coerced
                # GH 11836
                data = _maybe_cast_with_dtype(data, dtype, copy)
                dtype = data.dtype

            if data.dtype.kind in ["i", "u", "f"]:
                # maybe coerce to a sub-class
                arr = data
            else:
                arr = com.asarray_tuplesafe(data, dtype=object)

                if dtype is None:
                    arr = _maybe_cast_data_without_dtype(arr)
                    dtype = arr.dtype

                    if kwargs:
                        return cls(arr, dtype, copy=copy, name=name, **kwargs)

            klass = cls._dtype_to_subclass(arr.dtype)
            arr = klass._ensure_array(arr, dtype, copy)
            disallow_kwargs(kwargs)
            return klass._simple_new(arr, name)

        elif is_scalar(data):
            raise cls._scalar_data_error(data)
        elif hasattr(data, "__array__"):
            return Index(np.asarray(data), dtype=dtype, copy=copy, name=name, **kwargs)
        else:

            if tupleize_cols and is_list_like(data):
                # GH21470: convert iterable to list before determining if empty
                if is_iterator(data):
                    data = list(data)

                if data and all(isinstance(e, tuple) for e in data):
                    # we must be all tuples, otherwise don't construct
                    # 10697
                    from pandas.core.indexes.multi import MultiIndex

                    return MultiIndex.from_tuples(
                        data, names=name or kwargs.get("names")
                    )
            # other iterable of some kind
            subarr = com.asarray_tuplesafe(data, dtype=object)
            return Index(subarr, dtype=dtype, copy=copy, name=name, **kwargs)

    @classmethod
    def _ensure_array(cls, data, dtype, copy: bool):
        """
        Ensure we have a valid array to pass to _simple_new.
        """
        if data.ndim > 1:
            # GH#13601, GH#20285, GH#27125
            raise ValueError("Index data must be 1-dimensional")
        if copy:
            # asarray_tuplesafe does not always copy underlying data,
            #  so need to make sure that this happens
            data = data.copy()
        return data

    @final
    @classmethod
    def _dtype_to_subclass(cls, dtype: DtypeObj):
        # Delay import for perf. https://github.com/pandas-dev/pandas/pull/31423

        if isinstance(dtype, ExtensionDtype):
            if isinstance(dtype, DatetimeTZDtype):
                from pandas import DatetimeIndex

                return DatetimeIndex
            elif isinstance(dtype, CategoricalDtype):
                from pandas import CategoricalIndex

                return CategoricalIndex
            elif isinstance(dtype, IntervalDtype):
                from pandas import IntervalIndex

                return IntervalIndex
            elif isinstance(dtype, PeriodDtype):
                from pandas import PeriodIndex

                return PeriodIndex

            elif isinstance(dtype, SparseDtype):
                return cls._dtype_to_subclass(dtype.subtype)

            return Index

        if dtype.kind == "M":
            from pandas import DatetimeIndex

            return DatetimeIndex

        elif dtype.kind == "m":
            from pandas import TimedeltaIndex

            return TimedeltaIndex

        elif is_float_dtype(dtype):
            from pandas import Float64Index

            return Float64Index
        elif is_unsigned_integer_dtype(dtype):
            from pandas import UInt64Index

            return UInt64Index
        elif is_signed_integer_dtype(dtype):
            from pandas import Int64Index

            return Int64Index

        elif dtype == object:
            # NB: assuming away MultiIndex
            return Index

        elif issubclass(dtype.type, (str, bool, np.bool_)):
            return Index

        raise NotImplementedError(dtype)

    """
    NOTE for new Index creation:

    - _simple_new: It returns new Index with the same type as the caller.
      All metadata (such as name) must be provided by caller's responsibility.
      Using _shallow_copy is recommended because it fills these metadata
      otherwise specified.

    - _shallow_copy: It returns new Index with the same type (using
      _simple_new), but fills caller's metadata otherwise specified. Passed
      kwargs will overwrite corresponding metadata.

    See each method's docstring.
    """

    @property
    def asi8(self):
        """
        Integer representation of the values.

        Returns
        -------
        ndarray
            An ndarray with int64 dtype.
        """
        warnings.warn(
            "Index.asi8 is deprecated and will be removed in a future version",
            FutureWarning,
            stacklevel=2,
        )
        return None

    @classmethod
    def _simple_new(cls: Type[_IndexT], values, name: Hashable = None) -> _IndexT:
        """
        We require that we have a dtype compat for the values. If we are passed
        a non-dtype compat, then coerce using the constructor.

        Must be careful not to recurse.
        """
        assert isinstance(values, np.ndarray), type(values)

        result = object.__new__(cls)
        result._data = values
        # _index_data is a (temporary?) fix to ensure that the direct data
        # manipulation we do in `_libs/reduction.pyx` continues to work.
        # We need access to the actual ndarray, since we're messing with
        # data buffers and strides.
        result._index_data = values
        result._name = name
        result._cache = {}
        result._reset_identity()

        return result

    @cache_readonly
    def _constructor(self: _IndexT) -> Type[_IndexT]:
        return type(self)

    @final
    def _maybe_check_unique(self) -> None:
        """
        Check that an Index has no duplicates.

        This is typically only called via
        `NDFrame.flags.allows_duplicate_labels.setter` when it's set to
        True (duplicates aren't allowed).

        Raises
        ------
        DuplicateLabelError
            When the index is not unique.
        """
        if not self.is_unique:
            msg = """Index has duplicates."""
            duplicates = self._format_duplicate_message()
            msg += f"\n{duplicates}"

            raise DuplicateLabelError(msg)

    @final
    def _format_duplicate_message(self):
        """
        Construct the DataFrame for a DuplicateLabelError.

        This returns a DataFrame indicating the labels and positions
        of duplicates in an index. This should only be called when it's
        already known that duplicates are present.

        Examples
        --------
        >>> idx = pd.Index(['a', 'b', 'a'])
        >>> idx._format_duplicate_message()
            positions
        label
        a        [0, 2]
        """
        from pandas import Series

        duplicates = self[self.duplicated(keep="first")].unique()
        assert len(duplicates)

        out = Series(np.arange(len(self))).groupby(self).agg(list)[duplicates]
        if self.nlevels == 1:
            out = out.rename_axis("label")
        return out.to_frame(name="positions")

    # --------------------------------------------------------------------
    # Index Internals Methods

    @final
    def _get_attributes_dict(self) -> Dict[str_t, Any]:
        """
        Return an attributes dict for my class.
        """
        return {k: getattr(self, k, None) for k in self._attributes}

    def _shallow_copy(self: _IndexT, values, name: Hashable = no_default) -> _IndexT:
        """
        Create a new Index with the same class as the caller, don't copy the
        data, use the same object attributes with passed in attributes taking
        precedence.

        *this is an internal non-public method*

        Parameters
        ----------
        values : the values to create the new Index, optional
        name : Label, defaults to self.name
        """
        name = self.name if name is no_default else name

        return self._simple_new(values, name=name)

    def _view(self: _IndexT) -> _IndexT:
        """
        fastpath to make a shallow copy, i.e. new object with same data.
        """
        result = self._simple_new(self._values, name=self.name)

        result._cache = self._cache
        return result

    @final
    def _rename(self: _IndexT, name: Hashable) -> _IndexT:
        """
        fastpath for rename if new name is already validated.
        """
        result = self._view()
        result._name = name
        return result

    @final
    def is_(self, other) -> bool:
        """
        More flexible, faster check like ``is`` but that works through views.

        Note: this is *not* the same as ``Index.identical()``, which checks
        that metadata is also the same.

        Parameters
        ----------
        other : object
            Other object to compare against.

        Returns
        -------
        bool
            True if both have same underlying data, False otherwise.

        See Also
        --------
        Index.identical : Works like ``Index.is_`` but also checks metadata.
        """
        if self is other:
            return True
        elif not hasattr(other, "_id"):
            return False
        elif self._id is None or other._id is None:
            return False
        else:
            return self._id is other._id

    @final
    def _reset_identity(self) -> None:
        """
        Initializes or resets ``_id`` attribute with new object.
        """
        self._id = _Identity(object())

    @final
    def _cleanup(self) -> None:
        self._engine.clear_mapping()

    @cache_readonly
    def _engine(self) -> libindex.ObjectEngine:
        # property, for now, slow to look up

        # to avoid a reference cycle, bind `target_values` to a local variable, so
        # `self` is not passed into the lambda.
        target_values = self._get_engine_target()
        return self._engine_type(lambda: target_values, len(self))

    @cache_readonly
    def _dir_additions_for_owner(self) -> Set[str_t]:
        """
        Add the string-like labels to the owner dataframe/series dir output.

        If this is a MultiIndex, it's first level values are used.
        """
        return {
            c
            for c in self.unique(level=0)[:100]
            if isinstance(c, str) and c.isidentifier()
        }

    # --------------------------------------------------------------------
    # Array-Like Methods

    # ndarray compat
    def __len__(self) -> int:
        """
        Return the length of the Index.
        """
        return len(self._data)

    def __array__(self, dtype=None) -> np.ndarray:
        """
        The array interface, return my values.
        """
        return np.asarray(self._data, dtype=dtype)

    def __array_wrap__(self, result, context=None):
        """
        Gets called after a ufunc and other functions.
        """
        result = lib.item_from_zerodim(result)
        if is_bool_dtype(result) or lib.is_scalar(result) or np.ndim(result) > 1:
            return result

        attrs = self._get_attributes_dict()
        return Index(result, **attrs)

    @cache_readonly
    def dtype(self):
        """
        Return the dtype object of the underlying data.
        """
        return self._data.dtype

    @final
    def ravel(self, order="C"):
        """
        Return an ndarray of the flattened values of the underlying data.

        Returns
        -------
        numpy.ndarray
            Flattened array.

        See Also
        --------
        numpy.ndarray.ravel : Return a flattened array.
        """
        warnings.warn(
            "Index.ravel returning ndarray is deprecated; in a future version "
            "this will return a view on self.",
            FutureWarning,
            stacklevel=2,
        )
        values = self._get_engine_target()
        return values.ravel(order=order)

    def view(self, cls=None):

        # we need to see if we are subclassing an
        # index type here
        if cls is not None and not hasattr(cls, "_typ"):
            dtype = cls
            if isinstance(cls, str):
                dtype = pandas_dtype(cls)

            if isinstance(dtype, (np.dtype, ExtensionDtype)) and needs_i8_conversion(
                dtype
            ):
                if dtype.kind == "m" and dtype != "m8[ns]":
                    # e.g. m8[s]
                    return self._data.view(cls)

                arr = self._data.view("i8")
                idx_cls = self._dtype_to_subclass(dtype)
                arr_cls = idx_cls._data_cls
                arr = arr_cls(self._data.view("i8"), dtype=dtype)
                return idx_cls._simple_new(arr, name=self.name)

            result = self._data.view(cls)
        else:
            result = self._view()
        if isinstance(result, Index):
            result._id = self._id
        return result

    def astype(self, dtype, copy=True):
        """
        Create an Index with values cast to dtypes.

        The class of a new Index is determined by dtype. When conversion is
        impossible, a TypeError exception is raised.

        Parameters
        ----------
        dtype : numpy dtype or pandas type
            Note that any signed integer `dtype` is treated as ``'int64'``,
            and any unsigned integer `dtype` is treated as ``'uint64'``,
            regardless of the size.
        copy : bool, default True
            By default, astype always returns a newly allocated object.
            If copy is set to False and internal requirements on dtype are
            satisfied, the original data is used to create a new Index
            or the original Index is returned.

        Returns
        -------
        Index
            Index with values cast to specified dtype.
        """
        if dtype is not None:
            dtype = pandas_dtype(dtype)

        if is_dtype_equal(self.dtype, dtype):
            return self.copy() if copy else self

        elif is_categorical_dtype(dtype):
            from pandas.core.indexes.category import CategoricalIndex

            return CategoricalIndex(
                self._values, name=self.name, dtype=dtype, copy=copy
            )

        elif is_extension_array_dtype(dtype):
            return Index(np.asarray(self), name=self.name, dtype=dtype, copy=copy)

        try:
            casted = self._values.astype(dtype, copy=copy)
        except (TypeError, ValueError) as err:
            raise TypeError(
                f"Cannot cast {type(self).__name__} to dtype {dtype}"
            ) from err
        return Index(casted, name=self.name, dtype=dtype)

    _index_shared_docs[
        "take"
    ] = """
        Return a new %(klass)s of the values selected by the indices.

        For internal compatibility with numpy arrays.

        Parameters
        ----------
        indices : list
            Indices to be taken.
        axis : int, optional
            The axis over which to select values, always 0.
        allow_fill : bool, default True
        fill_value : bool, default None
            If allow_fill=True and fill_value is not None, indices specified by
            -1 is regarded as NA. If Index doesn't hold NA, raise ValueError.

        Returns
        -------
        numpy.ndarray
            Elements of given indices.

        See Also
        --------
        numpy.ndarray.take: Return an array formed from the
            elements of a at the given indices.
        """

    @Appender(_index_shared_docs["take"] % _index_doc_kwargs)
    def take(self, indices, axis=0, allow_fill=True, fill_value=None, **kwargs):
        if kwargs:
            nv.validate_take((), kwargs)
        indices = ensure_platform_int(indices)
        allow_fill = self._maybe_disallow_fill(allow_fill, fill_value, indices)

        # Note: we discard fill_value and use self._na_value, only relevant
        #  in the case where allow_fill is True and fill_value is not None
        taken = algos.take(
            self._values, indices, allow_fill=allow_fill, fill_value=self._na_value
        )
        return type(self)._simple_new(taken, name=self.name)

    @final
    def _maybe_disallow_fill(self, allow_fill: bool, fill_value, indices) -> bool:
        """
        We only use pandas-style take when allow_fill is True _and_
        fill_value is not None.
        """
        if allow_fill and fill_value is not None:
            # only fill if we are passing a non-None fill_value
            if self._can_hold_na:
                if (indices < -1).any():
                    raise ValueError(
                        "When allow_fill=True and fill_value is not None, "
                        "all indices must be >= -1"
                    )
            else:
                cls_name = type(self).__name__
                raise ValueError(
                    f"Unable to fill values because {cls_name} cannot contain NA"
                )
        else:
            allow_fill = False
        return allow_fill

    _index_shared_docs[
        "repeat"
    ] = """
        Repeat elements of a %(klass)s.

        Returns a new %(klass)s where each element of the current %(klass)s
        is repeated consecutively a given number of times.

        Parameters
        ----------
        repeats : int or array of ints
            The number of repetitions for each element. This should be a
            non-negative integer. Repeating 0 times will return an empty
            %(klass)s.
        axis : None
            Must be ``None``. Has no effect but is accepted for compatibility
            with numpy.

        Returns
        -------
        repeated_index : %(klass)s
            Newly created %(klass)s with repeated elements.

        See Also
        --------
        Series.repeat : Equivalent function for Series.
        numpy.repeat : Similar method for :class:`numpy.ndarray`.

        Examples
        --------
        >>> idx = pd.Index(['a', 'b', 'c'])
        >>> idx
        Index(['a', 'b', 'c'], dtype='object')
        >>> idx.repeat(2)
        Index(['a', 'a', 'b', 'b', 'c', 'c'], dtype='object')
        >>> idx.repeat([1, 2, 3])
        Index(['a', 'b', 'b', 'c', 'c', 'c'], dtype='object')
        """

    @Appender(_index_shared_docs["repeat"] % _index_doc_kwargs)
    def repeat(self, repeats, axis=None):
        repeats = ensure_platform_int(repeats)
        nv.validate_repeat((), {"axis": axis})
        res_values = self._values.repeat(repeats)

        return type(self)._simple_new(res_values, name=self.name)

    # --------------------------------------------------------------------
    # Copying Methods

    def copy(
        self: _IndexT,
        name: Optional[Hashable] = None,
        deep: bool = False,
        dtype: Optional[Dtype] = None,
        names: Optional[Sequence[Hashable]] = None,
    ) -> _IndexT:
        """
        Make a copy of this object.

        Name and dtype sets those attributes on the new object.

        Parameters
        ----------
        name : Label, optional
            Set name for new object.
        deep : bool, default False
        dtype : numpy dtype or pandas type, optional
            Set dtype for new object.

            .. deprecated:: 1.2.0
                use ``astype`` method instead.
        names : list-like, optional
            Kept for compatibility with MultiIndex. Should not be used.

        Returns
        -------
        Index
            Index refer to new object which is a copy of this object.

        Notes
        -----
        In most cases, there should be no functional difference from using
        ``deep``, but if ``deep`` is passed it will attempt to deepcopy.
        """
        name = self._validate_names(name=name, names=names, deep=deep)[0]
        if deep:
            new_data = self._data.copy()
            new_index = type(self)._simple_new(new_data, name=name)
        else:
            new_index = self._rename(name=name)

        if dtype:
            warnings.warn(
                "parameter dtype is deprecated and will be removed in a future "
                "version. Use the astype method instead.",
                FutureWarning,
                stacklevel=2,
            )
            new_index = new_index.astype(dtype)
        return new_index

    @final
    def __copy__(self, **kwargs):
        return self.copy(**kwargs)

    @final
    def __deepcopy__(self, memo=None):
        """
        Parameters
        ----------
        memo, default None
            Standard signature. Unused
        """
        return self.copy(deep=True)

    # --------------------------------------------------------------------
    # Rendering Methods

    @final
    def __repr__(self) -> str_t:
        """
        Return a string representation for this object.
        """
        klass_name = type(self).__name__
        data = self._format_data()
        attrs = self._format_attrs()
        space = self._format_space()
        attrs_str = [f"{k}={v}" for k, v in attrs]
        prepr = f",{space}".join(attrs_str)

        # no data provided, just attributes
        if data is None:
            data = ""

        return f"{klass_name}({data}{prepr})"

    def _format_space(self) -> str_t:

        # using space here controls if the attributes
        # are line separated or not (the default)

        # max_seq_items = get_option('display.max_seq_items')
        # if len(self) > max_seq_items:
        #    space = "\n%s" % (' ' * (len(klass) + 1))
        return " "

    @property
    def _formatter_func(self):
        """
        Return the formatter function.
        """
        return default_pprint

    def _format_data(self, name=None) -> str_t:
        """
        Return the formatted data as a unicode string.
        """
        # do we want to justify (only do so for non-objects)
        is_justify = True

        if self.inferred_type == "string":
            is_justify = False
        elif self.inferred_type == "categorical":
            self = cast("CategoricalIndex", self)
            if is_object_dtype(self.categories):
                is_justify = False

        return format_object_summary(
            self, self._formatter_func, is_justify=is_justify, name=name
        )

    def _format_attrs(self):
        """
        Return a list of tuples of the (attr,formatted_value).
        """
        return format_object_attrs(self)

    def _mpl_repr(self):
        # how to represent ourselves to matplotlib
        return self.values

    def format(
        self,
        name: bool = False,
        formatter: Optional[Callable] = None,
        na_rep: str_t = "NaN",
    ) -> List[str_t]:
        """
        Render a string representation of the Index.
        """
        header = []
        if name:
            header.append(
                pprint_thing(self.name, escape_chars=("\t", "\r", "\n"))
                if self.name is not None
                else ""
            )

        if formatter is not None:
            return header + list(self.map(formatter))

        return self._format_with_header(header, na_rep=na_rep)

    def _format_with_header(
        self, header: List[str_t], na_rep: str_t = "NaN"
    ) -> List[str_t]:
        from pandas.io.formats.format import format_array

        values = self._values

        if is_object_dtype(values.dtype):
            values = lib.maybe_convert_objects(values, safe=True)

            result = [pprint_thing(x, escape_chars=("\t", "\r", "\n")) for x in values]

            # could have nans
            mask = isna(values)
            if mask.any():
                result_arr = np.array(result)
                result_arr[mask] = na_rep
                result = result_arr.tolist()
        else:
            result = trim_front(format_array(values, None, justify="left"))
        return header + result

    @final
    def to_native_types(self, slicer=None, **kwargs):
        """
        Format specified values of `self` and return them.

        .. deprecated:: 1.2.0

        Parameters
        ----------
        slicer : int, array-like
            An indexer into `self` that specifies which values
            are used in the formatting process.
        kwargs : dict
            Options for specifying how the values should be formatted.
            These options include the following:

            1) na_rep : str
                The value that serves as a placeholder for NULL values
            2) quoting : bool or None
                Whether or not there are quoted values in `self`
            3) date_format : str
                The format used to represent date-like values.

        Returns
        -------
        numpy.ndarray
            Formatted values.
        """
        warnings.warn(
            "The 'to_native_types' method is deprecated and will be removed in "
            "a future version. Use 'astype(str)' instead.",
            FutureWarning,
            stacklevel=2,
        )
        values = self
        if slicer is not None:
            values = values[slicer]
        return values._format_native_types(**kwargs)

    def _format_native_types(self, na_rep="", quoting=None, **kwargs):
        """
        Actually format specific types of the index.
        """
        mask = isna(self)
        if not self.is_object() and not quoting:
            values = np.asarray(self).astype(str)
        else:
            values = np.array(self, dtype=object, copy=True)

        values[mask] = na_rep
        return values

    def _summary(self, name=None) -> str_t:
        """
        Return a summarized representation.

        Parameters
        ----------
        name : str
            name to use in the summary representation

        Returns
        -------
        String with a summarized representation of the index
        """
        if len(self) > 0:
            head = self[0]
            if hasattr(head, "format") and not isinstance(head, str):
                head = head.format()
            tail = self[-1]
            if hasattr(tail, "format") and not isinstance(tail, str):
                tail = tail.format()
            index_summary = f", {head} to {tail}"
        else:
            index_summary = ""

        if name is None:
            name = type(self).__name__
        return f"{name}: {len(self)} entries{index_summary}"

    # --------------------------------------------------------------------
    # Conversion Methods

    def to_flat_index(self):
        """
        Identity method.

        .. versionadded:: 0.24.0

        This is implemented for compatibility with subclass implementations
        when chaining.

        Returns
        -------
        pd.Index
            Caller.

        See Also
        --------
        MultiIndex.to_flat_index : Subclass implementation.
        """
        return self

    def to_series(self, index=None, name: Hashable = None) -> Series:
        """
        Create a Series with both index and values equal to the index keys.

        Useful with map for returning an indexer based on an index.

        Parameters
        ----------
        index : Index, optional
            Index of resulting Series. If None, defaults to original index.
        name : str, optional
            Name of resulting Series. If None, defaults to name of original
            index.

        Returns
        -------
        Series
            The dtype will be based on the type of the Index values.

        See Also
        --------
        Index.to_frame : Convert an Index to a DataFrame.
        Series.to_frame : Convert Series to DataFrame.

        Examples
        --------
        >>> idx = pd.Index(['Ant', 'Bear', 'Cow'], name='animal')

        By default, the original Index and original name is reused.

        >>> idx.to_series()
        animal
        Ant      Ant
        Bear    Bear
        Cow      Cow
        Name: animal, dtype: object

        To enforce a new Index, specify new labels to ``index``:

        >>> idx.to_series(index=[0, 1, 2])
        0     Ant
        1    Bear
        2     Cow
        Name: animal, dtype: object

        To override the name of the resulting column, specify `name`:

        >>> idx.to_series(name='zoo')
        animal
        Ant      Ant
        Bear    Bear
        Cow      Cow
        Name: zoo, dtype: object
        """
        from pandas import Series

        if index is None:
            index = self._view()
        if name is None:
            name = self.name

        return Series(self._values.copy(), index=index, name=name)

    def to_frame(self, index: bool = True, name=None) -> DataFrame:
        """
        Create a DataFrame with a column containing the Index.

        .. versionadded:: 0.24.0

        Parameters
        ----------
        index : bool, default True
            Set the index of the returned DataFrame as the original Index.

        name : object, default None
            The passed name should substitute for the index name (if it has
            one).

        Returns
        -------
        DataFrame
            DataFrame containing the original Index data.

        See Also
        --------
        Index.to_series : Convert an Index to a Series.
        Series.to_frame : Convert Series to DataFrame.

        Examples
        --------
        >>> idx = pd.Index(['Ant', 'Bear', 'Cow'], name='animal')
        >>> idx.to_frame()
               animal
        animal
        Ant       Ant
        Bear     Bear
        Cow       Cow

        By default, the original Index is reused. To enforce a new Index:

        >>> idx.to_frame(index=False)
            animal
        0   Ant
        1  Bear
        2   Cow

        To override the name of the resulting column, specify `name`:

        >>> idx.to_frame(index=False, name='zoo')
            zoo
        0   Ant
        1  Bear
        2   Cow
        """
        from pandas import DataFrame

        if name is None:
            name = self.name or 0
        result = DataFrame({name: self._values.copy()})

        if index:
            result.index = self
        return result

    # --------------------------------------------------------------------
    # Name-Centric Methods

    @property
    def name(self):
        """
        Return Index or MultiIndex name.
        """
        return self._name

    @name.setter
    def name(self, value):
        if self._no_setting_name:
            # Used in MultiIndex.levels to avoid silently ignoring name updates.
            raise RuntimeError(
                "Cannot set name on a level of a MultiIndex. Use "
                "'MultiIndex.set_names' instead."
            )
        maybe_extract_name(value, None, type(self))
        self._name = value

    @final
    def _validate_names(
        self, name=None, names=None, deep: bool = False
    ) -> List[Hashable]:
        """
        Handles the quirks of having a singular 'name' parameter for general
        Index and plural 'names' parameter for MultiIndex.
        """
        from copy import deepcopy

        if names is not None and name is not None:
            raise TypeError("Can only provide one of `names` and `name`")
        elif names is None and name is None:
            new_names = deepcopy(self.names) if deep else self.names
        elif names is not None:
            if not is_list_like(names):
                raise TypeError("Must pass list-like as `names`.")
            new_names = names
        elif not is_list_like(name):
            new_names = [name]
        else:
            new_names = name

        if len(new_names) != len(self.names):
            raise ValueError(
                f"Length of new names must be {len(self.names)}, got {len(new_names)}"
            )

        # All items in 'new_names' need to be hashable
        validate_all_hashable(*new_names, error_name=f"{type(self).__name__}.name")

        return new_names

    def _get_names(self) -> FrozenList:
        return FrozenList((self.name,))

    def _set_names(self, values, level=None) -> None:
        """
        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

        Raises
        ------
        TypeError if each name is not hashable.
        """
        if not is_list_like(values):
            raise ValueError("Names must be a list-like")
        if len(values) != 1:
            raise ValueError(f"Length of new names must be 1, got {len(values)}")

        # GH 20527
        # All items in 'name' need to be hashable:
        validate_all_hashable(*values, error_name=f"{type(self).__name__}.name")

        self._name = values[0]

    names = property(fset=_set_names, fget=_get_names)

    @final
    def set_names(self, names, level=None, inplace: bool = False):
        """
        Set Index or MultiIndex name.

        Able to set new names partially and by level.

        Parameters
        ----------

        names : label or list of label or dict-like for MultiIndex
            Name(s) to set.

            .. versionchanged:: 1.3.0

        level : int, label or list of int or label, optional
            If the index is a MultiIndex and names is not dict-like, level(s) to set
            (None for all levels). Otherwise level must be None.

            .. versionchanged:: 1.3.0

        inplace : bool, default False
            Modifies the object directly, instead of creating a new Index or
            MultiIndex.

        Returns
        -------
        Index or None
            The same type as the caller or None if ``inplace=True``.

        See Also
        --------
        Index.rename : Able to set new names without level.

        Examples
        --------
        >>> idx = pd.Index([1, 2, 3, 4])
        >>> idx
        Int64Index([1, 2, 3, 4], dtype='int64')
        >>> idx.set_names('quarter')
        Int64Index([1, 2, 3, 4], dtype='int64', name='quarter')

        >>> idx = pd.MultiIndex.from_product([['python', 'cobra'],
        ...                                   [2018, 2019]])
        >>> idx
        MultiIndex([('python', 2018),
                    ('python', 2019),
                    ( 'cobra', 2018),
                    ( 'cobra', 2019)],
                   )
        >>> idx.set_names(['kind', 'year'], inplace=True)
        >>> idx
        MultiIndex([('python', 2018),
                    ('python', 2019),
                    ( 'cobra', 2018),
                    ( 'cobra', 2019)],
                   names=['kind', 'year'])
        >>> idx.set_names('species', level=0)
        MultiIndex([('python', 2018),
                    ('python', 2019),
                    ( 'cobra', 2018),
                    ( 'cobra', 2019)],
                   names=['species', 'year'])

        When renaming levels with a dict, levels can not be passed.

        >>> idx.set_names({'kind': 'snake'})
        MultiIndex([('python', 2018),
                    ('python', 2019),
                    ( 'cobra', 2018),
                    ( 'cobra', 2019)],
                   names=['snake', 'year'])
        """
        if level is not None and not isinstance(self, ABCMultiIndex):
            raise ValueError("Level must be None for non-MultiIndex")

        elif level is not None and not is_list_like(level) and is_list_like(names):
            raise TypeError("Names must be a string when a single level is provided.")

        elif not is_list_like(names) and level is None and self.nlevels > 1:
            raise TypeError("Must pass list-like as `names`.")

        elif is_dict_like(names) and not isinstance(self, ABCMultiIndex):
            raise TypeError("Can only pass dict-like as `names` for MultiIndex.")

        elif is_dict_like(names) and level is not None:
            raise TypeError("Can not pass level for dictlike `names`.")

        if isinstance(self, ABCMultiIndex) and is_dict_like(names) and level is None:
            # Transform dict to list of new names and corresponding levels
            level, names_adjusted = [], []
            for i, name in enumerate(self.names):
                if name in names.keys():
                    level.append(i)
                    names_adjusted.append(names[name])
            names = names_adjusted

        if not is_list_like(names):
            names = [names]
        if level is not None and not is_list_like(level):
            level = [level]

        if inplace:
            idx = self
        else:
            idx = self._view()

        idx._set_names(names, level=level)
        if not inplace:
            return idx

    def rename(self, name, inplace=False):
        """
        Alter Index or MultiIndex name.

        Able to set new names without level. Defaults to returning new index.
        Length of names must match number of levels in MultiIndex.

        Parameters
        ----------
        name : label or list of labels
            Name(s) to set.
        inplace : bool, default False
            Modifies the object directly, instead of creating a new Index or
            MultiIndex.

        Returns
        -------
        Index or None
            The same type as the caller or None if ``inplace=True``.

        See Also
        --------
        Index.set_names : Able to set new names partially and by level.

        Examples
        --------
        >>> idx = pd.Index(['A', 'C', 'A', 'B'], name='score')
        >>> idx.rename('grade')
        Index(['A', 'C', 'A', 'B'], dtype='object', name='grade')

        >>> idx = pd.MultiIndex.from_product([['python', 'cobra'],
        ...                                   [2018, 2019]],
        ...                                   names=['kind', 'year'])
        >>> idx
        MultiIndex([('python', 2018),
                    ('python', 2019),
                    ( 'cobra', 2018),
                    ( 'cobra', 2019)],
                   names=['kind', 'year'])
        >>> idx.rename(['species', 'year'])
        MultiIndex([('python', 2018),
                    ('python', 2019),
                    ( 'cobra', 2018),
                    ( 'cobra', 2019)],
                   names=['species', 'year'])
        >>> idx.rename('species')
        Traceback (most recent call last):
        TypeError: Must pass list-like as `names`.
        """
        return self.set_names([name], inplace=inplace)

    # --------------------------------------------------------------------
    # Level-Centric Methods

    @property
    def nlevels(self) -> int:
        """
        Number of levels.
        """
        return 1

    def _sort_levels_monotonic(self: _IndexT) -> _IndexT:
        """
        Compat with MultiIndex.
        """
        return self

    @final
    def _validate_index_level(self, level) -> None:
        """
        Validate index level.

        For single-level Index getting level number is a no-op, but some
        verification must be done like in MultiIndex.

        """
        if isinstance(level, int):
            if level < 0 and level != -1:
                raise IndexError(
                    "Too many levels: Index has only 1 level, "
                    f"{level} is not a valid level number"
                )
            elif level > 0:
                raise IndexError(
                    f"Too many levels: Index has only 1 level, not {level + 1}"
                )
        elif level != self.name:
            raise KeyError(
                f"Requested level ({level}) does not match index name ({self.name})"
            )

    def _get_level_number(self, level) -> int:
        self._validate_index_level(level)
        return 0

    def sortlevel(self, level=None, ascending=True, sort_remaining=None):
        """
        For internal compatibility with the Index API.

        Sort the Index. This is for compat with MultiIndex

        Parameters
        ----------
        ascending : bool, default True
            False to sort in descending order

        level, sort_remaining are compat parameters

        Returns
        -------
        Index
        """
        if not isinstance(ascending, (list, bool)):
            raise TypeError(
                "ascending must be a single bool value or"
                "a list of bool values of length 1"
            )

        if isinstance(ascending, list):
            if len(ascending) != 1:
                raise TypeError("ascending must be a list of bool values of length 1")
            ascending = ascending[0]

        if not isinstance(ascending, bool):
            raise TypeError("ascending must be a bool value")

        return self.sort_values(return_indexer=True, ascending=ascending)

    def _get_level_values(self, level):
        """
        Return an Index of values for requested level.

        This is primarily useful to get an individual level of values from a
        MultiIndex, but is provided on Index as well for compatibility.

        Parameters
        ----------
        level : int or str
            It is either the integer position or the name of the level.

        Returns
        -------
        Index
            Calling object, as there is only one level in the Index.

        See Also
        --------
        MultiIndex.get_level_values : Get values for a level of a MultiIndex.

        Notes
        -----
        For Index, level should be 0, since there are no multiple levels.

        Examples
        --------
        >>> idx = pd.Index(list('abc'))
        >>> idx
        Index(['a', 'b', 'c'], dtype='object')

        Get level values by supplying `level` as integer:

        >>> idx.get_level_values(0)
        Index(['a', 'b', 'c'], dtype='object')
        """
        self._validate_index_level(level)
        return self

    get_level_values = _get_level_values

    @final
    def droplevel(self, level=0):
        """
        Return index with requested level(s) removed.

        If resulting index has only 1 level left, the result will be
        of Index type, not MultiIndex.

        Parameters
        ----------
        level : int, str, or list-like, default 0
            If a string is given, must be the name of a level
            If list-like, elements must be names or indexes of levels.

        Returns
        -------
        Index or MultiIndex

        Examples
        --------
        >>> mi = pd.MultiIndex.from_arrays(
        ... [[1, 2], [3, 4], [5, 6]], names=['x', 'y', 'z'])
        >>> mi
        MultiIndex([(1, 3, 5),
                    (2, 4, 6)],
                   names=['x', 'y', 'z'])

        >>> mi.droplevel()
        MultiIndex([(3, 5),
                    (4, 6)],
                   names=['y', 'z'])

        >>> mi.droplevel(2)
        MultiIndex([(1, 3),
                    (2, 4)],
                   names=['x', 'y'])

        >>> mi.droplevel('z')
        MultiIndex([(1, 3),
                    (2, 4)],
                   names=['x', 'y'])

        >>> mi.droplevel(['x', 'y'])
        Int64Index([5, 6], dtype='int64', name='z')
        """
        if not isinstance(level, (tuple, list)):
            level = [level]

        levnums = sorted(self._get_level_number(lev) for lev in level)[::-1]

        return self._drop_level_numbers(levnums)

    @final
    def _drop_level_numbers(self, levnums: List[int]):
        """
        Drop MultiIndex levels by level _number_, not name.
        """

        if not levnums and not isinstance(self, ABCMultiIndex):
            return self
        if len(levnums) >= self.nlevels:
            raise ValueError(
                f"Cannot remove {len(levnums)} levels from an index with "
                f"{self.nlevels} levels: at least one level must be left."
            )
        # The two checks above guarantee that here self is a MultiIndex
        self = cast("MultiIndex", self)

        new_levels = list(self.levels)
        new_codes = list(self.codes)
        new_names = list(self.names)

        for i in levnums:
            new_levels.pop(i)
            new_codes.pop(i)
            new_names.pop(i)

        if len(new_levels) == 1:

            # set nan if needed
            mask = new_codes[0] == -1
            result = new_levels[0].take(new_codes[0])
            if mask.any():
                result = result.putmask(mask, np.nan)

            result._name = new_names[0]
            return result
        else:
            from pandas.core.indexes.multi import MultiIndex

            return MultiIndex(
                levels=new_levels,
                codes=new_codes,
                names=new_names,
                verify_integrity=False,
            )

    def _get_grouper_for_level(self, mapper, level=None):
        """
        Get index grouper corresponding to an index level

        Parameters
        ----------
        mapper: Group mapping function or None
            Function mapping index values to groups
        level : int or None
            Index level

        Returns
        -------
        grouper : Index
            Index of values to group on.
        labels : ndarray of int or None
            Array of locations in level_index.
        uniques : Index or None
            Index of unique values for level.
        """
        assert level is None or level == 0
        if mapper is None:
            grouper = self
        else:
            grouper = self.map(mapper)

        return grouper, None, None

    # --------------------------------------------------------------------
    # Introspection Methods

    @final
    @property
    def is_monotonic(self) -> bool:
        """
        Alias for is_monotonic_increasing.
        """
        return self.is_monotonic_increasing

    @property
    def is_monotonic_increasing(self) -> bool:
        """
        Return if the index is monotonic increasing (only equal or
        increasing) values.

        Examples
        --------
        >>> Index([1, 2, 3]).is_monotonic_increasing
        True
        >>> Index([1, 2, 2]).is_monotonic_increasing
        True
        >>> Index([1, 3, 2]).is_monotonic_increasing
        False
        """
        return self._engine.is_monotonic_increasing

    @property
    def is_monotonic_decreasing(self) -> bool:
        """
        Return if the index is monotonic decreasing (only equal or
        decreasing) values.

        Examples
        --------
        >>> Index([3, 2, 1]).is_monotonic_decreasing
        True
        >>> Index([3, 2, 2]).is_monotonic_decreasing
        True
        >>> Index([3, 1, 2]).is_monotonic_decreasing
        False
        """
        return self._engine.is_monotonic_decreasing

    @final
    @property
    def _is_strictly_monotonic_increasing(self) -> bool:
        """
        Return if the index is strictly monotonic increasing
        (only increasing) values.

        Examples
        --------
        >>> Index([1, 2, 3])._is_strictly_monotonic_increasing
        True
        >>> Index([1, 2, 2])._is_strictly_monotonic_increasing
        False
        >>> Index([1, 3, 2])._is_strictly_monotonic_increasing
        False
        """
        return self.is_unique and self.is_monotonic_increasing

    @final
    @property
    def _is_strictly_monotonic_decreasing(self) -> bool:
        """
        Return if the index is strictly monotonic decreasing
        (only decreasing) values.

        Examples
        --------
        >>> Index([3, 2, 1])._is_strictly_monotonic_decreasing
        True
        >>> Index([3, 2, 2])._is_strictly_monotonic_decreasing
        False
        >>> Index([3, 1, 2])._is_strictly_monotonic_decreasing
        False
        """
        return self.is_unique and self.is_monotonic_decreasing

    @cache_readonly
    def is_unique(self) -> bool:
        """
        Return if the index has unique values.
        """
        return self._engine.is_unique

    @final
    @property
    def has_duplicates(self) -> bool:
        """
        Check if the Index has duplicate values.

        Returns
        -------
        bool
            Whether or not the Index has duplicate values.

        Examples
        --------
        >>> idx = pd.Index([1, 5, 7, 7])
        >>> idx.has_duplicates
        True

        >>> idx = pd.Index([1, 5, 7])
        >>> idx.has_duplicates
        False

        >>> idx = pd.Index(["Watermelon", "Orange", "Apple",
        ...                 "Watermelon"]).astype("category")
        >>> idx.has_duplicates
        True

        >>> idx = pd.Index(["Orange", "Apple",
        ...                 "Watermelon"]).astype("category")
        >>> idx.has_duplicates
        False
        """
        return not self.is_unique

    @final
    def is_boolean(self) -> bool:
        """
        Check if the Index only consists of booleans.

        Returns
        -------
        bool
            Whether or not the Index only consists of booleans.

        See Also
        --------
        is_integer : Check if the Index only consists of integers.
        is_floating : Check if the Index is a floating type.
        is_numeric : Check if the Index only consists of numeric data.
        is_object : Check if the Index is of the object dtype.
        is_categorical : Check if the Index holds categorical data.
        is_interval : Check if the Index holds Interval objects.
        is_mixed : Check if the Index holds data with mixed data types.

        Examples
        --------
        >>> idx = pd.Index([True, False, True])
        >>> idx.is_boolean()
        True

        >>> idx = pd.Index(["True", "False", "True"])
        >>> idx.is_boolean()
        False

        >>> idx = pd.Index([True, False, "True"])
        >>> idx.is_boolean()
        False
        """
        return self.inferred_type in ["boolean"]

    @final
    def is_integer(self) -> bool:
        """
        Check if the Index only consists of integers.

        Returns
        -------
        bool
            Whether or not the Index only consists of integers.

        See Also
        --------
        is_boolean : Check if the Index only consists of booleans.
        is_floating : Check if the Index is a floating type.
        is_numeric : Check if the Index only consists of numeric data.
        is_object : Check if the Index is of the object dtype.
        is_categorical : Check if the Index holds categorical data.
        is_interval : Check if the Index holds Interval objects.
        is_mixed : Check if the Index holds data with mixed data types.

        Examples
        --------
        >>> idx = pd.Index([1, 2, 3, 4])
        >>> idx.is_integer()
        True

        >>> idx = pd.Index([1.0, 2.0, 3.0, 4.0])
        >>> idx.is_integer()
        False

        >>> idx = pd.Index(["Apple", "Mango", "Watermelon"])
        >>> idx.is_integer()
        False
        """
        return self.inferred_type in ["integer"]

    @final
    def is_floating(self) -> bool:
        """
        Check if the Index is a floating type.

        The Index may consist of only floats, NaNs, or a mix of floats,
        integers, or NaNs.

        Returns
        -------
        bool
            Whether or not the Index only consists of only consists of floats, NaNs, or
            a mix of floats, integers, or NaNs.

        See Also
        --------
        is_boolean : Check if the Index only consists of booleans.
        is_integer : Check if the Index only consists of integers.
        is_numeric : Check if the Index only consists of numeric data.
        is_object : Check if the Index is of the object dtype.
        is_categorical : Check if the Index holds categorical data.
        is_interval : Check if the Index holds Interval objects.
        is_mixed : Check if the Index holds data with mixed data types.

        Examples
        --------
        >>> idx = pd.Index([1.0, 2.0, 3.0, 4.0])
        >>> idx.is_floating()
        True

        >>> idx = pd.Index([1.0, 2.0, np.nan, 4.0])
        >>> idx.is_floating()
        True

        >>> idx = pd.Index([1, 2, 3, 4, np.nan])
        >>> idx.is_floating()
        True

        >>> idx = pd.Index([1, 2, 3, 4])
        >>> idx.is_floating()
        False
        """
        return self.inferred_type in ["floating", "mixed-integer-float", "integer-na"]

    @final
    def is_numeric(self) -> bool:
        """
        Check if the Index only consists of numeric data.

        Returns
        -------
        bool
            Whether or not the Index only consists of numeric data.

        See Also
        --------
        is_boolean : Check if the Index only consists of booleans.
        is_integer : Check if the Index only consists of integers.
        is_floating : Check if the Index is a floating type.
        is_object : Check if the Index is of the object dtype.
        is_categorical : Check if the Index holds categorical data.
        is_interval : Check if the Index holds Interval objects.
        is_mixed : Check if the Index holds data with mixed data types.

        Examples
        --------
        >>> idx = pd.Index([1.0, 2.0, 3.0, 4.0])
        >>> idx.is_numeric()
        True

        >>> idx = pd.Index([1, 2, 3, 4.0])
        >>> idx.is_numeric()
        True

        >>> idx = pd.Index([1, 2, 3, 4])
        >>> idx.is_numeric()
        True

        >>> idx = pd.Index([1, 2, 3, 4.0, np.nan])
        >>> idx.is_numeric()
        True

        >>> idx = pd.Index([1, 2, 3, 4.0, np.nan, "Apple"])
        >>> idx.is_numeric()
        False
        """
        return self.inferred_type in ["integer", "floating"]

    @final
    def is_object(self) -> bool:
        """
        Check if the Index is of the object dtype.

        Returns
        -------
        bool
            Whether or not the Index is of the object dtype.

        See Also
        --------
        is_boolean : Check if the Index only consists of booleans.
        is_integer : Check if the Index only consists of integers.
        is_floating : Check if the Index is a floating type.
        is_numeric : Check if the Index only consists of numeric data.
        is_categorical : Check if the Index holds categorical data.
        is_interval : Check if the Index holds Interval objects.
        is_mixed : Check if the Index holds data with mixed data types.

        Examples
        --------
        >>> idx = pd.Index(["Apple", "Mango", "Watermelon"])
        >>> idx.is_object()
        True

        >>> idx = pd.Index(["Apple", "Mango", 2.0])
        >>> idx.is_object()
        True

        >>> idx = pd.Index(["Watermelon", "Orange", "Apple",
        ...                 "Watermelon"]).astype("category")
        >>> idx.is_object()
        False

        >>> idx = pd.Index([1.0, 2.0, 3.0, 4.0])
        >>> idx.is_object()
        False
        """
        return is_object_dtype(self.dtype)

    @final
    def is_categorical(self) -> bool:
        """
        Check if the Index holds categorical data.

        Returns
        -------
        bool
            True if the Index is categorical.

        See Also
        --------
        CategoricalIndex : Index for categorical data.
        is_boolean : Check if the Index only consists of booleans.
        is_integer : Check if the Index only consists of integers.
        is_floating : Check if the Index is a floating type.
        is_numeric : Check if the Index only consists of numeric data.
        is_object : Check if the Index is of the object dtype.
        is_interval : Check if the Index holds Interval objects.
        is_mixed : Check if the Index holds data with mixed data types.

        Examples
        --------
        >>> idx = pd.Index(["Watermelon", "Orange", "Apple",
        ...                 "Watermelon"]).astype("category")
        >>> idx.is_categorical()
        True

        >>> idx = pd.Index([1, 3, 5, 7])
        >>> idx.is_categorical()
        False

        >>> s = pd.Series(["Peter", "Victor", "Elisabeth", "Mar"])
        >>> s
        0        Peter
        1       Victor
        2    Elisabeth
        3          Mar
        dtype: object
        >>> s.index.is_categorical()
        False
        """
        return self.inferred_type in ["categorical"]

    @final
    def is_interval(self) -> bool:
        """
        Check if the Index holds Interval objects.

        Returns
        -------
        bool
            Whether or not the Index holds Interval objects.

        See Also
        --------
        IntervalIndex : Index for Interval objects.
        is_boolean : Check if the Index only consists of booleans.
        is_integer : Check if the Index only consists of integers.
        is_floating : Check if the Index is a floating type.
        is_numeric : Check if the Index only consists of numeric data.
        is_object : Check if the Index is of the object dtype.
        is_categorical : Check if the Index holds categorical data.
        is_mixed : Check if the Index holds data with mixed data types.

        Examples
        --------
        >>> idx = pd.Index([pd.Interval(left=0, right=5),
        ...                 pd.Interval(left=5, right=10)])
        >>> idx.is_interval()
        True

        >>> idx = pd.Index([1, 3, 5, 7])
        >>> idx.is_interval()
        False
        """
        return self.inferred_type in ["interval"]

    @final
    def is_mixed(self) -> bool:
        """
        Check if the Index holds data with mixed data types.

        Returns
        -------
        bool
            Whether or not the Index holds data with mixed data types.

        See Also
        --------
        is_boolean : Check if the Index only consists of booleans.
        is_integer : Check if the Index only consists of integers.
        is_floating : Check if the Index is a floating type.
        is_numeric : Check if the Index only consists of numeric data.
        is_object : Check if the Index is of the object dtype.
        is_categorical : Check if the Index holds categorical data.
        is_interval : Check if the Index holds Interval objects.

        Examples
        --------
        >>> idx = pd.Index(['a', np.nan, 'b'])
        >>> idx.is_mixed()
        True

        >>> idx = pd.Index([1.0, 2.0, 3.0, 5.0])
        >>> idx.is_mixed()
        False
        """
        warnings.warn(
            "Index.is_mixed is deprecated and will be removed in a future version. "
            "Check index.inferred_type directly instead.",
            FutureWarning,
            stacklevel=2,
        )
        return self.inferred_type in ["mixed"]

    @final
    def holds_integer(self) -> bool:
        """
        Whether the type is an integer type.
        """
        return self.inferred_type in ["integer", "mixed-integer"]

    @cache_readonly
    def inferred_type(self) -> str_t:
        """
        Return a string of the type inferred from the values.
        """
        return lib.infer_dtype(self._values, skipna=False)

    @cache_readonly
    def _is_all_dates(self) -> bool:
        """
        Whether or not the index values only consist of dates.
        """
        return is_datetime_array(ensure_object(self._values))

    @cache_readonly
    @final
    def is_all_dates(self):
        """
        Whether or not the index values only consist of dates.
        """
        warnings.warn(
            "Index.is_all_dates is deprecated, will be removed in a future version.  "
            "check index.inferred_type instead",
            FutureWarning,
            stacklevel=2,
        )
        return self._is_all_dates

    # --------------------------------------------------------------------
    # Pickle Methods

    def __reduce__(self):
        d = {"data": self._data}
        d.update(self._get_attributes_dict())
        return _new_Index, (type(self), d), None

    # --------------------------------------------------------------------
    # Null Handling Methods

    _na_value = np.nan
    """The expected NA value to use with this index."""

    @cache_readonly
    def _isnan(self):
        """
        Return if each value is NaN.
        """
        if self._can_hold_na:
            return isna(self)
        else:
            # shouldn't reach to this condition by checking hasnans beforehand
            values = np.empty(len(self), dtype=np.bool_)
            values.fill(False)
            return values

    @cache_readonly
    def hasnans(self) -> bool:
        """
        Return if I have any nans; enables various perf speedups.
        """
        if self._can_hold_na:
            return bool(self._isnan.any())
        else:
            return False

    @final
    def isna(self) -> np.ndarray:
        """
        Detect missing values.

        Return a boolean same-sized object indicating if the values are NA.
        NA values, such as ``None``, :attr:`numpy.NaN` or :attr:`pd.NaT`, get
        mapped to ``True`` values.
        Everything else get mapped to ``False`` values. Characters such as
        empty strings `''` or :attr:`numpy.inf` are not considered NA values
        (unless you set ``pandas.options.mode.use_inf_as_na = True``).

        Returns
        -------
        numpy.ndarray
            A boolean array of whether my values are NA.

        See Also
        --------
        Index.notna : Boolean inverse of isna.
        Index.dropna : Omit entries with missing values.
        isna : Top-level isna.
        Series.isna : Detect missing values in Series object.

        Examples
        --------
        Show which entries in a pandas.Index are NA. The result is an
        array.

        >>> idx = pd.Index([5.2, 6.0, np.NaN])
        >>> idx
        Float64Index([5.2, 6.0, nan], dtype='float64')
        >>> idx.isna()
        array([False, False,  True])

        Empty strings are not considered NA values. None is considered an NA
        value.

        >>> idx = pd.Index(['black', '', 'red', None])
        >>> idx
        Index(['black', '', 'red', None], dtype='object')
        >>> idx.isna()
        array([False, False, False,  True])

        For datetimes, `NaT` (Not a Time) is considered as an NA value.

        >>> idx = pd.DatetimeIndex([pd.Timestamp('1940-04-25'),
        ...                         pd.Timestamp(''), None, pd.NaT])
        >>> idx
        DatetimeIndex(['1940-04-25', 'NaT', 'NaT', 'NaT'],
                      dtype='datetime64[ns]', freq=None)
        >>> idx.isna()
        array([False,  True,  True,  True])
        """
        return self._isnan

    isnull = isna

    @final
    def notna(self) -> np.ndarray:
        """
        Detect existing (non-missing) values.

        Return a boolean same-sized object indicating if the values are not NA.
        Non-missing values get mapped to ``True``. Characters such as empty
        strings ``''`` or :attr:`numpy.inf` are not considered NA values
        (unless you set ``pandas.options.mode.use_inf_as_na = True``).
        NA values, such as None or :attr:`numpy.NaN`, get mapped to ``False``
        values.

        Returns
        -------
        numpy.ndarray
            Boolean array to indicate which entries are not NA.

        See Also
        --------
        Index.notnull : Alias of notna.
        Index.isna: Inverse of notna.
        notna : Top-level notna.

        Examples
        --------
        Show which entries in an Index are not NA. The result is an
        array.

        >>> idx = pd.Index([5.2, 6.0, np.NaN])
        >>> idx
        Float64Index([5.2, 6.0, nan], dtype='float64')
        >>> idx.notna()
        array([ True,  True, False])

        Empty strings are not considered NA values. None is considered a NA
        value.

        >>> idx = pd.Index(['black', '', 'red', None])
        >>> idx
        Index(['black', '', 'red', None], dtype='object')
        >>> idx.notna()
        array([ True,  True,  True, False])
        """
        return ~self.isna()

    notnull = notna

    def fillna(self, value=None, downcast=None):
        """
        Fill NA/NaN values with the specified value.

        Parameters
        ----------
        value : scalar
            Scalar value to use to fill holes (e.g. 0).
            This value cannot be a list-likes.
        downcast : dict, default is None
            A dict of item->dtype of what to downcast if possible,
            or the string 'infer' which will try to downcast to an appropriate
            equal type (e.g. float64 to int64 if possible).

        Returns
        -------
        Index

        See Also
        --------
        DataFrame.fillna : Fill NaN values of a DataFrame.
        Series.fillna : Fill NaN Values of a Series.
        """
        value = self._require_scalar(value)
        if self.hasnans:
            result = self.putmask(self._isnan, value)
            if downcast is None:
                # no need to care metadata other than name
                # because it can't have freq if
                return Index(result, name=self.name)
        return self._view()

    def dropna(self: _IndexT, how: str_t = "any") -> _IndexT:
        """
        Return Index without NA/NaN values.

        Parameters
        ----------
        how : {'any', 'all'}, default 'any'
            If the Index is a MultiIndex, drop the value when any or all levels
            are NaN.

        Returns
        -------
        Index
        """
        if how not in ("any", "all"):
            raise ValueError(f"invalid how option: {how}")

        if self.hasnans:
            res_values = self._values[~self._isnan]
            return type(self)._simple_new(res_values, name=self.name)
        return self._view()

    # --------------------------------------------------------------------
    # Uniqueness Methods

    def unique(self: _IndexT, level: Optional[Hashable] = None) -> _IndexT:
        """
        Return unique values in the index.

        Unique values are returned in order of appearance, this does NOT sort.

        Parameters
        ----------
        level : int or hashable, optional
            Only return values from specified level (for MultiIndex).
            If int, gets the level by integer position, else by level name.

        Returns
        -------
        Index

        See Also
        --------
        unique : Numpy array of unique values in that column.
        Series.unique : Return unique values of Series object.
        """
        if level is not None:
            self._validate_index_level(level)

        if self.is_unique:
            return self._view()

        result = super().unique()
        return self._shallow_copy(result)

    @final
    def drop_duplicates(self: _IndexT, keep: Union[str_t, bool] = "first") -> _IndexT:
        """
        Return Index with duplicate values removed.

        Parameters
        ----------
        keep : {'first', 'last', ``False``}, default 'first'
            - 'first' : Drop duplicates except for the first occurrence.
            - 'last' : Drop duplicates except for the last occurrence.
            - ``False`` : Drop all duplicates.

        Returns
        -------
        deduplicated : Index

        See Also
        --------
        Series.drop_duplicates : Equivalent method on Series.
        DataFrame.drop_duplicates : Equivalent method on DataFrame.
        Index.duplicated : Related method on Index, indicating duplicate
            Index values.

        Examples
        --------
        Generate an pandas.Index with duplicate values.

        >>> idx = pd.Index(['lama', 'cow', 'lama', 'beetle', 'lama', 'hippo'])

        The `keep` parameter controls  which duplicate values are removed.
        The value 'first' keeps the first occurrence for each
        set of duplicated entries. The default value of keep is 'first'.

        >>> idx.drop_duplicates(keep='first')
        Index(['lama', 'cow', 'beetle', 'hippo'], dtype='object')

        The value 'last' keeps the last occurrence for each set of duplicated
        entries.

        >>> idx.drop_duplicates(keep='last')
        Index(['cow', 'beetle', 'lama', 'hippo'], dtype='object')

        The value ``False`` discards all sets of duplicated entries.

        >>> idx.drop_duplicates(keep=False)
        Index(['cow', 'beetle', 'hippo'], dtype='object')
        """
        if self.is_unique:
            return self._view()

        return super().drop_duplicates(keep=keep)

    def duplicated(self, keep: Union[str_t, bool] = "first") -> np.ndarray:
        """
        Indicate duplicate index values.

        Duplicated values are indicated as ``True`` values in the resulting
        array. Either all duplicates, all except the first, or all except the
        last occurrence of duplicates can be indicated.

        Parameters
        ----------
        keep : {'first', 'last', False}, default 'first'
            The value or values in a set of duplicates to mark as missing.

            - 'first' : Mark duplicates as ``True`` except for the first
              occurrence.
            - 'last' : Mark duplicates as ``True`` except for the last
              occurrence.
            - ``False`` : Mark all duplicates as ``True``.

        Returns
        -------
        numpy.ndarray

        See Also
        --------
        Series.duplicated : Equivalent method on pandas.Series.
        DataFrame.duplicated : Equivalent method on pandas.DataFrame.
        Index.drop_duplicates : Remove duplicate values from Index.

        Examples
        --------
        By default, for each set of duplicated values, the first occurrence is
        set to False and all others to True:

        >>> idx = pd.Index(['lama', 'cow', 'lama', 'beetle', 'lama'])
        >>> idx.duplicated()
        array([False, False,  True, False,  True])

        which is equivalent to

        >>> idx.duplicated(keep='first')
        array([False, False,  True, False,  True])

        By using 'last', the last occurrence of each set of duplicated values
        is set on False and all others on True:

        >>> idx.duplicated(keep='last')
        array([ True, False,  True, False, False])

        By setting keep on ``False``, all duplicates are True:

        >>> idx.duplicated(keep=False)
        array([ True, False,  True, False,  True])
        """
        if self.is_unique:
            # fastpath available bc we are immutable
            return np.zeros(len(self), dtype=bool)
        return super().duplicated(keep=keep)

    def _get_unique_index(self: _IndexT) -> _IndexT:
        """
        Returns an index containing unique values.

        Returns
        -------
        Index
        """
        return self.unique()

    # --------------------------------------------------------------------
    # Arithmetic & Logical Methods

    def __iadd__(self, other):
        # alias for __add__
        return self + other

    @final
    def __and__(self, other):
        warnings.warn(
            "Index.__and__ operating as a set operation is deprecated, "
            "in the future this will be a logical operation matching "
            "Series.__and__.  Use index.intersection(other) instead",
            FutureWarning,
            stacklevel=2,
        )
        return self.intersection(other)

    @final
    def __or__(self, other):
        warnings.warn(
            "Index.__or__ operating as a set operation is deprecated, "
            "in the future this will be a logical operation matching "
            "Series.__or__.  Use index.union(other) instead",
            FutureWarning,
            stacklevel=2,
        )
        return self.union(other)

    @final
    def __xor__(self, other):
        warnings.warn(
            "Index.__xor__ operating as a set operation is deprecated, "
            "in the future this will be a logical operation matching "
            "Series.__xor__.  Use index.symmetric_difference(other) instead",
            FutureWarning,
            stacklevel=2,
        )
        return self.symmetric_difference(other)

    @final
    def __nonzero__(self):
        raise ValueError(
            f"The truth value of a {type(self).__name__} is ambiguous. "
            "Use a.empty, a.bool(), a.item(), a.any() or a.all()."
        )

    __bool__ = __nonzero__

    # --------------------------------------------------------------------
    # Set Operation Methods

    def _get_reconciled_name_object(self, other):
        """
        If the result of a set operation will be self,
        return self, unless the name changes, in which
        case make a shallow copy of self.
        """
        name = get_op_result_name(self, other)
        if self.name != name:
            return self.rename(name)
        return self

    @final
    def _validate_sort_keyword(self, sort):
        if sort not in [None, False]:
            raise ValueError(
                "The 'sort' keyword only takes the values of "
                f"None or False; {sort} was passed."
            )

    @final
    def union(self, other, sort=None):
        """
        Form the union of two Index objects.

        If the Index objects are incompatible, both Index objects will be
        cast to dtype('object') first.

            .. versionchanged:: 0.25.0

        Parameters
        ----------
        other : Index or array-like
        sort : bool or None, default None
            Whether to sort the resulting Index.

            * None : Sort the result, except when

              1. `self` and `other` are equal.
              2. `self` or `other` 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
        -------
        union : Index

        Examples
        --------
        Union matching dtypes

        >>> idx1 = pd.Index([1, 2, 3, 4])
        >>> idx2 = pd.Index([3, 4, 5, 6])
        >>> idx1.union(idx2)
        Int64Index([1, 2, 3, 4, 5, 6], dtype='int64')

        Union mismatched dtypes

        >>> idx1 = pd.Index(['a', 'b', 'c', 'd'])
        >>> idx2 = pd.Index([1, 2, 3, 4])
        >>> idx1.union(idx2)
        Index(['a', 'b', 'c', 'd', 1, 2, 3, 4], dtype='object')

        MultiIndex case

        >>> idx1 = pd.MultiIndex.from_arrays(
        ...     [[1, 1, 2, 2], ["Red", "Blue", "Red", "Blue"]]
        ... )
        >>> idx1
        MultiIndex([(1,  'Red'),
            (1, 'Blue'),
            (2,  'Red'),
            (2, 'Blue')],
           )
        >>> idx2 = pd.MultiIndex.from_arrays(
        ...     [[3, 3, 2, 2], ["Red", "Green", "Red", "Green"]]
        ... )
        >>> idx2
        MultiIndex([(3,   'Red'),
            (3, 'Green'),
            (2,   'Red'),
            (2, 'Green')],
           )
        >>> idx1.union(idx2)
        MultiIndex([(1,  'Blue'),
            (1,   'Red'),
            (2,  'Blue'),
            (2, 'Green'),
            (2,   'Red'),
            (3, 'Green'),
            (3,   'Red')],
           )
        >>> idx1.union(idx2, sort=False)
        MultiIndex([(1,   'Red'),
            (1,  'Blue'),
            (2,   'Red'),
            (2,  'Blue'),
            (3,   'Red'),
            (3, 'Green'),
            (2, 'Green')],
           )
        """
        self._validate_sort_keyword(sort)
        self._assert_can_do_setop(other)
        other, result_name = self._convert_can_do_setop(other)

        if not is_dtype_equal(self.dtype, other.dtype):
            if isinstance(self, ABCMultiIndex) and not is_object_dtype(
                unpack_nested_dtype(other)
            ):
                raise NotImplementedError(
                    "Can only union MultiIndex with MultiIndex or Index of tuples, "
                    "try mi.to_flat_index().union(other) instead."
                )

            dtype = find_common_type([self.dtype, other.dtype])
            if self._is_numeric_dtype and other._is_numeric_dtype:
                # Right now, we treat union(int, float) a bit special.
                # See https://github.com/pandas-dev/pandas/issues/26778 for discussion
                # We may change union(int, float) to go to object.
                # float | [u]int -> float  (the special case)
                # <T>   | <T>    -> T
                # <T>   | <U>    -> object
                if not (is_integer_dtype(self.dtype) and is_integer_dtype(other.dtype)):
                    dtype = "float64"
                else:
                    # one is int64 other is uint64
                    dtype = object

            left = self.astype(dtype, copy=False)
            right = other.astype(dtype, copy=False)
            return left.union(right, sort=sort)

        elif not len(other) or self.equals(other):
            # NB: whether this (and the `if not len(self)` check below) come before
            #  or after the is_dtype_equal check above affects the returned dtype
            return self._get_reconciled_name_object(other)

        elif not len(self):
            return other._get_reconciled_name_object(self)

        result = self._union(other, sort=sort)

        return self._wrap_setop_result(other, result)

    def _union(self, other: Index, sort):
        """
        Specific union logic should go here. In subclasses, union behavior
        should be overwritten here rather than in `self.union`.

        Parameters
        ----------
        other : Index or array-like
        sort : False or None, default False
            Whether to sort the resulting index.

            * False : do not sort the result.
            * None : sort the result, except when `self` and `other` are equal
              or when the values cannot be compared.

        Returns
        -------
        Index
        """
        # TODO(EA): setops-refactor, clean all this up
        lvals = self._values
        rvals = other._values

        if (
            sort is None
            and self.is_monotonic
            and other.is_monotonic
            and not (self.has_duplicates and other.has_duplicates)
        ):
            # Both are unique and monotonic, so can use outer join
            try:
                return self._outer_indexer(lvals, rvals)[0]
            except (TypeError, IncompatibleFrequency):
                # incomparable objects
                value_list = list(lvals)

                # worth making this faster? a very unusual case
                value_set = set(lvals)
                value_list.extend([x for x in rvals if x not in value_set])
                return Index(value_list)._values  # do type inference here

        elif not other.is_unique and not self.is_unique:
            # self and other both have duplicates
            result = algos.union_with_duplicates(lvals, rvals)
            return _maybe_try_sort(result, sort)

        # Either other or self is not unique
        # find indexes of things in "other" that are not in "self"
        if self.is_unique:
            indexer = self.get_indexer(other)
            missing = (indexer == -1).nonzero()[0]
        else:
            missing = algos.unique1d(self.get_indexer_non_unique(other)[1])

        if len(missing) > 0:
            other_diff = algos.take_nd(rvals, missing, allow_fill=False)
            result = concat_compat((lvals, other_diff))
        else:
            result = lvals

        if not self.is_monotonic or not other.is_monotonic:
            result = _maybe_try_sort(result, sort)

        return result

    @final
    def _wrap_setop_result(self, other, result):
        if needs_i8_conversion(self.dtype) and isinstance(result, np.ndarray):
            self = cast("DatetimeIndexOpsMixin", self)
            result = type(self._data)._simple_new(result, dtype=self.dtype)
        elif is_categorical_dtype(self.dtype) and isinstance(result, np.ndarray):
            result = Categorical(result, dtype=self.dtype)

        name = get_op_result_name(self, other)
        if isinstance(result, Index):
            if result.name != name:
                return result.rename(name)
            return result
        else:
            return self._shallow_copy(result, name=name)

    # TODO: standardize return type of non-union setops type(self vs other)
    @final
    def intersection(self, other, sort=False):
        """
        Form the intersection of two Index objects.

        This returns a new Index with elements common to the index and `other`.

        Parameters
        ----------
        other : Index or array-like
        sort : False or None, default False
            Whether to sort the resulting index.

            * False : do not sort the result.
            * None : sort the result, except when `self` and `other` are equal
              or when the values cannot be compared.

            .. versionadded:: 0.24.0

            .. versionchanged:: 0.24.1

               Changed the default from ``True`` to ``False``, to match
               the behaviour of 0.23.4 and earlier.

        Returns
        -------
        intersection : Index

        Examples
        --------
        >>> idx1 = pd.Index([1, 2, 3, 4])
        >>> idx2 = pd.Index([3, 4, 5, 6])
        >>> idx1.intersection(idx2)
        Int64Index([3, 4], dtype='int64')
        """
        self._validate_sort_keyword(sort)
        self._assert_can_do_setop(other)
        other, result_name = self._convert_can_do_setop(other)

        if self.equals(other):
            if self.has_duplicates:
                return self.unique()._get_reconciled_name_object(other)
            return self._get_reconciled_name_object(other)

        elif not self._should_compare(other):
            # We can infer that the intersection is empty.
            if isinstance(self, ABCMultiIndex):
                return self[:0].rename(result_name)
            return Index([], name=result_name)

        elif not is_dtype_equal(self.dtype, other.dtype):
            dtype = find_common_type([self.dtype, other.dtype])
            this = self.astype(dtype, copy=False)
            other = other.astype(dtype, copy=False)
            return this.intersection(other, sort=sort)

        result = self._intersection(other, sort=sort)
        return self._wrap_setop_result(other, result)

    def _intersection(self, other: Index, sort=False):
        """
        intersection specialized to the case with matching dtypes.
        """
        # TODO(EA): setops-refactor, clean all this up
        lvals = self._values
        rvals = other._values

        if self.is_monotonic and other.is_monotonic:
            try:
                result = self._inner_indexer(lvals, rvals)[0]
            except TypeError:
                pass
            else:
                # TODO: algos.unique1d should preserve DTA/TDA
                res = algos.unique1d(result)
                return ensure_wrapped_if_datetimelike(res)

        try:
            indexer = other.get_indexer(lvals)
        except InvalidIndexError:
            # InvalidIndexError raised by get_indexer if non-unique
            indexer, _ = other.get_indexer_non_unique(lvals)

        mask = indexer != -1
        indexer = indexer.take(mask.nonzero()[0])

        result = other.take(indexer).unique()._values
        result = _maybe_try_sort(result, sort)

        # Intersection has to be unique
        assert Index(result).is_unique

        return result

    @final
    def difference(self, other, sort=None):
        """
        Return a new Index with elements of index not in `other`.

        This is the set difference of two Index objects.

        Parameters
        ----------
        other : Index or array-like
        sort : False or None, default None
            Whether to sort the resulting index. By default, the
            values are attempted to be sorted, but any TypeError from
            incomparable elements is caught by pandas.

            * None : Attempt to sort the result, but catch any TypeErrors
              from comparing incomparable elements.
            * 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
        -------
        difference : Index

        Examples
        --------
        >>> idx1 = pd.Index([2, 1, 3, 4])
        >>> idx2 = pd.Index([3, 4, 5, 6])
        >>> idx1.difference(idx2)
        Int64Index([1, 2], dtype='int64')
        >>> idx1.difference(idx2, sort=False)
        Int64Index([2, 1], dtype='int64')
        """
        self._validate_sort_keyword(sort)
        self._assert_can_do_setop(other)
        other, result_name = self._convert_can_do_setop(other)

        if self.equals(other):
            # Note: we do not (yet) sort even if sort=None GH#24959
            return self[:0].rename(result_name)

        if len(other) == 0:
            # Note: we do not (yet) sort even if sort=None GH#24959
            return self.rename(result_name)

        result = self._difference(other, sort=sort)
        return self._wrap_setop_result(other, result)

    def _difference(self, other, sort):

        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)
        the_diff = this._values.take(label_diff)
        the_diff = _maybe_try_sort(the_diff, sort)

        return the_diff

    def symmetric_difference(self, other, result_name=None, sort=None):
        """
        Compute the symmetric difference of two Index objects.

        Parameters
        ----------
        other : Index or array-like
        result_name : str
        sort : False or None, default None
            Whether to sort the resulting index. By default, the
            values are attempted to be sorted, but any TypeError from
            incomparable elements is caught by pandas.

            * None : Attempt to sort the result, but catch any TypeErrors
              from comparing incomparable elements.
            * 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
        -------
        symmetric_difference : Index

        Notes
        -----
        ``symmetric_difference`` contains elements that appear in either
        ``idx1`` or ``idx2`` but not both. Equivalent to the Index created by
        ``idx1.difference(idx2) | idx2.difference(idx1)`` with duplicates
        dropped.

        Examples
        --------
        >>> idx1 = pd.Index([1, 2, 3, 4])
        >>> idx2 = pd.Index([2, 3, 4, 5])
        >>> idx1.symmetric_difference(idx2)
        Int64Index([1, 5], dtype='int64')

        You can also use the ``^`` operator:

        >>> idx1 ^ idx2
        Int64Index([1, 5], dtype='int64')
        """
        self._validate_sort_keyword(sort)
        self._assert_can_do_setop(other)
        other, result_name_update = self._convert_can_do_setop(other)
        if result_name is None:
            result_name = result_name_update

        if not self._should_compare(other):
            return self.union(other, sort=sort).rename(result_name)
        elif not is_dtype_equal(self.dtype, other.dtype):
            dtype = find_common_type([self.dtype, other.dtype])
            this = self.astype(dtype, copy=False)
            that = other.astype(dtype, copy=False)
            return this.symmetric_difference(that, sort=sort).rename(result_name)

        this = self._get_unique_index()
        other = other._get_unique_index()
        indexer = this.get_indexer_for(other)

        # {this} minus {other}
        common_indexer = indexer.take((indexer != -1).nonzero()[0])
        left_indexer = np.setdiff1d(
            np.arange(this.size), common_indexer, assume_unique=True
        )
        left_diff = this._values.take(left_indexer)

        # {other} minus {this}
        right_indexer = (indexer == -1).nonzero()[0]
        right_diff = other._values.take(right_indexer)

        the_diff = concat_compat([left_diff, right_diff])
        the_diff = _maybe_try_sort(the_diff, sort)

        return Index(the_diff, name=result_name)

    @final
    def _assert_can_do_setop(self, other) -> bool:
        if not is_list_like(other):
            raise TypeError("Input must be Index or array-like")
        return True

    def _convert_can_do_setop(self, other) -> Tuple[Index, Hashable]:
        if not isinstance(other, Index):
            other = Index(other, name=self.name)
            result_name = self.name
        else:
            result_name = get_op_result_name(self, other)
        return other, result_name

    # --------------------------------------------------------------------
    # Indexing Methods

    def get_loc(self, key, method=None, tolerance=None):
        """
        Get integer location, slice or boolean mask for requested label.

        Parameters
        ----------
        key : label
        method : {None, 'pad'/'ffill', 'backfill'/'bfill', 'nearest'}, optional
            * default: exact matches only.
            * pad / ffill: find the PREVIOUS index value if no exact match.
            * backfill / bfill: use NEXT index value if no exact match
            * nearest: use the NEAREST index value if no exact match. Tied
              distances are broken by preferring the larger index value.
        tolerance : int or float, optional
            Maximum distance from index value for inexact matches. The value of
            the index at the matching location must satisfy the equation
            ``abs(index[loc] - key) <= tolerance``.

        Returns
        -------
        loc : int if unique index, slice if monotonic index, else mask

        Examples
        --------
        >>> unique_index = pd.Index(list('abc'))
        >>> unique_index.get_loc('b')
        1

        >>> monotonic_index = pd.Index(list('abbc'))
        >>> monotonic_index.get_loc('b')
        slice(1, 3, None)

        >>> non_monotonic_index = pd.Index(list('abcb'))
        >>> non_monotonic_index.get_loc('b')
        array([False,  True, False,  True])
        """
        if method is None:
            if tolerance is not None:
                raise ValueError(
                    "tolerance argument only valid if using pad, "
                    "backfill or nearest lookups"
                )
            casted_key = self._maybe_cast_indexer(key)
            try:
                return self._engine.get_loc(casted_key)
            except KeyError as err:
                raise KeyError(key) from err

        if is_scalar(key) and isna(key) and not self.hasnans:
            raise KeyError(key)

        if tolerance is not None:
            tolerance = self._convert_tolerance(tolerance, np.asarray(key))

        indexer = self.get_indexer([key], method=method, tolerance=tolerance)
        if indexer.ndim > 1 or indexer.size > 1:
            raise TypeError("get_loc requires scalar valued input")
        loc = indexer.item()
        if loc == -1:
            raise KeyError(key)
        return loc

    _index_shared_docs[
        "get_indexer"
    ] = """
        Compute indexer and mask for new index given the current index. The
        indexer should be then used as an input to ndarray.take to align the
        current data to the new index.

        Parameters
        ----------
        target : %(target_klass)s
        method : {None, 'pad'/'ffill', 'backfill'/'bfill', 'nearest'}, optional
            * default: exact matches only.
            * pad / ffill: find the PREVIOUS index value if no exact match.
            * backfill / bfill: use NEXT index value if no exact match
            * nearest: use the NEAREST index value if no exact match. Tied
              distances are broken by preferring the larger index value.
        limit : int, optional
            Maximum number of consecutive labels in ``target`` to match for
            inexact matches.
        tolerance : optional
            Maximum distance between original and new labels for inexact
            matches. The values of the index at the matching locations must
            satisfy the equation ``abs(index[indexer] - target) <= tolerance``.

            Tolerance may be a scalar value, which applies the same tolerance
            to all values, or list-like, which applies variable tolerance per
            element. List-like includes list, tuple, array, Series, and must be
            the same size as the index and its dtype must exactly match the
            index's type.

        Returns
        -------
        indexer : ndarray of int
            Integers from 0 to n - 1 indicating that the index at these
            positions matches the corresponding target values. Missing values
            in the target are marked by -1.
        %(raises_section)s
        Examples
        --------
        >>> index = pd.Index(['c', 'a', 'b'])
        >>> index.get_indexer(['a', 'b', 'x'])
        array([ 1,  2, -1])

        Notice that the return value is an array of locations in ``index``
        and ``x`` is marked by -1, as it is not in ``index``.
        """

    @Appender(_index_shared_docs["get_indexer"] % _index_doc_kwargs)
    @final
    def get_indexer(
        self, target, method=None, limit=None, tolerance=None
    ) -> np.ndarray:

        method = missing.clean_reindex_fill_method(method)
        target = ensure_index(target)

        self._check_indexing_method(method)

        if not self._index_as_unique:
            raise InvalidIndexError(self._requires_unique_msg)

        if not self._should_compare(target) and not is_interval_dtype(self.dtype):
            # IntervalIndex get special treatment bc numeric scalars can be
            #  matched to Interval scalars
            return self._get_indexer_non_comparable(target, method=method, unique=True)

        pself, ptarget = self._maybe_promote(target)
        if pself is not self or ptarget is not target:
            return pself.get_indexer(
                ptarget, method=method, limit=limit, tolerance=tolerance
            )

        return self._get_indexer(target, method, limit, tolerance)

    def _get_indexer(
        self, target: Index, method=None, limit=None, tolerance=None
    ) -> np.ndarray:
        if tolerance is not None:
            tolerance = self._convert_tolerance(tolerance, target)

        if not is_dtype_equal(self.dtype, target.dtype):
            dtype = self._find_common_type_compat(target)

            this = self.astype(dtype, copy=False)
            target = target.astype(dtype, copy=False)
            return this.get_indexer(
                target, method=method, limit=limit, tolerance=tolerance
            )

        if method in ["pad", "backfill"]:
            indexer = self._get_fill_indexer(target, method, limit, tolerance)
        elif method == "nearest":
            indexer = self._get_nearest_indexer(target, limit, tolerance)
        else:
            if tolerance is not None:
                raise ValueError(
                    "tolerance argument only valid if doing pad, "
                    "backfill or nearest reindexing"
                )
            if limit is not None:
                raise ValueError(
                    "limit argument only valid if doing pad, "
                    "backfill or nearest reindexing"
                )

            indexer = self._engine.get_indexer(target._get_engine_target())

        return ensure_platform_int(indexer)

    @final
    def _check_indexing_method(self, method: Optional[str_t]) -> None:
        """
        Raise if we have a get_indexer `method` that is not supported or valid.
        """
        # GH#37871 for now this is only for IntervalIndex and CategoricalIndex
        if not (is_interval_dtype(self.dtype) or is_categorical_dtype(self.dtype)):
            return

        if method is None:
            return

        if method in ["bfill", "backfill", "pad", "ffill", "nearest"]:
            raise NotImplementedError(
                f"method {method} not yet implemented for {type(self).__name__}"
            )

        raise ValueError("Invalid fill method")

    def _convert_tolerance(
        self, tolerance, target: Union[np.ndarray, Index]
    ) -> np.ndarray:
        # override this method on subclasses
        tolerance = np.asarray(tolerance)
        if target.size != tolerance.size and tolerance.size > 1:
            raise ValueError("list-like tolerance size must match target index size")
        return tolerance

    @final
    def _get_fill_indexer(
        self, target: Index, method: str_t, limit=None, tolerance=None
    ) -> np.ndarray:

        target_values = target._get_engine_target()

        if self.is_monotonic_increasing and target.is_monotonic_increasing:
            engine_method = (
                self._engine.get_pad_indexer
                if method == "pad"
                else self._engine.get_backfill_indexer
            )
            indexer = engine_method(target_values, limit)
        else:
            indexer = self._get_fill_indexer_searchsorted(target, method, limit)
        if tolerance is not None and len(self):
            indexer = self._filter_indexer_tolerance(target_values, indexer, tolerance)
        return indexer

    @final
    def _get_fill_indexer_searchsorted(
        self, target: Index, method: str_t, limit=None
    ) -> np.ndarray:
        """
        Fallback pad/backfill get_indexer that works for monotonic decreasing
        indexes and non-monotonic targets.
        """
        if limit is not None:
            raise ValueError(
                f"limit argument for {repr(method)} method only well-defined "
                "if index and target are monotonic"
            )

        side = "left" if method == "pad" else "right"

        # find exact matches first (this simplifies the algorithm)
        indexer = self.get_indexer(target)
        nonexact = indexer == -1
        indexer[nonexact] = self._searchsorted_monotonic(target[nonexact], side)
        if side == "left":
            # searchsorted returns "indices into a sorted array such that,
            # if the corresponding elements in v were inserted before the
            # indices, the order of a would be preserved".
            # Thus, we need to subtract 1 to find values to the left.
            indexer[nonexact] -= 1
            # This also mapped not found values (values of 0 from
            # np.searchsorted) to -1, which conveniently is also our
            # sentinel for missing values
        else:
            # Mark indices to the right of the largest value as not found
            indexer[indexer == len(self)] = -1
        return indexer

    @final
    def _get_nearest_indexer(self, target: Index, limit, tolerance) -> np.ndarray:
        """
        Get the indexer for the nearest index labels; requires an index with
        values that can be subtracted from each other (e.g., not strings or
        tuples).
        """
        if not len(self):
            return self._get_fill_indexer(target, "pad")

        left_indexer = self.get_indexer(target, "pad", limit=limit)
        right_indexer = self.get_indexer(target, "backfill", limit=limit)

        target_values = target._get_engine_target()
        own_values = self._get_engine_target()
        left_distances = np.abs(own_values[left_indexer] - target_values)
        right_distances = np.abs(own_values[right_indexer] - target_values)

        op = operator.lt if self.is_monotonic_increasing else operator.le
        indexer = np.where(
            op(left_distances, right_distances) | (right_indexer == -1),
            left_indexer,
            right_indexer,
        )
        if tolerance is not None:
            indexer = self._filter_indexer_tolerance(target_values, indexer, tolerance)
        return indexer

    @final
    def _filter_indexer_tolerance(
        self,
        target: Union[Index, np.ndarray, ExtensionArray],
        indexer: np.ndarray,
        tolerance,
    ) -> np.ndarray:
        own_values = self._get_engine_target()
        distance = abs(own_values[indexer] - target)
        return np.where(distance <= tolerance, indexer, -1)

    # --------------------------------------------------------------------
    # Indexer Conversion Methods

    def _get_partial_string_timestamp_match_key(self, key: T) -> T:
        """
        Translate any partial string timestamp matches in key, returning the
        new key.

        Only relevant for MultiIndex.
        """
        # GH#10331
        return key

    @final
    def _validate_positional_slice(self, key: slice) -> None:
        """
        For positional indexing, a slice must have either int or None
        for each of start, stop, and step.
        """
        self._validate_indexer("positional", key.start, "iloc")
        self._validate_indexer("positional", key.stop, "iloc")
        self._validate_indexer("positional", key.step, "iloc")

    def _convert_slice_indexer(self, key: slice, kind: str_t):
        """
        Convert a slice indexer.

        By definition, these are labels unless 'iloc' is passed in.
        Floats are not allowed as the start, step, or stop of the slice.

        Parameters
        ----------
        key : label of the slice bound
        kind : {'loc', 'getitem'}
        """
        assert kind in ["loc", "getitem"], kind

        # potentially cast the bounds to integers
        start, stop, step = key.start, key.stop, key.step

        # figure out if this is a positional indexer
        def is_int(v):
            return v is None or is_integer(v)

        is_index_slice = is_int(start) and is_int(stop) and is_int(step)
        is_positional = is_index_slice and not (
            self.is_integer() or self.is_categorical()
        )

        if kind == "getitem":
            """
            called from the getitem slicers, validate that we are in fact
            integers
            """
            if self.is_integer() or is_index_slice:
                self._validate_indexer("slice", key.start, "getitem")
                self._validate_indexer("slice", key.stop, "getitem")
                self._validate_indexer("slice", key.step, "getitem")
                return key

        # convert the slice to an indexer here

        # if we are mixed and have integers
        if is_positional:
            try:
                # Validate start & stop
                if start is not None:
                    self.get_loc(start)
                if stop is not None:
                    self.get_loc(stop)
                is_positional = False
            except KeyError:
                pass

        if com.is_null_slice(key):
            # It doesn't matter if we are positional or label based
            indexer = key
        elif is_positional:
            if kind == "loc":
                # GH#16121, GH#24612, GH#31810
                warnings.warn(
                    "Slicing a positional slice with .loc is not supported, "
                    "and will raise TypeError in a future version.  "
                    "Use .loc with labels or .iloc with positions instead.",
                    FutureWarning,
                    stacklevel=6,
                )
            indexer = key
        else:
            indexer = self.slice_indexer(start, stop, step, kind=kind)

        return indexer

    def _convert_listlike_indexer(self, keyarr):
        """
        Parameters
        ----------
        keyarr : list-like
            Indexer to convert.

        Returns
        -------
        indexer : numpy.ndarray or None
            Return an ndarray or None if cannot convert.
        keyarr : numpy.ndarray
            Return tuple-safe keys.
        """
        if isinstance(keyarr, Index):
            pass
        else:
            keyarr = self._convert_arr_indexer(keyarr)

        indexer = self._convert_list_indexer(keyarr)
        return indexer, keyarr

    def _convert_arr_indexer(self, keyarr) -> np.ndarray:
        """
        Convert an array-like indexer to the appropriate dtype.

        Parameters
        ----------
        keyarr : array-like
            Indexer to convert.

        Returns
        -------
        converted_keyarr : array-like
        """
        return com.asarray_tuplesafe(keyarr)

    def _convert_list_indexer(self, keyarr):
        """
        Convert a list-like indexer to the appropriate dtype.

        Parameters
        ----------
        keyarr : Index (or sub-class)
            Indexer to convert.
        kind : iloc, loc, optional

        Returns
        -------
        positional indexer or None
        """
        return None

    @final
    def _invalid_indexer(self, form: str_t, key) -> TypeError:
        """
        Consistent invalid indexer message.
        """
        return TypeError(
            f"cannot do {form} indexing on {type(self).__name__} with these "
            f"indexers [{key}] of type {type(key).__name__}"
        )

    # --------------------------------------------------------------------
    # Reindex Methods

    @final
    def _validate_can_reindex(self, indexer: np.ndarray) -> None:
        """
        Check if we are allowing reindexing with this particular indexer.

        Parameters
        ----------
        indexer : an integer ndarray

        Raises
        ------
        ValueError if its a duplicate axis
        """
        # trying to reindex on an axis with duplicates
        if not self._index_as_unique and len(indexer):
            raise ValueError("cannot reindex from a duplicate axis")

    def reindex(self, target, method=None, level=None, limit=None, tolerance=None):
        """
        Create index with target's values.

        Parameters
        ----------
        target : an iterable

        Returns
        -------
        new_index : pd.Index
            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, "name")

        # GH7774: preserve dtype/tz if target is empty and not an Index.
        target = ensure_has_len(target)  # target may be an iterator

        if not isinstance(target, Index) and len(target) == 0:
            target = self[:0]
        else:
            target = ensure_index(target)

        if level is not None:
            if method is not None:
                raise TypeError("Fill method not supported if level passed")
            _, indexer, _ = self._join_level(
                target, level, how="right", return_indexers=True
            )
        else:
            if self.equals(target):
                indexer = None
            else:
                if self._index_as_unique:
                    indexer = self.get_indexer(
                        target, method=method, limit=limit, tolerance=tolerance
                    )
                else:
                    if method is not None or limit is not None:
                        raise ValueError(
                            "cannot reindex a non-unique index "
                            "with a method or limit"
                        )
                    indexer, _ = self.get_indexer_non_unique(target)

        if preserve_names and target.nlevels == 1 and target.name != self.name:
            target = target.copy()
            target.name = self.name

        return target, indexer

    def _reindex_non_unique(self, target):
        """
        Create a new index with target's values (move/add/delete values as
        necessary) use with non-unique Index and a possibly non-unique target.

        Parameters
        ----------
        target : an iterable

        Returns
        -------
        new_index : pd.Index
            Resulting index.
        indexer : np.ndarray or None
            Indices of output values in original index.

        """
        target = ensure_index(target)
        if len(target) == 0:
            # GH#13691
            return self[:0], np.array([], dtype=np.intp), None

        indexer, missing = self.get_indexer_non_unique(target)
        check = indexer != -1
        new_labels = self.take(indexer[check])
        new_indexer = None

        if len(missing):
            length = np.arange(len(indexer))

            missing = ensure_platform_int(missing)
            missing_labels = target.take(missing)
            missing_indexer = ensure_int64(length[~check])
            cur_labels = self.take(indexer[check]).values
            cur_indexer = ensure_int64(length[check])

            new_labels = np.empty((len(indexer),), dtype=object)
            new_labels[cur_indexer] = cur_labels
            new_labels[missing_indexer] = missing_labels

            # GH#38906
            if not len(self):

                new_indexer = np.arange(0)

            # a unique indexer
            elif target.is_unique:

                # see GH5553, make sure we use the right indexer
                new_indexer = np.arange(len(indexer))
                new_indexer[cur_indexer] = np.arange(len(cur_labels))
                new_indexer[missing_indexer] = -1

            # we have a non_unique selector, need to use the original
            # indexer here
            else:

                # need to retake to have the same size as the indexer
                indexer[~check] = -1

                # reset the new indexer to account for the new size
                new_indexer = np.arange(len(self.take(indexer)))
                new_indexer[~check] = -1

        if isinstance(self, ABCMultiIndex):
            new_index = type(self).from_tuples(new_labels, names=self.names)
        else:
            new_index = Index(new_labels, name=self.name)
        return new_index, indexer, new_indexer

    # --------------------------------------------------------------------
    # Join Methods

    def join(self, other, how="left", level=None, return_indexers=False, sort=False):
        """
        Compute join_index and indexers to conform data
        structures to the new index.

        Parameters
        ----------
        other : Index
        how : {'left', 'right', 'inner', 'outer'}
        level : int or level name, default None
        return_indexers : bool, default False
        sort : bool, default False
            Sort the join keys lexicographically in the result Index. If False,
            the order of the join keys depends on the join type (how keyword).

        Returns
        -------
        join_index, (left_indexer, right_indexer)
        """
        other = ensure_index(other)
        self_is_mi = isinstance(self, ABCMultiIndex)
        other_is_mi = isinstance(other, ABCMultiIndex)

        # try to figure out the join level
        # GH3662
        if level is None and (self_is_mi or other_is_mi):

            # have the same levels/names so a simple join
            if self.names == other.names:
                pass
            else:
                return self._join_multi(other, how=how, return_indexers=return_indexers)

        # join on the level
        if level is not None and (self_is_mi or other_is_mi):
            return self._join_level(
                other, level, how=how, return_indexers=return_indexers
            )

        if len(other) == 0 and how in ("left", "outer"):
            join_index = self._view()
            if return_indexers:
                rindexer = np.repeat(-1, len(join_index))
                return join_index, None, rindexer
            else:
                return join_index

        if len(self) == 0 and how in ("right", "outer"):
            join_index = other._view()
            if return_indexers:
                lindexer = np.repeat(-1, len(join_index))
                return join_index, lindexer, None
            else:
                return join_index

        if self._join_precedence < other._join_precedence:
            how = {"right": "left", "left": "right"}.get(how, how)
            result = other.join(
                self, how=how, level=level, return_indexers=return_indexers
            )
            if return_indexers:
                x, y, z = result
                result = x, z, y
            return result

        if not is_dtype_equal(self.dtype, other.dtype):
            this = self.astype("O")
            other = other.astype("O")
            return this.join(other, how=how, return_indexers=return_indexers)

        _validate_join_method(how)

        if not self.is_unique and not other.is_unique:
            return self._join_non_unique(
                other, how=how, return_indexers=return_indexers
            )
        elif not self.is_unique or not other.is_unique:
            if self.is_monotonic and other.is_monotonic:
                return self._join_monotonic(
                    other, how=how, return_indexers=return_indexers
                )
            else:
                return self._join_non_unique(
                    other, how=how, return_indexers=return_indexers
                )
        elif (
            self.is_monotonic
            and other.is_monotonic
            and (
                not isinstance(self, ABCMultiIndex)
                or not any(is_categorical_dtype(dtype) for dtype in self.dtypes)
            )
        ):
            # Categorical is monotonic if data are ordered as categories, but join can
            #  not handle this in case of not lexicographically monotonic GH#38502
            try:
                return self._join_monotonic(
                    other, how=how, return_indexers=return_indexers
                )
            except TypeError:
                pass

        if how == "left":
            join_index = self
        elif how == "right":
            join_index = other
        elif how == "inner":
            # TODO: sort=False here for backwards compat. It may
            # be better to use the sort parameter passed into join
            join_index = self.intersection(other, sort=False)
        elif how == "outer":
            # TODO: sort=True here for backwards compat. It may
            # be better to use the sort parameter passed into join
            join_index = self.union(other)

        if sort:
            join_index = join_index.sort_values()

        if return_indexers:
            if join_index is self:
                lindexer = None
            else:
                lindexer = self.get_indexer(join_index)
            if join_index is other:
                rindexer = None
            else:
                rindexer = other.get_indexer(join_index)
            return join_index, lindexer, rindexer
        else:
            return join_index

    @final
    def _join_multi(self, other, how, return_indexers=True):
        from pandas.core.indexes.multi import MultiIndex
        from pandas.core.reshape.merge import restore_dropped_levels_multijoin

        # figure out join names
        self_names_list = list(com.not_none(*self.names))
        other_names_list = list(com.not_none(*other.names))
        self_names_order = self_names_list.index
        other_names_order = other_names_list.index
        self_names = set(self_names_list)
        other_names = set(other_names_list)
        overlap = self_names & other_names

        # need at least 1 in common
        if not overlap:
            raise ValueError("cannot join with no overlapping index names")

        if isinstance(self, MultiIndex) and isinstance(other, MultiIndex):

            # Drop the non-matching levels from left and right respectively
            ldrop_names = sorted(self_names - overlap, key=self_names_order)
            rdrop_names = sorted(other_names - overlap, key=other_names_order)

            # if only the order differs
            if not len(ldrop_names + rdrop_names):
                self_jnlevels = self
                other_jnlevels = other.reorder_levels(self.names)
            else:
                self_jnlevels = self.droplevel(ldrop_names)
                other_jnlevels = other.droplevel(rdrop_names)

            # Join left and right
            # Join on same leveled multi-index frames is supported
            join_idx, lidx, ridx = self_jnlevels.join(
                other_jnlevels, how, return_indexers=True
            )

            # Restore the dropped levels
            # Returned index level order is
            # common levels, ldrop_names, rdrop_names
            dropped_names = ldrop_names + rdrop_names

            levels, codes, names = restore_dropped_levels_multijoin(
                self, other, dropped_names, join_idx, lidx, ridx
            )

            # Re-create the multi-index
            multi_join_idx = MultiIndex(
                levels=levels, codes=codes, names=names, verify_integrity=False
            )

            multi_join_idx = multi_join_idx.remove_unused_levels()

            if return_indexers:
                return multi_join_idx, lidx, ridx
            else:
                return multi_join_idx

        jl = list(overlap)[0]

        # Case where only one index is multi
        # make the indices into mi's that match
        flip_order = False
        if isinstance(self, MultiIndex):
            self, other = other, self
            flip_order = True
            # flip if join method is right or left
            how = {"right": "left", "left": "right"}.get(how, how)

        level = other.names.index(jl)
        result = self._join_level(
            other, level, how=how, return_indexers=return_indexers
        )

        if flip_order and isinstance(result, tuple):
            return result[0], result[2], result[1]
        return result

    @final
    def _join_non_unique(self, other, how="left", return_indexers=False):
        from pandas.core.reshape.merge import get_join_indexers

        # We only get here if dtypes match
        assert self.dtype == other.dtype

        lvalues = self._get_engine_target()
        rvalues = other._get_engine_target()

        left_idx, right_idx = get_join_indexers(
            [lvalues], [rvalues], how=how, sort=True
        )

        left_idx = ensure_platform_int(left_idx)
        right_idx = ensure_platform_int(right_idx)

        join_index = np.asarray(lvalues.take(left_idx))
        mask = left_idx == -1
        np.putmask(join_index, mask, rvalues.take(right_idx))

        join_index = self._wrap_joined_index(join_index, other)

        if return_indexers:
            return join_index, left_idx, right_idx
        else:
            return join_index

    @final
    def _join_level(
        self, other, level, how="left", return_indexers=False, keep_order=True
    ):
        """
        The join method *only* affects the level of the resulting
        MultiIndex. Otherwise it just exactly aligns the Index data to the
        labels of the level in the MultiIndex.

        If ```keep_order == True```, the order of the data indexed by the
        MultiIndex will not be changed; otherwise, it will tie out
        with `other`.
        """
        from pandas.core.indexes.multi import MultiIndex

        def _get_leaf_sorter(labels):
            """
            Returns sorter for the inner most level while preserving the
            order of higher levels.
            """
            if labels[0].size == 0:
                return np.empty(0, dtype="int64")

            if len(labels) == 1:
                return get_group_index_sorter(labels[0])

            # find indexers of beginning of each set of
            # same-key labels w.r.t all but last level
            tic = labels[0][:-1] != labels[0][1:]
            for lab in labels[1:-1]:
                tic |= lab[:-1] != lab[1:]

            starts = np.hstack(([True], tic, [True])).nonzero()[0]
            lab = ensure_int64(labels[-1])
            return lib.get_level_sorter(lab, ensure_int64(starts))

        if isinstance(self, MultiIndex) and isinstance(other, MultiIndex):
            raise TypeError("Join on level between two MultiIndex objects is ambiguous")

        left, right = self, other

        flip_order = not isinstance(self, MultiIndex)
        if flip_order:
            left, right = right, left
            how = {"right": "left", "left": "right"}.get(how, how)

        assert isinstance(left, MultiIndex)

        level = left._get_level_number(level)
        old_level = left.levels[level]

        if not right.is_unique:
            raise NotImplementedError(
                "Index._join_level on non-unique index is not implemented"
            )

        new_level, left_lev_indexer, right_lev_indexer = old_level.join(
            right, how=how, return_indexers=True
        )

        if left_lev_indexer is None:
            if keep_order or len(left) == 0:
                left_indexer = None
                join_index = left
            else:  # sort the leaves
                left_indexer = _get_leaf_sorter(left.codes[: level + 1])
                join_index = left[left_indexer]

        else:
            left_lev_indexer = ensure_int64(left_lev_indexer)
            rev_indexer = lib.get_reverse_indexer(left_lev_indexer, len(old_level))
            old_codes = left.codes[level]
            new_lev_codes = algos.take_nd(
                rev_indexer, old_codes[old_codes != -1], allow_fill=False
            )

            new_codes = list(left.codes)
            new_codes[level] = new_lev_codes

            new_levels = list(left.levels)
            new_levels[level] = new_level

            if keep_order:  # just drop missing values. o.w. keep order
                left_indexer = np.arange(len(left), dtype=np.intp)
                mask = new_lev_codes != -1
                if not mask.all():
                    new_codes = [lab[mask] for lab in new_codes]
                    left_indexer = left_indexer[mask]

            else:  # tie out the order with other
                if level == 0:  # outer most level, take the fast route
                    ngroups = 1 + new_lev_codes.max()
                    left_indexer, counts = libalgos.groupsort_indexer(
                        new_lev_codes, ngroups
                    )

                    # missing values are placed first; drop them!
                    left_indexer = left_indexer[counts[0] :]
                    new_codes = [lab[left_indexer] for lab in new_codes]

                else:  # sort the leaves
                    mask = new_lev_codes != -1
                    mask_all = mask.all()
                    if not mask_all:
                        new_codes = [lab[mask] for lab in new_codes]

                    left_indexer = _get_leaf_sorter(new_codes[: level + 1])
                    new_codes = [lab[left_indexer] for lab in new_codes]

                    # left_indexers are w.r.t masked frame.
                    # reverse to original frame!
                    if not mask_all:
                        left_indexer = mask.nonzero()[0][left_indexer]

            join_index = MultiIndex(
                levels=new_levels,
                codes=new_codes,
                names=left.names,
                verify_integrity=False,
            )

        if right_lev_indexer is not None:
            right_indexer = algos.take_nd(
                right_lev_indexer, join_index.codes[level], allow_fill=False
            )
        else:
            right_indexer = join_index.codes[level]

        if flip_order:
            left_indexer, right_indexer = right_indexer, left_indexer

        if return_indexers:
            left_indexer = (
                None if left_indexer is None else ensure_platform_int(left_indexer)
            )
            right_indexer = (
                None if right_indexer is None else ensure_platform_int(right_indexer)
            )
            return join_index, left_indexer, right_indexer
        else:
            return join_index

    @final
    def _join_monotonic(self, other, how="left", return_indexers=False):
        # We only get here with matching dtypes
        assert other.dtype == self.dtype

        if self.equals(other):
            ret_index = other if how == "right" else self
            if return_indexers:
                return ret_index, None, None
            else:
                return ret_index

        sv = self._get_engine_target()
        ov = other._get_engine_target()

        if self.is_unique and other.is_unique:
            # We can perform much better than the general case
            if how == "left":
                join_index = self
                lidx = None
                ridx = self._left_indexer_unique(sv, ov)
            elif how == "right":
                join_index = other
                lidx = self._left_indexer_unique(ov, sv)
                ridx = None
            elif how == "inner":
                join_index, lidx, ridx = self._inner_indexer(sv, ov)
                join_index = self._wrap_joined_index(join_index, other)
            elif how == "outer":
                join_index, lidx, ridx = self._outer_indexer(sv, ov)
                join_index = self._wrap_joined_index(join_index, other)
        else:
            if how == "left":
                join_index, lidx, ridx = self._left_indexer(sv, ov)
            elif how == "right":
                join_index, ridx, lidx = self._left_indexer(ov, sv)
            elif how == "inner":
                join_index, lidx, ridx = self._inner_indexer(sv, ov)
            elif how == "outer":
                join_index, lidx, ridx = self._outer_indexer(sv, ov)
            join_index = self._wrap_joined_index(join_index, other)

        if return_indexers:
            lidx = None if lidx is None else ensure_platform_int(lidx)
            ridx = None if ridx is None else ensure_platform_int(ridx)
            return join_index, lidx, ridx
        else:
            return join_index

    def _wrap_joined_index(
        self: _IndexT, joined: np.ndarray, other: _IndexT
    ) -> _IndexT:
        assert other.dtype == self.dtype

        if isinstance(self, ABCMultiIndex):
            name = self.names if self.names == other.names else None
        else:
            name = get_op_result_name(self, other)
        return self._constructor(joined, name=name)

    # --------------------------------------------------------------------
    # Uncategorized Methods

    @property
    def values(self) -> ArrayLike:
        """
        Return an array representing the data in the Index.

        .. warning::

           We recommend using :attr:`Index.array` or
           :meth:`Index.to_numpy`, depending on whether you need
           a reference to the underlying data or a NumPy array.

        Returns
        -------
        array: numpy.ndarray or ExtensionArray

        See Also
        --------
        Index.array : Reference to the underlying data.
        Index.to_numpy : A NumPy array representing the underlying data.
        """
        return self._data

    @cache_readonly
    @doc(IndexOpsMixin.array)
    def array(self) -> ExtensionArray:
        array = self._data
        if isinstance(array, np.ndarray):
            from pandas.core.arrays.numpy_ import PandasArray

            array = PandasArray(array)
        return array

    @property
    def _values(self) -> Union[ExtensionArray, np.ndarray]:
        """
        The best array representation.

        This is an ndarray or ExtensionArray.

        ``_values`` are consistent between ``Series`` and ``Index``.

        It may differ from the public '.values' method.

        index             | values          | _values       |
        ----------------- | --------------- | ------------- |
        Index             | ndarray         | ndarray       |
        CategoricalIndex  | Categorical     | Categorical   |
        DatetimeIndex     | ndarray[M8ns]   | DatetimeArray |
        DatetimeIndex[tz] | ndarray[M8ns]   | DatetimeArray |
        PeriodIndex       | ndarray[object] | PeriodArray   |
        IntervalIndex     | IntervalArray   | IntervalArray |

        See Also
        --------
        values : Values
        """
        return self._data

    def _get_engine_target(self) -> np.ndarray:
        """
        Get the ndarray that we can pass to the IndexEngine constructor.
        """
        return self._values

    @doc(IndexOpsMixin.memory_usage)
    def memory_usage(self, deep: bool = False) -> int:
        result = super().memory_usage(deep=deep)

        # include our engine hashtable
        result += self._engine.sizeof(deep=deep)
        return result

    @final
    def where(self, cond, other=None):
        """
        Replace values where the condition is False.

        The replacement is taken from other.

        Parameters
        ----------
        cond : bool array-like with the same length as self
            Condition to select the values on.
        other : scalar, or array-like, default None
            Replacement if the condition is False.

        Returns
        -------
        pandas.Index
            A copy of self with values replaced from other
            where the condition is False.

        See Also
        --------
        Series.where : Same method for Series.
        DataFrame.where : Same method for DataFrame.

        Examples
        --------
        >>> idx = pd.Index(['car', 'bike', 'train', 'tractor'])
        >>> idx
        Index(['car', 'bike', 'train', 'tractor'], dtype='object')
        >>> idx.where(idx.isin(['car', 'train']), 'other')
        Index(['car', 'other', 'train', 'other'], dtype='object')
        """
        if isinstance(self, ABCMultiIndex):
            raise NotImplementedError(
                ".where is not supported for MultiIndex operations"
            )
        cond = np.asarray(cond, dtype=bool)
        return self.putmask(~cond, other)

    # construction helpers
    @final
    @classmethod
    def _scalar_data_error(cls, data):
        # We return the TypeError so that we can raise it from the constructor
        #  in order to keep mypy happy
        return TypeError(
            f"{cls.__name__}(...) must be called with a collection of some "
            f"kind, {repr(data)} was passed"
        )

    @final
    @classmethod
    def _string_data_error(cls, data):
        raise TypeError(
            "String dtype not supported, you may need "
            "to explicitly cast to a numeric type"
        )

    def _validate_fill_value(self, value):
        """
        Check if the value can be inserted into our array without casting,
        and convert it to an appropriate native type if necessary.

        Raises
        ------
        TypeError
            If the value cannot be inserted into an array of this dtype.
        """
        if not can_hold_element(self.dtype, value):
            raise TypeError
        return value

    @final
    def _require_scalar(self, value):
        """
        Check that this is a scalar value that we can use for setitem-like
        operations without changing dtype.
        """
        if not is_scalar(value):
            raise TypeError(f"'value' must be a scalar, passed: {type(value).__name__}")
        return value

    @property
    def _has_complex_internals(self) -> bool:
        """
        Indicates if an index is not directly backed by a numpy array
        """
        # used to avoid libreduction code paths, which raise or require conversion
        return False

    def _is_memory_usage_qualified(self) -> bool:
        """
        Return a boolean if we need a qualified .info display.
        """
        return self.is_object()

    def is_type_compatible(self, kind: str_t) -> bool:
        """
        Whether the index type is compatible with the provided type.
        """
        return kind == self.inferred_type

    def __contains__(self, key: Any) -> bool:
        """
        Return a boolean indicating whether the provided key is in the index.

        Parameters
        ----------
        key : label
            The key to check if it is present in the index.

        Returns
        -------
        bool
            Whether the key search is in the index.

        Raises
        ------
        TypeError
            If the key is not hashable.

        See Also
        --------
        Index.isin : Returns an ndarray of boolean dtype indicating whether the
            list-like key is in the index.

        Examples
        --------
        >>> idx = pd.Index([1, 2, 3, 4])
        >>> idx
        Int64Index([1, 2, 3, 4], dtype='int64')

        >>> 2 in idx
        True
        >>> 6 in idx
        False
        """
        hash(key)
        try:
            return key in self._engine
        except (OverflowError, TypeError, ValueError):
            return False

    @final
    def __hash__(self):
        raise TypeError(f"unhashable type: {repr(type(self).__name__)}")

    @final
    def __setitem__(self, key, value):
        raise TypeError("Index does not support mutable operations")

    def __getitem__(self, key):
        """
        Override numpy.ndarray's __getitem__ method to work as desired.

        This function adds lists and Series as valid boolean indexers
        (ndarrays only supports ndarray with dtype=bool).

        If resulting ndim != 1, plain ndarray is returned instead of
        corresponding `Index` subclass.

        """
        # There's no custom logic to be implemented in __getslice__, so it's
        # not overloaded intentionally.
        getitem = self._data.__getitem__

        if is_scalar(key):
            key = com.cast_scalar_indexer(key, warn_float=True)
            return getitem(key)

        if isinstance(key, slice):
            # This case is separated from the conditional above to avoid
            # pessimization of basic indexing.
            result = getitem(key)
            # Going through simple_new for performance.
            return type(self)._simple_new(result, name=self.name)

        if com.is_bool_indexer(key):
            key = np.asarray(key, dtype=bool)

        result = getitem(key)
        if not is_scalar(result):
            if np.ndim(result) > 1:
                deprecate_ndim_indexing(result)
                return result
            # NB: Using _constructor._simple_new would break if MultiIndex
            #  didn't override __getitem__
            return self._constructor._simple_new(result, name=self.name)
        else:
            return result

    @final
    def _can_hold_identifiers_and_holds_name(self, name) -> bool:
        """
        Faster check for ``name in self`` when we know `name` is a Python
        identifier (e.g. in NDFrame.__getattr__, which hits this to support
        . key lookup). For indexes that can't hold identifiers (everything
        but object & categorical) we just return False.

        https://github.com/pandas-dev/pandas/issues/19764
        """
        if self.is_object() or self.is_categorical():
            return name in self
        return False

    def append(self, other):
        """
        Append a collection of Index options together.

        Parameters
        ----------
        other : Index or list/tuple of indices

        Returns
        -------
        appended : Index
        """
        to_concat = [self]

        if isinstance(other, (list, tuple)):
            to_concat += list(other)
        else:
            to_concat.append(other)

        for obj in to_concat:
            if not isinstance(obj, Index):
                raise TypeError("all inputs must be Index")

        names = {obj.name for obj in to_concat}
        name = None if len(names) > 1 else self.name

        return self._concat(to_concat, name)

    def _concat(self, to_concat: List[Index], name: Hashable) -> Index:
        """
        Concatenate multiple Index objects.
        """
        to_concat_vals = [x._values for x in to_concat]

        result = concat_compat(to_concat_vals)
        return Index(result, name=name)

    def putmask(self, mask, value):
        """
        Return a new Index of the values set with the mask.

        Returns
        -------
        Index

        See Also
        --------
        numpy.ndarray.putmask : Changes elements of an array
            based on conditional and input values.
        """
        mask, noop = validate_putmask(self._values, mask)
        if noop:
            return self.copy()

        if value is None and (self._is_numeric_dtype or self.dtype == object):
            value = self._na_value
        try:
            converted = self._validate_fill_value(value)
        except (ValueError, TypeError) as err:
            if is_object_dtype(self):
                raise err

            dtype = self._find_common_type_compat(value)
            return self.astype(dtype).putmask(mask, value)

        values = self._values.copy()
        converted = setitem_datetimelike_compat(values, mask.sum(), converted)
        np.putmask(values, mask, converted)

        return type(self)._simple_new(values, name=self.name)

    def equals(self, other: Any) -> bool:
        """
        Determine if two Index object are equal.

        The things that are being compared are:

        * The elements inside the Index object.
        * The order of the elements inside the Index object.

        Parameters
        ----------
        other : Any
            The other object to compare against.

        Returns
        -------
        bool
            True if "other" is an Index and it has the same elements and order
            as the calling index; False otherwise.

        Examples
        --------
        >>> idx1 = pd.Index([1, 2, 3])
        >>> idx1
        Int64Index([1, 2, 3], dtype='int64')
        >>> idx1.equals(pd.Index([1, 2, 3]))
        True

        The elements inside are compared

        >>> idx2 = pd.Index(["1", "2", "3"])
        >>> idx2
        Index(['1', '2', '3'], dtype='object')

        >>> idx1.equals(idx2)
        False

        The order is compared

        >>> ascending_idx = pd.Index([1, 2, 3])
        >>> ascending_idx
        Int64Index([1, 2, 3], dtype='int64')
        >>> descending_idx = pd.Index([3, 2, 1])
        >>> descending_idx
        Int64Index([3, 2, 1], dtype='int64')
        >>> ascending_idx.equals(descending_idx)
        False

        The dtype is *not* compared

        >>> int64_idx = pd.Int64Index([1, 2, 3])
        >>> int64_idx
        Int64Index([1, 2, 3], dtype='int64')
        >>> uint64_idx = pd.UInt64Index([1, 2, 3])
        >>> uint64_idx
        UInt64Index([1, 2, 3], dtype='uint64')
        >>> int64_idx.equals(uint64_idx)
        True
        """
        if self.is_(other):
            return True

        if not isinstance(other, Index):
            return False

        if is_object_dtype(self.dtype) and not is_object_dtype(other.dtype):
            # if other is not object, use other's logic for coercion
            return other.equals(self)

        if isinstance(other, ABCMultiIndex):
            # d-level MultiIndex can equal d-tuple Index
            return other.equals(self)

        if is_extension_array_dtype(other.dtype):
            # All EA-backed Index subclasses override equals
            return other.equals(self)

        return array_equivalent(self._values, other._values)

    @final
    def identical(self, other) -> bool:
        """
        Similar to equals, but checks that object attributes and types are also equal.

        Returns
        -------
        bool
            If two Index objects have equal elements and same type True,
            otherwise False.
        """
        return (
            self.equals(other)
            and all(
                getattr(self, c, None) == getattr(other, c, None)
                for c in self._comparables
            )
            and type(self) == type(other)
        )

    @final
    def asof(self, label):
        """
        Return the label from the index, or, if not present, the previous one.

        Assuming that the index is sorted, return the passed index label if it
        is in the index, or return the previous index label if the passed one
        is not in the index.

        Parameters
        ----------
        label : object
            The label up to which the method returns the latest index label.

        Returns
        -------
        object
            The passed label if it is in the index. The previous label if the
            passed label is not in the sorted index or `NaN` if there is no
            such label.

        See Also
        --------
        Series.asof : Return the latest value in a Series up to the
            passed index.
        merge_asof : Perform an asof merge (similar to left join but it
            matches on nearest key rather than equal key).
        Index.get_loc : An `asof` is a thin wrapper around `get_loc`
            with method='pad'.

        Examples
        --------
        `Index.asof` returns the latest index label up to the passed label.

        >>> idx = pd.Index(['2013-12-31', '2014-01-02', '2014-01-03'])
        >>> idx.asof('2014-01-01')
        '2013-12-31'

        If the label is in the index, the method returns the passed label.

        >>> idx.asof('2014-01-02')
        '2014-01-02'

        If all of the labels in the index are later than the passed label,
        NaN is returned.

        >>> idx.asof('1999-01-02')
        nan

        If the index is not sorted, an error is raised.

        >>> idx_not_sorted = pd.Index(['2013-12-31', '2015-01-02',
        ...                            '2014-01-03'])
        >>> idx_not_sorted.asof('2013-12-31')
        Traceback (most recent call last):
        ValueError: index must be monotonic increasing or decreasing
        """
        try:
            loc = self.get_loc(label, method="pad")
        except KeyError:
            return self._na_value
        else:
            if isinstance(loc, slice):
                loc = loc.indices(len(self))[-1]
            return self[loc]

    def asof_locs(self, where: Index, mask) -> np.ndarray:
        """
        Return the locations (indices) of labels in the index.

        As in the `asof` function, if the label (a particular entry in
        `where`) is not in the index, the latest index label up to the
        passed label is chosen and its index returned.

        If all of the labels in the index are later than a label in `where`,
        -1 is returned.

        `mask` is used to ignore NA values in the index during calculation.

        Parameters
        ----------
        where : Index
            An Index consisting of an array of timestamps.
        mask : array-like
            Array of booleans denoting where values in the original
            data are not NA.

        Returns
        -------
        numpy.ndarray
            An array of locations (indices) of the labels from the Index
            which correspond to the return values of the `asof` function
            for every element in `where`.
        """
        locs = self._values[mask].searchsorted(where._values, side="right")
        locs = np.where(locs > 0, locs - 1, 0)

        result = np.arange(len(self))[mask].take(locs)

        # TODO: overload return type of ExtensionArray.__getitem__
        first_value = cast(Any, self._values[mask.argmax()])
        result[(locs == 0) & (where._values < first_value)] = -1

        return result

    @final
    def sort_values(
        self,
        return_indexer: bool = False,
        ascending: bool = True,
        na_position: str_t = "last",
        key: Optional[Callable] = None,
    ):
        """
        Return a sorted copy of the index.

        Return a sorted copy of the index, and optionally return the indices
        that sorted the index itself.

        Parameters
        ----------
        return_indexer : bool, default False
            Should the indices that would sort the index be returned.
        ascending : bool, default True
            Should the index values be sorted in an ascending order.
        na_position : {'first' or 'last'}, default 'last'
            Argument 'first' puts NaNs at the beginning, 'last' puts NaNs at
            the end.

            .. versionadded:: 1.2.0

        key : callable, optional
            If not None, apply the key function to the index values
            before sorting. This is similar to the `key` argument in the
            builtin :meth:`sorted` function, with the notable difference that
            this `key` function should be *vectorized*. It should expect an
            ``Index`` and return an ``Index`` of the same shape.

            .. versionadded:: 1.1.0

        Returns
        -------
        sorted_index : pandas.Index
            Sorted copy of the index.
        indexer : numpy.ndarray, optional
            The indices that the index itself was sorted by.

        See Also
        --------
        Series.sort_values : Sort values of a Series.
        DataFrame.sort_values : Sort values in a DataFrame.

        Examples
        --------
        >>> idx = pd.Index([10, 100, 1, 1000])
        >>> idx
        Int64Index([10, 100, 1, 1000], dtype='int64')

        Sort values in ascending order (default behavior).

        >>> idx.sort_values()
        Int64Index([1, 10, 100, 1000], dtype='int64')

        Sort values in descending order, and also get the indices `idx` was
        sorted by.

        >>> idx.sort_values(ascending=False, return_indexer=True)
        (Int64Index([1000, 100, 10, 1], dtype='int64'), array([3, 1, 0, 2]))
        """
        idx = ensure_key_mapped(self, key)

        # GH 35584. Sort missing values according to na_position kwarg
        # ignore na_position for MultiIndex
        if not isinstance(self, ABCMultiIndex):
            _as = nargsort(
                items=idx, ascending=ascending, na_position=na_position, key=key
            )
        else:
            _as = idx.argsort()
            if not ascending:
                _as = _as[::-1]

        sorted_index = self.take(_as)

        if return_indexer:
            return sorted_index, _as
        else:
            return sorted_index

    @final
    def sort(self, *args, **kwargs):
        """
        Use sort_values instead.
        """
        raise TypeError("cannot sort an Index object in-place, use sort_values instead")

    def shift(self, periods=1, freq=None):
        """
        Shift index by desired number of time frequency increments.

        This method is for shifting the values of datetime-like indexes
        by a specified time increment a given number of times.

        Parameters
        ----------
        periods : int, default 1
            Number of periods (or increments) to shift by,
            can be positive or negative.
        freq : pandas.DateOffset, pandas.Timedelta or str, optional
            Frequency increment to shift by.
            If None, the index is shifted by its own `freq` attribute.
            Offset aliases are valid strings, e.g., 'D', 'W', 'M' etc.

        Returns
        -------
        pandas.Index
            Shifted index.

        See Also
        --------
        Series.shift : Shift values of Series.

        Notes
        -----
        This method is only implemented for datetime-like index classes,
        i.e., DatetimeIndex, PeriodIndex and TimedeltaIndex.

        Examples
        --------
        Put the first 5 month starts of 2011 into an index.

        >>> month_starts = pd.date_range('1/1/2011', periods=5, freq='MS')
        >>> month_starts
        DatetimeIndex(['2011-01-01', '2011-02-01', '2011-03-01', '2011-04-01',
                       '2011-05-01'],
                      dtype='datetime64[ns]', freq='MS')

        Shift the index by 10 days.

        >>> month_starts.shift(10, freq='D')
        DatetimeIndex(['2011-01-11', '2011-02-11', '2011-03-11', '2011-04-11',
                       '2011-05-11'],
                      dtype='datetime64[ns]', freq=None)

        The default value of `freq` is the `freq` attribute of the index,
        which is 'MS' (month start) in this example.

        >>> month_starts.shift(10)
        DatetimeIndex(['2011-11-01', '2011-12-01', '2012-01-01', '2012-02-01',
                       '2012-03-01'],
                      dtype='datetime64[ns]', freq='MS')
        """
        raise NotImplementedError(
            f"This method is only implemented for DatetimeIndex, PeriodIndex and "
            f"TimedeltaIndex; Got type {type(self).__name__}"
        )

    def argsort(self, *args, **kwargs) -> np.ndarray:
        """
        Return the integer indices that would sort the index.

        Parameters
        ----------
        *args
            Passed to `numpy.ndarray.argsort`.
        **kwargs
            Passed to `numpy.ndarray.argsort`.

        Returns
        -------
        numpy.ndarray
            Integer indices that would sort the index if used as
            an indexer.

        See Also
        --------
        numpy.argsort : Similar method for NumPy arrays.
        Index.sort_values : Return sorted copy of Index.

        Examples
        --------
        >>> idx = pd.Index(['b', 'a', 'd', 'c'])
        >>> idx
        Index(['b', 'a', 'd', 'c'], dtype='object')

        >>> order = idx.argsort()
        >>> order
        array([1, 0, 3, 2])

        >>> idx[order]
        Index(['a', 'b', 'c', 'd'], dtype='object')
        """
        # This works for either ndarray or EA, is overridden
        #  by RangeIndex, MultIIndex
        return self._data.argsort(*args, **kwargs)

    @final
    def get_value(self, series: Series, key):
        """
        Fast lookup of value from 1-dimensional ndarray.

        Only use this if you know what you're doing.

        Returns
        -------
        scalar or Series
        """
        warnings.warn(
            "get_value is deprecated and will be removed in a future version. "
            "Use Series[key] instead",
            FutureWarning,
            stacklevel=2,
        )

        self._check_indexing_error(key)

        try:
            # GH 20882, 21257
            # First try to convert the key to a location
            # If that fails, raise a KeyError if an integer
            # index, otherwise, see if key is an integer, and
            # try that
            loc = self.get_loc(key)
        except KeyError:
            if not self._should_fallback_to_positional():
                raise
            elif is_integer(key):
                # If the Index cannot hold integer, then this is unambiguously
                #  a locational lookup.
                loc = key
            else:
                raise

        return self._get_values_for_loc(series, loc, key)

    def _check_indexing_error(self, key):
        if not is_scalar(key):
            # if key is not a scalar, directly raise an error (the code below
            # would convert to numpy arrays and raise later any way) - GH29926
            raise InvalidIndexError(key)

    def _should_fallback_to_positional(self) -> bool:
        """
        Should an integer key be treated as positional?
        """
        return not self.holds_integer() and not self.is_boolean()

    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`

        key is included for MultiIndex compat.
        """
        if is_integer(loc):
            return series._values[loc]

        return series.iloc[loc]

    @final
    def set_value(self, arr, key, value):
        """
        Fast lookup of value from 1-dimensional ndarray.

        .. deprecated:: 1.0

        Notes
        -----
        Only use this if you know what you're doing.
        """
        warnings.warn(
            (
                "The 'set_value' method is deprecated, and "
                "will be removed in a future version."
            ),
            FutureWarning,
            stacklevel=2,
        )
        loc = self._engine.get_loc(key)
        validate_numeric_casting(arr.dtype, value)
        arr[loc] = value

    _index_shared_docs[
        "get_indexer_non_unique"
    ] = """
        Compute indexer and mask for new index given the current index. The
        indexer should be then used as an input to ndarray.take to align the
        current data to the new index.

        Parameters
        ----------
        target : %(target_klass)s

        Returns
        -------
        indexer : ndarray of int
            Integers from 0 to n - 1 indicating that the index at these
            positions matches the corresponding target values. Missing values
            in the target are marked by -1.
        missing : ndarray of int
            An indexer into the target of the values not found.
            These correspond to the -1 in the indexer array.
        """

    @Appender(_index_shared_docs["get_indexer_non_unique"] % _index_doc_kwargs)
    def get_indexer_non_unique(self, target):
        target = ensure_index(target)

        if not self._should_compare(target) and not is_interval_dtype(self.dtype):
            # IntervalIndex get special treatment bc numeric scalars can be
            #  matched to Interval scalars
            return self._get_indexer_non_comparable(target, method=None, unique=False)

        pself, ptarget = self._maybe_promote(target)
        if pself is not self or ptarget is not target:
            return pself.get_indexer_non_unique(ptarget)

        if not is_dtype_equal(self.dtype, target.dtype):
            # TODO: if object, could use infer_dtype to preempt costly
            #  conversion if still non-comparable?
            dtype = self._find_common_type_compat(target)

            this = self.astype(dtype, copy=False)
            that = target.astype(dtype, copy=False)
            return this.get_indexer_non_unique(that)

        tgt_values = target._get_engine_target()

        indexer, missing = self._engine.get_indexer_non_unique(tgt_values)
        return ensure_platform_int(indexer), missing

    @final
    def get_indexer_for(self, target, **kwargs):
        """
        Guaranteed return of an indexer even when non-unique.

        This dispatches to get_indexer or get_indexer_non_unique
        as appropriate.

        Returns
        -------
        numpy.ndarray
            List of indices.
        """
        if self._index_as_unique:
            return self.get_indexer(target, **kwargs)
        indexer, _ = self.get_indexer_non_unique(target)
        return indexer

    @final
    def _get_indexer_non_comparable(self, target: Index, method, unique: bool = True):
        """
        Called from get_indexer or get_indexer_non_unique when the target
        is of a non-comparable dtype.

        For get_indexer lookups with method=None, get_indexer is an _equality_
        check, so non-comparable dtypes mean we will always have no matches.

        For get_indexer lookups with a method, get_indexer is an _inequality_
        check, so non-comparable dtypes mean we will always raise TypeError.

        Parameters
        ----------
        target : Index
        method : str or None
        unique : bool, default True
            * True if called from get_indexer.
            * False if called from get_indexer_non_unique.

        Raises
        ------
        TypeError
            If doing an inequality check, i.e. method is not None.
        """
        if method is not None:
            other = unpack_nested_dtype(target)
            raise TypeError(f"Cannot compare dtypes {self.dtype} and {other.dtype}")

        no_matches = -1 * np.ones(target.shape, dtype=np.intp)
        if unique:
            # This is for get_indexer
            return no_matches
        else:
            # This is for get_indexer_non_unique
            missing = np.arange(len(target), dtype=np.intp)
            return no_matches, missing

    @property
    def _index_as_unique(self) -> bool:
        """
        Whether we should treat this as unique for the sake of
        get_indexer vs get_indexer_non_unique.

        For IntervalIndex compat.
        """
        return self.is_unique

    _requires_unique_msg = "Reindexing only valid with uniquely valued Index objects"

    @final
    def _maybe_promote(self, other: Index):
        """
        When dealing with an object-dtype Index and a non-object Index, see
        if we can upcast the object-dtype one to improve performance.
        """

        if isinstance(self, ABCDatetimeIndex) and isinstance(other, ABCDatetimeIndex):
            if (
                self.tz is not None
                and other.tz is not None
                and not tz_compare(self.tz, other.tz)
            ):
                # standardize on UTC
                return self.tz_convert("UTC"), other.tz_convert("UTC")

        elif self.inferred_type == "date" and isinstance(other, ABCDatetimeIndex):
            try:
                return type(other)(self), other
            except OutOfBoundsDatetime:
                return self, other
        elif self.inferred_type == "timedelta" and isinstance(other, ABCTimedeltaIndex):
            # TODO: we dont have tests that get here
            return type(other)(self), other
        elif self.inferred_type == "boolean":
            if not is_object_dtype(self.dtype):
                return self.astype("object"), other.astype("object")

        if not is_object_dtype(self.dtype) and is_object_dtype(other.dtype):
            # Reverse op so we dont need to re-implement on the subclasses
            other, self = other._maybe_promote(self)

        return self, other

    @final
    def _find_common_type_compat(self, target) -> DtypeObj:
        """
        Implementation of find_common_type that adjusts for Index-specific
        special cases.
        """
        if is_interval_dtype(self.dtype) and is_valid_na_for_dtype(target, self.dtype):
            # e.g. setting NA value into IntervalArray[int64]
            self = cast("IntervalIndex", self)
            return IntervalDtype(np.float64, closed=self.closed)

        target_dtype, _ = infer_dtype_from(target, pandas_dtype=True)
        dtype = find_common_type([self.dtype, target_dtype])
        if dtype.kind in ["i", "u"]:
            # TODO: what about reversed with self being categorical?
            if (
                isinstance(target, Index)
                and is_categorical_dtype(target.dtype)
                and target.hasnans
            ):
                # FIXME: find_common_type incorrect with Categorical GH#38240
                # FIXME: some cases where float64 cast can be lossy?
                dtype = np.dtype(np.float64)
        if dtype.kind == "c":
            dtype = np.dtype(object)
        return dtype

    @final
    def _should_compare(self, other: Index) -> bool:
        """
        Check if `self == other` can ever have non-False entries.
        """

        if (other.is_boolean() and self.is_numeric()) or (
            self.is_boolean() and other.is_numeric()
        ):
            # GH#16877 Treat boolean labels passed to a numeric index as not
            #  found. Without this fix False and True would be treated as 0 and 1
            #  respectively.
            return False

        other = unpack_nested_dtype(other)
        dtype = other.dtype
        return self._is_comparable_dtype(dtype) or is_object_dtype(dtype)

    def _is_comparable_dtype(self, dtype: DtypeObj) -> bool:
        """
        Can we compare values of the given dtype to our own?
        """
        return True

    @final
    def groupby(self, values) -> PrettyDict[Hashable, np.ndarray]:
        """
        Group the index labels by a given array of values.

        Parameters
        ----------
        values : array
            Values used to determine the groups.

        Returns
        -------
        dict
            {group name -> group labels}
        """
        # TODO: if we are a MultiIndex, we can do better
        # that converting to tuples
        if isinstance(values, ABCMultiIndex):
            values = values._values
        values = Categorical(values)
        result = values._reverse_indexer()

        # map to the label
        result = {k: self.take(v) for k, v in result.items()}

        return PrettyDict(result)

    def map(self, mapper, na_action=None):
        """
        Map values using input correspondence (a dict, Series, or function).

        Parameters
        ----------
        mapper : function, dict, or Series
            Mapping correspondence.
        na_action : {None, 'ignore'}
            If 'ignore', propagate NA values, without passing them to the
            mapping correspondence.

        Returns
        -------
        applied : Union[Index, MultiIndex], inferred
            The output of the mapping function applied to the index.
            If the function returns a tuple with more than one element
            a MultiIndex will be returned.
        """
        from pandas.core.indexes.multi import MultiIndex

        new_values = super()._map_values(mapper, na_action=na_action)

        attributes = self._get_attributes_dict()

        # we can return a MultiIndex
        if new_values.size and isinstance(new_values[0], tuple):
            if isinstance(self, MultiIndex):
                names = self.names
            elif attributes.get("name"):
                names = [attributes.get("name")] * len(new_values[0])
            else:
                names = None
            return MultiIndex.from_tuples(new_values, names=names)

        attributes["copy"] = False
        if not new_values.size:
            # empty
            attributes["dtype"] = self.dtype

        return Index(new_values, **attributes)

    # TODO: De-duplicate with map, xref GH#32349
    @final
    def _transform_index(self, func, level=None) -> Index:
        """
        Apply function to all values found in index.

        This includes transforming multiindex entries separately.
        Only apply function to one level of the MultiIndex if level is specified.
        """
        if isinstance(self, ABCMultiIndex):
            if level is not None:
                items = [
                    tuple(func(y) if i == level else y for i, y in enumerate(x))
                    for x in self
                ]
            else:
                items = [tuple(func(y) for y in x) for x in self]
            return type(self).from_tuples(items, names=self.names)
        else:
            items = [func(x) for x in self]
            return Index(items, name=self.name, tupleize_cols=False)

    def isin(self, values, level=None):
        """
        Return a boolean array where the index values are in `values`.

        Compute boolean array of whether each index value is found in the
        passed set of values. The length of the returned boolean array matches
        the length of the index.

        Parameters
        ----------
        values : set or list-like
            Sought values.
        level : str or int, optional
            Name or position of the index level to use (if the index is a
            `MultiIndex`).

        Returns
        -------
        is_contained : ndarray
            NumPy array of boolean values.

        See Also
        --------
        Series.isin : Same for Series.
        DataFrame.isin : Same method for DataFrames.

        Notes
        -----
        In the case of `MultiIndex` you must either specify `values` as a
        list-like object containing tuples that are the same length as the
        number of levels, or specify `level`. Otherwise it will raise a
        ``ValueError``.

        If `level` is specified:

        - if it is the name of one *and only one* index level, use that level;
        - otherwise it should be a number indicating level position.

        Examples
        --------
        >>> idx = pd.Index([1,2,3])
        >>> idx
        Int64Index([1, 2, 3], dtype='int64')

        Check whether each index value in a list of values.

        >>> idx.isin([1, 4])
        array([ True, False, False])

        >>> midx = pd.MultiIndex.from_arrays([[1,2,3],
        ...                                  ['red', 'blue', 'green']],
        ...                                  names=('number', 'color'))
        >>> midx
        MultiIndex([(1,   'red'),
                    (2,  'blue'),
                    (3, 'green')],
                   names=['number', 'color'])

        Check whether the strings in the 'color' level of the MultiIndex
        are in a list of colors.

        >>> midx.isin(['red', 'orange', 'yellow'], level='color')
        array([ True, False, False])

        To check across the levels of a MultiIndex, pass a list of tuples:

        >>> midx.isin([(1, 'red'), (3, 'red')])
        array([ True, False, False])

        For a DatetimeIndex, string values in `values` are converted to
        Timestamps.

        >>> dates = ['2000-03-11', '2000-03-12', '2000-03-13']
        >>> dti = pd.to_datetime(dates)
        >>> dti
        DatetimeIndex(['2000-03-11', '2000-03-12', '2000-03-13'],
        dtype='datetime64[ns]', freq=None)

        >>> dti.isin(['2000-03-11'])
        array([ True, False, False])
        """
        if level is not None:
            self._validate_index_level(level)
        return algos.isin(self._values, values)

    def _get_string_slice(self, key: str_t):
        # this is for partial string indexing,
        # overridden in DatetimeIndex, TimedeltaIndex and PeriodIndex
        raise NotImplementedError

    def slice_indexer(
        self,
        start: Optional[Hashable] = None,
        end: Optional[Hashable] = None,
        step: Optional[int] = None,
        kind: Optional[str_t] = None,
    ) -> slice:
        """
        Compute the slice indexer for input labels and step.

        Index needs to be ordered and unique.

        Parameters
        ----------
        start : label, default None
            If None, defaults to the beginning.
        end : label, default None
            If None, defaults to the end.
        step : int, default None
        kind : str, default None

        Returns
        -------
        indexer : slice

        Raises
        ------
        KeyError : If key does not exist, or key is not unique and index is
            not ordered.

        Notes
        -----
        This function assumes that the data is sorted, so use at your own peril

        Examples
        --------
        This is a method on all index types. For example you can do:

        >>> idx = pd.Index(list('abcd'))
        >>> idx.slice_indexer(start='b', end='c')
        slice(1, 3, None)

        >>> idx = pd.MultiIndex.from_arrays([list('abcd'), list('efgh')])
        >>> idx.slice_indexer(start='b', end=('c', 'g'))
        slice(1, 3, None)
        """
        start_slice, end_slice = self.slice_locs(start, end, step=step, kind=kind)

        # return a slice
        if not is_scalar(start_slice):
            raise AssertionError("Start slice bound is non-scalar")
        if not is_scalar(end_slice):
            raise AssertionError("End slice bound is non-scalar")

        return slice(start_slice, end_slice, step)

    def _maybe_cast_indexer(self, key):
        """
        If we have a float key and are not a floating index, then try to cast
        to an int if equivalent.
        """
        if not self.is_floating():
            return com.cast_scalar_indexer(key)
        return key

    @final
    def _validate_indexer(self, form: str_t, key, kind: str_t):
        """
        If we are positional indexer, validate that we have appropriate
        typed bounds must be an integer.
        """
        assert kind in ["getitem", "iloc"]

        if key is not None and not is_integer(key):
            raise self._invalid_indexer(form, key)

    def _maybe_cast_slice_bound(self, label, side: str_t, kind):
        """
        This function should be overloaded in subclasses that allow non-trivial
        casting on label-slice bounds, e.g. datetime-like indices allowing
        strings containing formatted datetimes.

        Parameters
        ----------
        label : object
        side : {'left', 'right'}
        kind : {'loc', 'getitem'} or None

        Returns
        -------
        label : object

        Notes
        -----
        Value of `side` parameter should be validated in caller.
        """
        assert kind in ["loc", "getitem", None]

        # We are a plain index here (sub-class override this method if they
        # wish to have special treatment for floats/ints, e.g. Float64Index and
        # datetimelike Indexes
        # reject them, if index does not contain label
        if (is_float(label) or is_integer(label)) and label not in self._values:
            raise self._invalid_indexer("slice", label)

        return label

    def _searchsorted_monotonic(self, label, side="left"):
        if self.is_monotonic_increasing:
            return self.searchsorted(label, side=side)
        elif self.is_monotonic_decreasing:
            # np.searchsorted expects ascending sort order, have to reverse
            # everything for it to work (element ordering, search side and
            # resulting value).
            pos = self[::-1].searchsorted(
                label, side="right" if side == "left" else "left"
            )
            return len(self) - pos

        raise ValueError("index must be monotonic increasing or decreasing")

    def get_slice_bound(self, label, side: str_t, kind) -> int:
        """
        Calculate slice bound that corresponds to given label.

        Returns leftmost (one-past-the-rightmost if ``side=='right'``) position
        of given label.

        Parameters
        ----------
        label : object
        side : {'left', 'right'}
        kind : {'loc', 'getitem'} or None

        Returns
        -------
        int
            Index of label.
        """
        assert kind in ["loc", "getitem", None]

        if side not in ("left", "right"):
            raise ValueError(
                "Invalid value for side kwarg, must be either "
                f"'left' or 'right': {side}"
            )

        original_label = label

        # For datetime indices label may be a string that has to be converted
        # to datetime boundary according to its resolution.
        label = self._maybe_cast_slice_bound(label, side, kind)

        # we need to look up the label
        try:
            slc = self.get_loc(label)
        except KeyError as err:
            try:
                return self._searchsorted_monotonic(label, side)
            except ValueError:
                # raise the original KeyError
                raise err

        if isinstance(slc, np.ndarray):
            # get_loc may return a boolean array or an array of indices, which
            # is OK as long as they are representable by a slice.
            if is_bool_dtype(slc):
                slc = lib.maybe_booleans_to_slice(slc.view("u1"))
            else:
                slc = lib.maybe_indices_to_slice(
                    slc.astype(np.intp, copy=False), len(self)
                )
            if isinstance(slc, np.ndarray):
                raise KeyError(
                    f"Cannot get {side} slice bound for non-unique "
                    f"label: {repr(original_label)}"
                )

        if isinstance(slc, slice):
            if side == "left":
                return slc.start
            else:
                return slc.stop
        else:
            if side == "right":
                return slc + 1
            else:
                return slc

    def slice_locs(self, start=None, end=None, step=None, kind=None):
        """
        Compute slice locations for input labels.

        Parameters
        ----------
        start : label, default None
            If None, defaults to the beginning.
        end : label, default None
            If None, defaults to the end.
        step : int, defaults None
            If None, defaults to 1.
        kind : {'loc', 'getitem'} or None

        Returns
        -------
        start, end : int

        See Also
        --------
        Index.get_loc : Get location for a single label.

        Notes
        -----
        This method only works if the index is monotonic or unique.

        Examples
        --------
        >>> idx = pd.Index(list('abcd'))
        >>> idx.slice_locs(start='b', end='c')
        (1, 3)
        """
        inc = step is None or step >= 0

        if not inc:
            # If it's a reverse slice, temporarily swap bounds.
            start, end = end, start

        # GH 16785: If start and end happen to be date strings with UTC offsets
        # attempt to parse and check that the offsets are the same
        if isinstance(start, (str, datetime)) and isinstance(end, (str, datetime)):
            try:
                ts_start = Timestamp(start)
                ts_end = Timestamp(end)
            except (ValueError, TypeError):
                pass
            else:
                if not tz_compare(ts_start.tzinfo, ts_end.tzinfo):
                    raise ValueError("Both dates must have the same UTC offset")

        start_slice = None
        if start is not None:
            start_slice = self.get_slice_bound(start, "left", kind)
        if start_slice is None:
            start_slice = 0

        end_slice = None
        if end is not None:
            end_slice = self.get_slice_bound(end, "right", kind)
        if end_slice is None:
            end_slice = len(self)

        if not inc:
            # Bounds at this moment are swapped, swap them back and shift by 1.
            #
            # slice_locs('B', 'A', step=-1): s='B', e='A'
            #
            #              s='A'                 e='B'
            # AFTER SWAP:    |                     |
            #                v ------------------> V
            #           -----------------------------------
            #           | | |A|A|A|A| | | | | |B|B| | | | |
            #           -----------------------------------
            #              ^ <------------------ ^
            # SHOULD BE:   |                     |
            #           end=s-1              start=e-1
            #
            end_slice, start_slice = start_slice - 1, end_slice - 1

            # i == -1 triggers ``len(self) + i`` selection that points to the
            # last element, not before-the-first one, subtracting len(self)
            # compensates that.
            if end_slice == -1:
                end_slice -= len(self)
            if start_slice == -1:
                start_slice -= len(self)

        return start_slice, end_slice

    def delete(self, loc):
        """
        Make new Index with passed location(-s) deleted.

        Parameters
        ----------
        loc : int or list of int
            Location of item(-s) which will be deleted.
            Use a list of locations to delete more than one value at the same time.

        Returns
        -------
        Index
            New Index with passed location(-s) deleted.

        See Also
        --------
        numpy.delete : Delete any rows and column from NumPy array (ndarray).

        Examples
        --------
        >>> idx = pd.Index(['a', 'b', 'c'])
        >>> idx.delete(1)
        Index(['a', 'c'], dtype='object')

        >>> idx = pd.Index(['a', 'b', 'c'])
        >>> idx.delete([0, 2])
        Index(['b'], dtype='object')
        """
        res_values = np.delete(self._data, loc)
        return type(self)._simple_new(res_values, name=self.name)

    def insert(self, loc: int, item):
        """
        Make new Index inserting new item at location.

        Follows Python list.append semantics for negative values.

        Parameters
        ----------
        loc : int
        item : object

        Returns
        -------
        new_index : Index
        """
        # Note: this method is overridden by all ExtensionIndex subclasses,
        #  so self is never backed by an EA.
        item = lib.item_from_zerodim(item)
        if is_valid_na_for_dtype(item, self.dtype) and self.dtype != object:
            item = self._na_value

        try:
            item = self._validate_fill_value(item)
        except TypeError:
            inferred, _ = infer_dtype_from(item)
            dtype = find_common_type([self.dtype, inferred])
            return self.astype(dtype).insert(loc, item)

        arr = np.asarray(self)

        # Use Index constructor to ensure we get tuples cast correctly.
        item = Index([item], dtype=self.dtype)._values
        idx = np.concatenate((arr[:loc], item, arr[loc:]))
        return Index(idx, name=self.name)

    def drop(self: _IndexT, labels, errors: str_t = "raise") -> _IndexT:
        """
        Make new Index with passed list of labels deleted.

        Parameters
        ----------
        labels : array-like
        errors : {'ignore', 'raise'}, default 'raise'
            If 'ignore', suppress error and existing labels are dropped.

        Returns
        -------
        dropped : Index

        Raises
        ------
        KeyError
            If not all of the labels are found in the selected axis
        """
        arr_dtype = "object" if self.dtype == "object" else None
        labels = com.index_labels_to_array(labels, dtype=arr_dtype)
        indexer = self.get_indexer_for(labels)
        mask = indexer == -1
        if mask.any():
            if errors != "ignore":
                raise KeyError(f"{labels[mask]} not found in axis")
            indexer = indexer[~mask]
        return self.delete(indexer)

    # --------------------------------------------------------------------
    # Generated Arithmetic, Comparison, and Unary Methods

    def _cmp_method(self, other, op):
        """
        Wrapper used to dispatch comparison operations.
        """
        if self.is_(other):
            # fastpath
            if op in {operator.eq, operator.le, operator.ge}:
                arr = np.ones(len(self), dtype=bool)
                if self._can_hold_na and not isinstance(self, ABCMultiIndex):
                    # TODO: should set MultiIndex._can_hold_na = False?
                    arr[self.isna()] = False
                return arr
            elif op in {operator.ne, operator.lt, operator.gt}:
                return np.zeros(len(self), dtype=bool)

        if isinstance(other, (np.ndarray, Index, ABCSeries, ExtensionArray)) and len(
            self
        ) != len(other):
            raise ValueError("Lengths must match to compare")

        if not isinstance(other, ABCMultiIndex):
            other = extract_array(other, extract_numpy=True)
        else:
            other = np.asarray(other)

        if is_object_dtype(self.dtype) and isinstance(other, ExtensionArray):
            # e.g. PeriodArray, Categorical
            with np.errstate(all="ignore"):
                result = op(self._values, other)

        elif is_object_dtype(self.dtype) and not isinstance(self, ABCMultiIndex):
            # don't pass MultiIndex
            with np.errstate(all="ignore"):
                result = ops.comp_method_OBJECT_ARRAY(op, self._values, other)

        else:
            with np.errstate(all="ignore"):
                result = ops.comparison_op(self._values, other, op)

        return result

    def _arith_method(self, other, op):
        """
        Wrapper used to dispatch arithmetic operations.
        """

        from pandas import Series

        result = op(Series(self), other)
        if isinstance(result, tuple):
            return (Index(result[0]), Index(result[1]))
        return Index(result)

    @final
    def _unary_method(self, op):
        result = op(self._values)
        return Index(result, name=self.name)

    def __abs__(self):
        return self._unary_method(operator.abs)

    def __neg__(self):
        return self._unary_method(operator.neg)

    def __pos__(self):
        return self._unary_method(operator.pos)

    def __inv__(self):
        # TODO: why not operator.inv?
        # TODO: __inv__ vs __invert__?
        return self._unary_method(lambda x: -x)

    def any(self, *args, **kwargs):
        """
        Return whether any element is Truthy.

        Parameters
        ----------
        *args
            Required for compatibility with numpy.
        **kwargs
            Required for compatibility with numpy.

        Returns
        -------
        any : bool or array_like (if axis is specified)
            A single element array_like may be converted to bool.

        See Also
        --------
        Index.all : Return whether all elements are True.
        Series.all : Return whether all elements are True.

        Notes
        -----
        Not a Number (NaN), positive infinity and negative infinity
        evaluate to True because these are not equal to zero.

        Examples
        --------
        >>> index = pd.Index([0, 1, 2])
        >>> index.any()
        True

        >>> index = pd.Index([0, 0, 0])
        >>> index.any()
        False
        """
        nv.validate_any(args, kwargs)
        self._maybe_disable_logical_methods("any")
        return np.any(self.values)

    def all(self, *args, **kwargs):
        """
        Return whether all elements are Truthy.

        Parameters
        ----------
        *args
            Required for compatibility with numpy.
        **kwargs
            Required for compatibility with numpy.

        Returns
        -------
        all : bool or array_like (if axis is specified)
            A single element array_like may be converted to bool.

        See Also
        --------
        Index.any : Return whether any element in an Index is True.
        Series.any : Return whether any element in a Series is True.
        Series.all : Return whether all elements in a Series are True.

        Notes
        -----
        Not a Number (NaN), positive infinity and negative infinity
        evaluate to True because these are not equal to zero.

        Examples
        --------
        **all**

        True, because nonzero integers are considered True.

        >>> pd.Index([1, 2, 3]).all()
        True

        False, because ``0`` is considered False.

        >>> pd.Index([0, 1, 2]).all()
        False

        **any**

        True, because ``1`` is considered True.

        >>> pd.Index([0, 0, 1]).any()
        True

        False, because ``0`` is considered False.

        >>> pd.Index([0, 0, 0]).any()
        False
        """
        nv.validate_all(args, kwargs)
        self._maybe_disable_logical_methods("all")
        return np.all(self.values)

    @final
    def _maybe_disable_logical_methods(self, opname: str_t):
        """
        raise if this Index subclass does not support any or all.
        """
        if (
            isinstance(self, ABCMultiIndex)
            or needs_i8_conversion(self.dtype)
            or is_interval_dtype(self.dtype)
            or is_categorical_dtype(self.dtype)
            or is_float_dtype(self.dtype)
        ):
            # This call will raise
            make_invalid_op(opname)(self)

    @property
    def shape(self) -> Shape:
        """
        Return a tuple of the shape of the underlying data.
        """
        # not using "(len(self), )" to return "correct" shape if the values
        # consists of a >1 D array (see GH-27775)
        # overridden in MultiIndex.shape to avoid materializing the values
        return self._values.shape


def ensure_index_from_sequences(sequences, names=None):
    """
    Construct an index from sequences of data.

    A single sequence returns an Index. Many sequences returns a
    MultiIndex.

    Parameters
    ----------
    sequences : sequence of sequences
    names : sequence of str

    Returns
    -------
    index : Index or MultiIndex

    Examples
    --------
    >>> ensure_index_from_sequences([[1, 2, 3]], names=["name"])
    Int64Index([1, 2, 3], dtype='int64', name='name')

    >>> ensure_index_from_sequences([["a", "a"], ["a", "b"]], names=["L1", "L2"])
    MultiIndex([('a', 'a'),
                ('a', 'b')],
               names=['L1', 'L2'])

    See Also
    --------
    ensure_index
    """
    from pandas.core.indexes.multi import MultiIndex

    if len(sequences) == 1:
        if names is not None:
            names = names[0]
        return Index(sequences[0], name=names)
    else:
        return MultiIndex.from_arrays(sequences, names=names)


def ensure_index(
    index_like: Union[AnyArrayLike, Sequence], copy: bool = False
) -> Index:
    """
    Ensure that we have an index from some index-like object.

    Parameters
    ----------
    index_like : sequence
        An Index or other sequence
    copy : bool, default False

    Returns
    -------
    index : Index or MultiIndex

    See Also
    --------
    ensure_index_from_sequences

    Examples
    --------
    >>> ensure_index(['a', 'b'])
    Index(['a', 'b'], dtype='object')

    >>> ensure_index([('a', 'a'),  ('b', 'c')])
    Index([('a', 'a'), ('b', 'c')], dtype='object')

    >>> ensure_index([['a', 'a'], ['b', 'c']])
    MultiIndex([('a', 'b'),
            ('a', 'c')],
           )
    """
    if isinstance(index_like, Index):
        if copy:
            index_like = index_like.copy()
        return index_like
    if hasattr(index_like, "name"):
        # https://github.com/python/mypy/issues/1424
        # error: Item "ExtensionArray" of "Union[ExtensionArray,
        # Sequence[Any]]" has no attribute "name"
        # error: Item "Sequence[Any]" of "Union[ExtensionArray, Sequence[Any]]"
        # has no attribute "name"
        # error: "Sequence[Any]" has no attribute "name"
        # error: Item "Sequence[Any]" of "Union[Series, Sequence[Any]]" has no
        # attribute "name"
        # error: Item "Sequence[Any]" of "Union[Any, Sequence[Any]]" has no
        # attribute "name"
        name = index_like.name  # type: ignore[union-attr, attr-defined]
        return Index(index_like, name=name, copy=copy)

    if is_iterator(index_like):
        index_like = list(index_like)

    # must check for exactly list here because of strict type
    # check in clean_index_list
    if isinstance(index_like, list):
        if type(index_like) != list:
            index_like = list(index_like)

        converted, all_arrays = lib.clean_index_list(index_like)

        if len(converted) > 0 and all_arrays:
            from pandas.core.indexes.multi import MultiIndex

            return MultiIndex.from_arrays(converted)
        else:
            if isinstance(converted, np.ndarray) and converted.dtype == np.int64:
                # Check for overflows if we should actually be uint64
                # xref GH#35481
                alt = np.asarray(index_like)
                if alt.dtype == np.uint64:
                    converted = alt

            index_like = converted
    else:
        # clean_index_list does the equivalent of copying
        # so only need to do this if not list instance
        if copy:
            index_like = copy_func(index_like)

    return Index(index_like)


def ensure_has_len(seq):
    """
    If seq is an iterator, put its values into a list.
    """
    try:
        len(seq)
    except TypeError:
        return list(seq)
    else:
        return seq


def trim_front(strings: List[str]) -> List[str]:
    """
    Trims zeros and decimal points.

    Examples
    --------
    >>> trim_front([" a", " b"])
    ['a', 'b']

    >>> trim_front([" a", " "])
    ['a', '']
    """
    if not strings:
        return strings
    while all(strings) and all(x[0] == " " for x in strings):
        strings = [x[1:] for x in strings]
    return strings


def _validate_join_method(method: str) -> None:
    if method not in ["left", "right", "inner", "outer"]:
        raise ValueError(f"do not recognize join method {method}")


def default_index(n: int) -> RangeIndex:
    from pandas.core.indexes.range import RangeIndex

    return RangeIndex(0, n, name=None)


def maybe_extract_name(name, obj, cls) -> Hashable:
    """
    If no name is passed, then extract it from data, validating hashability.
    """
    if name is None and isinstance(obj, (Index, ABCSeries)):
        # Note we don't just check for "name" attribute since that would
        #  pick up e.g. dtype.name
        name = obj.name

    # GH#29069
    if not is_hashable(name):
        raise TypeError(f"{cls.__name__}.name must be a hashable type")

    return name


def _maybe_cast_with_dtype(data: np.ndarray, dtype: np.dtype, copy: bool) -> np.ndarray:
    """
    If a dtype is passed, cast to the closest matching dtype that is supported
    by Index.

    Parameters
    ----------
    data : np.ndarray
    dtype : np.dtype
    copy : bool

    Returns
    -------
    np.ndarray
    """
    # we need to avoid having numpy coerce
    # things that look like ints/floats to ints unless
    # they are actually ints, e.g. '0' and 0.0
    # should not be coerced
    # GH 11836
    if is_integer_dtype(dtype):
        inferred = lib.infer_dtype(data, skipna=False)
        if inferred == "integer":
            data = maybe_cast_to_integer_array(data, dtype, copy=copy)
        elif inferred in ["floating", "mixed-integer-float"]:
            if isna(data).any():
                raise ValueError("cannot convert float NaN to integer")

            if inferred == "mixed-integer-float":
                data = maybe_cast_to_integer_array(data, dtype)

            # If we are actually all equal to integers,
            # then coerce to integer.
            try:
                data = _try_convert_to_int_array(data, copy, dtype)
            except ValueError:
                data = np.array(data, dtype=np.float64, copy=copy)

        elif inferred != "string":
            data = data.astype(dtype)
    elif is_float_dtype(dtype):
        inferred = lib.infer_dtype(data, skipna=False)
        if inferred != "string":
            data = data.astype(dtype)
    else:
        data = np.array(data, dtype=dtype, copy=copy)

    return data


def _maybe_cast_data_without_dtype(subarr):
    """
    If we have an arraylike input but no passed dtype, try to infer
    a supported dtype.

    Parameters
    ----------
    subarr : np.ndarray, Index, or Series

    Returns
    -------
    converted : np.ndarray or ExtensionArray
    dtype : np.dtype or ExtensionDtype
    """
    # Runtime import needed bc IntervalArray imports Index
    from pandas.core.arrays import (
        DatetimeArray,
        IntervalArray,
        PeriodArray,
        TimedeltaArray,
    )

    assert subarr.dtype == object, subarr.dtype
    inferred = lib.infer_dtype(subarr, skipna=False)

    if inferred == "integer":
        try:
            data = _try_convert_to_int_array(subarr, False, None)
            return data
        except ValueError:
            pass

        return subarr

    elif inferred in ["floating", "mixed-integer-float", "integer-na"]:
        # TODO: Returns IntegerArray for integer-na case in the future
        data = np.asarray(subarr).astype(np.float64, copy=False)
        return data

    elif inferred == "interval":
        try:
            data = IntervalArray._from_sequence(subarr, copy=False)
            return data
        except (ValueError, TypeError):
            # GH27172: mixed closed Intervals --> object dtype
            pass
    elif inferred == "boolean":
        # don't support boolean explicitly ATM
        pass
    elif inferred != "string":
        if inferred.startswith("datetime"):
            try:
                data = DatetimeArray._from_sequence(subarr, copy=False)
                return data
            except (ValueError, OutOfBoundsDatetime):
                # GH 27011
                # If we have mixed timezones, just send it
                # down the base constructor
                pass

        elif inferred.startswith("timedelta"):
            data = TimedeltaArray._from_sequence(subarr, copy=False)
            return data
        elif inferred == "period":
            try:
                data = PeriodArray._from_sequence(subarr)
                return data
            except IncompatibleFrequency:
                pass

    return subarr


def _try_convert_to_int_array(
    data: np.ndarray, copy: bool, dtype: np.dtype
) -> np.ndarray:
    """
    Attempt to convert an array of data into an integer array.

    Parameters
    ----------
    data : The data to convert.
    copy : bool
        Whether to copy the data or not.
    dtype : np.dtype

    Returns
    -------
    int_array : data converted to either an ndarray[int64] or ndarray[uint64]

    Raises
    ------
    ValueError if the conversion was not successful.
    """
    if not is_unsigned_integer_dtype(dtype):
        # skip int64 conversion attempt if uint-like dtype is passed, as
        # this could return Int64Index when UInt64Index is what's desired
        try:
            res = data.astype("i8", copy=False)
            if (res == data).all():
                return res  # TODO: might still need to copy
        except (OverflowError, TypeError, ValueError):
            pass

    # Conversion to int64 failed (possibly due to overflow) or was skipped,
    # so let's try now with uint64.
    try:
        res = data.astype("u8", copy=False)
        if (res == data).all():
            return res  # TODO: might still need to copy
    except (OverflowError, TypeError, ValueError):
        pass

    raise ValueError


def get_unanimous_names(*indexes: Index) -> Tuple[Hashable, ...]:
    """
    Return common name if all indices agree, otherwise None (level-by-level).

    Parameters
    ----------
    indexes : list of Index objects

    Returns
    -------
    list
        A list representing the unanimous 'names' found.
    """
    name_tups = [tuple(i.names) for i in indexes]
    name_sets = [{*ns} for ns in zip_longest(*name_tups)]
    names = tuple(ns.pop() if len(ns) == 1 else None for ns in name_sets)
    return names


def unpack_nested_dtype(other: _IndexT) -> _IndexT:
    """
    When checking if our dtype is comparable with another, we need
    to unpack CategoricalDtype to look at its categories.dtype.

    Parameters
    ----------
    other : Index

    Returns
    -------
    Index
    """
    dtype = other.dtype
    if is_categorical_dtype(dtype):
        # If there is ever a SparseIndex, this could get dispatched
        #  here too.
        return dtype.categories
    return other


def _maybe_try_sort(result, sort):
    if sort is None:
        try:
            result = algos.safe_sort(result)
        except TypeError as err:
            warnings.warn(
                f"{err}, sort order is undefined for incomparable objects",
                RuntimeWarning,
                stacklevel=4,
            )
    return result