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# Source code for torch.autograd.function

    import functools
    import warnings
    from collections import OrderedDict
    from typing import Any, List, Optional, Tuple

    import torch
    import torch._C as _C
    import torch._functorch as _functorch
    import torch.utils.hooks as hooks
    from torch._C import _functions
    from torch._functorch.autograd_function import custom_function_call

    __all__ = [
        "FunctionCtx",
        "BackwardCFunction",
        "FunctionMeta",
        "Function",
        "once_differentiable",
        "traceable",
        "InplaceFunction",
        "NestedIOFunction",
    ]

    # Formerly known as: _ContextMethodMixin
    class FunctionCtx:
        def save_for_backward(self, *tensors: torch.Tensor):
            r"""Saves given tensors for a future call to :func:`~Function.backward`.

            ``save_for_backward`` should be called at most once, only from inside the
            :func:`forward` method, and only with tensors.

            All tensors intended to be used in the backward pass should be saved
            with ``save_for_backward`` (as opposed to directly on ``ctx``) to prevent
            incorrect gradients and memory leaks, and enable the application of saved
            tensor hooks. See :class:`torch.autograd.graph.saved_tensors_hooks`.

            Note that if intermediary tensors, tensors that are neither inputs
            nor outputs of :func:`forward`, are saved for backward, your custom Function
            may not support double backward.
            Custom Functions that do not support double backward should decorate their
            :func:`backward` method with ``@once_differentiable`` so that performing
            double backward raises an error. If you'd like to support double backward,
            you can either recompute intermediaries based on the inputs during backward
            or return the intermediaries as the outputs of the custom Function. See the
            `double backward tutorial <https://pytorch.org/tutorials/intermediate/custom_function_double_backward_tutorial.html>`_
            for more details.

            In :func:`backward`, saved tensors can be accessed through the :attr:`saved_tensors`
            attribute. Before returning them to the user, a check is made to ensure
            they weren't used in any in-place operation that modified their content.

            Arguments can also be ``None``. This is a no-op.

            See :ref:`extending-autograd` for more details on how to use this method.

            Example::
                >>> # xdoctest: +REQUIRES(env:TORCH_DOCTEST_AUTOGRAD)
                >>> class Func(Function):
                >>>     @staticmethod
                >>>     def forward(ctx, x: torch.Tensor, y: torch.Tensor, z: int):
                >>>         w = x * z
                >>>         out = x * y + y * z + w * y
                >>>         ctx.save_for_backward(x, y, w, out)
                >>>         ctx.z = z  # z is not a tensor
                >>>         return out
                >>>
                >>>     @staticmethod
                >>>     @once_differentiable
                >>>     def backward(ctx, grad_out):
                >>>         x, y, w, out = ctx.saved_tensors
                >>>         z = ctx.z
                >>>         gx = grad_out * (y + y * z)
                >>>         gy = grad_out * (x + z + w)
                >>>         gz = None
                >>>         return gx, gy, gz
                >>>
                >>> a = torch.tensor(1., requires_grad=True, dtype=torch.double)
                >>> b = torch.tensor(2., requires_grad=True, dtype=torch.double)
                >>> c = 4
                >>> d = Func.apply(a, b, c)

            """
            self.to_save = tensors

        def save_for_forward(self, *tensors: torch.Tensor):
            r"""Saves given tensors for a future call to :func:`~Function.jvp`.

            ``save_for_forward`` should be only called once, from inside the :func:`forward`
            method, and only be called with tensors.

            In :func:`jvp`, saved objects can be accessed through the :attr:`saved_tensors`
            attribute.

            Arguments can also be ``None``. This is a no-op.

            See :ref:`extending-autograd` for more details on how to use this method.

            Example::
                >>> # xdoctest: +SKIP
                >>> class Func(torch.autograd.Function):
                >>>     @staticmethod
                >>>     def forward(ctx, x: torch.Tensor, y: torch.Tensor, z: int):
                >>>         ctx.save_for_backward(x, y)
                >>>         ctx.save_for_forward(x, y)
                >>>         ctx.z = z
                >>>         return x * y * z
                >>>
                >>>     @staticmethod
                >>>     def jvp(ctx, x_t, y_t, _):
                >>>         x, y = ctx.saved_tensors
                >>>         z = ctx.z
                >>>         return z * (y * x_t + x * y_t)
                >>>
                >>>     @staticmethod
                >>>     def vjp(ctx, grad_out):
                >>>         x, y = ctx.saved_tensors
                >>>         z = ctx.z
                >>>         return z * grad_out * y, z * grad_out * x, None
                >>>
                >>>     a = torch.tensor(1., requires_grad=True, dtype=torch.double)
                >>>     t = torch.tensor(1., dtype=torch.double)
                >>>     b = torch.tensor(2., requires_grad=True, dtype=torch.double)
                >>>     c = 4
                >>>
                >>>     with fwAD.dual_level():
                >>>         a_dual = fwAD.make_dual(a, t)
                >>>         d = Func.apply(a_dual, b, c)

            """
            for tensor in tensors:
                assert isinstance(tensor, torch.Tensor) or tensor is None, (
                    "save_for_forward expects all arguments to be tensors; you should "
                    "save non-tensors as attributes on ctx."
                )

            self.saved_for_forward = tensors

        def mark_dirty(self, *args: torch.Tensor):
            r"""Marks given tensors as modified in an in-place operation.

            **This should be called at most once, only from inside the**
            :func:`forward` **method, and all arguments should be inputs.**

            Every tensor that's been modified in-place in a call to :func:`forward`
            should be given to this function, to ensure correctness of our checks.
            It doesn't matter whether the function is called before or after
            modification.

            Examples::
                >>> # xdoctest: +REQUIRES(env:TORCH_DOCTEST_AUTOGRAD)
                >>> class Inplace(Function):
                >>>     @staticmethod
                >>>     def forward(ctx, x):
                >>>         x_npy = x.numpy() # x_npy shares storage with x
                >>>         x_npy += 1
                >>>         ctx.mark_dirty(x)
                >>>         return x
                >>>
                >>>     @staticmethod
                >>>     @once_differentiable
                >>>     def backward(ctx, grad_output):
                >>>         return grad_output
                >>>
                >>> a = torch.tensor(1., requires_grad=True, dtype=torch.double).clone()
                >>> b = a * a
                >>> Inplace.apply(a)  # This would lead to wrong gradients!
                >>>                   # but the engine would not know unless we mark_dirty
                >>> # xdoctest: +SKIP
                >>> b.backward() # RuntimeError: one of the variables needed for gradient
                >>>              # computation has been modified by an inplace operation

            """
            self.dirty_tensors = args

        def mark_shared_storage(self, *pairs):
            warnings.warn(
                "mark_shared_storage is deprecated. "
                "Tensors with shared storages are automatically tracked. Note "
                "that calls to `set_()` are not tracked"
            )

        def mark_non_differentiable(self, *args: torch.Tensor):
            r"""Marks outputs as non-differentiable.

            **This should be called at most once, only from inside the**
            :func:`forward` **method, and all arguments should be tensor outputs.**

            This will mark outputs as not requiring gradients, increasing the
            efficiency of backward computation. You still need to accept a gradient
            for each output in :meth:`~Function.backward`, but it's always going to
            be a zero tensor with the same shape as the shape of a corresponding
            output.

            This is used e.g. for indices returned from a sort. See example::
                >>> class Func(Function):
                >>>     @staticmethod
                >>>     def forward(ctx, x):
                >>>         sorted, idx = x.sort()
                >>>         ctx.mark_non_differentiable(idx)
                >>>         ctx.save_for_backward(x, idx)
                >>>         return sorted, idx
                >>>
                >>>     @staticmethod
                >>>     @once_differentiable
                >>>     def backward(ctx, g1, g2):  # still need to accept g2
                >>>         x, idx = ctx.saved_tensors
                >>>         grad_input = torch.zeros_like(x)
                >>>         grad_input.index_add_(0, idx, g1)
                >>>         return grad_input

            """
            self.non_differentiable = args

        def set_materialize_grads(self, value: bool):
            r"""Sets whether to materialize grad tensors. Default is ``True``.

            **This should be called only from inside the** :func:`forward` **method**

            If ``True``, undefined grad tensors will be expanded to tensors full of zeros
            prior to calling the :func:`backward` and :func:`jvp` methods.

            Example::
                >>> # xdoctest: +REQUIRES(env:TORCH_DOCTEST_AUTOGRAD)
                >>> class SimpleFunc(Function):
                >>>     @staticmethod
                >>>     def forward(ctx, x):
                >>>         return x.clone(), x.clone()
                >>>
                >>>     @staticmethod
                >>>     @once_differentiable
                >>>     def backward(ctx, g1, g2):
                >>>         return g1 + g2  # No check for None necessary
                >>>
                >>> # We modify SimpleFunc to handle non-materialized grad outputs
                >>> class Func(Function):
                >>>     @staticmethod
                >>>     def forward(ctx, x):
                >>>         ctx.set_materialize_grads(False)
                >>>         ctx.save_for_backward(x)
                >>>         return x.clone(), x.clone()
                >>>
                >>>     @staticmethod
                >>>     @once_differentiable
                >>>     def backward(ctx, g1, g2):
                >>>         x, = ctx.saved_tensors
                >>>         grad_input = torch.zeros_like(x)
                >>>         if g1 is not None:  # We must check for None now
                >>>             grad_input += g1
                >>>         if g2 is not None:
                >>>             grad_input += g2
                >>>         return grad_input
                >>>
                >>> a = torch.tensor(1., requires_grad=True)
                >>> b, _ = Func.apply(a)  # induces g2 to be undefined

            """
            self.materialize_grads = value

    # DO NOT USE: This is only defined to be able to load old serialized models
    _ContextMethodMixin = FunctionCtx

    class _HookMixin:
        @staticmethod
        def _register_hook(backward_hooks, hook):
            if backward_hooks is None:
                backward_hooks = OrderedDict()
            handle = hooks.RemovableHandle(backward_hooks)
            backward_hooks[handle.id] = hook
            return backward_hooks, handle

    class BackwardCFunction(_C._FunctionBase, FunctionCtx, _HookMixin):
        def apply(self, *args):
            # _forward_cls is defined by derived class
            # The user should define either backward or vjp but never both.
            backward_fn = self._forward_cls.backward  # type: ignore[attr-defined]
            vjp_fn = self._forward_cls.vjp  # type: ignore[attr-defined]
            if backward_fn is not Function.backward and vjp_fn is not Function.vjp:
                raise RuntimeError(
                    "Implementing both 'backward' and 'vjp' for a custom "
                    "Function is not allowed. You should only implement one "
                    "of them."
                )
            user_fn = vjp_fn if vjp_fn is not Function.vjp else backward_fn
            return user_fn(self, *args)

        def apply_jvp(self, *args):
            # _forward_cls is defined by derived class
            return self._forward_cls.jvp(self, *args)  # type: ignore[attr-defined]

        def _compiled_autograd_key(self):
            return self._forward_cls._compiled_autograd_key(self)  # type: ignore[attr-defined]

    class FunctionMeta(type):
        """Function metaclass.

        This metaclass sets up the following properties:
            _backward_cls: The Function class corresponding to the differentiated
                version of this function (which is generated on the fly by this
                metaclass).
        """

        def __init__(cls, name, bases, attrs):
            backward_fn = type(
                name + "Backward", (BackwardCFunction,), {"_forward_cls": cls}
            )
            cls._backward_cls = backward_fn

            super().__init__(name, bases, attrs)

    class _SingleLevelFunction(
        _C._FunctionBase, FunctionCtx, _HookMixin, metaclass=FunctionMeta
    ):
        @staticmethod
        def forward(ctx: Any, *args: Any, **kwargs: Any) -> Any:
            r"""
            This function is to be overridden by all subclasses. There are two ways
            to define forward:

            Usage 1 (Combined forward and ctx)::

                @staticmethod
                def forward(ctx: Any, *args: Any, **kwargs: Any) -> Any:
                    pass

            - It must accept a context ctx as the first argument, followed by any
              number of arguments (tensors or other types).
            - See :ref:`combining-forward-context` for more details

            Usage 2 (Separate forward and ctx)::

                @staticmethod
                def forward(*args: Any, **kwargs: Any) -> Any:
                    pass

                @staticmethod
                def setup_context(ctx: Any, inputs: Tuple[Any, ...], output: Any) -> None:
                    pass

            - The forward no longer accepts a ctx argument.
            - Instead, you must also override the :meth:`torch.autograd.Function.setup_context`
              staticmethod to handle setting up the ``ctx`` object.
              ``output`` is the output of the forward, ``inputs`` are a Tuple of inputs
              to the forward.
            - See :ref:`extending-autograd` for more details

            The context can be used to store arbitrary data that can be then
            retrieved during the backward pass. Tensors should not be stored
            directly on `ctx` (though this is not currently enforced for
            backward compatibility). Instead, tensors should be saved either with
            :func:`ctx.save_for_backward` if they are intended to be used in
            ``backward`` (equivalently, ``vjp``) or :func:`ctx.save_for_forward`
            if they are intended to be used for in ``jvp``.
            """
            raise NotImplementedError(
                "You must implement the forward function for custom" " autograd.Function."
            )

        @staticmethod
        def setup_context(ctx: Any, inputs: Tuple[Any, ...], output: Any) -> Any:
            r"""There are two ways to define the forward pass of an autograd.Function.

            Either:

            1. Override forward with the signature forward(ctx, *args, **kwargs).
               ``setup_context`` is not overridden. Setting up the ctx for backward
               happens inside the ``forward``.
            2. Override forward with the signature forward(*args, **kwargs) and
               override ``setup_context``. Setting up the ctx for backward happens
               inside ``setup_context`` (as opposed to inside the ``forward``)

            See :meth:`torch.autograd.Function.forward` and :ref:`extending-autograd` for more details.
            """
            raise NotImplementedError("setup_context is not implemented.")

        @staticmethod
        def backward(ctx: Any, *grad_outputs: Any) -> Any:
            r"""Defines a formula for differentiating the operation with backward mode
            automatic differentiation (alias to the vjp function).

            This function is to be overridden by all subclasses.

            It must accept a context :attr:`ctx` as the first argument, followed by
            as many outputs as the :func:`forward` returned (None will be passed in
            for non tensor outputs of the forward function),
            and it should return as many tensors, as there were inputs to
            :func:`forward`. Each argument is the gradient w.r.t the given output,
            and each returned value should be the gradient w.r.t. the
            corresponding input. If an input is not a Tensor or is a Tensor not
            requiring grads, you can just pass None as a gradient for that input.

            The context can be used to retrieve tensors saved during the forward
            pass. It also has an attribute :attr:`ctx.needs_input_grad` as a tuple
            of booleans representing whether each input needs gradient. E.g.,
            :func:`backward` will have ``ctx.needs_input_grad[0] = True`` if the
            first input to :func:`forward` needs gradient computed w.r.t. the
            output.
            """
            raise NotImplementedError(
                "You must implement either the backward or vjp method for "
                "your custom autograd.Function to use it with backward "
                "mode AD."
            )

        # vjp and backward are alias of each other
        vjp = backward

        @staticmethod
        def jvp(ctx: Any, *grad_inputs: Any) -> Any:
            r"""Defines a formula for differentiating the operation with forward mode
            automatic differentiation.
            This function is to be overridden by all subclasses.
            It must accept a context :attr:`ctx` as the first argument, followed by
            as many inputs as the :func:`forward` got (None will be passed in
            for non tensor inputs of the forward function),
            and it should return as many tensors as there were outputs to
            :func:`forward`. Each argument is the gradient w.r.t the given input,
            and each returned value should be the gradient w.r.t. the
            corresponding output. If an output is not a Tensor or the function is not
            differentiable with respect to that output, you can just pass None as a
            gradient for that input.

            You can use the :attr:`ctx` object to pass any value from the forward to this
            functions.
            """
            raise NotImplementedError(
                "You must implement the jvp function for custom "
                "autograd.Function to use it with forward mode AD."
            )

    class Function(_SingleLevelFunction):
        r"""Base class to create custom `autograd.Function`

        To create a custom `autograd.Function`, subclass this class and implement
        the :meth:`forward` and :meth:`backward` static methods. Then, to use your custom
        op in the forward pass, call the class method ``apply``. Do not call
        :meth:`forward` directly.

        To ensure correctness and best performance, make sure you are calling the
        correct methods on ``ctx`` and validating your backward function using
        :func:`torch.autograd.gradcheck`.

        See :ref:`extending-autograd` for more details on how to use this class.

        Examples::

            >>> # xdoctest: +REQUIRES(env:TORCH_DOCTEST_AUTOGRAD)
            >>> class Exp(Function):
            >>>     @staticmethod
            >>>     def forward(ctx, i):
            >>>         result = i.exp()
            >>>         ctx.save_for_backward(result)
            >>>         return result
            >>>
            >>>     @staticmethod
            >>>     def backward(ctx, grad_output):
            >>>         result, = ctx.saved_tensors
            >>>         return grad_output * result
            >>>
            >>> # Use it by calling the apply method:
            >>> # xdoctest: +SKIP
            >>> output = Exp.apply(input)
        """

        def __init__(self, *args, **kwargs):
            cls = self.__class__
            warnings.warn(
                f"{cls} should not be instantiated. Methods on autograd functions"
                "are all static, so you should invoke them on the class itself. "
                "Instantiating an autograd function will raise an "
                "error in a future version of PyTorch.",
                DeprecationWarning,
            )

        def __call__(self, *args, **kwargs):
            raise RuntimeError(
                "Legacy autograd function with non-static forward method is deprecated. "
                "Please use new-style autograd function with static forward method. "
                "(Example: https://pytorch.org/docs/stable/autograd.html#torch.autograd.Function)"
            )

        # for the tracer
        is_traceable = False

        """
        Bool that specifies if PyTorch should attempt to autogenerate
        :func:`torch.vmap` support for this autograd.Function. You may set this to
        True only if this autograd.Function's forward, backward, and jvp (if they
        exist) are written using PyTorch operations; otherwise, please override
        :meth:`torch.autograd.Function.vmap` to add support for :func:`torch.vmap`.

        Please see :ref:`func-autograd-function` for more details.
        """
        generate_vmap_rule = False

        @staticmethod
        def vmap(info, in_dims, *args):
            r"""Defines a rule for the behavior of this autograd.Function underneath
            :func:`torch.vmap`. For a :func:`torch.autograd.Function` to support
            :func:`torch.vmap`, you must either override this staticmethod, or set
            ``generate_vmap_rule`` to ``True`` (you may not do both).

            If you choose to override this staticmethod: it must accept

            - an ``info`` object as the first argument. ``info.batch_size``
              specifies the size of the dimension being vmapped over,
              while ``info.randomness`` is the randomness option passed to
              :func:`torch.vmap`.
            - an ``in_dims`` tuple as the second argument.
              For each arg in ``args``, ``in_dims`` has a corresponding
              ``Optional[int]``. It is ``None`` if the arg is not a Tensor or if
              the arg is not being vmapped over, otherwise, it is an integer
              specifying what dimension of the Tensor is being vmapped over.
            - ``*args``, which is the same as the args to :meth:`~Function.forward`.

            The return of the vmap staticmethod is a tuple of ``(output, out_dims)``.
            Similar to ``in_dims``, ``out_dims`` should be of the same structure as
            ``output`` and contain one ``out_dim`` per output that specifies if the
            output has the vmapped dimension and what index it is in.

            Please see :ref:`func-autograd-function` for more details.
            """
            raise NotImplementedError(
                "To use autograd.Function with vmap, you must either override the "
                "vmap staticmethod or set generate_vmap_rule=True."
            )

        @classmethod
        def apply(cls, *args, **kwargs):
            if not torch._C._are_functorch_transforms_active():
                # See NOTE: [functorch vjp and autograd interaction]
                args = _functorch.utils.unwrap_dead_wrappers(args)
                return super().apply(*args, **kwargs)  # type: ignore[misc]

            if cls.setup_context == _SingleLevelFunction.setup_context:
                raise RuntimeError(
                    "In order to use an autograd.Function with functorch transforms "
                    "(vmap, grad, jvp, jacrev, ...), it must override the setup_context "
                    "staticmethod. For more details, please see "
                    "https://pytorch.org/docs/master/notes/extending.func.html"
                )

            return custom_function_call(cls, *args, **kwargs)

    def once_differentiable(fn):
        @functools.wraps(fn)
        def wrapper(ctx, *args):
            with torch.no_grad():
                outputs = fn(ctx, *args)

            if not torch.is_grad_enabled():
                return outputs

            # If any of the inputs have requires_grad=True, we force the outputs
            # to have requires_grad=True but point to a grad_fn which throws an
            # error message during (double) back-propagation.
            # XXX: this is only an approximation of requires_grad - there's no way
            # to figure out if fn didn't use ctx.saved_tensors and as a result
            # some Tensors might require grad, even if no args do.
            # Unfortunately, this leads to unexpected error messages ("no nodes
            # require computing gradients"), but I don't have a better idea.
            # These functions would raise an error in backward anyway.
            requires_grad = any(
                isinstance(arg, torch.Tensor) and arg.requires_grad for arg in args
            )
            if not requires_grad:
                return outputs

            if not isinstance(outputs, tuple):
                outputs = (outputs,)

            err_fn = _functions.DelayedError(
                b"trying to differentiate twice a function that was marked "
                b"with @once_differentiable",
                len(outputs),
            )

            # Create aliases of each output that has requires_grad=True. We need
            # at least one of the inputs to err_fn to require grad so that the
            # output will have a grad_fn.
            def fake_requires_grad(var):
                if var is not None:
                    var = var.detach()
                    var.requires_grad = True
                return var

            return err_fn(*[fake_requires_grad(v) for v in outputs])

        return wrapper

    def traceable(fn_cls):
        r"""Marks Function as traceable for the JIT.

        Traceable functions have additional restrictions - they can't pass any
        data-dependent values to backward (e.g. Prod passes the output, which makes
        it non-traceable), and their backward should be implemented entirely in terms
        of operations on autograd Tensors in all cases.

        DON'T USE THIS DECORATOR. IT IS FOR INTERNAL USE ONLY AND SHOULD BE HANDLED WITH
        CARE (or can give incorrect results otherwise).
        """
        fn_cls.is_traceable = True
        return fn_cls

    class InplaceFunction(Function):
        def __init__(self, inplace=False):
            super().__init__()
            self.inplace = inplace

    def _nested_map(condition, fn, condition_msg=None):
        def _map(obj):
            if condition(obj):
                return fn(obj)
            elif obj is None:
                return None
            elif isinstance(obj, (list, tuple)):
                mapped = (_map(x) for x in obj)
                if hasattr(obj, "_fields"):
                    # obj is namedtuple
                    return type(obj)(*mapped)
                return type(obj)(mapped)
            elif isinstance(obj, dict):
                return {x: _map(obj[x]) for x in obj}
            else:
                raise ValueError(
                    "Auto nesting doesn't know how to process "
                    "an input object of type "
                    + torch.typename(obj)
                    + (
                        ". Accepted types: " + condition_msg + ", or lists/tuples of them"
                        if condition_msg
                        else ""
                    )
                )

        return _map

    def _jit_unwrap_structured(obj):
        if hasattr(obj, "_jit_unwrap"):
            return obj._jit_unwrap()
        return obj

    def _iter_filter(condition, allow_unknown=False, condition_msg=None, conversion=None):
        def _iter(obj):
            if conversion is not None:
                obj = conversion(obj)
            if condition(obj):
                yield obj
            elif obj is None:
                return
            elif isinstance(obj, (list, tuple)):
                for o in obj:
                    yield from _iter(o)
            elif isinstance(obj, dict):
                # We only accept primitive key types, so we needn't inspect them
                for o in obj.values():
                    yield from _iter(o)
            elif allow_unknown:
                yield obj
            else:
                raise ValueError(
                    "Auto nesting doesn't know how to process "
                    "an input object of type "
                    + torch.typename(obj)
                    + (
                        ". Accepted types: " + condition_msg + ", or lists/tuples of them"
                        if condition_msg
                        else ""
                    )
                )

        return _iter

    def _unflatten(input, proto):
        # unflatten a list or tuple input into a nested list/tuple structure
        # specified by proto
        def unflatten_helper(input, proto):
            res: List[Optional[torch.Tensor]] = []
            if hasattr(proto, "_jit_wrap"):
                return proto._jit_wrap(input)
            if not isinstance(proto, (list, tuple)):
                return input[0], input[1:]
            for e in proto:
                if e is None:
                    res.append(e)
                else:
                    res_e, input = unflatten_helper(input, e)
                    res.append(res_e)
            return type(proto)(res), input

        return unflatten_helper(input, proto)[0]

    _iter_jit_values = _iter_filter(
        lambda o: o is None or isinstance(o, torch._C.Value),
        condition_msg="jit's Values or None",
    )
    _iter_tensors = _iter_filter(
        lambda x: isinstance(x, torch.Tensor),
        condition_msg="Tensors",
        conversion=_jit_unwrap_structured,
    )
    _iter_tensors_permissive = _iter_filter(
        lambda x: isinstance(x, torch.Tensor),
        allow_unknown=True,
        condition_msg="Tensors (permissive)",
    )
    _iter_None_tensors = _iter_filter(
        lambda o: o is None or isinstance(o, torch.Tensor), condition_msg="Tensors or None"
    )
    _map_tensor_data = _nested_map(
        lambda x: isinstance(x, torch.Tensor), lambda o: o.data, condition_msg="Tensors"
    )

    class NestedIOFunction(Function):
        # The 'type: ignore' statements are needed here because these functions are declared as '@staticmethod' in the
        # superclass (Function) but are instance methods here, which mypy reports as incompatible.

        def _do_forward(self, *input):
            self._nested_input = input
            flat_input = tuple(_iter_tensors(input))
            flat_output = super()._do_forward(*flat_input)  # type: ignore[misc]
            nested_output = self._nested_output
            nested_tensors = _unflatten(flat_output, self._nested_output)
            return nested_tensors

        def _do_backward(self, gradients, retain_variables):
            self.retain_variables = retain_variables
            result = super()._do_backward(gradients, retain_variables)  # type: ignore[misc]
            if not retain_variables:
                del self._nested_output
                del self._to_save_nested
            return result

        def backward(self, *gradients: Any) -> Any:  # type: ignore[override]
            nested_gradients = _unflatten(gradients, self._nested_output)
            result = self.backward_extended(*nested_gradients)  # type: ignore[func-returns-value]
            return tuple(_iter_None_tensors(result))

        __call__ = _do_forward

        def forward(self, *args: Any) -> Any:  # type: ignore[override]
            nested_tensors = _map_tensor_data(self._nested_input)
            result = self.forward_extended(*nested_tensors)  # type: ignore[func-returns-value]
            del self._nested_input
            self._nested_output = result
            return tuple(_iter_tensors(result))

        def save_for_backward(self, *args: Any) -> None:
            self.to_save = tuple(_iter_tensors(args))
            self._to_save_nested = args

        @property
        def saved_tensors(self):
            flat_tensors = super().saved_tensors  # type: ignore[misc]
            return _unflatten(flat_tensors, self._to_save_nested)

        def mark_dirty(self, *args: Any, **kwargs: Any) -> None:
            self.dirty_tensors = tuple(_iter_tensors((args, kwargs)))

        def mark_non_differentiable(self, *args: Any, **kwargs: Any) -> None:
            self.non_differentiable = tuple(_iter_tensors((args, kwargs)))

        def forward_extended(self, *input: Any) -> None:
            raise NotImplementedError

        def backward_extended(self, *grad_output: Any) -> None:
            raise NotImplementedError

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