# Copyright 2018 The JAX Authors.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
#     https://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
from __future__ import annotations

import dataclasses
from functools import partial
from typing import (Any, Callable, Dict, Iterable, Optional, Sequence, Set,
                    Tuple, Type, Union)

import numpy as np

import jax
from jax import config
from jax._src import core
from jax._src import source_info_util
from jax._src import linear_util as lu
from jax._src.ad_util import (add_jaxvals, add_jaxvals_p, zeros_like_jaxval,
                              zeros_like_p, Zero, SymbolicZero,
                              replace_rule_output_symbolic_zeros, instantiate)
from jax._src.core import raise_to_shaped, Trace, Tracer, AxisName
from jax._src.interpreters import partial_eval as pe
from jax._src.tree_util import (tree_unflatten, tree_flatten,
                                register_pytree_node)
from jax._src.util import (unzip2, unzip3, safe_map, safe_zip, split_list,
                           canonicalize_axis, moveaxis, as_hashable_function,
                           curry, memoize, weakref_lru_cache)


Array = Any
map, unsafe_map = safe_map, map
zip, unsafe_zip = safe_zip, zip


# Piles

# i:(Fin 3) => f32[[3, 1, 4].i]
@dataclasses.dataclass(frozen=True)
class PileTy:
  binder: core.Var
  length: Union[int, Tracer, core.Var]
  elt_ty: core.DShapedArray
  def __repr__(self) -> str:
    return f'Var{id(self.binder)}:{self.length} => {self.elt_ty}'
  replace = dataclasses.replace

# [3, 1, 4].i
@dataclasses.dataclass(frozen=True)
class IndexedAxisSize:
  idx: core.Var
  lengths: Union[Array, core.Var, Tracer]
  def __repr__(self) -> str:
    return f'{str(self.lengths)}.Var{id(self.idx)}'
  replace = dataclasses.replace

# Pile(aval=a:3 => f32[[3 1 4].a],
#      data=DeviceArray([0., 1., 2., 0., 0., 1., 2., 3.], dtype=float32))
@dataclasses.dataclass(frozen=True)
class Pile:
  aval: PileTy
  data: Array

# To vmap over a pile, one must specify the axis as PileAxis.
class PileAxis: pass
pile_axis = PileAxis()

# As a temporary measure before we have more general JITable / ADable interfaces
# (analogues to vmappable), to enable Piles to be used with other
# transformations and higher-order primitives (primarily jit, though also grad
# with allow_int=True) we register them as pytrees.
# TODO(mattjj): add JITable / ADable interfaces, remove this pytree registration
def _pile_flatten(pile):
  lengths = []
  new_shape = [lengths.append(d.lengths) or d.replace(lengths=len(lengths))
               if type(d) is IndexedAxisSize else d
               for d in pile.aval.elt_ty.shape]
  elt_ty = pile.aval.elt_ty.update(shape=tuple(new_shape))
  aval = pile.aval.replace(elt_ty=elt_ty)
  return (lengths, pile.data), aval
def _pile_unflatten(aval, x):
  lengths, data = x
  new_shape = [d.replace(lengths=lengths[d.lengths - 1])
               if type(d) is IndexedAxisSize else d
               for d in aval.elt_ty.shape]
  elt_ty = aval.elt_ty.update(shape=tuple(new_shape))
  aval = aval.replace(elt_ty=elt_ty)
  return Pile(aval, data)
register_pytree_node(Pile, _pile_flatten, _pile_unflatten)

def _pile_result(axis_size, stacked_axis, ragged_axis, segment_lens, x):
  binder = core.Var(0, '', core.ShapedArray((), np.dtype('int32')))
  if stacked_axis != 0:
    raise NotImplementedError  # TODO Transpose x so the stacked axis is axis 0
  shape = list(x.shape)
  del shape[0]
  shape[ragged_axis-1] = IndexedAxisSize(binder, segment_lens)
  elt_ty = core.DShapedArray(tuple(shape), x.dtype, x.weak_type)
  return Pile(PileTy(binder, axis_size, elt_ty), x)


@dataclasses.dataclass(frozen=True)
class RaggedAxis:
  stacked_axis: int
  # TODO(mattjj,axch): Generalize to multiple ragged dimensions
  # e.g. `i:(Fin 3) => f32[lens1.i, lens2.i]`
  ragged_axis: int
  segment_lengths: Array


def _update_annotation(
    f: lu.WrappedFun, orig_type: Optional[core.InputType],
    axis_size: core.AxisSize, axis_name: AxisName,
    explicit_in_dims: Sequence[Optional[Union[int, RaggedAxis]]],
    segment_lens: Sequence[Array],
  ) -> lu.WrappedFun:
  if orig_type is None: return f
  # By convention, `explicit_in_dims` only accounts for explicit arguments.
  assert len(explicit_in_dims) == sum(explicit for _, explicit in orig_type)
  # We need to:
  #  * if `axis_size` is dynamic, add a new implicit binder (type) for it;
  #  * for each element of `segment_lengths`, add a new explicit binder for it;
  #  * drop other implicit binders, replacing DBIdx which refer to them with
  #    Name objects;
  #  * for each (aval, in_dim) pair: if int-valued in_dim, add batch axis (int
  #    size if `axis_size` is int, otherwise Name); if RaggedAxis-valued in_dim,
  #    add batch axis (int if corresponding segment_lengths is concrete, Name if
  #    not);
  #  * generate full in_type with implicit args too.

  class Name:
    def __init__(self, a): self.a = a
  names = [Name(a) for a, _  in orig_type]
  avals = [a.update(shape=tuple(names[d.val] if type(d) is pe.DBIdx else d  # type: ignore
                                for d in a.shape))
           if type(a) is core.DShapedArray else a for a, e in orig_type if e]

  new_avals = [core.raise_to_shaped(core.get_aval(s)) for s in segment_lens]
  sz = Name(axis_size.aval) if isinstance(axis_size, Tracer) else axis_size
  for a, d in zip(avals, explicit_in_dims):
    if isinstance(d, RaggedAxis):
      raise NotImplementedError
    else:
      new_avals.append(core.unmapped_aval(sz, axis_name, d, a))  # type: ignore

  mentioned = {d for a in new_avals if type(a) is core.DShapedArray
               for d in a.shape if type(d) is Name}
  expl_names = set(map(Name, new_avals))
  impl_names = mentioned - expl_names  # type: ignore
  impl_part = [(n.a, False) for n in impl_names]  # type: ignore
  name_map = {n: pe.DBIdx(i) for i, n in enumerate((*impl_names, *expl_names))}
  expl_part = [(a.update(shape=tuple(name_map.get(d, d) for d in a.shape))
                if type(a) is core.DShapedArray else a, True) for a in new_avals]
  return lu.annotate(f, (*impl_part, *expl_part))

### vmappable typeclass

Vmappable = Any
Elt = Any
MapSpec = Any
AxisSize = Any
GetIdx = Callable[[], Tracer]  # TODO(mattjj): revise this laziness
ToEltHandler = Callable[[Callable, GetIdx, Vmappable, MapSpec], Elt]
FromEltHandler = Callable[[Callable, AxisSize, Elt, MapSpec], Vmappable]
MakeIotaHandler = Callable[[AxisSize], Array]

def to_elt(trace: Trace, get_idx: GetIdx, x: Vmappable, spec: MapSpec) -> Elt:
  handler = to_elt_handlers.get(type(x))
  if handler:
    return handler(partial(to_elt, trace, get_idx), get_idx, x, spec)
  elif type(x) is Pile:
    if spec is not pile_axis:
      raise TypeError("pile input without using pile_axis in_axes spec")
    (d, ias), = ((i, sz) for i, sz in enumerate(x.aval.elt_ty.shape)
                 if type(sz) is IndexedAxisSize)
    return BatchTracer(trace, x.data, RaggedAxis(0, d+1, ias.lengths))  # type: ignore
  elif isinstance(spec, int) or spec is None:
    spec = spec and canonicalize_axis(spec, len(np.shape(x)))
    return (BatchTracer(trace, x, spec, source_info_util.current())
            if spec is not None else x)
  else:
    assert False
to_elt_handlers: Dict[Type, ToEltHandler] = {}

def from_elt(trace: 'BatchTrace', axis_size: AxisSize, x: Elt, spec: MapSpec
             ) -> Vmappable:
  handler = from_elt_handlers.get(type(x))
  if handler:
    return handler(partial(from_elt, trace), axis_size, x, spec)
  x_ = trace.full_raise(x)
  val, bdim = x_.val, x_.batch_dim
  if type(bdim) is RaggedAxis:
    if spec is not pile_axis:
      # TODO(mattjj): improve this error message
      raise TypeError("ragged output without using pile_axis out_axes spec")
    return _pile_result(axis_size, bdim.stacked_axis, bdim.ragged_axis, bdim.segment_lengths, val)
  else:
    return matchaxis(trace.axis_name, axis_size, x_.batch_dim, spec, x_.val)
from_elt_handlers: Dict[Type, FromEltHandler] = {}

def make_iota(axis_size: AxisSize) -> Array:
  handler = make_iota_handlers.get(type(axis_size))
  if handler:
    return handler(axis_size)
  else:
    return jax.lax.iota('int32', int(axis_size))
make_iota_handlers: Dict[Type, MakeIotaHandler] = {}

def register_vmappable(data_type: Type, spec_type: Type, axis_size_type: Type,
                       to_elt: Callable, from_elt: Callable,
                       make_iota: Optional[Callable]):
  vmappables[data_type] = (spec_type, axis_size_type)
  spec_types.add(spec_type)
  to_elt_handlers[data_type] = to_elt
  from_elt_handlers[data_type] = from_elt
  if make_iota: make_iota_handlers[axis_size_type] = make_iota
vmappables: Dict[Type, Tuple[Type, Type]] = {}
spec_types: Set[Type] = {PileAxis}

def unregister_vmappable(data_type: Type) -> None:
  spec_type, axis_size_type = vmappables.pop(data_type)
  spec_types.remove(spec_type)
  del to_elt_handlers[data_type]
  del from_elt_handlers[data_type]
  if axis_size_type in make_iota_handlers:
    del make_iota_handlers[axis_size_type]

def is_vmappable(x: Any) -> bool:
  return type(x) is Pile or type(x) in vmappables

@lu.transformation_with_aux
def flatten_fun_for_vmap(in_tree, *args_flat):
  py_args, py_kwargs = tree_unflatten(in_tree, args_flat)
  ans = yield py_args, py_kwargs
  yield tree_flatten(ans, is_leaf=is_vmappable)

### tracer

# TODO(mattjj): use a special sentinel type rather than None
NotMapped = type(None)
not_mapped = None


class BatchTracer(Tracer):
  __slots__ = ['val', 'batch_dim', 'source_info']

  def __init__(self, trace, val, batch_dim: Union[NotMapped, int, RaggedAxis],
               source_info: Optional[source_info_util.SourceInfo] = None):
    if config.jax_enable_checks:
      assert type(batch_dim) in (NotMapped, int, RaggedAxis)
      if type(batch_dim) is int:
        aval = raise_to_shaped(core.get_aval(val))
        assert 0 <= batch_dim < len(aval.shape)  # type: ignore
    self._trace = trace
    self.val = val
    self.batch_dim = batch_dim
    self.source_info = source_info

  @property
  def aval(self):
    aval = raise_to_shaped(core.get_aval(self.val))
    if self.batch_dim is not_mapped:
      return aval
    elif type(self.batch_dim) is int:
      return core.mapped_aval(aval.shape[self.batch_dim], self.batch_dim, aval)
    elif type(self.batch_dim) is RaggedAxis:
      new_aval = core.mapped_aval(
        aval.shape[self.batch_dim.stacked_axis], self.batch_dim.stacked_axis, aval)
      shape = list(new_aval.shape)  # type: ignore
      size_tracer = BatchTracer(self._trace, self.batch_dim.segment_lengths, 0)
      ragged_axis = self.batch_dim.ragged_axis
      if self.batch_dim.stacked_axis < self.batch_dim.ragged_axis:
        ragged_axis -= 1
      shape[ragged_axis] = size_tracer
      return core.DShapedArray(shape=tuple(shape), dtype=aval.dtype,
                               weak_type=aval.weak_type)

  def full_lower(self):
    if self.batch_dim is not_mapped:
      return core.full_lower(self.val)
    else:
      return self

  def _origin_msg(self):
    if self.source_info is None:
      return ""
    return (f"\nThis BatchTracer with object id {id(self)} was created on line:"
            f"\n  {source_info_util.summarize(self.source_info)}")

  def _contents(self):
    return [('val', self.val), ('batch_dim', self.batch_dim)]

  def get_referent(self):
    if self.batch_dim is None or type(self.batch_dim) is int:
      return core.get_referent(self.val)
    else:  # TODO(mattjj): could handle the RaggedAxis case?
      return self

class BatchTrace(Trace):

  def __init__(self, *args, axis_name, spmd_axis_name = None):
    super().__init__(*args)
    self.axis_name = axis_name
    self.spmd_axis_name = spmd_axis_name

  def pure(self, val):
    return BatchTracer(self, val, not_mapped, source_info_util.current())

  def lift(self, val):
    return BatchTracer(self, val, not_mapped, source_info_util.current())

  def sublift(self, val):
    return BatchTracer(self, val.val, val.batch_dim, source_info_util.current())

  def get_primitive_batcher(self, primitive, frame):
    if primitive in primitive_batchers:
      return primitive_batchers[primitive]
    elif self.spmd_axis_name is not None and primitive in spmd_axis_primitive_batchers:
      return partial(spmd_axis_primitive_batchers[primitive],
                     self.spmd_axis_name, frame.size, frame.name,
                     frame.main_trace.trace_type)
    elif primitive in axis_primitive_batchers:
      return self.get_axis_primitive_batcher(primitive, frame)
    msg = "Batching rule for '{}' not implemented"
    raise NotImplementedError(msg.format(primitive))

  def get_axis_primitive_batcher(self, primitive, frame):
    return partial(axis_primitive_batchers[primitive],
        frame.size, frame.name, frame.main_trace.trace_type)

  def get_frame(self, vals, dims) -> core.AxisEnvFrame:
    if any(d is not not_mapped for d in dims):
      sizes = (x.shape[d] if type(d) is int else len(d.segment_lengths)
               for x, d in zip(vals, dims) if d is not not_mapped)
      axis_size, = core.dedup_referents(sizes)
    else:
      axis_size = None  # can't be inferred from data
    if self.axis_name is core.no_axis_name:
      assert axis_size is not None  # must be inferrable from data
      return core.AxisEnvFrame(self.axis_name, axis_size, self.main)
    frame = core.axis_frame(self.axis_name, self.main)
    assert axis_size is None or axis_size == frame.size, (axis_size, frame.size)
    assert frame.main_trace is self.main
    return frame

  def process_primitive(self, primitive, tracers, params):
    vals_in, dims_in = unzip2((t.val, t.batch_dim) for t in tracers)
    is_axis_primitive = primitive in axis_primitive_batchers
    used_names = core.used_axis_names(primitive, params)
    if is_axis_primitive and _main_trace_for_axis_names(self.main, used_names):
      frame = self.get_frame(vals_in, dims_in)
      batcher_primitive = self.get_axis_primitive_batcher(primitive, frame)
      val_out, dim_out = batcher_primitive(vals_in, dims_in, **params)
    elif all(bdim is not_mapped for bdim in dims_in):
      return primitive.bind(*vals_in, **params)
    else:
      frame = self.get_frame(vals_in, dims_in)
      batched_primitive = self.get_primitive_batcher(primitive, frame)
      val_out, dim_out = batched_primitive(vals_in, dims_in, **params)
    src = source_info_util.current()
    if primitive.multiple_results:
      return [BatchTracer(self, x, d, src) for x, d in zip(val_out, dim_out)]
    else:
      return BatchTracer(self, val_out, dim_out, src)

  def process_call(self, call_primitive, f, tracers, params):
    assert call_primitive.multiple_results
    params = dict(params, name=params.get('name', f.__name__))
    vals, dims = unzip2((t.val, t.batch_dim) for t in tracers)
    if all(bdim is not_mapped for bdim in dims):
      return call_primitive.bind(f, *vals, **params)
    sizes = (x.shape[d] if type(d) is int else len(d.segment_lengths)
             for x, d in zip(vals, dims) if d is not not_mapped)
    axis_size, = core.dedup_referents(sizes)
    f_, dims_out = batch_subtrace(f, self.main, tuple(dims))
    f_ = _update_annotation(f_, f.in_type, axis_size, self.axis_name, dims, [])
    vals_out = call_primitive.bind(f_, *vals, **params)
    src = source_info_util.current()
    return [BatchTracer(self, v, d, src) for v, d in zip(vals_out, dims_out)]

  def post_process_call(self, call_primitive, out_tracers, params):
    vals, dims, srcs = unzip3((t.val, t.batch_dim, t.source_info)
                              for t in out_tracers)
    main = self.main
    def todo(vals):
      trace = main.with_cur_sublevel()
      return map(partial(BatchTracer, trace), vals, dims, srcs)
    return vals, todo

  def process_map(self, map_primitive, f: lu.WrappedFun, tracers, params):
    vals, dims = unzip2((t.val, t.batch_dim) for t in tracers)
    if all(dim is not_mapped for dim in dims):
      return map_primitive.bind(f, *vals, **params)
    else:
      assert len({x.shape[d] for x, d in zip(vals, dims) if d is not not_mapped}) == 1
      # The logic for the dimension math below is as follows:
      # ╔═════════════╦════════════════════════════════════════╦═══════════╗
      # ║ d / in_axis ║ None                                   ║ int       ║
      # ╠═════════════╬════════════════════════════════════════╩═══════════╣
      # ║ None        ║ No extra axis, so in_axis unaffected               ║
      # ╠═════════════╬════════════════════════════════════════╦═══════════╣
      # ║ int         ║ Not mapped, so batching dim unaffected ║ See below ║
      # ╚═════════════╩════════════════════════════════════════╩═══════════╝
      # When both d and in_axis are defined then:
      # - If `d <= in_axis`, we have to move the `in_axis` one dimension further;
      # - If `d >  in_axis`, we have to decrement `d` (as `in_axis` will get removed).
      def both_mapped(in_out_axis, d):
        return in_out_axis is not None and d is not not_mapped
      new_in_axes = tuple(
        in_axis + 1 if both_mapped(in_axis, d) and d <= in_axis else in_axis
        for d, in_axis in zip(dims, params['in_axes']))
      new_dims = tuple(
        d - 1 if both_mapped(in_axis, d) and in_axis < d else d
        for d, in_axis in zip(dims, params['in_axes']))
      f, dims_out = batch_subtrace(f, self.main, new_dims)
      out_axes_thunk = params['out_axes_thunk']
      # NOTE: This assumes that the choice of the dimensions over which outputs
      #       are batched is entirely dependent on the function and not e.g. on the
      #       data or its shapes.
      @as_hashable_function(closure=out_axes_thunk)
      def new_out_axes_thunk():
        return tuple(out_axis + 1 if both_mapped(out_axis, d) and d < out_axis else out_axis
                     for out_axis, d in zip(out_axes_thunk(), dims_out()))
      new_params = dict(params, in_axes=new_in_axes, out_axes_thunk=new_out_axes_thunk)
      vals_out = map_primitive.bind(f, *vals, **new_params)
      dims_out_ = [d + 1 if both_mapped(out_axis, d) and out_axis <= d else d
                   for d, out_axis in zip(dims_out(), out_axes_thunk())]
      src = source_info_util.current()
      return [BatchTracer(self, v, d, src) for v, d in zip(vals_out, dims_out_)]

  def post_process_map(self, call_primitive, out_tracers, params):
    vals, dims, srcs = unzip3((t.val, t.batch_dim, t.source_info)
                              for t in out_tracers)
    main = self.main
    def both_mapped(in_out_axis, d):
      return in_out_axis is not None and d is not not_mapped
    def todo(vals):
      trace = main.with_cur_sublevel()
      return [BatchTracer(trace, v, d + 1 if both_mapped(oa, d) and oa <= d else d, s)
              for v, d, oa, s in zip(vals, dims, params['out_axes_thunk'](), srcs)]
    if call_primitive.map_primitive:
      def out_axes_transform(out_axes):
        return tuple(out_axis + 1 if both_mapped(out_axis, d) and d < out_axis else out_axis
                     for out_axis, d in zip(out_axes, dims))
      todo = (todo, out_axes_transform)
    return vals, todo

  def process_custom_jvp_call(self, prim, fun, jvp, tracers, *, symbolic_zeros):
    in_vals, in_dims = unzip2((t.val, t.batch_dim) for t in tracers)
    fun, out_dims1 = batch_subtrace(fun, self.main, in_dims)
    jvp, out_dims2 = batch_custom_jvp_subtrace(jvp, self.main, in_dims)
    out_vals = prim.bind(fun, jvp, *in_vals, symbolic_zeros=symbolic_zeros)
    fst, out_dims = lu.merge_linear_aux(out_dims1, out_dims2)
    if not fst:
      assert out_dims == out_dims[:len(out_dims) // 2] * 2
      out_dims = out_dims[:len(out_dims) // 2]
    src = source_info_util.current()
    return [BatchTracer(self, v, d, src) for v, d in zip(out_vals, out_dims)]

  def post_process_custom_jvp_call(self, out_tracers, jvp_was_run):
    vals, dims, srcs = unzip3((t.val, t.batch_dim, t.source_info)
                              for t in out_tracers)
    main = self.main
    def todo(vals):
      trace = main.with_cur_sublevel()
      if jvp_was_run:
        primal_dims, tangent_dims = dims[:len(vals)], dims[len(vals):]
        assert primal_dims == tangent_dims
        primal_srcs = srcs[:len(vals)]
        return map(partial(BatchTracer, trace), vals, primal_dims, primal_srcs)
      else:
        return map(partial(BatchTracer, trace), vals, dims, srcs)
    return vals, todo

  def process_custom_vjp_call(self, prim, fun, fwd, bwd, tracers, *, out_trees,
                              symbolic_zeros):  # pytype: disable=signature-mismatch
    in_vals, in_dims = unzip2((t.val, t.batch_dim) for t in tracers)
    axis_size, = {x.shape[d] for x, d in zip(in_vals, in_dims)
                  if d is not not_mapped}
    fwd_in_dims = [d for in_dim in in_dims for d in [in_dim, not_mapped]]
    fun, out_dims1 = batch_subtrace(fun, self.main, in_dims)
    fwd, out_dims2 = batch_subtrace(fwd, self.main, fwd_in_dims)
    bwd = batch_custom_vjp_bwd(bwd, self.axis_name, axis_size,
                               out_dims2, in_dims, self.main.trace_type,
                               self.spmd_axis_name)
    out_vals = prim.bind(fun, fwd, bwd, *in_vals, out_trees=out_trees,
                         symbolic_zeros=symbolic_zeros)
    fst, out_dims = lu.merge_linear_aux(out_dims1, out_dims2)
    if not fst:
      _, res_tree = out_trees()
      _, out_dims = split_list(out_dims, [res_tree.num_leaves])
    src = source_info_util.current()
    return [BatchTracer(self, v, d, src) for v, d in zip(out_vals, out_dims)]

  def post_process_custom_vjp_call(self, out_tracers, _):
    vals, dims, srcs = unzip3((t.val, t.batch_dim, t.source_info)
                              for t in out_tracers)
    main = self.main
    def todo(vals):
      trace = main.with_cur_sublevel()
      return map(partial(BatchTracer, trace), vals, dims, srcs)
    return vals, todo

  def post_process_custom_vjp_call_fwd(self, out_tracers, out_trees):
    vals, dims, srcs = unzip3((t.val, t.batch_dim, t.source_info)
                              for t in out_tracers)
    axis_size, = {x.shape[d] for x, d in zip(vals, dims) if d is not not_mapped}
    main, trace_type = self.main, self.main.trace_type
    axis_name = self.axis_name
    _, res_tree = out_trees()
    num_res = res_tree.num_leaves
    res_dims, primal_dims = split_list(dims, [num_res])
    _, primal_srcs = split_list(srcs, [num_res])
    def todo(vals):
      trace = main.with_cur_sublevel()
      return map(partial(BatchTracer, trace), vals, primal_dims, primal_srcs)
    def bwd_transform(bwd):
      return batch_custom_vjp_bwd(bwd, axis_name, axis_size, dims, (None,),
                                  trace_type, self.spmd_axis_name)
    return vals, todo, bwd_transform

def _main_trace_for_axis_names(main_trace: core.MainTrace,
                               axis_name: Iterable[AxisName],
                               ) -> bool:
  # This function exists to identify whether a main trace corresponds to any of
  # the axis names used by a primitive. Axis names alone aren't enough because
  # axis names can shadow, so we use the main trace as a tag.
  return any(main_trace is core.axis_frame(n).main_trace for n in axis_name)

### API for batching callables with vmappable inputs and outputs

def batch(fun: lu.WrappedFun, axis_name: AxisName, axis_size,
          in_dims, out_dim_dests, main_type: Type[BatchTrace] = BatchTrace,
          spmd_axis_name: Optional[Tuple[AxisName, ...]] = None
          ) -> lu.WrappedFun:
  # we split up _batch_inner and _batch_outer for the leak checker
  f = _batch_inner(fun, axis_size, out_dim_dests)
  return _batch_outer(f, axis_name, axis_size, in_dims, main_type,
                      spmd_axis_name)

@lu.transformation
def _batch_outer(axis_name, axis_size, in_dims, main_type, spmd_axis_name,
                 *in_vals):
  with core.new_main(
      main_type, axis_name=axis_name, spmd_axis_name=spmd_axis_name) as main:
    with core.extend_axis_env(axis_name, axis_size, main):
      with source_info_util.transform_name_stack('vmap'):
        outs = yield (main, in_dims, *in_vals), {}
      del main
  yield outs

@lu.transformation
def _batch_inner(axis_size, out_dim_dests, main, in_dims, *in_vals):
  in_dims = in_dims() if callable(in_dims) else in_dims
  trace = main.with_cur_sublevel()
  idx = memoize(lambda: BatchTracer(trace, make_iota(axis_size), 0,
                                    source_info_util.current()))
  in_tracers = map(partial(to_elt, trace, idx), in_vals, in_dims)
  outs = yield in_tracers, {}
  out_dim_dests = out_dim_dests() if callable(out_dim_dests) else out_dim_dests
  out_vals = map(partial(from_elt, trace, axis_size), outs, out_dim_dests)
  yield out_vals

# NOTE: This divides the in_axes by the tile_size and multiplies the out_axes by it.
def vtile(f_flat: lu.WrappedFun,
          in_axes_flat: Tuple[Optional[int], ...],
          out_axes_flat: Tuple[Optional[int], ...],
          tile_size: Optional[int],
          axis_name: AxisName,
          main_type: Type[BatchTrace] = BatchTrace):
  @curry
  def tile_axis(arg, axis: Optional[int], tile_size):
    if axis is None:
      return arg
    shape = list(arg.shape)
    shape[axis:axis+1] = [tile_size, shape[axis] // tile_size]
    return arg.reshape(shape)

  def untile_axis(out, axis: Optional[int]):
    if axis is None:
      return out
    shape = list(out.shape)
    shape[axis:axis+2] = [shape[axis] * shape[axis+1]]
    return out.reshape(shape)

  @lu.transformation
  def _map_to_tile(*args_flat):
    sizes = (x.shape[i] for x, i in safe_zip(args_flat, in_axes_flat) if i is not None)
    tile_size_ = tile_size or next(sizes, None)
    assert tile_size_ is not None, "No mapped arguments?"
    outputs_flat = yield map(tile_axis(tile_size=tile_size_), args_flat, in_axes_flat), {}
    yield map(untile_axis, outputs_flat, out_axes_flat)

  return _map_to_tile(batch(
      f_flat, axis_name, tile_size, in_axes_flat, out_axes_flat, main_type=main_type))

### API for batching functions with jaxpr type inputs and outputs

@lu.transformation_with_aux
def batch_subtrace(main, in_dims, *in_vals):
  trace = main.with_cur_sublevel()
  in_dims = in_dims() if callable(in_dims) else in_dims
  in_tracers = [BatchTracer(trace, x, dim, source_info_util.current())
                if dim is not None else x for x, dim in zip(in_vals, in_dims)]
  outs = yield in_tracers, {}
  out_tracers = map(trace.full_raise, outs)
  out_vals, out_dims = unzip2((t.val, t.batch_dim) for t in out_tracers)
  yield out_vals, out_dims


### API for batching jaxprs

def batch_jaxpr2(closed_jaxpr: core.ClosedJaxpr,
                 axis_size: core.AxisSize,
                 in_axes: Tuple[Union[int, NotMapped], ...],
                 axis_name: AxisName,
                 spmd_axis_name: AxisName,
                 main_type: Type[BatchTrace],
                 ) -> Tuple[core.ClosedJaxpr, Tuple[Union[int, NotMapped], ...]]:
  return _batch_jaxpr2(closed_jaxpr, axis_size, tuple(in_axes), axis_name,
                       spmd_axis_name, main_type)

@weakref_lru_cache
def _batch_jaxpr2(closed_jaxpr: core.ClosedJaxpr,
                 axis_size: core.AxisSize,
                 in_axes: Tuple[Union[int, NotMapped], ...],
                 axis_name: AxisName,
                 spmd_axis_name: AxisName,
                 main_type: Type[BatchTrace],
                 ) -> Tuple[core.ClosedJaxpr, Tuple[Union[int, NotMapped], ...]]:
  f = lu.wrap_init(core.jaxpr_as_fun(closed_jaxpr))
  f, out_axes = _batch_jaxpr_inner(f, axis_size)
  f = _batch_jaxpr_outer(f, axis_name, spmd_axis_name, axis_size, in_axes,
                         main_type)
  avals_in = [core.unmapped_aval(axis_size, axis_name, b, aval)
              if b is not not_mapped else aval
              for aval, b in unsafe_zip(closed_jaxpr.in_avals, in_axes)]
  jaxpr_out, _, consts = pe.trace_to_jaxpr_dynamic(f, avals_in)
  return core.ClosedJaxpr(jaxpr_out, consts), out_axes()

def batch_jaxpr(closed_jaxpr, axis_size, in_batched, instantiate, axis_name,
                spmd_axis_name, main_type):
  inst = tuple(instantiate) if isinstance(instantiate, list) else instantiate
  return _batch_jaxpr(closed_jaxpr, axis_size, tuple(in_batched), inst,
                      axis_name, spmd_axis_name, main_type)

def _batch_jaxpr(closed_jaxpr, axis_size, in_batched, instantiate, axis_name,
                 spmd_axis_name, main_type):
  assert (isinstance(instantiate, bool) or
          isinstance(instantiate, (list, tuple)) and
          all(isinstance(b, bool) for b in instantiate))
  if isinstance(instantiate, bool):
    instantiate = [instantiate] * len(closed_jaxpr.out_avals)
  in_axes = [0 if b else not_mapped for b in in_batched]
  out_axes_dest = [0 if inst else zero_if_mapped for inst in instantiate]
  return batch_jaxpr_axes(closed_jaxpr, axis_size, in_axes, out_axes_dest,
                          axis_name, spmd_axis_name, main_type)

def batch_jaxpr_axes(closed_jaxpr, axis_size, in_axes, out_axes_dest, axis_name,
                     spmd_axis_name, main_type):
  return _batch_jaxpr_axes(closed_jaxpr, axis_size, tuple(in_axes),
                           tuple(out_axes_dest), axis_name, spmd_axis_name,
                           main_type)

@weakref_lru_cache
def _batch_jaxpr_axes(closed_jaxpr, axis_size, in_axes, out_axes_dest,
                      axis_name, spmd_axis_name, main_type):
  f = lu.wrap_init(core.jaxpr_as_fun(closed_jaxpr))
  f, out_axes = _batch_jaxpr_inner(f, axis_size)
  f, out_batched = _match_axes_jaxpr(f, axis_size, out_axes_dest, out_axes)
  f = _batch_jaxpr_outer(f, axis_name, spmd_axis_name, axis_size, in_axes,
                         main_type)
  avals_in = [core.unmapped_aval(axis_size, axis_name, b, aval) if b is not not_mapped
              else aval for aval, b in unsafe_zip(closed_jaxpr.in_avals, in_axes)]
  jaxpr_out, _, consts = pe.trace_to_jaxpr_dynamic(f, avals_in)
  return core.ClosedJaxpr(jaxpr_out, consts), out_batched()

@lu.transformation_with_aux
def _batch_jaxpr_inner(axis_size, main, in_axes, *in_vals):
  trace = main.with_cur_sublevel()
  in_tracers = [BatchTracer(trace, val, dim) if dim is not None else val
                for val, dim in zip(in_vals, in_axes)]
  outs = yield in_tracers, {}
  out_tracers = map(trace.full_raise, outs)
  out_vals, out_axes = unzip2((t.val, t.batch_dim) for t in out_tracers)
  yield out_vals, out_axes

@lu.transformation_with_aux
def _match_axes_jaxpr(axis_size, out_axes_dest, out_axes, main, in_axes,
                      *in_vals):
  trace = main.with_cur_sublevel()
  out_vals = yield (main, in_axes, *in_vals), {}
  out_axes = out_axes()
  out_axes_dest = [(None if src is not_mapped else 0)
                   if dst is zero_if_mapped else dst
                   for src, dst in unsafe_zip(out_axes, out_axes_dest)]
  if len(out_axes_dest) != len(out_axes):
    out_axis_dest, = out_axes_dest
    out_axes_dest = [out_axis_dest] * len(out_axes)
  out_vals = map(partial(matchaxis, trace.axis_name, axis_size),
                 out_axes, out_axes_dest, out_vals)
  out_batched = [dst is not None for dst in out_axes_dest]
  yield out_vals, out_batched

@lu.transformation
def _batch_jaxpr_outer(axis_name, spmd_axis_name, axis_size, in_dims, main_type,
                       *in_vals):
  if axis_size is None:
    axis_size, = {x.shape[d] for x, d in zip(in_vals, in_dims) if d is not not_mapped}
  in_dims = in_dims() if callable(in_dims) else in_dims
  in_dims = [canonicalize_axis(ax, np.ndim(x)) if isinstance(ax, int)
             else ax for x, ax in unsafe_zip(in_vals, in_dims)]
  with core.new_main(main_type, axis_name=axis_name,
                     spmd_axis_name=spmd_axis_name) as main:
    with core.extend_axis_env(axis_name, axis_size, main):
      out_vals = yield (main, in_dims, *in_vals), {}
      del main
  yield out_vals

def _merge_bdims(x, y):
  if x == y:
    return x
  elif x is not_mapped:
    return y
  elif y is not_mapped:
    return x
  else:
    return x  # arbitrary

class ZeroIfMapped: pass
zero_if_mapped = ZeroIfMapped()

### functions for handling custom_vjp

@lu.transformation_with_aux
def batch_custom_jvp_subtrace(main, in_dims, *in_vals):
  size, = {x.shape[d] for x, d in zip(in_vals, in_dims * 2)
           if d is not not_mapped}
  trace = main.with_cur_sublevel()
  in_tracers = [val if dim is None else
                SymbolicZero(core.mapped_aval(size, dim, val.aval))
                if type(val) is SymbolicZero else BatchTracer(trace, val, dim)
                for val, dim in zip(in_vals, in_dims * 2)]
  outs = yield in_tracers, {}
  # TODO(mattjj,frostig): instantiating any SymbolicZero output is easy, but can
  # be wasteful in the rare case it actually triggers; handle symbolically!
  outs = [instantiate(replace_rule_output_symbolic_zeros(x)) for x in outs]
  out_tracers = map(trace.full_raise, outs)
  out_vals, out_dims = unzip2((t.val, t.batch_dim) for t in out_tracers)
  out_primals, out_tangents = split_list(out_vals, [len(out_vals) // 2])
  out_primal_bds, out_tangent_bds = split_list(out_dims, [len(out_vals) // 2])
  out_dims = map(_merge_bdims, out_primal_bds, out_tangent_bds)
  out_primals  = map(partial(matchaxis, trace.axis_name, size),
                     out_primal_bds, out_dims,  out_primals)
  out_tangents = map(partial(matchaxis, trace.axis_name, size),
                     out_tangent_bds, out_dims, out_tangents)
  yield out_primals + out_tangents, out_dims * 2

def batch_custom_vjp_bwd(bwd, axis_name, axis_size, in_dims, out_dim_dests,
                         main_type, spmd_axis_name):
  def new_bwd(*args):
    in_dims_ = in_dims() if callable(in_dims) else in_dims
    args = [SymbolicZero(core.mapped_aval(axis_size, dim, x.aval))
            if type(x) is SymbolicZero else x
            for x, dim in zip(args, in_dims_)]
    in_dims_ = [None if type(x) is SymbolicZero else d
                for x, d in zip(args, in_dims_)]
    bwd_, out_dims_thunk = batch_subtrace(lu.wrap_init(bwd))
    bwd_ = _batch_outer(bwd_, axis_name, axis_size, in_dims_, main_type,
                        spmd_axis_name)
    bwd_ = _match_axes_and_sum(bwd_, axis_size, axis_name, out_dims_thunk,
                               out_dim_dests)
    return bwd_.call_wrapped(*args)
  return new_bwd

@lu.transformation
def _match_axes_and_sum(axis_size, axis_name, out_dims_thunk, out_dim_dests, *in_vals):
  # this is like _match_axes, but we do reduce-sums as needed
  out_vals = yield in_vals, {}
  yield map(partial(_matchaxis_symbolic_zeros, axis_name, axis_size, axis_name,
                    sum_match=True), out_dims_thunk(), out_dim_dests, out_vals)

def _matchaxis_symbolic_zeros(axis_name, sz, name, src, dst, x, sum_match=False):
  # Just like `matchaxis`, but handles symbolic zeros using ad_util.py
  # TODO(mattjj): dedup with matchaxis
  if isinstance(x, (Zero, SymbolicZero)):
    if src == dst:
      return x
    elif type(src) == type(dst) == int:
      aval = core.mapped_aval(sz, src, x.aval)
      return Zero(core.unmapped_aval(sz, name, dst, aval))
    elif src is not_mapped and dst is not not_mapped:
      return Zero(core.unmapped_aval(sz, name, dst, x.aval))
    elif dst is not_mapped and sum_match:
      return Zero(core.mapped_aval(sz, src, x.aval))
    else:
      raise ValueError((axis_name, x, src, dst))
  else:
    return matchaxis(axis_name, sz, src, dst, x, sum_match=sum_match)


### utilities for defining primitives' batching rules

BatchingRule = Callable[..., Tuple[Any, Union[None, int, Tuple[Union[None, int], ...]]]]
primitive_batchers : Dict[core.Primitive, BatchingRule] = {}
axis_primitive_batchers: Dict[core.Primitive, Callable] = {}
spmd_axis_primitive_batchers: Dict[core.Primitive, Callable] = {}

def defvectorized(prim):
  primitive_batchers[prim] = partial(vectorized_batcher, prim)

def vectorized_batcher(prim, batched_args, batch_dims, **params):
  assert all(batch_dims[0] == bd for bd in batch_dims[1:]), batch_dims
  return prim.bind(*batched_args, **params), batch_dims[0]

def defbroadcasting(prim):
  primitive_batchers[prim] = partial(broadcast_batcher, prim)

def broadcast_batcher(prim, args, dims, **params):
  """Process a primitive with built-in broadcasting.

  Args:
    args: the possibly-batched arguments
    dims: list or tuple of the same length as `args`, where each
      entry indicates the batching state of the corresponding entry to `args`:
      either an int indicating the batch dimension, or else `not_mapped`
      indicating no batching.
  """
  assert len(args) > 1
  shape, dim = next((x.shape, d) for x, d in zip(args, dims)
                    if d is not not_mapped)
  if all(core.symbolic_equal_shape(shape, x.shape) and d == dim
         for x, d in zip(args, dims) if np.ndim(x)):
    # if there's only agreeing batch dims and scalars, just call the primitive
    out = prim.bind(*args, **params)
    return (out, (dim,) * len(out)) if prim.multiple_results else (out, dim)
  else:
    # We pass size of 1 here because (1) at least one argument has a real batch
    # dimension and (2) all unmapped axes can have a singleton axis inserted and
    # then rely on the primitive's built-in broadcasting.
    args = [bdim_at_front(x, d, 1) if np.ndim(x) else x
            for x, d in zip(args, dims)]
    ndim = max(np.ndim(x) for x in args)  # special-case scalar broadcasting
    args = [_handle_scalar_broadcasting(ndim, x, d) for x, d in zip(args, dims)]
    out = prim.bind(*args, **params)
    return (out, (0,) * len(out)) if prim.multiple_results else (out, 0)

def _handle_scalar_broadcasting(nd, x, d):
  if d is not_mapped or nd == np.ndim(x):
    return x
  else:
    return jax.lax.expand_dims(x, tuple(range(np.ndim(x), nd)))

def defreducer(prim, ident):
  primitive_batchers[prim] = partial(reducer_batcher, prim, ident)

def reducer_batcher(prim, ident, batched_args, batch_dims, axes, **params):
  def out_axis(axes, axis):
    return int(list(np.delete(np.arange(operand.ndim), axes)).index(axis))
  operand, = batched_args
  bdim, = batch_dims
  if isinstance(bdim, int):
    axes = tuple(np.where(np.less(axes, bdim), axes, np.add(axes, 1)))
    bdim_out = out_axis(axes, bdim)
    if 'input_shape' in params:
      params = dict(params, input_shape=operand.shape)
    return prim.bind(operand, axes=axes, **params), bdim_out
  elif isinstance(bdim, RaggedAxis):
    assert ident is not None, "TODO Ragged batching a reduction requires an identity"
    axes = tuple(np.where(np.less(axes, bdim.stacked_axis), axes, np.add(axes, 1)))
    bdim_out = out_axis(axes, bdim.stacked_axis)
    if bdim.ragged_axis in axes:
      operand = mask_ragged_axis(operand, ident, bdim)
      result = prim.bind(operand, axes=axes, **params)
      return result, bdim_out
    else:
      result = prim.bind(operand, axes=axes, **params)
      return result, RaggedAxis(bdim_out, out_axis(axes, bdim.ragged_axis), bdim.segment_lengths)
  else:
    assert False

def mask_ragged_axis(operand, ident, axis_spec):
  value = ident(operand.dtype)
  positions = jax.lax.broadcasted_iota('int32', operand.shape, axis_spec.ragged_axis)
  # TODO(mattjj, axch) cant get ._data, need to convert it
  lengths = jax.lax.convert_element_type(axis_spec.segment_lengths._data, 'int32')
  limits = jax.lax.broadcast_in_dim(
      lengths, operand.shape, [axis_spec.stacked_axis])
  mask = positions < limits
  return jax.lax.select(mask, operand, jax.lax.broadcast(value, operand.shape))

### general utilities for manipulating axes on jaxpr types (not vmappables)

def broadcast(x, sz, axis):
  shape = list(np.shape(x))
  shape.insert(axis, sz)
  broadcast_dims = tuple(np.delete(np.arange(len(shape)), axis))
  return jax.lax.broadcast_in_dim(x, shape, broadcast_dims)

def matchaxis(axis_name, sz, src, dst, x, sum_match=False):
  if dst == pile_axis:
    x = bdim_at_front(x, src, sz)
    elt_ty = x.aval.update(shape=x.shape[1:])
    aval = PileTy(core.Var(0, '', core.ShapedArray((), np.dtype('int32'))),
                  x.shape[0], elt_ty)
    return Pile(aval, x)
  try:
    _ = core.get_aval(x)
  except TypeError as e:
    raise TypeError(f"Output from batched function {repr(x)} with type "
                    f"{type(x)} is not a valid JAX type") from e
  if src == dst:
    return x
  elif type(src) == type(dst) == int:
    return moveaxis(x, src, dst)
  elif src is not_mapped and dst is not not_mapped:
    return broadcast(x, sz, canonicalize_axis(dst, np.ndim(x) + 1))
  elif dst is not_mapped and sum_match:
    return x.sum(src)
  else:
    if (not isinstance(axis_name, core._TempAxisName) and
        axis_name is not core.no_axis_name):
      raise ValueError(f'vmap has mapped output ({axis_name=}) but out_axes is {dst}')
    else:
      raise ValueError(f'vmap has mapped output but out_axes is {dst}')

def bdim_at_front(x, bdim, size):
  if bdim is not_mapped:
    return broadcast(x, size, 0)
  else:
    return moveaxis(x, bdim, 0)

# sets up primitive batchers for ad_util and xla primitives

def add_batched(batched_args, batch_dims):
  bdx, bdy = batch_dims
  x, y = batched_args
  if bdx == bdy:
    return add_jaxvals(x, y), bdx
  elif bdx is not_mapped:
    x = broadcast(x, y.shape[bdy], bdy)
    return add_jaxvals(x, y), bdy
  elif bdy is not_mapped:
    y = broadcast(y, x.shape[bdx], bdx)
    return add_jaxvals(x, y), bdx
  else:
    x = moveaxis(x, bdx, bdy)
    return add_jaxvals(x, y), bdy
primitive_batchers[add_jaxvals_p] = add_batched

def zeros_like_batched(batched_args, batch_dims):
  val, = batched_args
  bdim, = batch_dims
  return zeros_like_jaxval(val), bdim
primitive_batchers[zeros_like_p] = zeros_like_batched