rocm_jax/jax/interpreters/partial_eval.py

# Copyright 2018 Google LLC
#
# 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

from collections import namedtuple
import contextlib
from dataclasses import dataclass
import functools
from functools import partial
import inspect
import itertools as it
import operator as op
from typing import (Any, Callable, Dict, NamedTuple, Optional, Sequence, Tuple,
                    List, Union, Hashable, cast)
from weakref import ref

import numpy as np

from jax import core
from jax._src import dtypes
from jax import linear_util as lu
from jax._src import profiler
from jax._src.ad_util import Zero
from jax._src.api_util import flattened_fun_in_tree, flatten_fun_nokwargs
from jax._src.tree_util import (PyTreeDef, treedef_tuple, tree_unflatten,
                                tree_leaves)
from jax._src.util import (unzip2, safe_zip, safe_map, toposort, split_list,
                           merge_lists, partition_list, OrderedSet,
                           as_hashable_function, weakref_lru_cache)
from jax.core import (Trace, Tracer, Jaxpr, Literal, get_aval, AbstractValue,
                      ClosedJaxpr, new_jaxpr_eqn, ConcreteArray,
                      raise_to_shaped, Var, DropVar, Atom, JaxprEqn, Primitive,
                      ShapedArray, DShapedArray, AbstractBInt, mapped_aval,
                      unmapped_aval)
from jax._src import source_info_util
from jax.config import config

map, unsafe_map = safe_map, map
zip, unsafe_zip = safe_zip, zip
def identity(x): return x

def _update_annotation(
    f: lu.WrappedFun,
    orig_type: Optional[Tuple[Tuple[AbstractValue, bool], ...]],
    in_knowns: List[bool]) -> lu.WrappedFun:
  if orig_type is None:
    return f
  return lu.annotate(f, tuple([ty for k, ty in zip(in_knowns, orig_type) if k]))

class PartialVal(tuple):
  """Partial value: either a known value or an unknown (abstract) value.

  Represented as a pair `(aval_opt, const)` of one of two kinds:
  * `(None, <Constant>)` indicates a known value, either a Python regular
    value, or a Tracer.
  * `(<AbstractValue>, *)` indicates an unknown value characterized by an
    abstract value.
  """
  def __new__(cls, xs: Tuple[Optional[AbstractValue], core.Value]):
    pv, const = xs
    if config.jax_enable_checks:
      # type checks
      assert isinstance(pv, (AbstractValue, type(None))), xs
      assert isinstance(const, core.Tracer) or type(const) is Zero or core.valid_jaxtype(const), xs
      # invariant checks
      if isinstance(pv, AbstractValue):
        assert get_aval(const) == core.abstract_unit, xs
    return tuple.__new__(cls, xs)

  @classmethod
  def known(cls, const: core.Value) -> PartialVal:
    return PartialVal((None, const))

  @classmethod
  def unknown(cls, aval: AbstractValue) -> PartialVal:
    return PartialVal((aval, core.unit))

  def is_known(self) -> bool:
    return self[0] is None

  def get_known(self) -> Optional[core.Value]:
    """Get the known value, if known, else None."""
    return self[1] if self[0] is None else None

  def get_aval(self) -> AbstractValue:
    """Get AbstractValue directly (if unknown) or from the constant (known)."""
    known = self.get_known()
    if known is not None:
      return get_aval(known)
    else:
      return self[0]


class JaxprTrace(Trace):

  def __init__(self, *args, name_stack: source_info_util.NameStack):
    super().__init__(*args)
    self.name_stack = name_stack

  def pure(self, val) -> JaxprTracer:
    return self.new_const(val)

  def lift(self, val) -> JaxprTracer:
    return self.new_const(val)

  def sublift(self, val) -> JaxprTracer:
    return JaxprTracer(self, val.pval, FreeVar(val))

  def new_const(self, val) -> JaxprTracer:
    if isinstance(val, Tracer) and val._trace.level == self.level:
      raise Exception
    return JaxprTracer(self, PartialVal.known(val), core.unit)

  def new_instantiated_literal(self, val) -> JaxprTracer:
    aval = get_aval(val)
    return JaxprTracer(self, PartialVal.unknown(aval),
                       Literal(val, raise_to_shaped(aval)))

  def new_instantiated_const(self, val) -> JaxprTracer:
    return JaxprTracer(self, PartialVal.unknown(get_aval(val)), ConstVar(val))

  def new_arg(self, pval: PartialVal) -> JaxprTracer:
    const = pval.get_known()
    # XXX: Think twice before changing this constant argument pruning!
    # This has really important consequences for partial_eval_jaxpr.
    # Most importantly, this guarantees that the unknown jaxpr never uses
    # known inputs (if it needs them, then they get passed through residuals).
    if const is None:
      return JaxprTracer(self, pval, LambdaBinding())
    else:
      return self.new_const(const)

  def instantiate_const(self, tracer) -> Tracer:
    const = tracer.pval.get_known()
    if const is None:
      return tracer
    else:
      if type(const) in core.literalable_types and np.shape(const) == ():
        return self.new_instantiated_literal(const)
      else:
        return self.new_instantiated_const(const)

  def instantiate_const_abstracted(self, tracer) -> JaxprTracer:
    const = tracer.pval.get_known()
    if const is None:
      return tracer
    else:
      aval = raise_to_shaped(get_aval(const), np.isscalar(const))
      return JaxprTracer(self, PartialVal.unknown(aval), ConstVar(const))

  def process_primitive(self, primitive, tracers, params):
    if primitive in custom_partial_eval_rules:
      return custom_partial_eval_rules[primitive](self, *tracers, **params)
    else:
      return self.default_process_primitive(primitive, tracers, params)

  def default_process_primitive(self, primitive, tracers, params):
    # By default, if all the input tracers are known, then bind the primitive
    # and consider all outputs known. Otherwise, stage the application into the
    # jaxpr and consider all outputs unknown.
    consts = [t.pval.get_known() for t in tracers]
    if all(c is not None for c in consts):
      return primitive.bind(*consts, **params)
    tracers = map(self.instantiate_const, tracers)
    avals = [t.aval for t in tracers]
    out_aval, effects = primitive.abstract_eval(*avals, **params)
    name_stack = self._current_truncated_name_stack()
    source = source_info_util.current().replace(name_stack=name_stack)
    if primitive.multiple_results:
      out_tracers = [JaxprTracer(self, PartialVal.unknown(aval), None)
                     for aval in out_aval]
      eqn = new_eqn_recipe(tracers, out_tracers, primitive, params, effects, source)
      for t in out_tracers: t.recipe = eqn
      return out_tracers
    else:
      out_tracer = JaxprTracer(self, PartialVal.unknown(out_aval), None)
      out_tracer.recipe = new_eqn_recipe(tracers, [out_tracer], primitive,
                                         params, effects, source)
      return out_tracer

  def process_call(self, primitive, f, tracers, params):
    if primitive in call_partial_eval_rules:
      return call_partial_eval_rules[primitive](self, primitive, f, tracers, params)

    update_params = call_param_updaters.get(primitive) or (lambda p, _, __: p)
    in_knowns, in_avals, in_consts = partition_pvals([t.pval for t in tracers])

    # We want to partially evaluate this call into two calls: one evaluated now
    # taking known values (in_consts) as inputs and producing known values
    # (out_consts) as outputs, and the other staged out as an eqn into the jaxpr
    # being built. The latter takes as input residuals (res) produced as outputs
    # of the first call, shared closed-over values (env), and explicit arguments
    # which were unknown to the first call (corresponding to in_avals).

    # Wrap f to perform the partial evaluation and plumb out aux data.
    f_ = trace_to_subjaxpr_nounits(f, self.main, False)
    f_, aux = partial_eval_wrapper_nounits(f_, tuple(in_knowns), tuple(in_avals))
    # Adjust parameters (e.g. donated_invars) for the call to be evaluated now.
    const_params = update_params(params, in_knowns, 0)

    # Run the call, getting known out vals and aux data used for staged-out call
    out = primitive.bind(_update_annotation(f_, f.in_type, in_knowns),
                         *in_consts, **const_params)
    out_knowns, out_avals, jaxpr, env = aux()
    # Split apart known outputs from the original call and residuals.
    out_consts, res = split_list(out, [len(out) - len(jaxpr.constvars)])

    # Create the input tracers for the staged-out (unknown-value) call.
    const_tracers = map(self.new_instantiated_const, res)
    env_tracers = map(self.full_raise, env)
    unknown_arg_tracers = [t for t in tracers if not t.is_known()]
    # Adjust parameters (e.g. donated_invars) for the staged-out call's args.
    num_new_args = len(const_tracers) + len(env_tracers)
    staged_params = update_params(params, map(op.not_, in_knowns), num_new_args)
    staged_params = dict(staged_params, call_jaxpr=convert_constvars_jaxpr(jaxpr))
    # The outputs of the staged-out call are Tracers with the new eqn as recipe.
    out_tracers = [JaxprTracer(self, PartialVal.unknown(a), None)
                   for a in out_avals]
    name_stack = self._current_truncated_name_stack()
    source = source_info_util.current().replace(name_stack=name_stack)
    eqn = new_eqn_recipe((*const_tracers, *env_tracers, *unknown_arg_tracers),
                         out_tracers, primitive, staged_params, jaxpr.effects, source)
    for t in out_tracers: t.recipe = eqn
    return merge_lists(out_knowns, out_tracers, out_consts)

  def process_map(self, primitive, f: lu.WrappedFun, tracers, params):
    update_params = call_param_updaters.get(primitive) or (lambda p, _, __: p)
    in_knowns, in_avals, in_consts = partition_pvals([t.pval for t in tracers])

    # This method is like process_call above, except:
    #   1. we delete an axis from mapped-over input avals' shapes, and
    #      analogously add an axis to mapped-over output avals' shapes;
    #   2. we update the in_axes and out_axes/out_axes_thunk parameters to
    #      reflect the inputs and outputs pruned from the unknown/known sides.

    # Map (delete an axis from) unknown inputs' avals as dictated by in_axes.
    unk_in_axes, const_in_axes = partition_list(in_knowns, params['in_axes'])
    in_avals_mapped = [mapped_aval(params['axis_size'], ax, aval)
                       for ax, aval in zip(unk_in_axes, in_avals)]

    # Wrap f to perform partial evaluation and plumb out aux data.
    f = trace_to_subjaxpr_nounits(f, self.main, False)
    f, aux = partial_eval_wrapper_nounits(f, tuple(in_knowns),
                                          tuple(in_avals_mapped))
    # Adjust params for knowns (e.g. donated_invars, in_axes, out_axes_thunk)
    const_params = update_params(params, in_knowns, 0)  # handles donated_invars
    out_axes_thunk = params['out_axes_thunk']
    @as_hashable_function(closure=out_axes_thunk)
    def const_out_axes_thunk():
      out_knowns, _, jaxpr, _ = aux()
      _, out_axes = partition_list(out_knowns, out_axes_thunk())
      return tuple(out_axes) + (0,) * len(jaxpr.constvars)  # res mapped axis 0
    const_params = dict(const_params, in_axes=tuple(const_in_axes),
                        out_axes_thunk=const_out_axes_thunk)

    # Run the map, getting known out vals and aux data used for staged-out map.
    out = primitive.bind(f, *in_consts, **const_params)
    out_knowns, out_avals_mapped, jaxpr, env = aux()
    # Split apart known outputs from the original call and residuals.
    out_consts, res = split_list(out, [len(out) - len(jaxpr.constvars)])

    # We can only check_jaxpr with the dynamic axis environment extended:
    with core.extend_axis_env(params['axis_name'], params['axis_size'], None):
      call_jaxpr = convert_constvars_jaxpr(jaxpr)

    # Compute staged and const out_axes, taking into account residuals.
    out_axes = params['out_axes_thunk']()
    staged_out_axes, _ = partition_list(out_knowns, out_axes)
    staged_in_axes = (0,) * len(res) + (None,) * len(env) + (*unk_in_axes,)

    # Create the input tracers for the staged-out (unkonwn-value) call.
    const_tracers = map(self.new_instantiated_const, res)
    env_tracers = map(self.full_raise, env)
    unknown_arg_tracers = [t for t in tracers if not t.is_known()]
    # Adjust params for staged-out call on unknown values.
    num_new_args = len(const_tracers) + len(env_tracers)
    staged_params = update_params(params, map(op.not_, in_knowns), num_new_args)
    staged_params = dict(staged_params, in_axes=staged_in_axes,
                         out_axes=tuple(staged_out_axes), call_jaxpr=call_jaxpr)
    # The outputs of the staged-out call are Tracers with the new eqn as recipe.
    out_avals = [unmapped_aval(params['axis_size'], params['axis_name'], ax, a)
                 for ax, a in zip(staged_out_axes, out_avals_mapped)]
    out_tracers = [JaxprTracer(self, PartialVal.unknown(a), None)
                   for a in out_avals]
    eqn = new_eqn_recipe((*const_tracers, *env_tracers, *unknown_arg_tracers),
                         out_tracers, primitive, staged_params,
                         jaxpr.effects,
                         source_info_util.current())
    for t in out_tracers: t.recipe = eqn

    return merge_lists(out_knowns, out_tracers, out_consts)

  def post_process_call(self, primitive, out_tracers, params):
    unknown_out_tracers = [t for t in out_tracers if not t.is_known()]
    jaxpr, res, env = tracers_to_jaxpr([], unknown_out_tracers)
    out_pvals = [t.pval for t in out_tracers]
    out_knowns, out_avals, out_consts = partition_pvals(out_pvals)
    out = [*out_consts, *res]
    main = self.main

    def todo(out):
      trace = main.with_cur_sublevel()
      out_consts, res = split_list(out, [len(out) - len(jaxpr.constvars)])
      const_tracers = map(trace.new_instantiated_const, res)
      in_tracers = (*const_tracers, *map(trace.full_raise, env))
      out_tracers = [JaxprTracer(trace, PartialVal.unknown(a), None)
                     for a in out_avals]
      update_params = call_param_updaters.get(primitive) or (lambda p, _, __: p)
      new_params = update_params(params, [], len(in_tracers))
      new_params = dict(new_params, call_jaxpr=convert_constvars_jaxpr(jaxpr))
      name_stack = self._current_truncated_name_stack()
      source = source_info_util.current().replace(name_stack=name_stack)
      eqn = new_eqn_recipe(in_tracers, out_tracers, primitive, new_params,
          jaxpr.effects, source)
      for t in out_tracers: t.recipe = eqn
      return merge_lists(out_knowns, out_tracers, out_consts)

    return out, todo

  def post_process_map(self, primitive, out_tracers, params):
    unknown_out_tracers = [t for t in out_tracers if not t.is_known()]
    jaxpr, res, env = tracers_to_jaxpr([], unknown_out_tracers)
    out_pvals = [t.pval for t in out_tracers]
    out_knowns, out_avals_mapped, out_consts = partition_pvals(out_pvals)
    out = [*out_consts, *res]
    main = self.main

    with core.extend_axis_env(params['axis_name'], params['axis_size'], None):
      call_jaxpr = convert_constvars_jaxpr(jaxpr)

    def todo(out):
      trace = main.with_cur_sublevel()
      out_consts, res = split_list(out, [len(out) - len(jaxpr.constvars)])
      const_tracers = map(trace.new_instantiated_const, res)
      env_tracers = map(trace.full_raise, env)

      staged_out_axes = tuple(out_axes_unknown)  # set by out_axes_transform
      staged_in_axes = (0,) * len(res) + (None,) * len(env)

      update_params = call_param_updaters.get(primitive) or (lambda p, _, __: p)
      staged_params = update_params(params, [], len(res) + len(env))
      staged_params = dict(staged_params, in_axes=staged_in_axes,
                           out_axes=tuple(staged_out_axes),
                           call_jaxpr=call_jaxpr)

      out_avals = [unmapped_aval(params['axis_size'], params['axis_name'], d, a)
                   for d, a in zip(staged_out_axes, out_avals_mapped)]
      out_tracers = [JaxprTracer(trace, PartialVal.unknown(a), None)
                     for a in out_avals]
      name_stack = self._current_truncated_name_stack()
      source = source_info_util.current().replace(name_stack=name_stack)
      eqn = new_eqn_recipe((*const_tracers, *env_tracers), out_tracers,
                           primitive, staged_params, jaxpr.effects, source)
      for t in out_tracers: t.recipe = eqn
      return merge_lists(out_knowns, out_tracers, out_consts)

    def out_axes_transform(out_axes):
      nonlocal out_axes_unknown
      out_axes_unknown, out_axes_known = partition_list(out_knowns, out_axes)
      return tuple(out_axes_known) + (0,) * len(jaxpr.constvars)
    out_axes_unknown: Optional[list] = None

    return out, (todo, out_axes_transform)

  def _current_truncated_name_stack(self):
    return source_info_util.current_name_stack()[len(self.name_stack):]

  def partial_eval(self, f: lu.WrappedFun, pvals: Sequence[PartialVal],
                   app: Callable[[lu.WrappedFun, Tuple[core.Value, ...]], Tuple[core.Value]],
                   instantiate: bool):
    """Partially evaluate f on a sequence of PartialVals."""
    in_avals, in_consts = unzip2(pvals)
    f = trace_to_subjaxpr(f, self.main, instantiate)
    f, aux = partial_eval_wrapper(f, tuple(in_avals))
    out_flat, (out_avals, jaxpr, env) = app(f, *in_consts), aux()
    out_consts, consts = split_list(out_flat, [len(out_flat)-len(jaxpr.constvars)])
    out_pvs = map(PartialVal, zip(out_avals, out_consts))
    env_tracers = map(self.full_raise, env)
    return jaxpr, out_pvs, consts, env_tracers

  def process_custom_jvp_call(self, prim, fun, jvp, tracers):
    tracers = map(self.instantiate_const_abstracted, tracers)
    in_avals, in_consts = unzip2(t.pval for t in tracers)  # in_consts are units
    fun = trace_to_subjaxpr(fun, self.main, True)
    fun, aux = partial_eval_wrapper(fun, tuple(in_avals))
    out_flat = prim.bind(fun, jvp, *in_consts)
    out_avals, jaxpr, env = aux()
    out_consts, consts = split_list(out_flat, [len(out_flat)-len(jaxpr.constvars)])
    out_pvals = map(PartialVal, zip(out_avals, out_consts))  # out_consts are units
    env_tracers = map(self.full_raise, env)
    out_tracers = [JaxprTracer(self, pval, None) for pval in out_pvals]
    const_tracers = map(self.new_instantiated_const, consts)
    in_tracers = (*const_tracers, *env_tracers, *tracers)
    closed_jaxpr = core.ClosedJaxpr(convert_constvars_jaxpr(jaxpr), ())

    @_memoize
    def jvp_jaxpr_thunk():
      jvp_ = trace_to_subjaxpr(jvp, self.main, True)
      jvp_, aux = partial_eval_wrapper(jvp_, tuple(in_avals) * 2)
      with core.new_sublevel():
        out_flat = jvp_.call_wrapped(*(in_consts * 2))  # in_consts are units
      out_avals, jaxpr, env = aux()
      _, consts = split_list(out_flat, [len(out_flat)-len(jaxpr.constvars)])
      converted_jaxpr = convert_envvars_to_constvars(jaxpr, len(env))
      return converted_jaxpr, (*consts, *env)

    name_stack = self._current_truncated_name_stack()
    source = source_info_util.current().replace(name_stack=name_stack)
    eqn = new_eqn_recipe(in_tracers, out_tracers, prim.initial_style,
                         dict(fun_jaxpr=closed_jaxpr,
                              jvp_jaxpr_thunk=jvp_jaxpr_thunk,
                              num_consts=len(consts) + len(env)),
                         jaxpr.effects,
                         source)
    for t in out_tracers: t.recipe = eqn
    return out_tracers

  def post_process_custom_jvp_call(self, out_tracers, _):
    # This path should only be reachable if we expose a partial eval API
    # unrelated to autodiff, since we raise an error when differentiation with
    # respect to values over which a custom_jvp function closes is detected.
    raise NotImplementedError  # TODO(mattjj)

  def process_custom_transpose(self, prim, call, tracers, **params):
    res_ts, lin_ts = split_list(tracers, [params['res_tree'].num_leaves])
    assert all(t.is_known()     for t in res_ts)
    lin_all_known   = all(t.is_known()     for t in lin_ts)
    if lin_all_known:
      res_cvals = [t.pval[1] for t in res_ts]
      lin_cvals = [t.pval[1] for t in lin_ts]
      return prim.bind(call, *res_cvals, *lin_cvals, **params)
    else:
      out_tracers = [JaxprTracer(self, PartialVal.unknown(aval), None)
                     for aval in params['out_types']]
      in_tracers = map(self.instantiate_const, tracers)
      new_params = dict(params, call=call)
      eqn = new_eqn_recipe(in_tracers, out_tracers, prim, new_params,
          core.no_effects, source_info_util.current())
      for t in out_tracers: t.recipe = eqn
      return out_tracers

  def process_custom_vjp_call(self, prim, fun, fwd, bwd, tracers, out_trees):
    tracers = map(self.instantiate_const_abstracted, tracers)
    in_avals, in_consts = unzip2(t.pval for t in tracers)  # in_consts are units
    fun = trace_to_subjaxpr(fun, self.main, True)
    fun, aux = partial_eval_wrapper(fun, tuple(in_avals))
    out_flat = prim.bind(fun, fwd, bwd, *in_consts, out_trees=out_trees)
    out_avals, jaxpr, env = aux()
    out_consts, consts = split_list(out_flat, [len(out_flat)-len(jaxpr.constvars)])
    out_pvals = map(PartialVal, zip(out_avals, out_consts))  # out_consts are units
    env_tracers = map(self.full_raise, env)
    out_tracers = [JaxprTracer(self, pval, None) for pval in out_pvals]
    const_tracers = map(self.new_instantiated_const, consts)
    in_tracers = (*const_tracers, *env_tracers, *tracers)
    closed_jaxpr = core.ClosedJaxpr(convert_constvars_jaxpr(jaxpr), ())

    @_memoize
    def fwd_jaxpr_thunk():
      fwd_ = trace_to_subjaxpr(fwd, self.main, True)
      fwd_, aux = partial_eval_wrapper(fwd_, tuple(in_avals))
      with core.new_sublevel():
        out_flat = fwd_.call_wrapped(*in_consts)  # in_consts are units
      out_avals, jaxpr, env = aux()
      _, consts = split_list(out_flat, [len(out_flat)-len(jaxpr.constvars)])
      converted_jaxpr = convert_envvars_to_constvars(jaxpr, len(env))
      return converted_jaxpr, (*consts, *env)

    name_stack = self._current_truncated_name_stack()
    source = source_info_util.current().replace(name_stack=name_stack)
    eqn = new_eqn_recipe(in_tracers, out_tracers, prim.initial_style,
                         dict(fun_jaxpr=closed_jaxpr,
                              fwd_jaxpr_thunk=fwd_jaxpr_thunk,
                              num_consts=len(consts) + len(env),
                              bwd=bwd, out_trees=out_trees),
                         jaxpr.effects,
                         source)
    for t in out_tracers: t.recipe = eqn
    return out_tracers

  def post_process_custom_vjp_call(self, out_tracers, _):
    # This path should only be reachable if we expose a partial eval API
    # unrelated to autodiff, since we raise an error when differentiation with
    # respect to values over which a custom_vjp function closes is detected.
    raise NotImplementedError  # TODO(mattjj)

def partition_pvals(
    pvals: List[PartialVal]
  ) -> Tuple[List[bool], List[AbstractValue], List[Any]]:
  knowns = [pval.is_known()  for pval in pvals                       ]
  avals  = [pval.get_aval()  for pval in pvals if not pval.is_known()]
  consts = [pval.get_known() for pval in pvals if     pval.is_known()]
  return knowns, avals, consts

@lu.transformation_with_aux
def partial_eval_wrapper_nounits(
    in_knowns: Sequence[bool], in_avals: Sequence[AbstractValue],
    *in_consts: Any):
  in_avals_, in_consts_ = iter(in_avals), iter(in_consts)
  in_pvals = [PartialVal.known(next(in_consts_)) if known else
              PartialVal.unknown(next(in_avals_)) for known in in_knowns]
  sentinel = object()
  assert next(in_avals_, sentinel) is next(in_consts_, sentinel) is sentinel
  jaxpr, (out_pvals, res, env) = yield (in_pvals,), {}
  out_knowns, out_avals, out_consts = partition_pvals(out_pvals)
  yield (*out_consts, *res), (out_knowns, out_avals, jaxpr, env)

custom_partial_eval_rules: Dict[Primitive, Callable] = {}
call_partial_eval_rules: Dict[Primitive, Callable] = {}
call_param_updaters: Dict[Primitive, Callable] = {}


def abstract_eval_fun(fun, *avals, debug_info=None, **params):
  _, avals_out, _ = trace_to_jaxpr_dynamic(
      lu.wrap_init(fun, params), avals, debug_info)
  assert all(isinstance(aval, AbstractValue) for aval in avals_out)
  return avals_out


JaxprTracerRecipe = Union['JaxprEqnRecipe', 'LambdaBinding', 'FreeVar',
                          'ConstVar', Literal, core.Unit]

class JaxprTracer(Tracer):
  __slots__ = ['pval', 'recipe']

  def __init__(self, trace: JaxprTrace, pval: PartialVal,
               recipe: Optional[JaxprTracerRecipe]):
    assert isinstance(pval, PartialVal)
    pv, const = pval
    if isinstance(const, Tracer) and const._trace.level >= trace.level:
      raise core.escaped_tracer_error(
          const, f"Tracer from a higher level: {const} in trace {trace}")
    self._trace = trace
    self.pval = pval
    self.recipe = recipe

  def __repr__(self):
    return 'Traced<{}:{}>'.format(self.aval, self._trace)

  @property
  def aval(self) -> AbstractValue:
    return self.pval.get_aval()

  @property
  def parents(self) -> Sequence[JaxprTracer]:
    if isinstance(self.recipe, JaxprEqnRecipe):
      return self.recipe.in_tracers
    else:
      return []

  def full_lower(self):
    known = self.pval.get_known()
    if known is not None:
      return core.full_lower(known)
    else:
      return self

  def is_known(self):
    return self.pval.is_known()

@profiler.annotate_function
def trace_to_jaxpr(
    fun: lu.WrappedFun, pvals: Sequence[PartialVal],
    instantiate: Union[bool, Sequence[bool]] = False,
  ) -> Tuple[Jaxpr, List[PartialVal], List[core.Value]]:
  """
  Partially evaluate a function, building a jaxpr for un-evaluated computation.

  Args:
    fun: lu.WrappedFun representing the function to be partially evaluated. The
      function must be flattened, in the sense of accepting jaxpr type arguments
      and returning a flat list of jaxpr type outputs.
    pvals: sequence of PartialVals of length equal to the number of inputs to
      `fun` indicating which inputs are known or unknown.
    instantiate: optional bool or sequence of bools of length equal to the
      number of outputs of `fun` indicating which outputs should be forced to be
      treated as unknown and hence instantiated in the jaxpr. If a single bool,
      the value is applied to all outputs. Default False.

  Returns:
    A triple where the first element is a jaxpr representing the computation
    which depends on unknown inputs; the second element is a list of PartialVals
    of length equal to the length of the output of `fun` representing which
    outputs are known and unknown (along with their values and abstract values,
    respectively); the third element is a list of known residual values. The
    returned jaxpr takes as inputs the known residual values followed by values
    of the originally unknown inputs.
  """
  current_name_stack = source_info_util.current_name_stack()
  with core.new_main(JaxprTrace, name_stack=current_name_stack) as main:
    fun = trace_to_subjaxpr(fun, main, instantiate)
    jaxpr, (out_pvals, consts, env) = fun.call_wrapped(pvals)
    assert not env
    del main, fun, env

  return jaxpr, out_pvals, consts

@profiler.annotate_function
def trace_to_jaxpr_nounits(
    fun: lu.WrappedFun, pvals: Sequence[PartialVal],
    instantiate: Union[bool, Sequence[bool]] = False,
  ) -> Tuple[Jaxpr, List[PartialVal], List[core.Value]]:
  current_name_stack = source_info_util.current_name_stack()
  with core.new_main(JaxprTrace, name_stack=current_name_stack) as main:
    fun = trace_to_subjaxpr_nounits(fun, main, instantiate)
    jaxpr, (out_pvals, consts, env) = fun.call_wrapped(pvals)
    assert not env
    del main, fun, env
  return jaxpr, out_pvals, consts


@lu.transformation
def trace_to_subjaxpr_nounits(
    main: core.MainTrace, instantiate: Union[bool, Sequence[bool]],
    in_pvals: Sequence[PartialVal]):
  assert all([isinstance(pv, PartialVal) for pv in in_pvals]), in_pvals
  trace = main.with_cur_sublevel()
  in_knowns  = [pval.is_known()     for pval in in_pvals]
  in_consts  = [pval.get_known()    for pval in in_pvals if     pval.is_known()]
  in_tracers = [trace.new_arg(pval) for pval in in_pvals if not pval.is_known()]
  in_args = merge_lists(in_knowns, in_tracers, in_consts)
  ans = yield in_args, {}
  assert isinstance(ans, (list, tuple)), (
      f"Got unexpected return type when tracing function to jaxpr: {ans}")
  assert all(isinstance(x, core.Tracer) or core.valid_jaxtype(x) for x in ans), (
      f"Got unexpected return type when tracing function to jaxpr: {ans}")
  if isinstance(instantiate, bool):
    instantiate = [instantiate] * len(ans)
  out_tracers = map(trace.full_raise, map(core.full_lower, ans))
  out_tracers = map(partial(instantiate_const_at, trace), instantiate, out_tracers)
  out_pvals = [t.pval for t in out_tracers]
  out_tracers_ = [t for t in out_tracers if not t.is_known()]
  jaxpr, consts, env = tracers_to_jaxpr(in_tracers, out_tracers_)
  del trace, in_tracers, out_tracers, out_tracers_
  yield jaxpr, (out_pvals, consts, env)

def instantiate_const_at(trace: JaxprTrace, instantiate: bool, tracer):
  if instantiate:
    return trace.instantiate_const(trace.full_raise(tracer))
  else:
    return tracer


FreeVar = namedtuple('FreeVar', ['val'])
ConstVar = namedtuple('ConstVar', ['val'])
LambdaBinding = namedtuple('LambdaBinding', [])
class JaxprEqnRecipe(NamedTuple):
  eqn_id: Any
  in_tracers: Sequence[JaxprTracer]
  out_tracer_refs: Sequence[ref[JaxprTracer]]
  out_avals: Sequence[core.AbstractValue]
  primitive: Primitive
  params: Dict[str, Any]
  effects: core.Effects
  source_info: source_info_util.SourceInfo

def new_eqn_recipe(in_tracers: Sequence[JaxprTracer],
                   out_tracers: Sequence[JaxprTracer],
                   primitive: Primitive,
                   params: Dict[str, Any],
                   effects: core.Effects,
                   source_info: source_info_util.SourceInfo
                  ) -> JaxprEqnRecipe:
  # TODO(necula): move these checks to core.check_jaxpr, and call in more places
  if primitive.call_primitive or primitive.map_primitive:
    assert "call_jaxpr" in params
    # assert len(invars) == len(params["call_jaxpr"].invars)  # TODO constvars?
    assert len(out_tracers) == len(params["call_jaxpr"].outvars)
    assert ("donated_invars" not in params or
            len(params["donated_invars"]) == len(params["call_jaxpr"].invars))
  if primitive.map_primitive:
    assert ("in_axes" in params and
            len(params["in_axes"]) == len(params["call_jaxpr"].invars))
    assert ("donated_invars" in params and
            len(params["donated_invars"]) == len(params["call_jaxpr"].invars))
  out_avals = [core.raise_to_shaped(t.aval) for t in out_tracers]
  return JaxprEqnRecipe(object(), tuple(in_tracers), map(ref, out_tracers),
                        out_avals, primitive, params, effects, source_info)


def recipe_to_eqn(getvar: Callable[[JaxprTracer], Atom],
                  recipe: JaxprEqnRecipe) -> core.JaxprEqn:
  (_, in_tracers, out_tracer_refs, out_avals, prim, params, eff, src) = recipe
  invars  = [getvar(t) for t in in_tracers]
  out_tracers = [t_ref() for t_ref in out_tracer_refs]
  outvars = [DropVar(a) if t is None else getvar(t)  # type: ignore
             for a, t in zip(out_avals, out_tracers)]
  return new_jaxpr_eqn(invars, outvars, prim, params, eff, src)

def tracers_to_jaxpr(
  in_tracers: Sequence[JaxprTracer],
  out_tracers: Sequence[JaxprTracer]
  ) -> Tuple[Jaxpr, Tuple[Any, ...], Tuple[Any, ...]]:
  """Constructs Jaxpr given tracers for inputs and outputs.

  Params:
    in_tracers: the tracers that were created for the function inputs
    out_tracers: the tracers that were output by the function.

  Returns: a triple of a `Jaxpr`, a list of constant values corresponding to
    the `constvars` in the returned Jaxps, and a list of environment values.
    The vars for the environment values have been prepended to the Jaxpr's
    `invars`.
  """
  newvar = core.gensym()
  t_to_var: Dict[int, Atom] = {}
  def getvar(t: JaxprTracer) -> Atom:
    var = t_to_var.get(id(t))
    if var is None:
      aval = t.pval.get_aval() if not t.pval.is_known() else core.abstract_unit
      var = t_to_var[id(t)] = newvar(aval)
    return var
  sorted_tracers = toposort(out_tracers)
  invars = map(getvar, in_tracers)
  eqns: List[core.JaxprEqn] = []
  env: Dict[Var, Any] = {}
  consts: Dict[Var, Any] = {}
  const_to_var: Dict[int, Var] = {}
  def getconstvar(c):
    var = const_to_var.get(id(c))
    if var is None:
      var = const_to_var[id(c)] = newvar(get_aval(c))
    return var
  processed_eqn_ids = set()
  for t in sorted_tracers:
    recipe = t.recipe
    if isinstance(recipe, JaxprEqnRecipe):
      if recipe.eqn_id not in processed_eqn_ids:
        eqns.append(recipe_to_eqn(getvar, recipe))
        processed_eqn_ids.add(recipe.eqn_id)
    elif isinstance(recipe, LambdaBinding):
      if not any(t is in_tracer for in_tracer in in_tracers):
        raise core.escaped_tracer_error(
            t, "Tracer not among input tracers {}".format(t))
      assert in_tracers, "Lambda binding with no args"
    elif isinstance(recipe, FreeVar):
      env[getvar(t)] = recipe.val  # type: ignore
    elif isinstance(recipe, ConstVar):
      v = t_to_var[id(t)] = getconstvar(recipe.val)
      consts[v] = recipe.val
    elif isinstance(recipe, Literal):
      t_to_var[id(t)] = recipe
    elif recipe is core.unit:
      t_to_var[id(t)] = core.unitvar
    else:
      raise TypeError(recipe)

  env_vars, env_vals = unzip2(env.items())
  const_vars, const_vals = unzip2(consts.items())
  # The env_vars are pre-pended to the invars
  effects = core.join_effects(*(eqn.effects for eqn in eqns))
  jaxpr = Jaxpr(const_vars, [*env_vars, *invars], map(getvar, out_tracers),
                eqns, effects)
  config.jax_enable_checks and core.check_jaxpr(jaxpr)
  return jaxpr, const_vals, env_vals

@weakref_lru_cache
def convert_constvars_jaxpr(jaxpr: Jaxpr) -> Jaxpr:
  """Moves the constvars to the start of invars."""
  config.jax_enable_checks and core.check_jaxpr(jaxpr)
  lifted_jaxpr = Jaxpr(constvars=(),
                       invars=jaxpr.constvars + jaxpr.invars,
                       outvars=jaxpr.outvars, eqns=jaxpr.eqns,
                       effects=jaxpr.effects)
  config.jax_enable_checks and core.check_jaxpr(lifted_jaxpr)
  return lifted_jaxpr

def convert_envvars_to_constvars(jaxpr: Jaxpr, num_env_vars: int) -> Jaxpr:
  config.jax_enable_checks and core.check_jaxpr(jaxpr)
  env_vars, invars = split_list(jaxpr.invars, [num_env_vars])
  converted_jaxpr = Jaxpr(constvars=jaxpr.constvars + env_vars,
                          invars=invars, outvars=jaxpr.outvars, eqns=jaxpr.eqns,
                          effects=jaxpr.effects)
  config.jax_enable_checks and core.check_jaxpr(converted_jaxpr)
  return converted_jaxpr


def _split_aval(unknown: bool, aval: AbstractValue) -> Tuple[AbstractValue, AbstractValue]:
  return (core.abstract_unit, aval) if unknown else (aval, core.abstract_unit)


def partial_eval_jaxpr_nounits(
    jaxpr: ClosedJaxpr, unknowns: Sequence[bool],
    instantiate: Union[bool, Sequence[bool]],
  ) -> Tuple[ClosedJaxpr, ClosedJaxpr, List[bool], List[AbstractValue]]:
  """Unzip a jaxpr in two by data dependence into 'known' and 'unknown' parts.

  That is, given a jaxpr and a sequence of booleans indicating which jaxpr
  inputs (i.e. invars) are considered unknown, produce two jaxprs, a list of
  booleans representing which of the original jaxpr's outputs are unknown (i.e.
  have a data dependence on an unknown input), and a list of abstract values
  representing residuals (part of the first jaxpr's output and the second
  jaxpr's input). The two jaxprs result from partitioning the original jaxpr's
  first-order primitive applications based on whether all the inputs to the
  application are known (in which case the application is represented in the
  'known' jaxpr and its result is considered known) or whether any inputs to the
  application are unknown (in which case the application is represented in the
  'unknown' jaxpr and its result is considered unknown). Higher-order primitives
  are recursively unzipped in two.

  The `instantiate` argument can be used to ensure some outputs are lifted into
  the 'unknown' jaxpr.

  For example, give an input jaxpr:

    { lambda ; a:f32[] b:f32[]. let
        c:f32[] = cos a
        d:f32[] = sin a
        e:f32[] = neg d
        f:f32[] = mul e b
      in (c, f) }

  then applying this function with `unknowns=[False, True]` and
  `instantiate=False` produces as an output triple:

    # jaxpr_known
    { lambda ; a:f32[]. let
       b:f32[] = cos a
       c:f32[] = sin a
       d:f32[] = neg c
     in (b, d) }

    # jaxpr_unknown
    { lambda ; a:f32[] b:f32[]. let c:f32[] = mul b a in (c,) }

    # out_unknowns
    [False, True]

  Notice in particular that the first output (jaxpr_known) contains all the
  primitive applications which do not have a data dependence on an unknown
  input. Also notice the input and output types: the input type of the first
  jaxpr produced represents the type of the known inputs of the original jaxpr,
  and the output type of the second jaxpr produced represents the type of the
  unknown outputs of the original jaxpr.

  In the above example, the output of jaxpr_known named `d` is a _residual_
  output, and corresponds to the input named `a` in jaxpr_unknown. In general,
  jaxpr_known will produce extra outputs (at the end of its output list)
  corresponding to intermeidate values of the original jaxpr which must be
  passed to jaxpr_unknown (as leading inputs).
  """
  instantiate = tuple(instantiate) if isinstance(instantiate, list) else instantiate
  return _partial_eval_jaxpr_nounits(jaxpr, tuple(unknowns), instantiate)

@weakref_lru_cache
def _partial_eval_jaxpr_nounits(jaxpr, in_unknowns, instantiate):
  f = lu.wrap_init(core.jaxpr_as_fun(jaxpr))

  cell = []
  def fun(*known_vals_in):
    known_vals_in = iter(known_vals_in)
    unknown_avals = (a for a, uk in zip(jaxpr.in_avals, in_unknowns) if uk)
    in_pvals = [PartialVal.unknown(next(unknown_avals)) if uk
                else PartialVal.known(next(known_vals_in)) for uk in in_unknowns]
    assert next(known_vals_in, None) is next(unknown_avals, None) is None
    jaxpr_unknown_, out_pvals, residuals = trace_to_jaxpr_nounits(
        f, in_pvals, instantiate=instantiate)
    jaxpr_unknown = convert_constvars_jaxpr(jaxpr_unknown_)
    out_unknowns = [not pval.is_known() for pval in out_pvals]
    res_avals = [core.raise_to_shaped(core.get_aval(r)) for r in residuals]
    cell.append((out_unknowns, jaxpr_unknown, res_avals))
    known_vals_out = [pval.get_known() for pval in out_pvals if pval.is_known()]
    return [*known_vals_out, *residuals]

  known_avals = [a for a, uk in zip(jaxpr.in_avals, in_unknowns) if not uk]
  jaxpr_known, _, consts_known = trace_to_jaxpr_dynamic(lu.wrap_init(fun), known_avals)
  (out_unknowns, jaxpr_unknown, res_avals), = cell

  # check jaxpr_known and jaxpr_unknown in isolation
  if config.jax_enable_checks:
    core.check_jaxpr(jaxpr_known)
    core.check_jaxpr(jaxpr_unknown)
  # check jaxpr_known has input type corresponding to known inputs of jaxpr
  assert ([v.aval for v in jaxpr_known.invars] ==
          [a for a, uk in zip(jaxpr.in_avals, in_unknowns) if not uk])
  # check jaxpr_known has out type corresponding to known outs of jaxpr plus res
  # TODO(mattjj): enable weak type checking here
  assert ([v.aval.strip_weak_type() for v in jaxpr_known.outvars] ==
          [a.strip_weak_type() for a, uk in zip(jaxpr.out_avals, out_unknowns)
           if not uk] + [a.strip_weak_type() for a in res_avals])
  # check jaxpr_unknown has input type corresponding to unknown inputs plus res
  assert ([v.aval for v in jaxpr_unknown.invars] ==
          res_avals + [a for a, uk in zip(jaxpr.in_avals, in_unknowns) if uk])
  # check jaxpr_unknown has output type corresponding to unknown outputs
  # TODO(mattjj): enable weak type checking here
  assert ([v.aval.strip_weak_type() for v in jaxpr_unknown.outvars] ==
          [a.strip_weak_type() for a, uk in zip(jaxpr.out_avals, out_unknowns)
           if uk])

  closed_jaxpr_known = ClosedJaxpr(jaxpr_known, consts_known)
  closed_jaxpr_unknown = ClosedJaxpr(jaxpr_unknown, ())
  return closed_jaxpr_known, closed_jaxpr_unknown, out_unknowns, res_avals


remat_call_p: Primitive = core.CallPrimitive('remat_call')
remat_call = remat_call_p.bind
remat_call_p.def_impl(core.call_impl)

def _remat_partial_eval(trace, _, f, tracers, params):
  concrete = params['concrete']

  # Unlike JaxprTrace.process_call, we want to form a jaxpr for the entirety of
  # the function being called, not just for the unknown parts. To do that, we
  # instantiate all the input tracers as constants in the jaxpr being formed.
  # Those tracers might have concrete avals, and doing abstract interpretation
  # on concrete avals engenders a tradeoff: it allows data-dependent Python
  # control flow to work, but it can in some cases lead to redundant FLOPs (done
  # both in the `bind` call below and the `core.jaxpr_as_fun` call). We use the
  # `concrete` parameter to switch this behavior, and if `concrete` is False
  # then we raise the avals to the Shaped level.
  if concrete:
    instantiated_tracers = map(trace.instantiate_const, tracers)
  else:
    instantiated_tracers = map(trace.instantiate_const_abstracted, tracers)

  # Using the instantiated tracers, run call_bind like JaxprTrace.process_call.
  # This way we record all primitives applied to the inputs (all treated as
  # unknown/instantiated) to produce the output. In the context of autodiff,
  # that means we record primal, residual, and tangent computations (e.g. sine,
  # cosine, and multiply).
  in_pvals = [t.pval for t in instantiated_tracers]
  in_knowns, in_avals, () = partition_pvals(in_pvals)  # all are unknown
  assert not any(in_knowns)
  f = trace_to_subjaxpr_nounits(f, trace.main, True)
  f, aux = partial_eval_wrapper_nounits(f, tuple(in_knowns), tuple(in_avals))
  consts = remat_call_p.bind(f, **params)  # no known inputs
  _, out_avals, jaxpr, env = aux()
  env_tracers = map(trace.full_raise, env)
  jaxpr = convert_constvars_jaxpr(jaxpr)
  if jaxpr.effects: raise NotImplementedError
  del in_pvals, in_knowns, in_avals, out_avals, f, aux, env
  # When concrete=True, we could avoid some redundant computation by extracting
  # values from any ConcreteArrays in `out_avals`, but we eschew that
  # optimization.

  # We're done with `f`, and in the steps to follow we work with `jaxpr`. To
  # that end, we want a list of which inputs to `jaxpr` are known/unknown.
  in_unknowns = ([False] * len(consts) +
                 [not t.is_known() for t in it.chain(env_tracers, tracers)])

  if params['policy']:
    # unzip into jaxpr_known and jaxpr_unknown
    jaxpr_known, jaxpr_unknown, out_unknowns, out_inst, _ = _partial_eval_jaxpr_custom(
        jaxpr, in_unknowns, params['policy'])
    jaxpr_known, in_used_known = dce_jaxpr(jaxpr_known, [True] * len(jaxpr_known.outvars))
    _, used_outs_unknown = partition_list(out_inst, out_unknowns)
    jaxpr_unknown, in_used_unknown = dce_jaxpr(jaxpr_unknown, used_outs_unknown)

    # compute known outputs and residuals (hoisted out of a remat_call)
    _, in_consts_ = unzip2(t.pval for t in it.chain(env_tracers, tracers)
                           if t.pval.is_known())
    _, known_inputs = partition_list(in_used_known, [*consts, *in_consts_])
    outs = core.eval_jaxpr(jaxpr_known, (), *known_inputs)
    known_outputs, res = split_list(outs, [len(out_unknowns)-sum(out_unknowns)])

    # set up unknown outputs with a recipe to call remat
    res_tracers = map(trace.new_instantiated_const, res)
    const_tracers = map(trace.new_instantiated_const, consts)
    in_jaxpr_tracers = [*res_tracers, *const_tracers, *env_tracers,
                        *instantiated_tracers]
    _, in_jaxpr_tracers = partition_list(in_used_unknown, in_jaxpr_tracers)
    unknown_outputs = [JaxprTracer(trace, PartialVal.unknown(x.aval), None)
                       for x in jaxpr_unknown.outvars]
    new_params = dict(params, call_jaxpr=jaxpr_unknown, differentiated=True)
    recipe = new_eqn_recipe(in_jaxpr_tracers, unknown_outputs, remat_call_p,
                            new_params, jaxpr_unknown.effects, source_info_util.current())
    for t in unknown_outputs: t.recipe = recipe
    return merge_lists(out_unknowns, known_outputs, unknown_outputs)
  else:
    # TODO(mattjj): this is an old parallel code path, to be deleted once the
    # new path is fully functional

    # Now that we have a `jaxpr` which represents as much as `f` as possible, we
    # want to actually compute known output values. To do that, we first extract
    # a `jaxpr_known`, and compute which outputs of `jaxpr` are known/unknown.
    jaxpr_known_, _, out_unknowns, res_avals = partial_eval_jaxpr_nounits(
        core.ClosedJaxpr(jaxpr, ()), in_unknowns, instantiate=False)  # type: ignore
    jaxpr_known, () = jaxpr_known_.jaxpr, jaxpr_known_.consts
    num_res = len(res_avals)
    # Next, we need values for known outputs. To get them, we need to evaluate
    # jaxpr_known, minus the residual outputs that we don't need. In addition to
    # eliminating residual outputs, we should perform DCE to eliminate the
    # computation of those residuals; for example, if the primal program
    # includes a sine, jaxpr_known includes both the sine and cosine, yet we
    # don't want to compute the cosine here.
    known_inputs = consts + [t for t in it.chain(env_tracers, tracers)
                             if t.pval.is_known()]
    num_known_outputs = len(out_unknowns) - sum(out_unknowns)
    jaxpr_known, kept_inputs = dce_jaxpr(
        jaxpr_known, [True] * num_known_outputs + [False] * num_res)
    known_inputs = [x for x, kept in zip(known_inputs, kept_inputs) if kept]
    known_outputs = core.eval_jaxpr(jaxpr_known, (), *known_inputs)
    del jaxpr_known, res_avals, num_res, num_known_outputs, kept_inputs

    # We compute unknown outputs by using the full `jaxpr`, though we can prune
    # out of it any known outputs and computations and only keep those
    # operations we need to compute the unknown outputs.
    jaxpr, kept_inputs = dce_jaxpr(jaxpr, out_unknowns)
    const_tracers = map(trace.instantiate_const, map(trace.full_raise, consts))
    unknown_inputs = [*const_tracers, *env_tracers, *instantiated_tracers]
    unknown_inputs = [x for x, kept in zip(unknown_inputs, kept_inputs) if kept]
    unknown_outputs = [JaxprTracer(trace, PartialVal.unknown(x.aval), None)
                       for x in jaxpr.outvars]
    eqn = new_eqn_recipe(unknown_inputs, unknown_outputs, remat_call_p,
                         dict(params, call_jaxpr=jaxpr,
                              differentiated=True),
                         jaxpr.effects, source_info_util.current())
    for t in unknown_outputs: t.recipe = eqn

    return merge_lists(out_unknowns, known_outputs, unknown_outputs)
call_partial_eval_rules[remat_call_p] = _remat_partial_eval

def _partition_knowns(pvals, unknowns: Sequence[bool]):
  return ([e for e, unknown in zip(pvals, unknowns) if not unknown],
          [e for e, unknown in zip(pvals, unknowns) if unknown])

def _zip_knowns(known_list, unknown_list, which_unknown: Sequence[bool]):
  assert len(known_list) + len(unknown_list) == len(which_unknown)
  known_iter, unknown_iter = iter(known_list), iter(unknown_list)
  return [next(unknown_iter) if uk else next(known_iter) for uk in which_unknown]


def _partial_eval_jaxpr_custom(
    jaxpr: Jaxpr, in_unknowns: Sequence[bool], saveable: Callable[..., bool],
  ) -> Tuple[Jaxpr, Jaxpr, Sequence[bool], Sequence[bool], int]:
  if jaxpr.constvars: raise NotImplementedError  # TODO(mattjj)
  env: Dict[Var, Tuple[bool, bool]] = {}
  residuals: OrderedSet[Var] = OrderedSet()

  def read(x: Atom) -> Tuple[bool, bool]:
    if type(x) is Var:
      return env[x]
    return (False, True)

  def write(unk: bool, inst: bool, v: Var) -> None:
    assert (unk, inst) != (True, False)
    env[v] = (unk, inst)

  def ensure_instantiated(inst: bool, x: Atom) -> Atom:
    if type(x) is Var and not inst:
      residuals.add(x)
    return x

  known_eqns, staged_eqns = [], []
  write(False, True, core.unitvar)
  map(write, in_unknowns, [True] * len(in_unknowns), jaxpr.invars)
  for eqn in jaxpr.eqns:
    unks_in, inst_in = unzip2(map(read, eqn.invars))
    rule = partial_eval_jaxpr_custom_rules.get(eqn.primitive)
    if rule:
      eqn1, eqn2, unks_out, inst_out, res = rule(saveable, unks_in, inst_in, eqn)
      eqn1 and known_eqns.append(eqn1); eqn2 and staged_eqns.append(eqn2)  # type: ignore
      residuals.update(res)
      map(write, unks_out, inst_out, eqn.outvars)
    elif any(unks_in):
      inputs = map(ensure_instantiated, inst_in, eqn.invars)
      staged_eqns.append(eqn.replace(invars=inputs))
      map(partial(write, True, True), eqn.outvars)
    else:
      known_eqns.append(eqn)
      if saveable(eqn.primitive, *[x.aval for x in eqn.invars], **eqn.params):
        map(partial(write, False, False), eqn.outvars)
      else:
        inputs = map(ensure_instantiated, inst_in, eqn.invars)
        staged_eqns.append(eqn.replace(invars=inputs))
        map(partial(write, False, True), eqn.outvars)
  out_unknowns, out_inst = unzip2(map(read, jaxpr.outvars))
  assert all(type(v) is Var for v in residuals), residuals

  ins_known, _ = partition_list(in_unknowns, jaxpr.invars)
  outs_known_, _ = partition_list(out_unknowns, jaxpr.outvars)
  outs_known = [x for x in outs_known_ if x.aval is not core.abstract_unit]
  known_effects = core.join_effects(*(eqn.effects for eqn in known_eqns))
  jaxpr_known = Jaxpr((), ins_known, [*outs_known, *residuals], known_eqns,
                      known_effects)
  config.jax_enable_checks and core.check_jaxpr(jaxpr_known)

  _, outs_staged = partition_list(out_inst, jaxpr.outvars)
  staged_effects = core.join_effects(*(eqn.effects for eqn in staged_eqns))
  jaxpr_staged = Jaxpr((), [*residuals, *jaxpr.invars], outs_staged,
                       staged_eqns, staged_effects)
  config.jax_enable_checks and core.check_jaxpr(jaxpr_staged)

  return jaxpr_known, jaxpr_staged, out_unknowns, out_inst, len(residuals)

# A primitive rule for policy-driven partial evaluation returns a 5-tuple
# with the components representing, respectively:
#  * the JaxprEqn for the 'known' side (or None if there is no known component),
#  * the JaxprEqn for the 'unknown' side (or None),
#  * a list of booleans indicating which of the original outputs are unknown,
#  * a list of booleans indicating which of the original outputs are
#    instantiated (i.e. available) in the 'unknown' side,
#  * a list of Var instances representing residuals to be added (i.e. to be
#    plumbed as outputs of the 'known' side jaxpr and added as input binders to
#    the 'unknown' jaxpr).
PartialEvalCustomResult = Tuple[Optional[JaxprEqn], Optional[JaxprEqn],
                                Sequence[bool], Sequence[bool], List[Var]]
PartialEvalCustomRule = Callable[
    [Callable[..., bool], Sequence[bool], Sequence[bool], JaxprEqn],
    PartialEvalCustomResult]
partial_eval_jaxpr_custom_rules: Dict[Primitive, PartialEvalCustomRule] = {}

def partial_eval_jaxpr_custom_rule_not_implemented(
    name: str, saveable: Callable[..., bool], unks_in: Sequence[bool],
    inst_in: Sequence[bool], eqn: JaxprEqn) -> PartialEvalCustomResult:
  msg = (f'custom-policy remat rule not implemented for {name}, '
         'open a feature request at https://github.com/google/jax/issues!')
  raise NotImplementedError(msg)


ParamsUpdater = Callable[[Sequence[bool], Sequence[bool], Sequence[bool],
                          int, dict, dict], Tuple[dict, dict]]

def call_partial_eval_custom_rule(
    jaxpr_param_name: str, params_updater: ParamsUpdater,
    saveable: Callable[..., bool], unks_in: List[bool], inst_in: List[bool],
    eqn: JaxprEqn
  ) -> Tuple[JaxprEqn, JaxprEqn, Sequence[bool], Sequence[bool], List[Var]]:
  jaxpr = eqn.params[jaxpr_param_name]
  jaxpr_known, jaxpr_staged, unks_out, inst_out, num_res = \
      _partial_eval_jaxpr_custom(jaxpr, unks_in, saveable)
  ins_known, _ = partition_list(unks_in, eqn.invars)
  # by convention, _partial_eval_jaxpr_custom drops units on known outputs
  known_units_out = [v.aval is core.abstract_unit for v in jaxpr.outvars]
  dropped_outs_known = map(op.or_, unks_out, known_units_out)
  kept_outs_known = [not d for d in dropped_outs_known]
  out_binders_known, _ = partition_list(dropped_outs_known, eqn.outvars)
  _, out_binders_staged = partition_list(inst_out, eqn.outvars)
  kept_outs_staged = inst_out
  newvar = core.gensym([jaxpr_known, jaxpr_staged])
  residuals = [newvar(v.aval) for v in jaxpr_staged.invars[:num_res]]
  params_known = {**eqn.params, jaxpr_param_name: jaxpr_known}
  params_staged = {**eqn.params, jaxpr_param_name: jaxpr_staged}
  params_known, params_staged = params_updater(
      unks_in, kept_outs_known, kept_outs_staged, num_res, params_known, params_staged)
  eqn_known = new_jaxpr_eqn(ins_known, [*out_binders_known, *residuals],
                            eqn.primitive, params_known, jaxpr_known.effects, eqn.source_info)
  eqn_staged = new_jaxpr_eqn([*residuals, *eqn.invars], out_binders_staged,
                             eqn.primitive, params_staged,
                             jaxpr_staged.effects, eqn.source_info)
  assert len(eqn_staged.invars) == len(jaxpr_staged.invars)
  new_inst = [x for x, inst in zip(eqn.invars, inst_in)
              if type(x) is Var and not inst]
  return eqn_known, eqn_staged, unks_out, inst_out, new_inst + residuals
partial_eval_jaxpr_custom_rules[core.call_p] = \
    partial(call_partial_eval_custom_rule, 'call_jaxpr',
            lambda _, __, ___, ____, x, y: (x, y))
partial_eval_jaxpr_custom_rules[core.named_call_p] = \
    partial(call_partial_eval_custom_rule, 'call_jaxpr',
            lambda _, __, ___, ____, x, y: (x, y))
partial_eval_jaxpr_custom_rules[remat_call_p] = \
    partial(call_partial_eval_custom_rule, 'call_jaxpr',
            lambda _, __, ___, ____, p1, p2: (p1, dict(p2, differentiated=True)))


def _jaxpr_forwarding(jaxpr: Jaxpr) -> List[Optional[int]]:
  # Compute which inputs are just forwarded to outputs.
  fwds: Dict[Var, Var] = dict(zip(jaxpr.invars, jaxpr.invars))
  for eqn in jaxpr.eqns:
    if eqn.primitive in forwarding_rules:
      eqn = eqn.replace(invars=[fwds.get(v, v) for v in eqn.invars])  # type: ignore
      fwd_vars, _ = forwarding_rules[eqn.primitive](eqn)
      for v_orig, v_new in zip(eqn.outvars, fwd_vars):
        if v_new is not None:
          fwds[v_orig] = v_new
  idxs: Dict[Var, int] = {v: i for i, v in enumerate(jaxpr.invars)}
  return [None if type(v) is Literal else idxs.get(fwds.get(v))  # type: ignore
          for v in jaxpr.outvars]


def dce_jaxpr(jaxpr: Jaxpr, used_outputs: Sequence[bool]
              ) -> Tuple[Jaxpr, List[bool]]:
  return _dce_jaxpr(jaxpr, tuple(used_outputs))

@weakref_lru_cache
def _dce_jaxpr(jaxpr: Jaxpr, used_outputs: Tuple[bool, ...]
               ) -> Tuple[Jaxpr, List[bool]]:
  env: Dict[Var, bool] = {}

  def read(v: Var) -> bool:
    return env.get(v, False)

  def write(x: Atom, b: bool) -> None:
    if type(x) is Var:
      env[x] = read(x) or b

  new_eqns = []
  map(write, jaxpr.outvars, used_outputs)
  for eqn in jaxpr.eqns[::-1]:
    used_outs = map(read, eqn.outvars)
    # If any outputs are used, then we need to keep a version of the eqn and
    # potentially mark some inputs as used. Otherwise mark all inputs as unused.
    if any(used_outs) or core.primitive_uses_outfeed(eqn.primitive, eqn.params):
      # If there's a rule for modifying the eqn and computing used inputs, apply
      # it. Otherwise, keep the eqn unmodified and mark all inputs as used.
      rule = dce_rules.get(eqn.primitive)
      if rule:
        used_ins, new_eqn = rule(used_outs, eqn)
      else:
        used_ins = [True] * len(eqn.invars)
        new_eqn = eqn
      new_eqns.append(new_eqn)
    else:
      used_ins = [False] * len(eqn.invars)
    map(write, eqn.invars, used_ins)
  used_inputs = map(read, jaxpr.invars)

  new_jaxpr = Jaxpr(jaxpr.constvars,
                    [v for v, b in zip(jaxpr.invars, used_inputs)   if b],
                    [v for v, b in zip(jaxpr.outvars, used_outputs) if b],
                    new_eqns[::-1], jaxpr.effects)
  config.jax_enable_checks and core.check_jaxpr(new_jaxpr)

  return new_jaxpr, used_inputs

DCERule = Callable[[List[bool], JaxprEqn], Tuple[List[bool], JaxprEqn]]
dce_rules: Dict[Primitive, DCERule] = {}


def dce_jaxpr_call_rule(used_outputs: List[bool], eqn: JaxprEqn
                        ) -> Tuple[List[bool], JaxprEqn]:
  new_jaxpr, used_inputs = dce_jaxpr(eqn.params['call_jaxpr'], used_outputs)
  new_params = dict(eqn.params, call_jaxpr=new_jaxpr)
  update_params = call_param_updaters.get(eqn.primitive)
  if update_params:
    new_params = update_params(new_params, used_inputs, 0)
  new_eqn = new_jaxpr_eqn([v for v, used in zip(eqn.invars, used_inputs) if used],
                          [v for v, used in zip(eqn.outvars, used_outputs) if used],
                          eqn.primitive, new_params, new_jaxpr.effects, eqn.source_info)
  return used_inputs, new_eqn
dce_rules[core.call_p] = dce_jaxpr_call_rule
dce_rules[core.named_call_p] = dce_jaxpr_call_rule
dce_rules[remat_call_p] = dce_jaxpr_call_rule


def move_binders_to_front(closed_jaxpr: ClosedJaxpr, to_move: Sequence[bool]
                          ) -> ClosedJaxpr:
  """Reorder `invars` by moving those indicated in `to_move` to the front."""
  return _move_binders_to_front(closed_jaxpr, tuple(to_move))

@weakref_lru_cache
def _move_binders_to_front(closed_jaxpr: ClosedJaxpr, to_move: Tuple[bool, ...]
                           ) -> ClosedJaxpr:
  assert len(closed_jaxpr.in_avals) == len(to_move)
  new_invars = _move_to_front(closed_jaxpr.jaxpr.invars, to_move)
  new_jaxpr = Jaxpr(closed_jaxpr.jaxpr.constvars, new_invars,
                    closed_jaxpr.jaxpr.outvars, closed_jaxpr.jaxpr.eqns,
                    closed_jaxpr.jaxpr.effects)
  new_closed_jaxpr = core.ClosedJaxpr(new_jaxpr, closed_jaxpr.consts)
  return new_closed_jaxpr

def _move_to_front(lst: Sequence, to_move: Sequence[bool]) -> Sequence:
  return ([elt for elt, move in zip(lst, to_move) if move] +
          [elt for elt, move in zip(lst, to_move) if not move])

def move_binders_to_back(closed_jaxpr: ClosedJaxpr, to_move: Sequence[bool]
                         ) -> ClosedJaxpr:
  """Reorder `invars` by moving those indicated in `to_move` to the back."""
  return move_binders_to_front(closed_jaxpr, map(op.not_, to_move))

class DynamicJaxprTracer(core.Tracer):
  __slots__ = ['aval']

  def __init__(self, trace, aval, line_info=None):
    self._trace = trace
    self._line_info = line_info
    self.aval = aval

  def full_lower(self):
    return self

  def _contents(self):
    return ()

  def _origin_msg(self):
    if not self._trace.main.jaxpr_stack:  # type: ignore
      # If this Tracer has been leaked the jaxpr stack may no longer be
      # available. So we can't print as much origin information.
      return ("\nThis Tracer was created on line "
              f"{source_info_util.summarize(self._line_info)}")
    else:
      invar_pos, progenitor_eqns = self._trace.frame.find_progenitors(self)
    dbg = self._trace.main.debug_info
    if dbg is None:
      return ""
    if invar_pos:
      origin = (f"While tracing the function {dbg.func_src_info} "
                f"for {dbg.traced_for}, "
                "this concrete value was not available in Python because it "
                f"depends on the value{'s' if len(invar_pos) > 1 else ''} "
                f"of {dbg.arg_info(invar_pos)}.")
    elif progenitor_eqns:
      msts = ["  operation "
              f"{core.pp_eqn(eqn, core.JaxprPpContext(), core.JaxprPpSettings(print_shapes=True))}\n"
              f"    from line {source_info_util.summarize(eqn.source_info)}"
              for eqn in progenitor_eqns[:5]]  # show at most 5
      origin = (f"While tracing the function {dbg.func_src_info} "
                f"for {dbg.traced_for}, "
                "this value became a tracer due to JAX operations on these lines:"
                "\n\n" + "\n\n".join(msts))
      if len(progenitor_eqns) > 5:
        origin += "\n\n(Additional originating lines are not shown.)"
    else:
      origin = (f"The error occured while tracing the function {dbg.func_src_info} "
                f"for {dbg.traced_for}.")
    return "\n" + origin

  def _assert_live(self) -> None:
    if not self._trace.main.jaxpr_stack:  # type: ignore
      raise core.escaped_tracer_error(self, None)

TracerId = int
AvalId = int
ConstId = int
class JaxprStackFrame:
  gensym: Callable[[AbstractValue], Var]
  tracer_to_var: Dict[TracerId, Var]
  constid_to_tracer: Dict[ConstId, Tracer]
  constvar_to_val: Dict[Var, Any]
  tracers: List[DynamicJaxprTracer]  # hold onto strong refs for all tracers
  eqns: List[JaxprEqn]
  invars: List[Var]
  effects: core.Effects

  def __init__(self):
    self.gensym = core.gensym()
    self.tracer_to_var = {}
    self.constid_to_tracer = {}
    self.constvar_to_val = {}
    self.tracers = []   # circ refs, frame->tracer->trace->main->frame,
    self.eqns = []      # cleared when we pop frame from main
    self.invars = []
    self.effects = set()

  def add_eqn(self, eqn: core.JaxprEqn):
    self.eqns.append(eqn)
    self.effects |= eqn.effects

  def to_jaxpr(self, out_tracers):
    # It's not necessary, but we keep the tracer-to-var mapping injective:
    assert len(self.tracer_to_var) == len(set(self.tracer_to_var.values()))
    outvars = [self.tracer_to_var[id(t)] for t in out_tracers]
    constvars, constvals = unzip2(self.constvar_to_val.items())
    jaxpr = Jaxpr(constvars, self.invars, outvars, self.eqns, self.effects)
    jaxpr, constvals = _const_folding_and_forwarding(jaxpr, constvals)
    jaxpr, constvals = _inline_literals(jaxpr, constvals)
    return jaxpr, constvals

  def newvar(self, aval):
    if isinstance(aval, DShapedArray):
      # this aval may have tracers in it, so we replace those with variables
      new_shape = [self.tracer_to_var[id(d)] if isinstance(d, Tracer) else d
                   for d in aval.shape]
      aval = aval.update(shape=tuple(new_shape))
    return self.gensym(aval)

  def find_progenitors(self, tracer):
    var = self.tracer_to_var.get(id(tracer))
    if not var:
      return None, None
    active_vars = {var}
    for eqn in self.eqns[::-1]:
      produced = set(eqn.outvars) & active_vars
      if produced:
        active_vars.difference_update(produced)
        active_vars.update(eqn.invars)
    invar_positions = [i for i, v in enumerate(self.invars) if v in active_vars]
    constvars = active_vars & set(self.constvar_to_val)
    const_eqns = [eqn for eqn in self.eqns if set(eqn.invars) & constvars]
    return invar_positions, const_eqns

def _const_folding_and_forwarding(jaxpr, constvals):
  consts: Dict[Var, Any] = dict(zip(jaxpr.constvars, constvals))
  var_subs: Dict[Var, Var] = {}  # not Dict[Var, Atom] b/c literals not inlined
  new_eqns = []
  for eqn in jaxpr.eqns:
    # always apply invar substitutions
    eqn = eqn.replace(invars=[var_subs.get(v, v) for v in eqn.invars])
    # if any inputs are constants and we have a constant-folding rule, apply it
    if eqn.primitive in const_fold_rules and any(v in consts for v in eqn.invars):
      consts_in = [consts.get(v) for v in eqn.invars]
      consts_out, new_eqn = const_fold_rules[eqn.primitive](consts_in, eqn)
      assert (new_eqn is None) == all(c is not None for c in consts_out)
      for v, c in zip(eqn.outvars, consts_out):
        if c is not None: consts[v] = c
      if new_eqn is None: continue
      else: eqn = new_eqn
    # if the application trivially maps some inputs to outputs, simplify
    if eqn.primitive in forwarding_rules:
      fwd_vars, new_eqn = forwarding_rules[eqn.primitive](eqn)
      assert (new_eqn is None) == all(v is not None for v in fwd_vars)
      for v_orig, v_new in zip(eqn.outvars, fwd_vars):
        if v_new is not None: var_subs[v_orig] = v_new
      if new_eqn is None: continue
      else: eqn = new_eqn
    new_eqns.append(eqn)
  new_constvars, new_constvals = unzip2(consts.items())
  new_outvars = [var_subs.get(v, v) for v in jaxpr.outvars]
  new_jaxpr = Jaxpr(new_constvars, jaxpr.invars, new_outvars, new_eqns, jaxpr.effects)
  return new_jaxpr, new_constvals

ConstFoldRule = Callable[[List[Optional[Any]], JaxprEqn],
                         Tuple[List[Optional[Any]], Optional[JaxprEqn]]]
const_fold_rules: Dict[Primitive, ConstFoldRule] = {}

ForwardingRule = Callable[[JaxprEqn],
                          Tuple[List[Optional[Var]], Optional[JaxprEqn]]]
forwarding_rules: Dict[Primitive, ForwardingRule] = {}

def _inline_literals(jaxpr, constvals):
  # This function also ensures variables are labeled in a canonical ordering,
  # prunes unused constants, and inserts `dropvar` symbols.
  lits = {v: Literal(c, v.aval) for v, c in zip(jaxpr.constvars, constvals)
          if type(c) in core.literalable_types and not np.shape(c)}
  lit: Callable[[Var], Optional[Literal]] = lits.get
  newname: Callable[[AbstractValue], Var] = core.gensym()
  newvars: Dict[Var, Var] = {}
  newvar = lambda aval: newname(_substitute_vars_in_type(lits, newvars, aval))
  var = lambda v: newvars.get(v) or newvars.setdefault(v, newvar(v.aval))
  dropvar = lambda aval: DropVar(_substitute_vars_in_type(lits, newvars, aval))

  def vars_in_shape(aval: AbstractValue) -> Sequence[Var]:
    if isinstance(aval, DShapedArray):
      return [d for d in aval.shape if isinstance(d, Var)]
    return []

  used = {v for eqn in jaxpr.eqns for invar in eqn.invars
          for v in it.chain([invar], vars_in_shape(invar.aval))}
  used |= {v for outvar in jaxpr.outvars
           for v in it.chain([outvar], vars_in_shape(outvar.aval))}
  new_constvars = [var(v) for v in jaxpr.constvars if v in used and not lit(v)]
  new_constvals = [c for v, c in zip(jaxpr.constvars, constvals)
                   if v in used and not lit(v)]
  new_invars = [var(v) for v in jaxpr.invars]
  new_eqns = []
  for eqn in jaxpr.eqns:
    invars = [lit(v) or var(v) for v in eqn.invars]
    outvars = [var(v) if v in used else dropvar(v.aval) for v in eqn.outvars]
    new_eqns.append(eqn.replace(invars=invars, outvars=outvars))
  new_outvars = [lit(v) or var(v) for v in jaxpr.outvars]
  new_jaxpr = Jaxpr(new_constvars, new_invars, new_outvars, new_eqns,
      jaxpr.effects)
  return new_jaxpr, new_constvals

class DynamicJaxprTrace(core.Trace):
  __slots__ = []  # type: ignore

  @property
  def frame(self):
    return self.main.jaxpr_stack[-1]  # pytype: disable=attribute-error

  def new_arg(self, aval):
    tracer = DynamicJaxprTracer(self, aval, source_info_util.current())
    self.frame.tracers.append(tracer)
    self.frame.tracer_to_var[id(tracer)] = var = self.frame.newvar(aval)
    self.frame.invars.append(var)
    return tracer

  def new_const(self, c):
    # TODO(mattjj): for ints, or hashable consts, don't rely on id
    tracer = self.frame.constid_to_tracer.get(id(c))
    if tracer is None:
      aval = raise_to_shaped(get_aval(c), weak_type=dtypes.is_weakly_typed(c))
      tracer = self._new_const(aval, c)
    return tracer

  pure = lift = new_const

  def _new_const(self, aval, c):
    tracer = DynamicJaxprTracer(self, aval, source_info_util.current())
    self.frame.tracers.append(tracer)
    self.frame.tracer_to_var[id(tracer)] = var = self.frame.newvar(aval)
    self.frame.constid_to_tracer[id(c)] = tracer
    self.frame.constvar_to_val[var] = c
    return tracer

  def sublift(self, t):
    # When lifting closed-over tracers corresponding to this same trace, the
    # variable to lift could have tracers (representing axis size variables) in
    # its shape. We must lift those too!
    tracer = self.frame.constid_to_tracer.get(id(t))
    if tracer is None:
      aval = raise_to_shaped(get_aval(t), weak_type=dtypes.is_weakly_typed(t))
      aval = self._lift_tracers_in_aval(aval)
      tracer = self._new_const(aval, t)
    return tracer

  def _lift_tracers_in_aval(self, aval):
    if (not isinstance(aval, DShapedArray) or
        not any(isinstance(d, Tracer) for d in aval.shape)):
      return aval
    shape = [self.full_raise(d) if isinstance(d, Tracer) else d
             for d in aval.shape]
    return aval.update(shape=tuple(shape))

  def getvar(self, tracer):
    var = self.frame.tracer_to_var.get(id(tracer))
    if var is None:
      raise core.escaped_tracer_error(tracer)
    return var

  def makevar(self, tracer):
    var = self.frame.tracer_to_var.get(id(tracer))
    assert var is None, "a jaxpr variable must be created only once per tracer"
    self.frame.tracers.append(tracer)
    var = self.frame.tracer_to_var[id(tracer)] = self.frame.newvar(tracer.aval)
    return var

  def instantiate_const(self, val):
    if (isinstance(val, Tracer) and val._trace.main is self.main
        and val._trace.sublevel == self.sublevel):
      return val
    else:
      return self.new_const(val)

  def process_primitive(self, primitive, tracers, params):
    if primitive in custom_staging_rules:
      return custom_staging_rules[primitive](self, *tracers, **params)
    return self.default_process_primitive(primitive, tracers, params)

  def default_process_primitive(self, primitive, tracers, params):
    avals = [t.aval for t in tracers]
    out_avals, effects = primitive.abstract_eval(*avals, **params)
    out_avals = [out_avals] if not primitive.multiple_results else out_avals
    source_info = source_info_util.current()
    out_tracers = [DynamicJaxprTracer(self, a, source_info) for a in out_avals]
    invars = map(self.getvar, tracers)
    outvars = map(self.makevar, out_tracers)
    eqn = new_jaxpr_eqn(invars, outvars, primitive, params, effects, source_info)
    self.frame.add_eqn(eqn)
    return out_tracers if primitive.multiple_results else out_tracers.pop()

  def process_call(self, call_primitive, f, explicit_tracers, params):
    if f.in_type is None:
      in_avals = [core.raise_to_shaped(get_aval(x)) for x in explicit_tracers]
      keep_inputs = [True] * len(explicit_tracers)
      im_tracers = []
    else:
      im_tracers = _extract_implicit_args(self, f.in_type, explicit_tracers)
      in_avals, keep_inputs = unzip2(f.in_type)
    with core.new_sublevel():
      jaxpr, out_avals, consts = trace_to_subjaxpr_dynamic(
        f, self.main, in_avals, keep_inputs=keep_inputs)
    if jaxpr.effects:
      raise NotImplementedError('Effects not supported for call primitives.')
    tracers = [*im_tracers, *explicit_tracers]
    if params.get('inline', False):
      return core.eval_jaxpr(jaxpr, consts, *tracers)
    env = {v: t for v, t in zip(jaxpr.constvars, consts) if isinstance(t, Tracer)}
    env.update(zip(jaxpr.invars, tracers))
    out_avals_ = [_substitute_tracers_in_type(env, a) for a in out_avals]
    source_info = source_info_util.current()
    out_tracers = [DynamicJaxprTracer(self, a, source_info) for a in out_avals_]
    invars = map(self.getvar, tracers)
    constvars = map(self.getvar, map(self.instantiate_const, consts))
    outvars = map(self.makevar, out_tracers)
    new_params = dict(params, call_jaxpr=convert_constvars_jaxpr(jaxpr))
    update_params = call_param_updaters.get(call_primitive)
    if update_params:
      new_params = update_params(new_params, [True] * len(explicit_tracers),
                                 len(consts) + len(im_tracers))
    eqn = new_jaxpr_eqn([*constvars, *invars], outvars,
                        call_primitive, new_params,
                        new_params['call_jaxpr'].effects, source_info)
    self.frame.add_eqn(eqn)
    return out_tracers

  def post_process_call(self, call_primitive, out_tracers, params):
    assert False  # unreachable

  def process_map(self, map_primitive, f, tracers, params):
    in_avals = [t.aval for t in tracers]
    axis_name, axis_size = params['axis_name'], params['axis_size']
    reduced_in_avals = [core.mapped_aval(axis_size, in_axis, a)
                        if in_axis is not None else a
                        for a, in_axis in zip(in_avals, params['in_axes'])]
    with core.extend_axis_env(axis_name, axis_size, None):  # type: ignore
      with core.new_sublevel():
        jaxpr, reduced_out_avals, consts = trace_to_subjaxpr_dynamic(
            f, self.main, reduced_in_avals)
      if jaxpr.effects:
        raise NotImplementedError('Effects not supported for map primitives.')
      out_axes = params['out_axes_thunk']()
      out_avals = [core.unmapped_aval(axis_size, axis_name, out_axis, a)
                  if out_axis is not None else a
                  for a, out_axis in zip(reduced_out_avals, out_axes)]
      source_info = source_info_util.current()
      out_tracers = [DynamicJaxprTracer(self, a, source_info) for a in out_avals]
      invars = map(self.getvar, tracers)
      constvars = map(self.getvar, map(self.instantiate_const, consts))
      outvars = map(self.makevar, out_tracers)
      new_in_axes = (None,) * len(consts) + params['in_axes']
      new_params = dict(params, in_axes=new_in_axes, out_axes=out_axes,
                        call_jaxpr=convert_constvars_jaxpr(jaxpr))
      del new_params['out_axes_thunk']
      update_params = call_param_updaters.get(map_primitive)
      if update_params:
        new_params = update_params(new_params, [True] * len(tracers), len(consts))
      eqn = new_jaxpr_eqn([*constvars, *invars], outvars, map_primitive,
                          new_params, new_params['call_jaxpr'].effects, source_info)
      self.frame.add_eqn(eqn)
    return out_tracers

  def post_process_map(self, map_primitive, out_tracers, params):
    assert False  # unreachable

  def process_custom_jvp_call(self, prim, fun, jvp, tracers):
    in_avals = [t.aval for t in tracers]
    with core.new_sublevel():
      fun_jaxpr, out_avals, consts = trace_to_subjaxpr_dynamic(fun, self.main, in_avals)
    if fun_jaxpr.effects:
      raise NotImplementedError('Effects not supported in `custom_jvp`.')
    closed_fun_jaxpr = core.ClosedJaxpr(convert_constvars_jaxpr(fun_jaxpr), ())
    main_ = ref(self.main)
    jvp_jaxpr_thunk = _memoize(
        lambda: trace_to_subjaxpr_dynamic(jvp, main_(), 2 * in_avals)[::2])
    out_tracers = [DynamicJaxprTracer(self, a) for a in out_avals]
    invars = map(self.getvar, tracers)
    constvars = map(self.getvar, map(self.instantiate_const, consts))
    outvars = map(self.makevar, out_tracers)
    eqn = new_jaxpr_eqn([*constvars, *invars], outvars, prim.initial_style,
                        dict(fun_jaxpr=closed_fun_jaxpr,
                             jvp_jaxpr_thunk=jvp_jaxpr_thunk,
                             num_consts=len(consts)),
                        fun_jaxpr.effects,
                        source_info_util.current())
    self.frame.add_eqn(eqn)
    return out_tracers

  def post_process_custom_jvp_call(self, out_tracers, _):
    assert False  # unreachable

  def process_custom_vjp_call(self, prim, fun, fwd, bwd, tracers, out_trees):
    in_avals = [t.aval for t in tracers]
    with core.new_sublevel():
      fun_jaxpr, out_avals, consts = trace_to_subjaxpr_dynamic(fun, self.main, in_avals)
    if fun_jaxpr.effects:
      raise NotImplementedError('Effects not supported in `custom_vjp`.')
    closed_fun_jaxpr = core.ClosedJaxpr(convert_constvars_jaxpr(fun_jaxpr), ())
    main_ = ref(self.main)
    fwd_jaxpr_thunk = _memoize(
        lambda: trace_to_subjaxpr_dynamic(fwd, main_(), in_avals)[::2])
    out_tracers = [DynamicJaxprTracer(self, a) for a in out_avals]
    invars = map(self.getvar, tracers)
    constvars = map(self.getvar, map(self.instantiate_const, consts))
    outvars = map(self.makevar, out_tracers)
    eqn = new_jaxpr_eqn([*constvars, *invars], outvars, prim.initial_style,
                        dict(fun_jaxpr=closed_fun_jaxpr,
                             fwd_jaxpr_thunk=fwd_jaxpr_thunk,
                             num_consts=len(consts),
                             bwd=bwd, out_trees=out_trees),
                        fun_jaxpr.effects,
                        source_info_util.current())
    self.frame.add_eqn(eqn)
    return out_tracers

  def post_process_custom_vjp_call(self, out_tracers, _):
    assert False  # unreachable

  def process_custom_transpose(self, prim, call, tracers,
                               transpose, out_types,
                               lin_tree, res_tree, out_tree):
    tracers_res, tracers_lin = split_list(tracers, [res_tree.num_leaves])

    in_avals_p = [t.aval for t in tracers]
    in_avals_t = [*[t.aval for t in tracers_res], *out_types]

    with core.new_sublevel():
      call_jaxpr, out_avals, call_consts = trace_to_subjaxpr_dynamic(
          call, self.main, in_avals_p)
    closed_call_jaxpr = core.ClosedJaxpr(
        convert_constvars_jaxpr(call_jaxpr), ())

    transpose_flat, in_tree2 = flatten_fun_nokwargs(
        lu.wrap_init(transpose), treedef_tuple((res_tree, out_tree)))

    main_ = ref(self.main)
    # the following thunk evaluates to a pair: transpose_jaxpr, transpose_consts
    transpose_jaxpr_thunk = _memoize(
        lambda: trace_to_subjaxpr_dynamic(
            transpose_flat, main_(), in_avals_t)[::2])

    out_tracers = [DynamicJaxprTracer(self, a) for a in out_avals]
    invars = map(self.getvar, tracers)
    constvars = map(self.getvar, map(self.instantiate_const, call_consts))
    outvars = map(self.makevar, out_tracers)
    eqn = new_jaxpr_eqn([*constvars, *invars], outvars, prim,
                        dict(call_jaxpr=closed_call_jaxpr,
                             transpose_jaxpr_thunk=transpose_jaxpr_thunk,
                             out_types=out_types, res_tree=res_tree,
                             lin_tree=lin_tree, out_tree=out_tree),
                        closed_call_jaxpr.effects,
                        source_info_util.current())
    self.frame.add_eqn(eqn)
    return out_tracers


custom_staging_rules: Dict[Primitive, Callable] = {}

def _memoize(thunk):
  if config.jax_check_tracer_leaks:
    return thunk

  cell = []
  saved_state = core.thread_local_state.trace_state.copy()
  def memoized():
    if not cell:
      prev_state = core.thread_local_state.trace_state
      core.thread_local_state.trace_state = saved_state
      try:
        cell.append(thunk())
      finally:
        core.thread_local_state.trace_state = prev_state
    return cell[0]
  return memoized


class DebugInfo(NamedTuple):
  func_src_info: str
  traced_for: str
  arg_info: Callable[[int], str]


def debug_info_final(fn: lu.WrappedFun, traced_for: str) -> DebugInfo:
  in_tree, has_kwargs = flattened_fun_in_tree(fn) or (None, False)
  return debug_info(fn.f, in_tree, has_kwargs, traced_for)

def debug_info(fn: Callable, in_tree: Optional[PyTreeDef], has_kwargs: bool,
               traced_for: str) -> DebugInfo:
  func_src_info = fun_sourceinfo(fn)
  if in_tree is not None:
    arg_info = partial(arg_info_pytree, fn, in_tree, has_kwargs)
  else:
    arg_info = arg_info_flattened  # type: ignore
  return DebugInfo(func_src_info, traced_for, arg_info)

def fun_sourceinfo(fun: Callable):
  while isinstance(fun, functools.partial):
    fun = fun.func
  fun = inspect.unwrap(fun)
  try:
    filename = fun.__code__.co_filename
    lineno = fun.__code__.co_firstlineno
    line_info = f"{fun.__name__} at {filename}:{lineno}"
    return line_info
  except AttributeError:
    return "<unknown>"

def arg_info_pytree(fn: Callable, in_tree: PyTreeDef, has_kwargs: bool,
                    flat_pos: List[int]) -> str:
  dummy_args = [False] * in_tree.num_leaves
  for i in flat_pos: dummy_args[i] = True
  if has_kwargs:
    args, kwargs = tree_unflatten(in_tree, dummy_args)
  else:
    args, kwargs = tree_unflatten(in_tree, dummy_args), {}
  try:
    ba = inspect.signature(fn).bind(*args, **kwargs)
  except (TypeError, ValueError):
    return arg_info_flattened(flat_pos)
  arg_names = [f"'{name}'" for name, x in ba.arguments.items()
               if any(tree_leaves(x))]
  if len(arg_names) == 1:
    return f"the argument {arg_names[0]}"
  elif len(arg_names) == 2:
    return f"the arguments {arg_names[0]} and {arg_names[1]}"
  else:
    *rest, last = arg_names
    return f"the arguments {', '.join(rest)}, and {last}"

def arg_info_flattened(flat_pos: List[int]) -> str:
  if len(flat_pos) > 1:
    return f"the argument passed at flattened positions {flat_pos}"
  else:
    return f"the argument passed at flattened position {flat_pos[0]}"


@profiler.annotate_function
def trace_to_jaxpr_dynamic(fun: lu.WrappedFun,
                           in_avals: Sequence[AbstractValue],
                           debug_info: Optional[DebugInfo] = None,
                           *,
                           keep_inputs: Optional[List[bool]] = None):
  with core.new_main(DynamicJaxprTrace, dynamic=True) as main:  # type: ignore
    main.debug_info = debug_info  # type: ignore
    main.jaxpr_stack = ()  # type: ignore
    jaxpr, out_avals, consts = trace_to_subjaxpr_dynamic(
      fun, main, in_avals, keep_inputs=keep_inputs)
    del main, fun
  return jaxpr, out_avals, consts

def trace_to_subjaxpr_dynamic(fun: lu.WrappedFun, main: core.MainTrace,
                              in_avals: Sequence[AbstractValue], *,
                              keep_inputs: Optional[Sequence[bool]] = None):
  # In general, the Tracers passed to ther Python callable underlying `fun` may
  # correspond to a subset of `in_avals` (i.e. a subset of the input binders in
  # the jaxpr). For example:
  #
  #   n = core.DShapedArray((), jnp.dtype('int32'), weak_type=False)
  #   a = core.DShapedArray((n,), jnp.dtype('float32'), weak_type=False)
  #   b = core.DShapedArray((n,), jnp.dtype('float32'), weak_type=False)
  #
  #   @lu.wrap_init
  #   def f(x, y):
  #     return x, y
  #
  #   jaxpr, _, _ = pe.trace_to_jaxpr_dynamic(f, [n, a, b],
  #                                           keep_inputs=[False, True, True])
  #   print(jaxpr)
  #   # { lambda ; a:i32[] b:f32[a] c:f32[a]. let  in (b, c) }
  #
  # The abstract values passed to trace_to_jaxpr_dynamic are in direct
  # correspondence to the input binders (invars) of the jaxpr it returns. But in
  # general the Tracers passed to the function f correspond only to a subset of
  # those abstract values. That's because axis size variables may not be
  # explicit arguments to f, while we make everything explicit in the jaxpr.
  keep_inputs = [True] * len(in_avals) if keep_inputs is None else keep_inputs

  frame = JaxprStackFrame()
  with extend_jaxpr_stack(main, frame), source_info_util.reset_name_stack():
    trace = DynamicJaxprTrace(main, core.cur_sublevel())
    in_tracers = _input_type_to_tracers(trace, in_avals)
    in_tracers_ = [t for t, keep in zip(in_tracers, keep_inputs) if keep]
    ans = fun.call_wrapped(*in_tracers_)
    out_tracers = map(trace.full_raise, ans)
    jaxpr, consts = frame.to_jaxpr(out_tracers)
    del fun, main, trace, frame, in_tracers, out_tracers, ans
  if not config.jax_dynamic_shapes:
    # TODO(frostig,mattjj): check_jaxpr is incomplete under dynamic
    # shapes; remove this guard when it is
    config.jax_enable_checks and core.check_jaxpr(jaxpr)
  return jaxpr, [v.aval for v in jaxpr.outvars], consts

@contextlib.contextmanager
def extend_jaxpr_stack(main, frame):
  main.jaxpr_stack = main.jaxpr_stack + (frame,)
  try:
    yield
  finally:
    assert frame is main.jaxpr_stack[-1]
    main.jaxpr_stack = main.jaxpr_stack[:-1]

@profiler.annotate_function
def trace_to_jaxpr_final(fun: lu.WrappedFun,
                         in_avals: Sequence[AbstractValue],
                         debug_info: Optional[DebugInfo] = None,
                         keep_inputs: Optional[Sequence[bool]] = None):
  with core.new_base_main(DynamicJaxprTrace) as main:  # type: ignore
    main.debug_info = debug_info  # type: ignore
    main.jaxpr_stack = ()  # type: ignore
    with core.new_sublevel():
      jaxpr, out_avals, consts = trace_to_subjaxpr_dynamic(
        fun, main, in_avals, keep_inputs=keep_inputs)
    del fun, main
  return jaxpr, out_avals, consts

def partial_eval_to_jaxpr_dynamic(fun: lu.WrappedFun, in_pvals: Sequence[PartialVal]):
  # This function provides a partial evaluation behavior used by Flax. We can't
  # use trace_to_jaxpr directly because of an interaction with the curent
  # custom_derivatives.py, which we work around by adding the EvalTrace.
  # TODO(mattjj): alias to trace_to_jaxpr after revising custom_derivatives.py
  with core.new_main(core.EvalTrace, dynamic=True) as _:  # type: ignore
    return trace_to_jaxpr(fun, in_pvals)


AbstractedAxisName = Hashable
AbstractedAxesSpec = Union[Dict[int, AbstractedAxisName], Tuple[AbstractedAxisName, ...]]

class DBIdx(NamedTuple):
  val: int

@dataclass(frozen=True)
class Bound:
  name: AbstractedAxisName
  bound: int

InputType = Tuple[Tuple[AbstractValue, bool], ...]

def infer_lambda_input_type(
    axes_specs: Optional[Sequence[AbstractedAxesSpec]],
    args: Sequence[Any]
  ) -> InputType:
  partial_specs = _canonicalize_specs(map(np.ndim, args), axes_specs)
  specs = _complete_specs(args, partial_specs)
  idxs, implicit_names = _collect_implicit(args, specs)
  implicit_inputs = [(_implicit_arg_type(n), False) for n in implicit_names]
  explicit_inputs = [(_arg_type(idxs, x, s), True) for x, s in zip(args, specs)]
  return (*implicit_inputs, *explicit_inputs)

def _canonicalize_specs(
    ndims: Sequence[int], specs: Optional[Sequence[AbstractedAxesSpec]]
  ) -> List[Dict[int, AbstractedAxisName]]:
  if specs is None:
    return [{}] * len(ndims)
  else:
    return [{i: d for i, d in enumerate(s) if d is not None} if type(s) is tuple
            else s for n, s in zip(ndims, specs)]

def _complete_specs(
    args: Sequence[Any], partial_specs: List[Dict[int, AbstractedAxisName]]
  ) -> List[Dict[int, AbstractedAxisName]]:
  # Identify each user-supplied name in partial_specs with a size.
  sizes: Dict[AbstractedAxisName, Union[int, DynamicJaxprTracer]] = {}
  for x, spec in zip(args, partial_specs):
    for i, name in spec.items():
      d = sizes.setdefault(name, x.shape[i])
      if d is not x.shape[i] and d != x.shape[i]: raise TypeError
  # Introduce new names as needed for Tracers in shapes.
  named_tracers: Dict[TracerId, AbstractedAxisName] = {
      id(d): name for name, d in sizes.items() if isinstance(d, Tracer)}
  specs: List[Dict[int, AbstractedAxisName]] = []
  for x, spec in zip(args, partial_specs):
    if isinstance(get_aval(x), DShapedArray):
      spec = dict(spec)
      for i, d in enumerate(x.shape):
        if isinstance(d, Tracer):
          spec[i] = named_tracers.get(id(d), TracerAsName(d))
    specs.append(spec)
  assert all(not spec or not any(isinstance(d, Tracer) and i not in spec
                                 for i, d in enumerate(x.shape))
             for x, spec in zip(args, specs))
  return specs

def _collect_implicit(
    args: Sequence[Any], specs: List[Dict[int, AbstractedAxisName]]
  ) -> Tuple[Dict[AbstractedAxisName, DBIdx], List[AbstractedAxisName]]:
  idxs: Dict[AbstractedAxisName, DBIdx] = {}
  explicit_tracers: Dict[TracerId, int] = {}
  counter = (DBIdx(i) for i in it.count())
  # Add implicit arguments to idxs.
  for explicit_idx, (x, spec) in enumerate(zip(args, specs)):
    for i, name in spec.items():
      if name not in idxs and id(x.shape[i]) not in explicit_tracers:
        idxs[name] = next(counter)
    if isinstance(x, Tracer):
      explicit_tracers[id(x)] = explicit_idx
  implicit_names: List[AbstractedAxisName] = list(idxs)

  # Now that we know the implicit args, add explicit args to idxs.
  offset = len(implicit_names)
  for x, spec in zip(args, specs):
    for i, name in spec.items():
      if id(x.shape[i]) in explicit_tracers:
        idxs[name] = DBIdx(offset + explicit_tracers[id(x.shape[i])])

  return idxs, implicit_names

def _implicit_arg_type(name: AbstractedAxisName) -> AbstractValue:
  if type(name) is Bound:
    return AbstractBInt(name.bound)
  else:
    return ShapedArray((), dtypes.dtype('int32'))

def _arg_type(
    idxs: Dict[AbstractedAxisName, DBIdx], x: Any,
    spec: Dict[int, AbstractedAxisName]
  ) -> AbstractValue:
  aval = get_aval(x)  # aval.shape could contain Tracers
  if not spec: return core.raise_to_shaped(aval)
  shape: List[Union[int, DBIdx]] = [idxs[spec[i]] if i in spec else d
                                    for i, d in enumerate(aval.shape)]
  assert not any(isinstance(d, Tracer) for d in shape)
  return DShapedArray(tuple(shape), aval.dtype, False)

class TracerAsName:
  tracer: DynamicJaxprTracer
  def __init__(self, tracer):
    trace = core.thread_local_state.trace_state.trace_stack.dynamic
    self.tracer = trace.with_cur_sublevel().full_raise(tracer)
  def __eq__(self, other):
    return isinstance(other, TracerAsName) and self.tracer is other.tracer
  def __hash__(self):
    return id(self.tracer)

def _extract_implicit_args(
    trace: DynamicJaxprTrace, in_type: Sequence[Tuple[AbstractValue, bool]],
    explicit_tracers: Sequence[DynamicJaxprTracer]
  ) -> Sequence[DynamicJaxprTracer]:
  # First, construct a list to represent the full argument list, leaving the
  # implicit arguments as Nones for now.
  explicit_tracers_ = iter(explicit_tracers)
  tracers = [next(explicit_tracers_) if expl else None for _, expl in in_type]
  assert next(explicit_tracers_, None) is None
  del explicit_tracers_

  # Next, populate the implicit arguments using DBIdxs in in_type.
  for i, (aval, explicit) in enumerate(in_type):
    if not explicit or not isinstance(aval, DShapedArray):
      continue  # can't populate an implicit argument
    tracer = tracers[i]
    assert tracer is not None
    for d1, d2 in zip(aval.shape, tracer.aval.shape):
      if isinstance(d1, DBIdx):
        if tracers[d1.val] is None:
          tracers[d1.val] = trace.instantiate_const(d2)
        assert tracers[d1.val] is trace.instantiate_const(d2)
  assert all(t is not None for t in tracers)
  return [t for t, (_, e) in zip(tracers, in_type) if not e]

def _in_avals_from_tracers(
    tracers: List[DynamicJaxprTracer]
  ) -> List[AbstractValue]:
  # Returned AbstractValues contain DBIdx indices. Uses Tracer obj id as name.
  dbidxs: Dict[TracerId, DBIdx] = {id(t): DBIdx(i) for i, t in enumerate(tracers)}
  in_avals: List[AbstractValue] = []
  for t in tracers:
    a = t.aval
    if isinstance(a, DShapedArray) and any(isinstance(d, Tracer) for d in a.shape):
      shape = [dbidxs[id(d)] if isinstance(d, Tracer) else d for d in a.shape]
      a = a.update(shape=tuple(shape))
    in_avals.append(a)
  return in_avals

def _input_type_to_tracers(
    trace: DynamicJaxprTrace, in_avals: Sequence[AbstractValue]
  )  -> Sequence[Tracer]:
  # Create input Tracers given input AbstractValues, each of which can contain
  # DeBruijn indices which refer to positions in the input argument list. That
  # is, each element `a` of `in_avals` can have DBIdx instances in its shape,
  # which must refer to positions left of `a`'s.
  in_tracers: List[Tracer] = []

  def _substitute_tracers_in_aval(a: AbstractValue) -> AbstractValue:
    if isinstance(a, DShapedArray) and any(type(d) is DBIdx for d in a.shape):
      shape = [in_tracers[d.val] if type(d) is DBIdx else d for d in a.shape]  # type: ignore
      return a.update(shape=tuple(shape))
    return a

  for a in in_avals:
    in_tracers.append(trace.new_arg(_substitute_tracers_in_aval(a)))
  return in_tracers

def _substitute_vars_in_type(
    consts: Dict[Var, Literal], env: Dict[Var, Var], a: AbstractValue
  ) -> AbstractValue:
  if isinstance(a, DShapedArray) and any(isinstance(d, Var) for d in a.shape):
    shape = [consts[d].val if d in consts else env[d]  # type: ignore
             if isinstance(d, Var) else d for d in a.shape]
    return a.update(shape=tuple(shape))
  else:
    return a

def _substitute_tracers_in_type(
    env: Dict[Var, Tracer], a: AbstractValue
  ) -> AbstractValue:
  # Substitutes variables into a given AbstractValue using given environment.
  # That is, the input is an AbstractValue possibly containing Vars, and the
  # output is an aval possibly containing Tracers.
  if isinstance(a, DShapedArray) and any(isinstance(d, Var) for d in a.shape):
    shape = [env[d] if isinstance(d, Var) else d for d in a.shape]
    return a.update(shape=tuple(shape))
  else:
    return a

Const = Any
Val = Any

def pad_jaxpr(jaxpr: Jaxpr, consts: Sequence[Const]
              ) -> Tuple[Jaxpr, List[Const]]:
  bounds = {v: v.aval.bound for v in jaxpr.invars
            if type(v.aval) is AbstractBInt}
  idxs = {v: DBIdx(i) for i, v in enumerate(jaxpr.invars)}

  def substitute(aval: AbstractValue) -> AbstractValue:
    if isinstance(aval, AbstractBInt):
      return ShapedArray((), np.dtype('int32'))
    elif isinstance(aval, DShapedArray):
      shape = [bounds.get(d, idxs.get(d, d)) for d in aval.shape]  # type: ignore
      typ = ShapedArray if all(type(d) is int for d in shape) else DShapedArray
      return typ(tuple(shape), aval.dtype, aval.weak_type)
    else:
      return aval

  in_avals = [substitute(v.aval) for v in jaxpr.invars]
  eval_padded = lu.wrap_init(partial(_eval_jaxpr_padded, jaxpr, consts))
  padded_jaxpr, _, padded_consts = trace_to_jaxpr_dynamic(eval_padded, in_avals)
  return padded_jaxpr, padded_consts

class BoundedAxisSize(NamedTuple):
  val: Union[int, DynamicJaxprTracer]
  bound: int

def _eval_jaxpr_padded(
    jaxpr: Jaxpr, consts: List[Const], *args: DynamicJaxprTracer
  ) -> List[Union[Const, DynamicJaxprTracer]]:
  env: Dict[Var, Val] = {}

  def read(x):
    return x.val if type(x) is Literal else env[x]

  def write(v, val) -> None:
    env[v] = val

  write(core.unitvar, core.unit)
  map(write, jaxpr.constvars, consts)
  map(write, jaxpr.invars, args)
  for eqn in jaxpr.eqns:
    rule = padding_rules[eqn.primitive]
    in_avals  = [_substitute_axis_sizes(env, v.aval) for v in eqn.invars]
    out_avals = [_substitute_axis_sizes(env, v.aval) for v in eqn.outvars]
    outs = rule(in_avals, out_avals, *map(read, eqn.invars), **eqn.params)
    map(write, eqn.outvars, outs)
  return map(read, jaxpr.outvars)

def _substitute_axis_sizes(env: Dict, aval: AbstractValue) -> AbstractValue:
  if isinstance(aval, DShapedArray):
    shp = [BoundedAxisSize(env[d], d.aval.bound) if type(d) is Var and
           type(d.aval) is AbstractBInt else env.get(d, d) for d in aval.shape]
    return DShapedArray(tuple(shp), aval.dtype, aval.weak_type)
  else:
    return aval

padding_rules: Dict[Primitive, Callable] = {}

def call_padding_rule(prim, in_avals, out_avals, *args, call_jaxpr, **params):
  if call_jaxpr.constvars: raise NotImplementedError
  padded_jaxpr, padded_consts = pad_jaxpr(call_jaxpr, ())
  if padded_consts: raise NotImplementedError
  new_params = dict(params, call_jaxpr=padded_jaxpr)
  subfuns, bind_params = prim.get_bind_params(new_params)
  return prim.bind(*subfuns, *args, **bind_params)


# TODO(mattjj): the following are deprecated; update callers to _nounits version
# See https://github.com/google/jax/pull/9498
@lu.transformation
def trace_to_subjaxpr(main: core.MainTrace, instantiate: Union[bool, Sequence[bool]],
                      pvals: Sequence[PartialVal]):
  assert all([isinstance(pv, PartialVal) for pv in pvals]), pvals
  trace = main.with_cur_sublevel()
  in_tracers = map(trace.new_arg, pvals)
  ans = yield in_tracers, {}
  assert isinstance(ans, (list, tuple)), (
      f"Got unexpected return type when tracing function to jaxpr: {ans}")
  assert all(isinstance(x, core.Tracer) or core.valid_jaxtype(x) for x in ans), (
      f"Got unexpected return type when tracing function to jaxpr: {ans}")
  instantiate = [instantiate] * len(ans) if isinstance(instantiate, bool) else instantiate
  out_tracers = map(trace.full_raise, map(core.full_lower, ans))
  out_tracers = map(partial(instantiate_const_at, trace), instantiate, out_tracers)
  jaxpr, consts, env = tracers_to_jaxpr(in_tracers, out_tracers)
  out_pvals = [t.pval for t in out_tracers]
  del trace, in_tracers, out_tracers
  yield jaxpr, (out_pvals, consts, env)

@lu.transformation_with_aux
def partial_eval_wrapper(pvs: Sequence[Optional[AbstractValue]], *consts):
  py_args = map(PartialVal, zip(pvs, consts))
  jaxpr, (out_pvals, consts, env) = yield (py_args,), {}
  out_pvs, out_consts = unzip2(out_pvals)
  out = tuple(out_consts) + tuple(consts)
  yield out, (out_pvs, jaxpr, env)

def partial_eval_jaxpr(jaxpr: ClosedJaxpr, unknowns: Sequence[bool],
                       instantiate: Union[bool, Sequence[bool]],
                       ) -> Tuple[ClosedJaxpr, ClosedJaxpr, Sequence[bool]]:
  instantiate = tuple(instantiate) if isinstance(instantiate, list) else instantiate
  return _partial_eval_jaxpr(jaxpr, tuple(unknowns), instantiate)

@weakref_lru_cache
def _partial_eval_jaxpr(jaxpr, unknowns, instantiate):
  f = lu.wrap_init(core.jaxpr_as_fun(jaxpr))

  cell = []
  def fun(*vals):
    pvals = [PartialVal.unknown(aval) if uk else PartialVal.known(val)
            for aval, val, uk in zip(jaxpr.in_avals, vals, unknowns)]
    jaxpr_2, out_pvals_2, consts_2 = trace_to_jaxpr(f, pvals, instantiate=instantiate)
    out_pvs_2, out_consts_2 = unzip2(out_pvals_2)
    cell.append((out_pvs_2, jaxpr_2, len(consts_2)))
    return out_consts_2 + consts_2

  # For jaxpr_known we pass core.unit for the unknown inputs, and known
  # PartialVal for the known inputs.
  in_avals = [core.abstract_unit if uk else a
              for a, uk in zip(jaxpr.in_avals, unknowns)]
  jaxpr_1, out_avals, consts_1 = trace_to_jaxpr_dynamic(lu.wrap_init(fun), in_avals)
  (out_pvs_2, jaxpr_2, num_res), = cell
  assert len(jaxpr_2.constvars) == num_res

  #   jaxpr :: a -> b
  # jaxpr_1 :: a1 -> [b1, res]
  # jaxpr_2 :: res | a2 -> b2
  # jaxpr_2 :: [a2, res] -> b2
  jaxpr_2 = convert_constvars_jaxpr(jaxpr_2)
  jaxpr_2.invars = jaxpr_2.invars[num_res:] + jaxpr_2.invars[:num_res]
  for var, unknown in zip(jaxpr_2.invars[:len(unknowns)], unknowns):
    if not unknown:
      var.aval = core.abstract_unit

  uk_out = [pv is not None for pv in out_pvs_2]

  in_avals_1, in_avals_2 = unzip2(map(_split_aval, unknowns, jaxpr.in_avals))
  out_avals_1, out_avals_2 = unzip2(map(_split_aval, uk_out, jaxpr.out_avals))
  # out_avals_1 and in_avals_2 need the residuals added
  res_avals = out_avals[len(jaxpr.out_avals):]
  assert len(res_avals) == num_res
  out_avals_1 = [*out_avals_1, *res_avals]
  in_avals_2 = [*in_avals_2, *res_avals]

  return ClosedJaxpr(jaxpr_1, consts_1), ClosedJaxpr(jaxpr_2, ()), uk_out

@weakref_lru_cache
def _drop_vars(jaxpr: Jaxpr, drop_ins: Tuple[bool, ...], drop_outs: Tuple[bool, ...]):
  return Jaxpr(jaxpr.constvars,
               [v for v, d in zip(jaxpr.invars, drop_ins) if not d],
               [v for v, d in zip(jaxpr.outvars, drop_outs) if not d],
               jaxpr.eqns, jaxpr.effects)

@weakref_lru_cache
def _dce_open_jaxpr(jaxpr: Jaxpr, outputs: Tuple[bool, ...], drop_outputs=False) -> Jaxpr:
  # This dead-code elimination is pretty rudimentary, and in particular doesn't
  # nontrivially DCE through scan, call, or other higher-order primitives.
  # TODO(mattjj): better DCE (i.e. use above dce_jaxpr)
  if drop_outputs:
    new_outvars = [var for var, output in zip(jaxpr.outvars, outputs) if output]
  else:
    new_outvars = [var if output else core.unitvar
                   for var, output in zip(jaxpr.outvars, outputs)]

  needed_vars = {v for v in new_outvars if type(v) is not Literal}
  new_eqns = []
  for eqn in jaxpr.eqns[::-1]:
    if set(eqn.outvars) & needed_vars:
      new_eqns.append(eqn)
      needed_vars.update(v for v in eqn.invars if type(v) is not Literal)
  new_eqns = new_eqns[::-1]
  return Jaxpr(jaxpr.constvars, jaxpr.invars, new_outvars, new_eqns,
               jaxpr.effects)