text: # coding=utf-8
  # Copyright 2019 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.
  """
  Control flow primitives.
  """
  
  from __future__ import absolute_import
  from __future__ import division
  from __future__ import print_function
  
  import collections
  import itertools
  import operator
  import threading
  
  import numpy as onp
  import six
  
  from jax import api
  from jax import core
  from jax import dtypes
  from jax.lax import lax
  from jax import linear_util as lu
  from jax.abstract_arrays import ShapedArray, raise_to_shaped
  from jax.api_util import flatten_fun_nokwargs, apply_flat_fun_nokwargs
  from jax.interpreters import ad
  from jax.interpreters import partial_eval as pe
  from jax.interpreters import xla
  from jax.interpreters import batching
  from jax.interpreters import masking
  from jax.lib import xla_bridge as xb
  from jax.lib import xla_client
  from jax.util import (partial, unzip2, safe_map, safe_zip, split_list,
                        split_dict, cache)
  from jax.tree_util import (tree_flatten, tree_unflatten, treedef_is_leaf,
                             treedef_children, treedef_tuple)
  from jax import ad_util
  
  _map = safe_map
  zip = safe_zip
  _reduce = six.moves.reduce
  
  
  @cache()
  def _initial_style_jaxpr(fun, in_tree, in_avals):
    in_pvals = [pe.PartialVal((aval, core.unit)) for aval in in_avals]
    fun, out_tree = flatten_fun_nokwargs(lu.wrap_init(fun), in_tree)
    jaxpr, out_pvals, consts = pe.trace_to_jaxpr(fun, in_pvals, instantiate=True,
                                                 stage_out_calls=True)
    out_avals = _map(raise_to_shaped, unzip2(out_pvals)[0])
    const_avals = tuple(raise_to_shaped(core.get_aval(c)) for c in consts)
    typed_jaxpr = core.TypedJaxpr(pe.closure_convert_jaxpr(jaxpr),
                                  (), const_avals + in_avals, out_avals)
    return typed_jaxpr, consts, out_tree()
  
  def _abstractify(x):
    return raise_to_shaped(core.get_aval(x))
  
  def typecheck(aval, x):
    aval = raise_to_shaped(aval).strip_weak_type()
    try:
      return aval == core.lattice_join(aval, core.get_aval(x)).strip_weak_type()
    except TypeError:
      return False
  
  def typematch(aval1, aval2):
    return (raise_to_shaped(aval1).strip_weak_type() ==
            raise_to_shaped(aval2).strip_weak_type())
  
  class FixedPointError(Exception): pass
  
  
  ### fori_loop and while_loop
  
  def _fori_cond_fun(loop_carry):
    i, upper, _ = loop_carry
    return lax.lt(i, upper)
  
  @cache()
  def _fori_body_fun(body_fun):
    def while_body_fun(loop_carry):
      i, upper, x = loop_carry
      return lax.add(i, lax._const(i, 1)), upper, body_fun(i, x)
    return while_body_fun
  
  def fori_loop(lower, upper, body_fun, init_val):
    """Loop from ``lower`` to ``upper`` by reduction to ``while_loop``.
  
    The type signature in brief is
  
    .. code-block:: haskell
  
      fori_loop :: Int -> Int -> ((int, a) -> a) -> a -> a
  
    The semantics of ``fori_loop`` are given by this Python implementation::
  
      def fori_loop(lower, upper, body_fun, init_val):
        val = init_val
        for i in range(lower, upper):
          val = body_fun(i, val)
        return val
  
    Unlike that Python version, ``fori_loop`` is implemented in terms of a call to
    ``while_loop``. See the docstring for ``while_loop`` for more information.
  
    Also unlike the Python analogue, the loop-carried value ``val`` must hold a
    fixed shape and dtype across all iterations (and not just be consistent up to
    NumPy rank/shape broadcasting and dtype promotion rules, for example). In
    other words, the type ``a`` in the type signature above represents an array
    with a fixed shape and dtype (or a nested tuple/list/dict container data
    structure with a fixed structure and arrays with fixed shape and dtype at the
    leaves).
  
    Args:
      lower: an integer representing the loop index lower bound (inclusive)
      upper: an integer representing the loop index upper bound (exclusive)
      body_fun: function of type ``(int, a) -> a``.
      init_val: initial loop carry value of type ``a``.
  
    Returns:
      Loop value from the final iteration, of type ``a``.
    """
    # TODO: perhaps do more type checking here, for better error messages.
    lower_dtype = dtypes.canonicalize_dtype(lax.dtype(lower))
    upper_dtype = dtypes.canonicalize_dtype(lax.dtype(upper))
    if lower_dtype != upper_dtype:
      msg = ("lower and upper arguments to fori_loop must have equal types, "
             "got {} and {}")
      raise TypeError(msg.format(lower_dtype.name, upper_dtype.name))
    _, _, result = while_loop(_fori_cond_fun, _fori_body_fun(body_fun),
                              (lower, upper, init_val))
    return result
  
  
  def while_loop(cond_fun, body_fun, init_val):
    """Call ``body_fun`` repeatedly in a loop while ``cond_fun`` is True.
  
    The type signature in brief is
  
    .. code-block:: haskell
  
      while_loop :: (a -> Bool) -> (a -> a) -> a -> a
  
    The semantics of ``while_loop`` are given by this Python implementation::
  
      def while_loop(cond_fun, body_fun, init_val):
        val = init_val
        while cond_fun(val):
          val = body_fun(val)
        return val
  
    Unlike that Python version, ``while_loop`` is a JAX primitive and is lowered
    to a single XLA While HLO. That makes it useful for reducing compilation times
    for jit-compiled functions, since native Python loop constructs in an ``@jit``
    function are unrolled, leading to large XLA computations.
  
    Also unlike the Python analogue, the loop-carried value ``val`` must hold a
    fixed shape and dtype across all iterations (and not just be consistent up to
    NumPy rank/shape broadcasting and dtype promotion rules, for example). In
    other words, the type ``a`` in the type signature above represents an array
    with a fixed shape and dtype (or a nested tuple/list/dict container data
    structure with a fixed structure and arrays with fixed shape and dtype at the
    leaves).
  
    Another difference from using Python-native loop constructs is that
    ``while_loop`` is not reverse-mode differentiable because XLA computations
    require static bounds on memory requirements.
  
    Args:
      cond_fun: function of type ``a -> Bool``.
      body_fun: function of type ``a -> a``.
      init_val: value of type ``a``, a type that can be a scalar, array, or any
        pytree (nested Python tuple/list/dict) thereof, representing the initial
        loop carry value.
  
    Returns:
      The output from the final iteration of body_fun, of type ``a``.
    """
    init_vals, in_tree = tree_flatten((init_val,))
    init_avals = tuple(_map(_abstractify, init_vals))
    cond_jaxpr, cond_consts, cond_tree = _initial_style_jaxpr(cond_fun, in_tree, init_avals)
    body_jaxpr, body_consts, body_tree = _initial_style_jaxpr(body_fun, in_tree, init_avals)
    if not treedef_is_leaf(cond_tree) or len(cond_jaxpr.out_avals) != 1:
      msg = "cond_fun must return a boolean scalar, but got pytree {}."
      raise TypeError(msg.format(cond_tree))
    if cond_jaxpr.out_avals[0].strip_weak_type() != ShapedArray((), onp.bool_):
      msg = "cond_fun must return a boolean scalar, but got output type(s) {}."
      raise TypeError(msg.format(cond_jaxpr.out_avals))
  
    in_tree_children = in_tree.children()
    assert len(in_tree_children) == 1
    _check_tree_and_avals("body_fun output and input",
                          # Extract the subtree and avals for the first element of the return tuple
                          body_tree, body_jaxpr.out_avals,
                          in_tree_children[0], init_avals)
    outs = while_p.bind(*itertools.chain(cond_consts, body_consts, init_vals),
                        cond_nconsts=len(cond_consts), cond_jaxpr=cond_jaxpr,
                        body_nconsts=len(body_consts), body_jaxpr=body_jaxpr)
    return tree_unflatten(body_tree, outs)
  
  def _while_loop_abstract_eval(*args, **kwargs):
    return kwargs["body_jaxpr"].out_avals
  
  def _while_loop_translation_rule(c, axis_env, *args, **kwargs):
    backend = kwargs.pop('backend')
    cond_jaxpr, body_jaxpr, cond_nconsts, body_nconsts = split_dict(
        kwargs, ["cond_jaxpr", "body_jaxpr", "cond_nconsts", "body_nconsts"])
    cond_consts, body_consts, init_vals = split_list(args, [cond_nconsts, body_nconsts])
    batched = bool(cond_jaxpr.out_avals[0].shape)
  
    # Since jaxprs don't have tuples and have multiple return values, but we need
    # the HLO While loop to take a single tuple input and output a single boolean
    # (for the cond computation) or a single tuple output (for the body
    # computation), we build XLA computations that handle the tuple munging before
    # generating a Call into the computations formed from the jaxprs.
  
    init_carry = c.Tuple(*(cond_consts + body_consts + init_vals))
  
    cond_c = xb.make_computation_builder("cond_computation")
    cond_carry = cond_c.ParameterWithShape(c.GetShape(init_carry))
    cond_carry_elts = [cond_c.GetTupleElement(cond_carry, i) for i in range(len(args))]
    x, _, z = split_list(cond_carry_elts, [cond_nconsts, body_nconsts])
    pred, = xla.jaxpr_subcomp(cond_c, cond_jaxpr.jaxpr, backend, axis_env,
                              _map(cond_c.Constant, cond_jaxpr.literals), (), *(x + z))
    if batched:
      scalar = ShapedArray((), onp.bool_)
      or_ = xla.primitive_subcomputation(lax.or_p, scalar, scalar)
      pred = cond_c.Reduce(pred, cond_c.Constant(onp.array(False)), or_,
                           list(range(cond_jaxpr.out_avals[0].ndim)))
  
    body_c = xb.make_computation_builder("body_computation")
    body_carry = body_c.ParameterWithShape(c.GetShape(init_carry))
    body_carry_elts = [body_c.GetTupleElement(body_carry, i) for i in range(len(args))]
    x, y, z = split_list(body_carry_elts, [cond_nconsts, body_nconsts])
    new_z = xla.jaxpr_subcomp(body_c, body_jaxpr.jaxpr, backend, axis_env,
                              _map(body_c.Constant, body_jaxpr.literals), (), *(y + z))
    if batched:
      body_pred, = xla.jaxpr_subcomp(body_c, cond_jaxpr.jaxpr, backend, axis_env,
                                     _map(body_c.Constant, cond_jaxpr.literals), (), *(x + z))
      new_z = _map(partial(_pred_bcast_select, body_c, body_pred), new_z, z)
      assert _map(body_c.GetShape, new_z) == _map(body_c.GetShape, z) # no broadcast
    new_carry = body_c.Tuple(*itertools.chain(x, y, new_z))
  
    ans = c.While(cond_c.Build(pred), body_c.Build(new_carry), init_carry)
    ans_elts = [c.GetTupleElement(ans, i) for i in range(len(args))]
    _,  _, z = split_list(ans_elts, [cond_nconsts, body_nconsts])
    return c.Tuple(*z)
  
  def _pred_bcast_select(c, pred, x, y):
    pred_shape = c.GetShape(pred).dimensions()
    x_shape = c.GetShape(x).dimensions()
    y_shape = c.GetShape(y).dimensions()
    assert x_shape == y_shape
    assert pred_shape == x_shape[:len(pred_shape)] == y_shape[:len(pred_shape)]
    bcast_pred = c.BroadcastInDim(pred, x_shape, list(range(len(pred_shape))))
    return c.Select(bcast_pred, x, y)
  
  def _while_loop_batching_rule(args, dims, cond_nconsts, cond_jaxpr,
                                body_nconsts, body_jaxpr):
    size, = {x.shape[d] for x, d in zip(args, dims) if d is not batching.not_mapped}
    orig_batched = [d is not batching.not_mapped for d in dims]
    cconst_bat, bconst_bat, init_bat = split_list(orig_batched, [cond_nconsts, body_nconsts])
  
    # Fixpoint computation of which carry are batched: either
    # batched from init, or the carry out is batched. Each iteration promotes
    # at least one carry to batched. We need at most len(carry) iterations,
    # but we need one last iteration to prepare the jaxpr based on the final
    # carry_bat.
    carry_bat = init_bat
    for _ in range(1 + len(carry_bat)):
      batched = bconst_bat + carry_bat
      body_jaxpr_batched, carry_bat_out = batching.batch_jaxpr(
          body_jaxpr, size, batched, instantiate=carry_bat)
      cond_jaxpr_batched, (pred_bat,) = batching.batch_jaxpr(
          cond_jaxpr, size, cconst_bat + carry_bat, instantiate=False)
      carry_bat_out = _map(partial(operator.or_, pred_bat), carry_bat_out)
      if carry_bat_out == carry_bat:
        break
      else:
        carry_bat = _map(operator.or_, carry_bat, carry_bat_out)
    else:
      assert False, "Fixpoint not reached"
  
    consts, init = split_list(args, [cond_nconsts + body_nconsts])
    const_dims, init_dims = split_list(dims, [cond_nconsts + body_nconsts])
    new_consts = [batching.moveaxis(x, d, 0) if d is not batching.not_mapped and d != 0
                  else x for x, d in zip(consts, const_dims)]
    new_init = [batching.broadcast(x, size, 0) if now_bat and not was_bat
                else batching.moveaxis(x, d, 0) if now_bat else x
                for x, d, was_bat, now_bat in zip(init, init_dims, init_bat, carry_bat)]
  
    outs = while_p.bind(*(new_consts + new_init),
                        cond_nconsts=cond_nconsts, cond_jaxpr=cond_jaxpr_batched,
                        body_nconsts=body_nconsts, body_jaxpr=body_jaxpr_batched)
    out_bdims = [0 if b else batching.not_mapped for b in carry_bat]
    return outs, out_bdims
  
  while_p = lax.Primitive('while')
  while_p.multiple_results = True
  while_p.def_impl(partial(xla.apply_primitive, while_p))
  while_p.def_abstract_eval(_while_loop_abstract_eval)
  xla.initial_style_translations[while_p] = _while_loop_translation_rule
  batching.primitive_batchers[while_p] = _while_loop_batching_rule
  
  
  ### cond
  
  def cond(pred, true_operand, true_fun, false_operand, false_fun):
    """Conditionally apply ``true_fun`` or ``false_fun``.
  
    Has equivalent semantics to this Python implementation::
  
      def cond(pred, true_operand, true_fun, false_operand, false_fun):
        if pred:
          return true_fun(true_operand)
        else:
          return false_fun(false_operand)
  
    Pred has to be a scalar type, collection types (list, tuple) are not supported
  
    """
  
    if len(onp.shape(pred)) != 0:
      raise TypeError("Pred must be a scalar, got {} of shape {}.".format(pred, onp.shape(pred)))
  
    try:
      pred_dtype = dtypes.result_type(pred)
    except TypeError:
      msg = ("Pred type must be either boolean or number, got {}.")
      raise TypeError(msg.format(pred))
  
    if pred_dtype.kind != 'b':
      if pred_dtype.kind in 'iuf':
        pred = pred != 0
      else:
        msg = ("Pred type must be either boolean or number, got {}.")
        raise TypeError(msg.format(pred_dtype))
    true_ops, true_tree = tree_flatten((true_operand,))
    true_avals = tuple(_map(_abstractify, true_ops))
    true_jaxpr, true_consts, true_out_tree = _initial_style_jaxpr(true_fun, true_tree, true_avals)
    false_ops, false_tree = tree_flatten((false_operand,))
    false_avals = tuple(_map(_abstractify, false_ops))
    false_jaxpr, false_consts, false_out_tree = _initial_style_jaxpr(false_fun, false_tree, false_avals)
    _check_tree_and_avals("true_fun and false_fun output",
                          true_out_tree, true_jaxpr.out_avals,
                          false_out_tree, false_jaxpr.out_avals)
    out = cond_p.bind(
        *itertools.chain([pred], true_consts, true_ops, false_consts, false_ops),
        true_jaxpr=true_jaxpr, false_jaxpr=false_jaxpr,
        true_nconsts=len(true_consts), false_nconsts=len(false_consts))
    return tree_unflatten(true_out_tree, out)
  
  def _cond_abstract_eval(*args, **kwargs):
    return kwargs["true_jaxpr"].out_avals
  
  def _cond_translation_rule(c, axis_env, pred, *args, **kwargs):
    backend = kwargs.pop("backend", None)
    true_jaxpr, false_jaxpr, true_nconsts, false_nconsts = split_dict(
        kwargs, ["true_jaxpr", "false_jaxpr", "true_nconsts", "false_nconsts"])
    true_nops = len(true_jaxpr.in_avals) - true_nconsts
    true_consts, true_ops, false_consts, false_ops = split_list(
        args, [true_nconsts, true_nops, false_nconsts])
  
    def make_computation(name, jaxpr, op_shape):
      c = xb.make_computation_builder(name)
      op = c.ParameterWithShape(op_shape)
      ops = [c.GetTupleElement(op, i) for i in range(len(jaxpr.in_avals))]
      outs = xla.jaxpr_subcomp(c, jaxpr.jaxpr, backend, axis_env,
                               _map(c.Constant, jaxpr.literals), (), *ops)
      return c.Build(c.Tuple(*outs))
  
    true_op = c.Tuple(*(true_consts + true_ops))
    true_c = make_computation("true_comp", true_jaxpr, c.GetShape(true_op))
  
    false_op = c.Tuple(*(false_consts + false_ops))
    false_c = make_computation("false_comp", false_jaxpr, c.GetShape(false_op))
  
    return c.Conditional(pred, true_op, true_c, false_op, false_c)
  
  def _cond_pred_bcast_select(pred, x, y):
    bcast_pred = lax.broadcast_in_dim(pred, onp.shape(x), list(range(onp.ndim(pred))))
    return lax.select(bcast_pred, x, y)
  
  def _cond_batching_rule(args, dims, true_jaxpr, false_jaxpr, true_nconsts,
                          false_nconsts):
    # TODO: maybe avoid moving arg axes to front if we're promoting to select?
    args = [batching.moveaxis(x, d, 0) if d is not batching.not_mapped and d != 0
            else x for x, d in zip(args, dims)]
    true_nops = len(true_jaxpr.in_avals) - true_nconsts
    (pred,), true_consts, true_ops, false_consts, false_ops = split_list(
        args, [1, true_nconsts, true_nops, false_nconsts])
    size, = {x.shape[d] for x, d in zip(args, dims) if d is not batching.not_mapped}
    orig_bat = [d is not batching.not_mapped for d in dims]
    (pred_bat,), tconst_bat, t_bat, fconst_bat, f_bat = split_list(
      orig_bat, [1, true_nconsts, true_nops, false_nconsts])
  
    _, true_out_bat = batching.batch_jaxpr(true_jaxpr, size, tconst_bat + t_bat, False)
    _, false_out_bat = batching.batch_jaxpr(false_jaxpr, size, fconst_bat + f_bat, False)
    out_bat = [a or b for a, b in zip(true_out_bat, false_out_bat)]
  
    true_jaxpr_batched, _ = batching.batch_jaxpr(true_jaxpr, size, tconst_bat + t_bat, out_bat)
    false_jaxpr_batched, _ = batching.batch_jaxpr(false_jaxpr, size, fconst_bat + f_bat, out_bat)
  
    if pred_bat:
      true_out = core.jaxpr_as_fun(true_jaxpr_batched)(*(true_consts + true_ops))
      false_out = core.jaxpr_as_fun(false_jaxpr_batched)(*(false_consts + false_ops))
      true_out = [batching.broadcast(x, size, 0) if not b else x
                  for x, b in zip(true_out, out_bat)]
      false_out = [batching.broadcast(x, size, 0) if not b else x
                   for x, b in zip(false_out, out_bat)]
      return [_cond_pred_bcast_select(pred, t, f)
              for t, f in zip(true_out, false_out)], [0] * len(true_out)
    else:
      out_dims = [0 if b else batching.not_mapped for b in out_bat]
      return cond_p.bind(
        *itertools.chain([pred], true_consts, true_ops, false_consts, false_ops),
        true_jaxpr=true_jaxpr_batched, false_jaxpr=false_jaxpr_batched,
        true_nconsts=len(true_consts), false_nconsts=len(false_consts)), out_dims
  
  cond_p = lax.Primitive('cond')
  cond_p.multiple_results = True
  cond_p.def_impl(partial(xla.apply_primitive, cond_p))
  cond_p.def_abstract_eval(_cond_abstract_eval)
  batching.primitive_batchers[cond_p] = _cond_batching_rule
  xla.initial_style_translations[cond_p] = _cond_translation_rule
  
  
  ### scan
  
  def scan(f, init, xs, length=None):
    """Scan a function over leading array axes while carrying along state.
  
    The type signature in brief is
  
    .. code-block:: haskell
  
      scan :: (c -> a -> (c, b)) -> c -> [a] -> (c, [b])
  
    where we use [t] here to denote the type t with an additional leading axis.
    That is, if t is an array type then [t] represents the type with an additional
    leading axis, and if t is a pytree (container) type with array leaves then [t]
    represents the type with the same pytree structure and corresponding leaves
    each with an additional leading axis.
  
    When ``a`` is an array type or None, and ``b`` is an array type, the semantics
    of ``scan`` are given roughly by this Python implementation::
  
      def scan(f, init, xs, length=None):
        if xs is None:
          xs = [None] * length
        carry = init
        ys = []
        for x in xs:
          carry, y = f(carry, x)
          ys.append(y)
        return carry, np.stack(ys)
  
    Unlike that Python version, both ``a`` and ``b`` may be arbitrary pytree
    types, and so multiple arrays can be scanned over at once and produce multiple
    output arrays. (None is actually an empty pytree.)
  
    Also unlike that Python version, ``scan`` is a JAX primitive and is lowered to
    a single XLA While HLO. That makes it useful for reducing compilation times
    for jit-compiled functions, since native Python loop constructs in an ``@jit``
    function are unrolled, leading to large XLA computations.
  
    Finally, the loop-carried value ``carry`` must hold a fixed shape and dtype
    across all iterations (and not just be consistent up to NumPy rank/shape
    broadcasting and dtype promotion rules, for example). In other words, the type
    ``c`` in the type signature above represents an array with a fixed shape and
    dtype (or a nested tuple/list/dict container data structure with a fixed
    structure and arrays with fixed shape and dtype at the leaves).
  
    Args:
      f: a Python function to be scanned of type ``c -> a -> (c, b)``, meaning
        that ``f`` accepts two arguments where the first is a value of the loop
        carry and the second is a slice of ``xs`` along its leading axis, and that
        ``f`` returns a pair where the first element represents a new value for
        the loop carry and the second represents a slice of the output.
      init: an initial loop carry value of type ``c``, which can be a scalar,
        array, or any pytree (nested Python tuple/list/dict) thereof, representing
        the initial loop carry value. This value must have the same structure as
        the first element of the pair returned by ``f``.
      xs: the value of type ``[a]`` over which to scan along the leading axis,
        where ``[a]`` can be an array or any pytree (nested Python
        tuple/list/dict) thereof with consistent leading axis sizes.
      length: optional integer specifying the number of loop iterations, which
        must agree with the sizes of leading axes of the arrays in ``xs`` (but can
        be used to perform scans where no input ``xs`` are needed).
  
    Returns:
      A pair of type ``(c, [b])`` where the first element represents the final
      loop carry value and the second element represents the stacked outputs of
      the second output of ``f`` when scanned over the leading axis of the inputs.
    """
    init_flat, init_tree = tree_flatten(init)
    xs_flat, _ = tree_flatten(xs)
    in_flat, in_tree = tree_flatten((init, xs))
  
    try:
      lengths = [x.shape[0] for x in xs_flat]
    except AttributeError:
      msg = "scan got value with no leading axis to scan over: {}."
      raise ValueError(msg.format(', '.join(str(x) for x in xs_flat
                                            if not hasattr(x, 'shape'))))
  
    if length is not None:
      length = int(length)
      if not all(length == l for l in lengths):
        msg = ("scan got `length` argument of {} which disagrees with "
               "leading axis sizes {}.")
        raise ValueError(msg.format(length, [x.shape[0] for x in xs_flat]))
    else:
      unique_lengths = set(lengths)
      if len(unique_lengths) > 1:
        msg = "scan got values with different leading axis sizes: {}."
        raise ValueError(msg.format(', '.join(str(x.shape[0]) for x in xs_flat)))
      elif len(unique_lengths) == 0:
        msg = "scan got no values to scan over and `length` not provided."
        raise ValueError(msg)
      else:
        length, = unique_lengths
  
    carry_avals = tuple(_map(_abstractify, init_flat))
    x_shapes = [masking.padded_shape_as_value(x.shape[1:]) for x in xs_flat]
    x_dtypes = [x.dtype for x in xs_flat]
    x_avals = tuple(_map(ShapedArray, x_shapes, x_dtypes))
    jaxpr, consts, out_tree = _initial_style_jaxpr(f, in_tree, carry_avals + x_avals)
    out_tree_children = out_tree.children()
    if len(out_tree_children) != 2:
      msg = "scan body output must be a pair, got {}."
      raise TypeError(msg.format(tree_unflatten(out_tree, jaxpr.out_avals)))
    _check_tree_and_avals("scan carry output and input",
                          # Extract the subtree and avals for the first element of the return tuple
                          out_tree_children[0], jaxpr.out_avals[:out_tree_children[0].num_leaves],
                          init_tree, carry_avals)
  
    out = scan_p.bind(*itertools.chain(consts, in_flat),
                      forward=True, length=length, jaxpr=jaxpr,
                      num_consts=len(consts), num_carry=len(init_flat),
                      linear=(False,) * (len(consts) + len(in_flat)))
    return tree_unflatten(out_tree, out)
  
  def _scan_impl(*args, **kwargs):
    forward, length, num_consts, num_carry, jaxpr, linear = split_dict(
        kwargs, ["forward", "length", "num_consts", "num_carry", "jaxpr", "linear"])
  
    consts, init, xs = split_list(args, [num_consts, num_carry])
    _, _, x_avals = split_list(jaxpr.in_avals, [num_consts, num_carry])
    _, y_avals = split_list(jaxpr.out_avals, [num_carry])
  
    def body_fun(i, vals):
      i = i if forward else length - i - 1
      carry, ys = split_list(vals, [num_carry])
      x = _map(partial(_index_array, i), x_avals, xs)
      out_flat = core.jaxpr_as_fun(jaxpr)(*(consts + carry + x))
      carry_out, y_updates = split_list(out_flat, [num_carry])
      ys_out = _map(partial(_update_array, i), y_avals, ys, y_updates)
      return carry_out + ys_out
  
    ys_init = _map(partial(_empty_array, length), y_avals)
    return fori_loop(lax._const(length, 0), length, body_fun, init + ys_init)
  
  def _index_array(i, aval, x):
    if aval is core.abstract_unit:
      return core.unit
    else:
      return lax.dynamic_index_in_dim(x, i, keepdims=False)
  
  def _empty_array(sz, aval):
    if aval is core.abstract_unit:
      return core.unit
    else:
      return lax.full((sz,) + aval.shape, 0, aval.dtype)
  
  def _update_array(i, aval, xs, x):
    if aval is core.abstract_unit:
      return core.unit
    else:
      return lax.dynamic_update_index_in_dim(xs, x, i, 0)
  
  def _scan_jvp(primals, tangents, forward, length, jaxpr, num_consts, num_carry,
                linear):
    num_xs = len(jaxpr.in_avals) - num_carry - num_consts
    num_ys = len(jaxpr.out_avals) - num_carry
    nonzeros = [t is not ad_util.zero for t in tangents]
    const_nz, init_nz, xs_nz = split_list(nonzeros, [num_consts, num_carry])
  
    # Fixpoint computation of which carry are not ad.zero: either
    # non-zero from init, or the carry out is non-zero. Each iteration promotes
    # at least one carry to non-zero. We need at most len(carry) iterations,
    # but we need one last iteration to prepare the jaxpr based on the final
    # carry_nz.
    carry_nz = init_nz
    for _ in range(1 + len(carry_nz)):
      nonzeros = const_nz + carry_nz + xs_nz
      jaxpr_jvp, nonzeros_out = ad.jvp_jaxpr(
          jaxpr, nonzeros, instantiate=carry_nz + [False] * num_ys)
      carry_nz_out, ys_nz = nonzeros_out[:num_carry], nonzeros_out[num_carry:]
      if carry_nz_out == carry_nz:
        break
      else:
        carry_nz = _map(operator.or_, carry_nz, carry_nz_out)
    else:
      assert False, "Fixpoint not reached"
  
    tangents = [ad.instantiate_zeros(x, t) if t is ad_util.zero and nz else t
                for x, t, nz in zip(primals, tangents, nonzeros)]
  
    consts, init, xs = split_list(primals, [num_consts, num_carry])
    all_tangents = split_list(tangents, [num_consts, num_carry])
    consts_dot, init_dot, xs_dot = _map(_prune_zeros, all_tangents)
  
    jaxpr_jvp_rearranged = ad.rearrange_binders(
        jaxpr_jvp,
        [num_consts, num_carry, num_xs], [len(consts_dot), len(init_dot), len(xs_dot)],
        [num_carry, num_ys], [len(init_dot), sum(nonzeros_out) - len(init_dot)])
  
    consts_linear, init_linear, xs_linear = split_list(linear, [num_consts, num_carry])
    jaxpr_jvp_linear = (consts_linear + [True] * len(consts_dot)
                        + init_linear + [True] * len(init_dot)
                        + xs_linear + [True] * len(xs_dot))
  
    out_flat = scan_p.bind(
        *(consts + consts_dot + init + init_dot + xs + xs_dot),
        forward=forward, length=length, jaxpr=jaxpr_jvp_rearranged,
        num_consts=num_consts+len(consts_dot), num_carry=num_carry+len(init_dot),
        linear=jaxpr_jvp_linear)
  
    carry, carry_dot, ys, ys_dot = split_list(out_flat, [num_carry, len(init_dot), num_ys])
    primals_out = carry + ys
    tangents_out = iter(carry_dot + ys_dot)
    tangents_out = [next(tangents_out) if nz else ad_util.zero for nz in nonzeros_out]
    return primals_out, tangents_out
  
  def _prune_zeros(ts):
    return [t for t in ts if t is not ad_util.zero]
  
  def _scan_partial_eval(trace, *tracers, **kwargs):
    forward, length, num_consts, num_carry, jaxpr, linear = split_dict(
        kwargs, ["forward", "length", "num_consts", "num_carry", "jaxpr", "linear"])
    num_xs = len(jaxpr.in_avals) - num_carry - num_consts
    num_ys = len(jaxpr.out_avals) - num_carry
  
    unknowns = [t.pval[0] is not None for t in tracers]
    const_uk, init_uk, xs_uk = split_list(unknowns, [num_consts, num_carry])
  
    # Fixpoint computation of which carry are unknown (not a constant): either
    # unknown from init, or the carry out is unknown. Each iteration promotes
    # at least one carry to unknown. We need at most len(carry) iterations,
    # but we need one last iteration to prepare the jaxpr based on the final
    # carry_uk.
    carry_uk = init_uk
    for _ in range(1 + len(carry_uk)):
      unknowns = const_uk + carry_uk + xs_uk
      jaxpr_1, jaxpr_2, out_uk = pe.partial_eval_jaxpr(
          jaxpr, unknowns, instantiate=carry_uk + [False] * num_ys)
      carry_uk_out, ys_uk = out_uk[:num_carry], out_uk[num_carry:]
      if carry_uk_out == carry_uk:
        break
      else:
        carry_uk = _map(operator.or_, carry_uk, carry_uk_out)
    else:
      assert False, "Fixpoint not reached"
    num_res = len(jaxpr_1.out_avals) - len(jaxpr_2.out_avals)
  
    # The residuals are treated as extensive outputs of jaxpr_1 (and extensive
    # inputs to jaxpr_2), but residuals that are loop-invariant can be hoisted.
    # TODO(mattjj): hoist other loop-invariant values here too (instantiate=False)
    invariant_pvals = [pe.PartialVal((None, core.unit if uk else t.pval[1]))
                       for uk, t in zip(unknowns[:num_consts], tracers[:num_consts])]
    other_pvals = [pe.PartialVal((a, core.unit)) for a in jaxpr_1.in_avals[num_consts:]]
    in_pvals_1 = invariant_pvals + other_pvals
    untyped_jaxpr_1, out_pvals_1, consts_1 = pe.trace_to_jaxpr(
        lu.wrap_init(core.jaxpr_as_fun(jaxpr_1)), in_pvals_1,
        instantiate=[True] * (num_carry + num_ys) + [False] * num_res)
    const_avals_1 = [raise_to_shaped(core.get_aval(c)) for c in consts_1]
    in_avals_1 = [core.abstract_unit] * num_consts + jaxpr_1.in_avals[num_consts:]
    out_avals_1 = [core.abstract_unit if pv is None else pv for pv, c in out_pvals_1]
    jaxpr_1_opt = pe.TypedJaxpr(pe.closure_convert_jaxpr(untyped_jaxpr_1),
                                (), const_avals_1 + in_avals_1, out_avals_1)
    num_consts_1 = num_consts + len(consts_1)
    # any now-known residuals are intensive, so we want to revise jaxpr_2 to take
    # those inputs as constants rather than as extensive inputs
    _, _, res_pvals = split_list(out_pvals_1, [num_carry, num_ys])
    intensive_residuals = [const for pv, const in res_pvals if pv is None]
    move = [False] * len(jaxpr_1.in_avals) + [pv is None for pv, _ in res_pvals]
    jaxpr_2_opt = pe.move_binders_to_front(jaxpr_2, move)
    num_consts_2 = num_consts + len(intensive_residuals)
  
    in_consts = (list(consts_1) + [core.unit] * num_consts +
                 [core.unit if uk else t.pval[1]
                  for uk, t in zip(unknowns[num_consts:], tracers[num_consts:])])
    linear_1 = ([False] * len(consts_1) + [True] * num_consts +
                [lin or uk for uk, lin in zip(unknowns[num_consts:], linear[num_consts:])])
    out_flat = scan_p.bind(
        *in_consts, forward=forward, length=length, jaxpr=jaxpr_1_opt,
        num_consts=num_consts_1, num_carry=num_carry, linear=linear_1)
    out_carry, ys, res_and_units = split_list(out_flat, [num_carry, num_ys])
    extensive_residuals = [r for r, (pv, _) in zip(res_and_units, res_pvals) if pv is not None]
  
    new_tracers = [trace.instantiate_const(t) if uk else trace.new_instantiated_literal(core.unit)
                   for uk, t in zip(unknowns, tracers)]
    carry_avals, y_avals = split_list(jaxpr.out_avals, [num_carry])
    ys_avals = _map(partial(_promote_aval_rank, length), y_avals)
    out_avals = carry_avals + ys_avals
    out_pvs = [aval if uk else None for aval, uk in zip(out_avals, out_uk)]
  
    out_consts = out_carry + ys
    int_res_tracers = _map(trace.new_instantiated_const, intensive_residuals)
    ext_res_tracers = _map(trace.new_instantiated_const, extensive_residuals)
    out_tracers = [pe.JaxprTracer(trace, pe.PartialVal((pv, const)), None)
                   for pv, const in zip(out_pvs, out_consts)]
    linear_2 = ([False] * len(int_res_tracers) +
                [lin or not uk for uk, lin in zip(unknowns, linear)] +
                [False] * len(ext_res_tracers))
    eqn = pe.new_eqn_recipe(int_res_tracers + new_tracers + ext_res_tracers,
                            out_tracers, scan_p, (),
                            dict(forward=forward, length=length, jaxpr=jaxpr_2_opt,
                                 num_consts=num_consts_2,
                                 num_carry=num_carry, linear=linear_2))
    for t in out_tracers: t.recipe = eqn
    return out_tracers
  
  def _promote_aval_rank(sz, aval):
    if aval is core.abstract_unit:
      return core.abstract_unit
    else:
      return ShapedArray((sz,) + aval.shape, aval.dtype)
  
  def _scan_transpose(cts, *args, **kwargs):
    forward, length, num_consts, num_carry, jaxpr, linear = split_dict(
        kwargs, ["forward", "length", "num_consts", "num_carry", "jaxpr", "linear"])
  
    # we've only implemented transposing scans with specific lin/nonlin patterns
    consts_lin, init_lin, xs_lin = split_list(linear, [num_consts, num_carry])
    num_ires = len(consts_lin) - sum(consts_lin)
    num_eres = len(xs_lin) - sum(xs_lin)
    if consts_lin != [False] * num_ires + [True] * (len(consts_lin) - num_ires):
      raise NotImplementedError
    if xs_lin != [True] * (len(xs_lin) - num_eres) + [False] * num_eres:
      raise NotImplementedError
    if not all(init_lin):
      raise NotImplementedError
  
    consts, init, xs = split_list(args, [num_consts, num_carry])
    ires, consts = split_list(consts, [num_ires])
    xs, eres = split_list(xs, [sum(xs_lin)])
    assert not any(r is ad.undefined_primal for r in ires)
    assert not any(r is ad.undefined_primal for r in eres)
  
    carry_avals, y_avals = split_list(jaxpr.out_avals, [num_carry])
    ys_avals = _map(partial(_promote_aval_rank, length), y_avals)
    ct_carry, ct_ys = split_list(cts, [num_carry])
    ct_carry = _map(ad.instantiate_zeros_aval, carry_avals, ct_carry)
    ct_ys = _map(ad.instantiate_zeros_aval, ys_avals, ct_ys)
    ct_consts = _map(ad_util.zeros_like_aval, jaxpr.in_avals[num_ires:num_consts])
  
    #       jaxpr :: [ires, T d] -> [T c] -> [T a, eres] -> ([T c], [T b])
    # jaxpr_trans :: [ires] -> [CT d, CT c] -> [CT b, eres] -> ([CT d, CT c], [CT a])
    jaxpr_trans = _transpose_jaxpr(num_ires, num_consts - num_ires, num_eres, jaxpr)
    linear_trans = ([False] * num_ires +
                    [True] * (len(ct_consts) + len(ct_carry) + len(ct_ys)) +
                    [False] * num_eres)
  
    outs = scan_p.bind(
        *(ires + ct_consts + ct_carry + ct_ys + eres), forward=not forward,
        length=length, jaxpr=jaxpr_trans, num_consts=num_ires,
        num_carry=num_consts-num_ires+num_carry, linear=linear_trans)
    ct_consts, ct_init, ct_xs = split_list(outs, [num_consts - num_ires, num_carry])
    return [None] * num_ires + ct_consts + ct_init + ct_xs + [None] * num_eres
  
  # transpose_jaxpr :: ([res1, c, a, res2] -> b)
  #                    -> ([res1, CT c, CT b, res2] -> [CT c, CT a])
  def _transpose_jaxpr(num_res1, num_c, num_res2, jaxpr):
    num_a = len(jaxpr.in_avals) - num_res1 - num_c - num_res2
    res1_avals, c_avals, a_avals, res2_avals = split_list(
        jaxpr.in_avals, [num_res1, num_c, num_a])
    num_b = len(jaxpr.out_avals)
    b_avals = list(jaxpr.out_avals)
  
    @lu.wrap_init
    def transposed(*res1_cbar_bbar_res2):
      res1, c_bar, b_bar, res2 = split_list(
          res1_cbar_bbar_res2, [num_res1, num_c, num_b])
      primals = res1 + [ad.undefined_primal] * (num_c + num_a) + res2
      _, cbar_abar = ad.backward_pass(jaxpr.jaxpr, jaxpr.literals, (), primals,
                                      b_bar)
      _, new_c_bar, a_bar, _ = split_list(cbar_abar, [num_res1, num_c, num_a])
      a_bar = _map(ad.instantiate_zeros_aval, a_avals, a_bar)
      c_bar = _map(ad.instantiate_zeros_aval, c_avals,
                  _map(ad.add_tangents, c_bar, new_c_bar))
      return c_bar + a_bar
    return _make_typed_jaxpr(transposed, res1_avals + c_avals + b_avals + res2_avals)
  
  def _make_typed_jaxpr(traceable, in_avals):
    pvals = [pe.PartialVal((aval, core.unit)) for aval in in_avals]
    jaxpr, pvals_out, consts = pe.trace_to_jaxpr(traceable, pvals, instantiate=True)
    out_avals, _ = unzip2(pvals_out)
    return core.TypedJaxpr(jaxpr, consts, in_avals, out_avals)
  
  
  def _scan_batching_rule(args, dims, forward, length, jaxpr, num_consts,
                          num_carry, linear):
    num_ys = len(jaxpr.out_avals) - num_carry
    size, = {x.shape[d] for x, d in zip(args, dims) if d is not batching.not_mapped}
    orig_batched = [d is not batching.not_mapped for d in dims]
    const_batched, init_batched, xs_batched = split_list(orig_batched, [num_consts, num_carry])
  
    # Fixpoint computation of which carry are batched: either
    # batched from init, or the carry out is batched. Each iteration promotes
    # at least one carry to batched. We need at most len(carry) iterations,
    # but we need one last iteration to prepare the jaxpr based on the final
    # carry_batched.
    carry_batched = init_batched
    for _ in range(1 + len(carry_batched)):
      batched = const_batched + carry_batched + xs_batched
      jaxpr_batched, batched_out = batching.batch_jaxpr(
          jaxpr, size, batched, instantiate=carry_batched + [False] * num_ys)
      carry_batched_out, ys_batched = batched_out[:num_carry], batched_out[num_carry:]
      if carry_batched_out == carry_batched:
        break
      else:
        carry_batched = _map(operator.or_, carry_batched, carry_batched_out)
    else:
      assert False, "Fixpoint not reached"
  
    consts, init, xs = split_list(args, [num_consts, num_carry])
    consts_bdims, init_bdims, xs_bdims = split_list(dims, [num_consts, num_carry])
    new_consts = [batching.moveaxis(x, d, 0) if d is not batching.not_mapped and d != 0
                  else x for x, d in zip(consts, consts_bdims)]
    new_init = [batching.broadcast(x, size, 0) if now_batched and not was_batched
                else batching.moveaxis(x, d, 0) if now_batched else x
                for x, d, was_batched, now_batched in
                zip(init, init_bdims, init_batched, carry_batched)]
    new_xs = [batching.moveaxis(x, d, 1) if d is not batching.not_mapped and d != 1
              else x for x, d in zip(xs, xs_bdims)]
    new_args = new_consts + new_init + new_xs
  
    outs = scan_p.bind(*new_args, forward=forward, length=length, jaxpr=jaxpr_batched,
                       num_consts=num_consts, num_carry=num_carry, linear=linear)
    carry_bdims = [0 if b else batching.not_mapped for b in carry_batched]
    ys_bdims = [1 if b else batching.not_mapped for b in ys_batched]
    return outs, carry_bdims + ys_bdims
  
  def _scan_polymorphic_shape_rule(shape_exprs, forward, length, jaxpr,
                                   num_consts, num_carry, linear):
    const_shexprs, init_shexprs, xs_shexprs = split_list(shape_exprs, [num_consts, num_carry])
    _, y_avals = split_list(jaxpr.out_avals, [num_carry])
    ys_shapes = [ShapeExpr(length, *y_aval.shape) for y_aval in y_avals]
    return init_shexprs + ys_shapes
  
  def _scan_masking_rule(shape_envs, padded_vals, shape_exprs, forward, length,
                         jaxpr, num_consts, num_carry, linear):
    out_shape = _scan_polymorphic_shape_rule(shape_exprs, forward, length, jaxpr,
                                             num_consts, num_carry, linear)
    dynamic_length = masking.eval_dim_expr(shape_envs.logical, length)
    masked_jaxpr = _masked_scan_jaxpr(jaxpr, num_consts, num_carry)
    consts, init, xs = split_list(padded_vals, [num_consts, num_carry])
    max_length, = {x.shape[0] for x in xs}
    const_linear, init_linear, xs_linear = split_list(linear, [num_consts, num_carry])
    out_vals = scan_p.bind(
        *itertools.chain([dynamic_length] + consts, [0], init, xs),
        forward=forward, length=max_length, jaxpr=masked_jaxpr,
        num_consts=1 + num_consts, num_carry=1 + num_carry,
        linear=[False] + const_linear + [False] + init_linear + xs_linear)
    return out_vals[1:], out_shape
  
  def _masked_scan_jaxpr(jaxpr, num_consts, num_carry):
    fun = core.jaxpr_as_fun(jaxpr)
  
    @lu.wrap_init
    def masked(*args):
      [dynamic_length], consts, [i], carry, xs = split_list(
          args, [1, num_consts, 1, num_carry])
      out = fun(*(consts + carry + xs))
      new_carry, ys = split_list(out, [num_carry])
      new_carry = [lax.select(i < dynamic_length, new_c, c)
                   for new_c, c in zip(new_carry, carry)]
      return [i + 1] + new_carry + ys
  
    aval = ShapedArray((), dtypes.int_)
    const_avals, carry_avals, x_avals = split_list(jaxpr.in_avals, [num_consts, num_carry])
    return _make_typed_jaxpr(masked, [aval] + const_avals + [aval] + carry_avals + x_avals)
  
  def scan_bind(*args, **kwargs):
    forward, length, num_consts, num_carry, jaxpr, linear = split_dict(
        kwargs, ["forward", "length", "num_consts", "num_carry", "jaxpr", "linear"])
    consts, init, xs = split_list(args, [num_consts, num_carry])
    assert len(linear) == len(args)
  
    # check that args match input types
    consts_avals, init_avals, x_avals = split_list(jaxpr.in_avals, [num_consts, num_carry])
    xs_avals = _map(partial(_promote_aval_rank, length), x_avals)
    assert all(_map(typecheck, consts_avals, consts)), (consts, consts_avals)
    assert all(_map(typecheck, init_avals, init))
    # assert all(_map(typecheck, xs_avals, xs))
    # check that output carry type matches input carry type
    carry_avals, _ = split_list(jaxpr.out_avals, [num_carry])
    assert all(_map(typematch, init_avals, carry_avals))
  
    # check that the data flow is sensible
    core.check_jaxpr(jaxpr.jaxpr)
  
    return core.Primitive.bind(scan_p, *args, forward=forward, length=length,
                               jaxpr=jaxpr, num_consts=num_consts,
                               num_carry=num_carry, linear=linear)
  
  scan_p = core.Primitive("scan")
  scan_p.multiple_results = True
  scan_p.def_custom_bind(scan_bind)
  scan_p.def_impl(_scan_impl)
  ad.primitive_jvps[scan_p] = _scan_jvp
  ad.primitive_transposes[scan_p] = _scan_transpose
  pe.custom_partial_eval_rules[scan_p] = _scan_partial_eval
  xla.initial_style_translations[scan_p] = xla.lower_fun(_scan_impl, initial_style=True)
  batching.primitive_batchers[scan_p] = _scan_batching_rule
  masking.shape_parameterized_primitive_rules[scan_p] = _scan_masking_rule
  
  
  def map(f, xs):
    """Map a function over leading array axes.
  
    Like Python's builtin map, except inputs and outputs are in the form of
    stacked arrays. Consider using the ``jax.vmap`` transform instead, unless you
    need to apply a function element by element for reduced memory usage or
    heterogeneous computation with other control flow primitives.
  
    When ``xs`` is an array type, the semantics of ``map`` are given by this
    Python implementation::
  
      def map(f, xs):
        return np.stack([f(x) for x in xs])
  
    Like ``scan``, ``map`` is implemented in terms of JAX primitives so many of
    the same advantages over a Python loop apply: ``xs`` may be an arbitrary
    nested pytree type, and the mapped computation is compiled only once.
  
    Args:
      f: a Python function to apply element-wise over the first axis or axes of
        ``xs``.
      xs: values over which to map along the leading axis.
  
    Returns:
      Mapped values.
    """
    g = lambda _, x: ((), f(x))
    _, ys = scan(g, (), xs)
    return ys
  
  
  def _concat_masking_rule(padded_vals, logical_shapes, dimension, operand_shapes):
    del operand_shapes  # Unused.
    result = lax.concatenate(padded_vals, dimension)  # fragmented
    offset = 0
    for padded_val, logical_shape in zip(padded_vals, logical_shapes):
      result = _memcpy(dimension, logical_shape[dimension], padded_val,
                       result, offset)
      offset = offset + logical_shape[dimension]
    return result
  
  def _memcpy(axis, num, src, dst, offset):
    def body(i, dst):
      update = lax.dynamic_index_in_dim(src, i, axis)
      return lax.dynamic_update_index_in_dim(dst, update, i + offset, axis)
    return fori_loop(0, num, body, dst)
  
  masking.masking_rules[lax.concatenate_p] = _concat_masking_rule
  
  
  def _check_tree(func_name, expected_name, actual_tree, expected_tree):
    if actual_tree != expected_tree:
      raise TypeError(
          "{}() output pytree structure must match {}, got {} and {}."
          .format(func_name, expected_name, actual_tree, expected_tree))
  
  
  def _check_tree_and_avals(what, tree1, avals1, tree2, avals2):
    """Raises TypeError if (tree1, avals1) does not match (tree2, avals2).
  
    Corresponding `tree` and `avals` must match in the sense that the number of leaves in
    `tree` must be equal to the length of `avals`.
    `what` will be prepended to details of the mismatch in TypeError.
    """
    if tree1 != tree2:
      msg = ("{} must have same type structure, got {} and {}.")
      raise TypeError(msg.format(what, tree1, tree2))
    if not all(safe_map(typematch, avals1, avals2)):
      msg = ("{} must have identical types, "
             "got\n{}\nand\n{}.")
      raise TypeError(msg.format(what, tree_unflatten(tree1, avals1),
                                 tree_unflatten(tree2, avals2)))
  
  
  def _stop_gradient_fun(f):
    """Create a version of f() that stops all gradients."""
    def wrapper(*args, **kwargs):
      args_flat, in_args_tree = tree_flatten((args, kwargs))
      args_avals = tuple(_map(_abstractify, args_flat))
      g = lambda a, b: f(*a, **b)
      jaxpr, consts, out_tree = _initial_style_jaxpr(g, in_args_tree, args_avals)
      out = core.jaxpr_as_fun(jaxpr)(*lax.stop_gradient(consts + tuple(args_flat)))
      return tree_unflatten(out_tree, out)
    return wrapper
  
  
  _RootTuple = collections.namedtuple('_RootTuple', 'f, solve, l_and_s')
  
  
  def _split_root_args(args, const_lengths):
    params_list = split_list(args, list(const_lengths))
    return _RootTuple(*params_list[:-1]), params_list[-1]
  
  
  def custom_root(f, initial_guess, solve, tangent_solve):
    """Differentiably solve for a roots of a function.
  
    This is a low-level routine, mostly intended for internal use in JAX.
    Gradients of custom_root() are defined with respect to closed-over variables
    from the provided function ``f`` via the implicit function theorem:
    https://en.wikipedia.org/wiki/Implicit_function_theorem
  
    Args:
      f: function for which to find a root. Should accept a single argument,
        return a tree of arrays with the same structure as its input.
      initial_guess: initial guess for a zero of f.
      solve: function to solve for the roots of f. Should take two positional
        arguments, f and initial_guess, and return a solution with the same
        structure as initial_guess such that func(solution) = 0. In other words,
        the following is assumed to be true (but not checked)::
  
          solution = solve(f, initial_guess)
          error = f(solution)
          assert all(error == 0)
  
      tangent_solve: function to solve the tangent system. Should take two
        positional arguments, a linear function ``g`` (the function ``f``
        linearized at its root) and a tree of array(s) ``y`` with the same
        structure as initial_guess, and return a solution ``x`` such that
        ``g(x)=y``:
  
        - For scalar ``y``, use ``lambda g, y: y / g(1.0)``.
        - For vector ``y``, you could use a linear solve with the Jacobian, if
          dimensionality of ``y`` is not too large:
          ``lambda g, y: np.linalg.solve(jacobian(g)(y), y)``.
  
    Returns:
      The result of calling solve(f, initial_guess) with gradients defined via
      implicit differentiation assuming ``f(solve(f, initial_guess)) == 0``.
    """
    guess_flat, in_args_tree = tree_flatten((initial_guess,))
    guess_avals = tuple(_map(_abstractify, guess_flat))
    f_jaxpr, f_consts, out_tree = _initial_style_jaxpr(
        f, in_args_tree, guess_avals)
  
    in_tree, = treedef_children(in_args_tree)
    _check_tree("f", "initial_guess", out_tree, in_tree)
  
    solve_jaxpr, solve_consts, solution_tree = _initial_style_jaxpr(
        partial(solve, _stop_gradient_fun(f)), in_args_tree, guess_avals)
    _check_tree("solve", "initial_guess", solution_tree, in_tree)
  
    def linearize_and_solve(x, b):
      unchecked_zeros, f_jvp = api.linearize(f, x)
      return tangent_solve(f_jvp, b)
  
    l_and_s_jaxpr, l_and_s_consts, out_tree = _initial_style_jaxpr(
        linearize_and_solve, treedef_tuple((in_tree,) * 2), guess_avals * 2)
    _check_tree("tangent_solve", "x", out_tree, in_tree)
  
    all_consts = [f_consts, solve_consts, l_and_s_consts]
    const_lengths = _RootTuple(*_map(len, all_consts))
    jaxprs = _RootTuple(f_jaxpr, solve_jaxpr, l_and_s_jaxpr)
  
    out_flat = root_p.bind(
        *(_flatten(all_consts) + guess_flat),
        const_lengths=const_lengths, jaxprs=jaxprs)
    return tree_unflatten(out_tree, out_flat)
  
  
  def _root_abstract_eval(*args, **kwargs):
    return _map(raise_to_shaped, args[sum(kwargs['const_lengths']):])
  
  
  def _root_impl(*args, **kwargs):
    const_lengths, jaxprs = split_dict(kwargs, ['const_lengths', 'jaxprs'])
    params, initial_guess = _split_root_args(args, const_lengths)
    solution = core.jaxpr_as_fun(jaxprs.solve)(*(params.solve + initial_guess))
    return solution
  
  
  def _root_jvp(primals, tangents, const_lengths, jaxprs):
    params, _ = _split_root_args(primals, const_lengths)
    solution = tuple(root_p.bind(
        *primals, const_lengths=const_lengths, jaxprs=jaxprs))
  
    params_dot, _ = _split_root_args(tangents, const_lengths)
  
    # F(m, u) = 0      # system of equations in u, parameterized by m
    #                  # solution is u*(m) defined in a neighborhood
    # F(m, u*(m)) = 0  # satisfied in a neighborhood
    #
    # ∂_0 F(m, u*(m)) + ∂_1 F(m, u*(m)) ∂ u*(m) = 0       # implied by line above
    # ∂ u*(m) = - (∂_1 F(m, u*(m)))^{-1} ∂_0 F(m, u*(m))  # rearrange
    #
    # ∂ u*(m)[v] = - (∂_1 F(m, u*(m)))^{-1} [∂_0 F(m, u*(m))[v]]  # jvp
  
    f = core.jaxpr_as_fun(jaxprs.f)
    linearize_and_solve = partial(
        core.jaxpr_as_fun(jaxprs.l_and_s), *params.l_and_s)
    f_at_solution = lambda *params: f(*itertools.chain(params, solution))
    _, rhs = ad.jvp(lu.wrap_init(f_at_solution)).call_wrapped(
        params.f, params_dot.f)
    solution_dot = _map(
        operator.neg, linearize_and_solve(*itertools.chain(solution, rhs)))
  
    return solution, solution_dot
  
  
  root_p = core.Primitive('root')
  root_p.multiple_results = True
  root_p.def_impl(_root_impl)
  root_p.def_abstract_eval(_root_abstract_eval)
  ad.primitive_jvps[root_p] = _root_jvp
  xla.initial_style_translations[root_p] = xla.lower_fun(
      _root_impl, initial_style=True)
  # TODO(shoyer): write batching rule
  
  
  class _LinearSolveTuple(collections.namedtuple(
      '_LinearSolveTuple', 'matvec, vecmat, solve, transpose_solve')):
  
    def transpose(self):
      return type(self)(self.vecmat, self.matvec, self.transpose_solve, self.solve)
  
  
  def _split_linear_solve_args(args, const_lengths):
    params_list = split_list(args, list(const_lengths))
    return _LinearSolveTuple(*params_list[:-1]), params_list[-1]
  
  
  def _transpose_function(linear_fun, primals):
    """Transpose a linear function."""
    # TODO(shoyer): can we use something more direct than the vjp machinery?
    # It's particularly awkward that we need the second argument to give
    # particular values of the primals, which are entirely arbitrary.
    _, vjp_fun = api.vjp(linear_fun, primals)
  
    def transposed_fun(x):
      (y,) = vjp_fun(x)
      return y
  
    return transposed_fun
  
  
  def _flatten(args):
    return [x for arg in args for x in arg]
  
  
  def _check_shapes(func_name, expected_name, actual, expected, tree):
    actual_shapes = _map(onp.shape, actual)
    expected_shapes = _map(onp.shape, expected)
    if actual_shapes != expected_shapes:
      actual_shape_tree = tree_unflatten(tree, actual_shapes)
      act_shape_tree = tree_unflatten(tree, actual_shapes)
      raise ValueError('{}() output shapes must match {}, got {} and {}'
                       .format(func_name, expected_name,
                               tree_unflatten(tree, actual_shapes),
                               tree_unflatten(tree, expected_shapes)))
  
  
  def custom_linear_solve(
      matvec, b, solve, transpose_solve=None, symmetric=False):
    """Perform a matrix-free linear solve with implicitly defined gradients.
  
    This function allows for overriding or defining gradients for a linear
    solve directly via implicit differentiation at the solution, rather than by
    differenting *through* the solve operation. This can sometimes be much faster
    or more numerically stable, or differentiating through the solve operation
    may not even be implemented (e.g., if ``solve`` uses ``lax.while_loop``).
  
    Required invariant::
  
        x = solve(matvec, b)  # solve the linear equation
        assert matvec(x) == b  # not checked
  
    Args:
      matvec: linear function to invert. Must be differentiable.
      b: constant right handle side of the equation. May be any nested structure
        of arrays.
      solve: higher level function that solves for solution to the linear
        equation, i.e., ``solve(matvec, x)) == x`` for all ``x`` of the same form
        as ``b``. This function need not be differenatiable.
      transpose_solve: higher level function for solving the transpose linear
        equation, i.e., ``transpose_solve(vecmat, x) == x``, where ``vecmat`` is
        the transpose of the linear map ``matvec`` (computed automatically with
        autodiff). Required for backwards mode automatic differentiation, unless
        ``symmetric=True``, in which case ``solve`` provides the default value.
      symmetric: bool indicating if it is safe to assume the linear map
        corresponds to a symmetric matrix, i.e., ``matvec == vecmat``.
  
    Returns:
      Result of ``solve(matvec, b)``, with gradients defined assuming that the
      solution ``x`` satisfies the linear equation ``matvec(x) == b``.
    """
    if transpose_solve is None and symmetric:
      transpose_solve = solve
  
    b_flat, in_args_tree = tree_flatten((b,))
    b_avals = tuple(_map(_abstractify, b_flat))
    matvec_jaxpr, matvec_consts, out_tree = _initial_style_jaxpr(
        matvec, in_args_tree, b_avals)
  
    tree, = treedef_children(in_args_tree)
    _check_tree("matvec", "b", out_tree, tree)
  
    solve_jaxpr, solve_consts, out_tree = _initial_style_jaxpr(
        partial(solve, matvec), in_args_tree, b_avals)
    _check_tree("solve", "b", out_tree, tree)
  
    if transpose_solve is None:
      vecmat_jaxpr = tr_solve_jaxpr = None
      vecmat_consts = tr_solve_consts = []
    else:
      if symmetric:
        vecmat = matvec
        vecmat_jaxpr = matvec_jaxpr
        vecmat_consts = matvec_consts
      else:
        vecmat = _transpose_function(matvec, b)
        vecmat_jaxpr, vecmat_consts, out_tree = _initial_style_jaxpr(
            vecmat, in_args_tree, b_avals)
        assert out_tree == tree
  
      tr_solve_jaxpr, tr_solve_consts, out_tree = _initial_style_jaxpr(
          partial(transpose_solve, vecmat), in_args_tree, b_avals)
      _check_tree("transpose_solve", "b", out_tree, tree)
  
    all_consts = [matvec_consts, vecmat_consts, solve_consts, tr_solve_consts]
    const_lengths = _LinearSolveTuple(*_map(len, all_consts))
    jaxprs = _LinearSolveTuple(
        matvec_jaxpr, vecmat_jaxpr, solve_jaxpr, tr_solve_jaxpr)
  
    out_flat = linear_solve_p.bind(
        *(_flatten(all_consts) + b_flat),
        const_lengths=const_lengths, jaxprs=jaxprs, tree=tree)
    return tree_unflatten(tree, out_flat)
  
  
  def _linear_solve_abstract_eval(*args, **kwargs):
    return _map(raise_to_shaped, args[sum(kwargs['const_lengths']):])
  
  
  def _custom_linear_solve_impl(*args, **kwargs):
    const_lengths, jaxprs, tree = split_dict(
        kwargs, ['const_lengths', 'jaxprs', 'tree'])
    params, b = _split_linear_solve_args(args, const_lengths)
    x = core.jaxpr_as_fun(jaxprs.solve)(*(params.solve + b))
    _check_shapes('solve', 'b', x, b, tree)
    return x
  
  
  def _tangent_linear_map(func, params, params_dot, *x):
    """Compute the tangent of a linear map.
  
    Assuming ``func(*params, *x)`` is linear in ``x`` and computes ``A @ x``,
    this function computes ``∂A @ x``.
    """
    assert any(p is not ad_util.zero for p in params_dot)
    zeros = [ad_util.zero] * len(x)
    _, out_tangent = ad.jvp(lu.wrap_init(func)).call_wrapped(
        params + list(x), params_dot + zeros)
    return out_tangent
  
  
  def _custom_linear_solve_jvp(primals, tangents, const_lengths, jaxprs, tree):
    # A x - b = 0
    # ∂A x + A ∂x - ∂b = 0
    # ∂x = A^{-1} (∂b - ∂A x)
  
    kwargs = dict(const_lengths=const_lengths, jaxprs=jaxprs, tree=tree)
    x = linear_solve_p.bind(*primals, **kwargs)
  
    params, _ = _split_linear_solve_args(primals, const_lengths)
    params_dot, b_dot = _split_linear_solve_args(tangents, const_lengths)
  
    if all(p is ad_util.zero for p in params_dot.matvec):
      # no need to evaluate matvec_tangents
      rhs = b_dot
    else:
      matvec_tangents = _tangent_linear_map(
          core.jaxpr_as_fun(jaxprs.matvec), params.matvec, params_dot.matvec, *x)
      _check_shapes("matvec", "b", matvec_tangents, x, tree)
      rhs = _map(ad.add_tangents, b_dot, _map(operator.neg, matvec_tangents))
  
    x_dot = linear_solve_p.bind(*(_flatten(params) + rhs), **kwargs)
  
    return x, x_dot
  
  
  def _linear_solve_transpose_rule(cotangent, *primals, **kwargs):
    const_lengths, jaxprs, tree = split_dict(
        kwargs, ['const_lengths', 'jaxprs', 'tree'])
  
    if jaxprs.transpose_solve is None:
      raise TypeError('transpose_solve required for backwards mode automatic '
                      'differentiation of custom_linear_solve')
  
    params, b = _split_linear_solve_args(primals, const_lengths)
    assert b == [ad.undefined_primal] * len(b)
    cotangent_b = linear_solve_p.bind(
        *(_flatten(params.transpose()) + cotangent),
        const_lengths=const_lengths.transpose(), jaxprs=jaxprs.transpose(),
        tree=tree)
    return [None] * sum(const_lengths) + cotangent_b
  
  
  linear_solve_p = core.Primitive('custom_linear_solve')
  linear_solve_p.multiple_results = True
  linear_solve_p.def_impl(_custom_linear_solve_impl)
  linear_solve_p.def_abstract_eval(_linear_solve_abstract_eval)
  ad.primitive_jvps[linear_solve_p] = _custom_linear_solve_jvp
  xla.initial_style_translations[linear_solve_p] = xla.lower_fun(
      _custom_linear_solve_impl, initial_style=True)
  ad.primitive_transposes[linear_solve_p] = _linear_solve_transpose_rule
  # TODO(shoyer): write batching rule
  