PEP: 533 Title: Deterministic cleanup for iterators Version: $Revision$ Last-Modified: $Date$ Author: Nathaniel J. Smith BDFL-Delegate: Yury Selivanov Status: Draft Type: Standards Track Content-Type: text/x-rst Created: 18-Oct-2016 Post-History: 18-Oct-2016 Abstract ======== We propose to extend the iterator protocol with a new ``__(a)iterclose__`` slot, which is called automatically on exit from ``(async) for`` loops, regardless of how they exit. This allows for convenient, deterministic cleanup of resources held by iterators without reliance on the garbage collector. This is especially valuable for asynchronous generators. Note on timing ============== In practical terms, the proposal here is divided into two separate parts: the handling of async iterators, which should ideally be implemented ASAP, and the handling of regular iterators, which is a larger but more relaxed project that can't start until 3.7 at the earliest. But since the changes are closely related, and we probably don't want to end up with async iterators and regular iterators diverging in the long run, it seems useful to look at them together. Background and motivation ========================= Python iterables often hold resources which require cleanup. For example: ``file`` objects need to be closed; the `WSGI spec `_ adds a ``close`` method on top of the regular iterator protocol and demands that consumers call it at the appropriate time (though forgetting to do so is a `frequent source of bugs `_); and PEP 342 (based on PEP 325) extended generator objects to add a ``close`` method to allow generators to clean up after themselves. Generally, objects that need to clean up after themselves also define a ``__del__`` method to ensure that this cleanup will happen eventually, when the object is garbage collected. However, relying on the garbage collector for cleanup like this causes serious problems in several cases: - In Python implementations that do not use reference counting (e.g. PyPy, Jython), calls to ``__del__`` may be arbitrarily delayed -- yet many situations require *prompt* cleanup of resources. Delayed cleanup produces problems like crashes due to file descriptor exhaustion, or WSGI timing middleware that collects bogus times. - Async generators (PEP 525) can only perform cleanup under the supervision of the appropriate coroutine runner. ``__del__`` doesn't have access to the coroutine runner; indeed, the coroutine runner might be garbage collected before the generator object. So relying on the garbage collector is effectively impossible without some kind of language extension. (PEP 525 does provide such an extension, but it has a number of limitations that this proposal fixes; see the "alternatives" section below for discussion.) .. XX add discussion of: - Causality preservation, context preservation - Exception swallowing Fortunately, Python provides a standard tool for doing resource cleanup in a more structured way: ``with`` blocks. For example, this code opens a file but relies on the garbage collector to close it:: def read_newline_separated_json(path): for line in open(path): yield json.loads(line) for document in read_newline_separated_json(path): ... and recent versions of CPython will point this out by issuing a ``ResourceWarning``, nudging us to fix it by adding a ``with`` block:: def read_newline_separated_json(path): with open(path) as file_handle: # <-- with block for line in file_handle: yield json.loads(line) for document in read_newline_separated_json(path): # <-- outer for loop ... But there's a subtlety here, caused by the interaction of ``with`` blocks and generators. ``with`` blocks are Python's main tool for managing cleanup, and they're a powerful one, because they pin the lifetime of a resource to the lifetime of a stack frame. But this assumes that someone will take care of cleaning up the stack frame... and for generators, this requires that someone ``close`` them. In this case, adding the ``with`` block *is* enough to shut up the ``ResourceWarning``, but this is misleading -- the file object cleanup here is still dependent on the garbage collector. The ``with`` block will only be unwound when the ``read_newline_separated_json`` generator is closed. If the outer ``for`` loop runs to completion then the cleanup will happen immediately; but if this loop is terminated early by a ``break`` or an exception, then the ``with`` block won't fire until the generator object is garbage collected. The correct solution requires that all *users* of this API wrap every ``for`` loop in its own ``with`` block:: with closing(read_newline_separated_json(path)) as genobj: for document in genobj: ... This gets even worse if we consider the idiom of decomposing a complex pipeline into multiple nested generators:: def read_users(path): with closing(read_newline_separated_json(path)) as gen: for document in gen: yield User.from_json(document) def users_in_group(path, group): with closing(read_users(path)) as gen: for user in gen: if user.group == group: yield user In general if you have N nested generators then you need N+1 ``with`` blocks to clean up 1 file. And good defensive programming would suggest that any time we use a generator, we should assume the possibility that there could be at least one ``with`` block somewhere in its (potentially transitive) call stack, either now or in the future, and thus always wrap it in a ``with``. But in practice, basically nobody does this, because programmers would rather write buggy code than tiresome repetitive code. In simple cases like this there are some workarounds that good Python developers know (e.g. in this simple case it would be idiomatic to pass in a file handle instead of a path and move the resource management to the top level), but in general we cannot avoid the use of ``with``/``finally`` inside of generators, and thus dealing with this problem one way or another. When beauty and correctness fight then beauty tends to win, so it's important to make correct code beautiful. Still, is this worth fixing? Until async generators came along I would have argued yes, but that it was a low priority, since everyone seems to be muddling along okay -- but async generators make it much more urgent. Async generators cannot do cleanup *at all* without some mechanism for deterministic cleanup that people will actually use, and async generators are particularly likely to hold resources like file descriptors. (After all, if they weren't doing I/O, they'd be generators, not async generators.) So we have to do something, and it might as well be a comprehensive fix to the underlying problem. And it's much easier to fix this now when async generators are first rolling out, then it will be to fix it later. The proposal itself is simple in concept: add a ``__(a)iterclose__`` method to the iterator protocol, and have (async) ``for`` loops call it when the loop is exited, even if this occurs via ``break`` or exception unwinding. Effectively, we're taking the current cumbersome idiom (``with`` block + ``for`` loop) and merging them together into a fancier ``for``. This may seem non-orthogonal, but makes sense when you consider that the existence of generators means that ``with`` blocks actually depend on iterator cleanup to work reliably, plus experience showing that iterator cleanup is often a desirable feature in its own right. Alternatives ============ PEP 525 asyncgen hooks ---------------------- PEP 525 proposes a `set of global thread-local hooks managed by new ``sys.{get/set}_asyncgen_hooks()`` functions `_, which allow event loops to integrate with the garbage collector to run cleanup for async generators. In principle, this proposal and PEP 525 are complementary, in the same way that ``with`` blocks and ``__del__`` are complementary: this proposal takes care of ensuring deterministic cleanup in most cases, while PEP 525's GC hooks clean up anything that gets missed. But ``__aiterclose__`` provides a number of advantages over GC hooks alone: - The GC hook semantics aren't part of the abstract async iterator protocol, but are instead restricted `specifically to the async generator concrete type `_. If you have an async iterator implemented using a class, like:: class MyAsyncIterator: async def __anext__(): ... then you can't refactor this into an async generator without changing its semantics, and vice-versa. This seems very unpythonic. (It also leaves open the question of what exactly class-based async iterators are supposed to do, given that they face exactly the same cleanup problems as async generators.) ``__aiterclose__``, on the other hand, is defined at the protocol level, so it's duck-type friendly and works for all iterators, not just generators. - Code that wants to work on non-CPython implementations like PyPy cannot in general rely on GC for cleanup. Without ``__aiterclose__``, it's more or less guaranteed that developers who develop and test on CPython will produce libraries that leak resources when used on PyPy. Developers who do want to target alternative implementations will either have to take the defensive approach of wrapping every ``for`` loop in a ``with`` block, or else carefully audit their code to figure out which generators might possibly contain cleanup code and add ``with`` blocks around those only. With ``__aiterclose__``, writing portable code becomes easy and natural. - An important part of building robust software is making sure that exceptions always propagate correctly without being lost. One of the most exciting things about async/await compared to traditional callback-based systems is that instead of requiring manual chaining, the runtime can now do the heavy lifting of propagating errors, making it *much* easier to write robust code. But, this beautiful new picture has one major gap: if we rely on the GC for generator cleanup, then exceptions raised during cleanup are lost. So, again, with ``__aiterclose__``, developers who care about this kind of robustness will either have to take the defensive approach of wrapping every ``for`` loop in a ``with`` block, or else carefully audit their code to figure out which generators might possibly contain cleanup code. ``__aiterclose__`` plugs this hole by performing cleanup in the caller's context, so writing more robust code becomes the path of least resistance. - The WSGI experience suggests that there exist important iterator-based APIs that need prompt cleanup and cannot rely on the GC, even in CPython. For example, consider a hypothetical WSGI-like API based around async/await and async iterators, where a response handler is an async generator that takes request headers + an async iterator over the request body, and yields response headers + the response body. (This is actually the use case that got me interested in async generators in the first place, i.e. this isn't hypothetical.) If we follow WSGI in requiring that child iterators must be closed properly, then without ``__aiterclose__`` the absolute most minimalistic middleware in our system looks something like:: async def noop_middleware(handler, request_header, request_body): async with aclosing(handler(request_body, request_body)) as aiter: async for response_item in aiter: yield response_item Arguably in regular code one can get away with skipping the ``with`` block around ``for`` loops, depending on how confident one is that one understands the internal implementation of the generator. But here we have to cope with arbitrary response handlers, so without ``__aiterclose__``, this ``with`` construction is a mandatory part of every middleware. ``__aiterclose__`` allows us to eliminate the mandatory boilerplate and an extra level of indentation from every middleware:: async def noop_middleware(handler, request_header, request_body): async for response_item in handler(request_header, request_body): yield response_item So the ``__aiterclose__`` approach provides substantial advantages over GC hooks. This leaves open the question of whether we want a combination of GC hooks + ``__aiterclose__``, or just ``__aiterclose__`` alone. Since the vast majority of generators are iterated over using a ``for`` loop or equivalent, ``__aiterclose__`` handles most situations before the GC has a chance to get involved. The case where GC hooks provide additional value is in code that does manual iteration, e.g.:: agen = fetch_newline_separated_json_from_url(...) while True: document = await type(agen).__anext__(agen) if document["id"] == needle: break # doesn't do 'await agen.aclose()' If we go with the GC-hooks + ``__aiterclose__`` approach, this generator will eventually be cleaned up by GC calling the generator ``__del__`` method, which then will use the hooks to call back into the event loop to run the cleanup code. If we go with the no-GC-hooks approach, this generator will eventually be garbage collected, with the following effects: - its ``__del__`` method will issue a warning that the generator was not closed (similar to the existing "coroutine never awaited" warning). - The underlying resources involved will still be cleaned up, because the generator frame will still be garbage collected, causing it to drop references to any file handles or sockets it holds, and then those objects's ``__del__`` methods will release the actual operating system resources. - But, any cleanup code inside the generator itself (e.g. logging, buffer flushing) will not get a chance to run. The solution here -- as the warning would indicate -- is to fix the code so that it calls ``__aiterclose__``, e.g. by using a ``with`` block:: async with aclosing(fetch_newline_separated_json_from_url(...)) as agen: while True: document = await type(agen).__anext__(agen) if document["id"] == needle: break Basically in this approach, the rule would be that if you want to manually implement the iterator protocol, then it's your responsibility to implement all of it, and that now includes ``__(a)iterclose__``. GC hooks add non-trivial complexity in the form of (a) new global interpreter state, (b) a somewhat complicated control flow (e.g., async generator GC always involves resurrection, so the details of PEP 442 are important), and (c) a new public API in asyncio (``await loop.shutdown_asyncgens()``) that users have to remember to call at the appropriate time. (This last point in particular somewhat undermines the argument that GC hooks provide a safe backup to guarantee cleanup, since if ``shutdown_asyncgens()`` isn't called correctly then I *think* it's possible for generators to be silently discarded without their cleanup code being called; compare this to the ``__aiterclose__``-only approach where in the worst case we still at least get a warning printed. This might be fixable.) All this considered, GC hooks arguably aren't worth it, given that the only people they help are those who want to manually call ``__anext__`` yet don't want to manually call ``__aiterclose__``. But Yury disagrees with me on this :-). And both options are viable. Always inject resources, and do all cleanup at the top level ------------------------------------------------------------ Several commentators on python-dev and python-ideas have suggested that a pattern to avoid these problems is to always pass resources in from above, e.g. ``read_newline_separated_json`` should take a file object rather than a path, with cleanup handled at the top level:: def read_newline_separated_json(file_handle): for line in file_handle: yield json.loads(line) def read_users(file_handle): for document in read_newline_separated_json(file_handle): yield User.from_json(document) with open(path) as file_handle: for user in read_users(file_handle): ... This works well in simple cases; here it lets us avoid the "N+1 ``with`` blocks problem". But unfortunately, it breaks down quickly when things get more complex. Consider if instead of reading from a file, our generator was reading from a streaming HTTP GET request -- while handling redirects and authentication via OAUTH. Then we'd really want the sockets to be managed down inside our HTTP client library, not at the top level. Plus there are other cases where ``finally`` blocks embedded inside generators are important in their own right: db transaction management, emitting logging information during cleanup (one of the major motivating use cases for WSGI ``close``), and so forth. So this is really a workaround for simple cases, not a general solution. More complex variants of __(a)iterclose__ ----------------------------------------- The semantics of ``__(a)iterclose__`` are somewhat inspired by ``with`` blocks, but context managers are more powerful: ``__(a)exit__`` can distinguish between a normal exit versus exception unwinding, and in the case of an exception it can examine the exception details and optionally suppress propagation. ``__(a)iterclose__`` as proposed here does not have these powers, but one can imagine an alternative design where it did. However, this seems like unwarranted complexity: experience suggests that it's common for iterables to have ``close`` methods, and even to have ``__exit__`` methods that call ``self.close()``, but I'm not aware of any common cases that make use of ``__exit__``'s full power. I also can't think of any examples where this would be useful. And it seems unnecessarily confusing to allow iterators to affect flow control by swallowing exceptions -- if you're in a situation where you really want that, then you should probably use a real ``with`` block anyway. Specification ============= This section describes where we want to eventually end up, though there are some backwards compatibility issues that mean we can't jump directly here. A later section describes the transition plan. Guiding principles ------------------ Generally, ``__(a)iterclose__`` implementations should: - be idempotent, - perform any cleanup that is appropriate on the assumption that the iterator will not be used again after ``__(a)iterclose__`` is called. In particular, once ``__(a)iterclose__`` has been called then calling ``__(a)next__`` produces undefined behavior. And generally, any code which starts iterating through an iterable with the intention of exhausting it, should arrange to make sure that ``__(a)iterclose__`` is eventually called, whether or not the iterator is actually exhausted. Changes to iteration -------------------- The core proposal is the change in behavior of ``for`` loops. Given this Python code:: for VAR in ITERABLE: LOOP-BODY else: ELSE-BODY we desugar to the equivalent of:: _iter = iter(ITERABLE) _iterclose = getattr(type(_iter), "__iterclose__", lambda: None) try: traditional-for VAR in _iter: LOOP-BODY else: ELSE-BODY finally: _iterclose(_iter) where the "traditional-for statement" here is meant as a shorthand for the classic 3.5-and-earlier ``for`` loop semantics. Besides the top-level ``for`` statement, Python also contains several other places where iterators are consumed. For consistency, these should call ``__iterclose__`` as well using semantics equivalent to the above. This includes: - ``for`` loops inside comprehensions - ``*`` unpacking - functions which accept and fully consume iterables, like ``list(it)``, ``tuple(it)``, ``itertools.product(it1, it2, ...)``, and others. In addition, a ``yield from`` that successfully exhausts the called generator should as a last step call its ``__iterclose__`` method. (Rationale: ``yield from`` already links the lifetime of the calling generator to the called generator; if the calling generator is closed when half-way through a ``yield from``, then this will already automatically close the called generator.) Changes to async iteration -------------------------- We also make the analogous changes to async iteration constructs, except that the new slot is called ``__aiterclose__``, and it's an async method that gets ``await``\ed. Modifications to basic iterator types ------------------------------------- Generator objects (including those created by generator comprehensions): - ``__iterclose__`` calls ``self.close()`` - ``__del__`` calls ``self.close()`` (same as now), and additionally issues a ``ResourceWarning`` if the generator wasn't exhausted. This warning is hidden by default, but can be enabled for those who want to make sure they aren't inadvertently relying on CPython-specific GC semantics. Async generator objects (including those created by async generator comprehensions): - ``__aiterclose__`` calls ``self.aclose()`` - ``__del__`` issues a ``RuntimeWarning`` if ``aclose`` has not been called, since this probably indicates a latent bug, similar to the "coroutine never awaited" warning. QUESTION: should file objects implement ``__iterclose__`` to close the file? On the one hand this would make this change more disruptive; on the other hand people really like writing ``for line in open(...): ...``, and if we get used to iterators taking care of their own cleanup then it might become very weird if files don't. New convenience functions ------------------------- The ``operator`` module gains two new functions, with semantics equivalent to the following:: def iterclose(it): if not isinstance(it, collections.abc.Iterator): raise TypeError("not an iterator") if hasattr(type(it), "__iterclose__"): type(it).__iterclose__(it) async def aiterclose(ait): if not isinstance(it, collections.abc.AsyncIterator): raise TypeError("not an iterator") if hasattr(type(ait), "__aiterclose__"): await type(ait).__aiterclose__(ait) The ``itertools`` module gains a new iterator wrapper that can be used to selectively disable the new ``__iterclose__`` behavior:: # QUESTION: I feel like there might be a better name for this one? class preserve(iterable): def __init__(self, iterable): self._it = iter(iterable) def __iter__(self): return self def __next__(self): return next(self._it) def __iterclose__(self): # Swallow __iterclose__ without passing it on pass Example usage (assuming that file objects implements ``__iterclose__``):: with open(...) as handle: # Iterate through the same file twice: for line in itertools.preserve(handle): ... handle.seek(0) for line in itertools.preserve(handle): ... :: @contextlib.contextmanager def iterclosing(iterable): it = iter(iterable) try: yield preserve(it) finally: iterclose(it) __iterclose__ implementations for iterator wrappers --------------------------------------------------- Python ships a number of iterator types that act as wrappers around other iterators: ``map``, ``zip``, ``itertools.accumulate``, ``csv.reader``, and others. These iterators should define a ``__iterclose__`` method which calls ``__iterclose__`` in turn on their underlying iterators. For example, ``map`` could be implemented as:: # Helper function map_chaining_exceptions(fn, items, last_exc=None): for item in items: try: fn(item) except BaseException as new_exc: if new_exc.__context__ is None: new_exc.__context__ = last_exc last_exc = new_exc if last_exc is not None: raise last_exc class map: def __init__(self, fn, *iterables): self._fn = fn self._iters = [iter(iterable) for iterable in iterables] def __iter__(self): return self def __next__(self): return self._fn(*[next(it) for it in self._iters]) def __iterclose__(self): map_chaining_exceptions(operator.iterclose, self._iters) def chain(*iterables): try: while iterables: for element in iterables.pop(0): yield element except BaseException as e: def iterclose_iterable(iterable): operations.iterclose(iter(iterable)) map_chaining_exceptions(iterclose_iterable, iterables, last_exc=e) In some cases this requires some subtlety; for example, ```itertools.tee`` `_ should not call ``__iterclose__`` on the underlying iterator until it has been called on *all* of the clone iterators. Example / Rationale ------------------- The payoff for all this is that we can now write straightforward code like:: def read_newline_separated_json(path): for line in open(path): yield json.loads(line) and be confident that the file will receive deterministic cleanup *without the end-user having to take any special effort*, even in complex cases. For example, consider this silly pipeline:: list(map(lambda key: key.upper(), doc["key"] for doc in read_newline_separated_json(path))) If our file contains a document where ``doc["key"]`` turns out to be an integer, then the following sequence of events will happen: 1. ``key.upper()`` raises an ``AttributeError``, which propagates out of the ``map`` and triggers the implicit ``finally`` block inside ``list``. 2. The ``finally`` block in ``list`` calls ``__iterclose__()`` on the map object. 3. ``map.__iterclose__()`` calls ``__iterclose__()`` on the generator comprehension object. 4. This injects a ``GeneratorExit`` exception into the generator comprehension body, which is currently suspended inside the comprehension's ``for`` loop body. 5. The exception propagates out of the ``for`` loop, triggering the ``for`` loop's implicit ``finally`` block, which calls ``__iterclose__`` on the generator object representing the call to ``read_newline_separated_json``. 6. This injects an inner ``GeneratorExit`` exception into the body of ``read_newline_separated_json``, currently suspended at the ``yield``. 7. The inner ``GeneratorExit`` propagates out of the ``for`` loop, triggering the ``for`` loop's implicit ``finally`` block, which calls ``__iterclose__()`` on the file object. 8. The file object is closed. 9. The inner ``GeneratorExit`` resumes propagating, hits the boundary of the generator function, and causes ``read_newline_separated_json``'s ``__iterclose__()`` method to return successfully. 10. Control returns to the generator comprehension body, and the outer ``GeneratorExit`` continues propagating, allowing the comprehension's ``__iterclose__()`` to return successfully. 11. The rest of the ``__iterclose__()`` calls unwind without incident, back into the body of ``list``. 12. The original ``AttributeError`` resumes propagating. (The details above assume that we implement ``file.__iterclose__``; if not then add a ``with`` block to ``read_newline_separated_json`` and essentially the same logic goes through.) Of course, from the user's point of view, this can be simplified down to just: 1. ``int.upper()`` raises an ``AttributeError`` 1. The file object is closed. 2. The ``AttributeError`` propagates out of ``list`` So we've accomplished our goal of making this "just work" without the user having to think about it. Transition plan =============== While the majority of existing ``for`` loops will continue to produce identical results, the proposed changes will produce backwards-incompatible behavior in some cases. Example:: def read_csv_with_header(lines_iterable): lines_iterator = iter(lines_iterable) for line in lines_iterator: column_names = line.strip().split("\t") break for line in lines_iterator: values = line.strip().split("\t") record = dict(zip(column_names, values)) yield record This code used to be correct, but after this proposal is implemented will require an ``itertools.preserve`` call added to the first ``for`` loop. [QUESTION: currently, if you close a generator and then try to iterate over it then it just raises ``Stop(Async)Iteration``, so code the passes the same generator object to multiple ``for`` loops but forgets to use ``itertools.preserve`` won't see an obvious error -- the second ``for`` loop will just exit immediately. Perhaps it would be better if iterating a closed generator raised a ``RuntimeError``? Note that files don't have this problem -- attempting to iterate a closed file object already raises ``ValueError``.] Specifically, the incompatibility happens when all of these factors come together: - The automatic calling of ``__(a)iterclose__`` is enabled - The iterable did not previously define ``__(a)iterclose__`` - The iterable does now define ``__(a)iterclose__`` - The iterable is re-used after the ``for`` loop exits So the problem is how to manage this transition, and those are the levers we have to work with. First, observe that the only async iterables where we propose to add ``__aiterclose__`` are async generators, and there is currently no existing code using async generators (though this will start changing very soon), so the async changes do not produce any backwards incompatibilities. (There is existing code using async iterators, but using the new async for loop on an old async iterator is harmless, because old async iterators don't have ``__aiterclose__``.) In addition, PEP 525 was accepted on a provisional basis, and async generators are by far the biggest beneficiary of this PEP's proposed changes. Therefore, I think we should strongly consider enabling ``__aiterclose__`` for ``async for`` loops and async generators ASAP, ideally for 3.6.0 or 3.6.1. For the non-async world, things are harder, but here's a potential transition path: In 3.7: Our goal is that existing unsafe code will start emitting warnings, while those who want to opt-in to the future can do that immediately: - We immediately add all the ``__iterclose__`` methods described above. - If ``from __future__ import iterclose`` is in effect, then ``for`` loops and ``*`` unpacking call ``__iterclose__`` as specified above. - If the future is *not* enabled, then ``for`` loops and ``*`` unpacking do *not* call ``__iterclose__``. But they do call some other method instead, e.g. ``__iterclose_warning__``. - Similarly, functions like ``list`` use stack introspection (!!) to check whether their direct caller has ``__future__.iterclose`` enabled, and use this to decide whether to call ``__iterclose__`` or ``__iterclose_warning__``. - For all the wrapper iterators, we also add ``__iterclose_warning__`` methods that forward to the ``__iterclose_warning__`` method of the underlying iterator or iterators. - For generators (and files, if we decide to do that), ``__iterclose_warning__`` is defined to set an internal flag, and other methods on the object are modified to check for this flag. If they find the flag set, they issue a ``PendingDeprecationWarning`` to inform the user that in the future this sequence would have led to a use-after-close situation and the user should use ``preserve()``. In 3.8: - Switch from ``PendingDeprecationWarning`` to ``DeprecationWarning`` In 3.9: - Enable the ``__future__`` unconditionally and remove all the ``__iterclose_warning__`` stuff. I believe that this satisfies the normal requirements for this kind of transition -- opt-in initially, with warnings targeted precisely to the cases that will be effected, and a long deprecation cycle. Probably the most controversial / risky part of this is the use of stack introspection to make the iterable-consuming functions sensitive to a ``__future__`` setting, though I haven't thought of any situation where it would actually go wrong yet... Acknowledgements ================ Thanks to Yury Selivanov, Armin Rigo, and Carl Friedrich Bolz for helpful discussion on earlier versions of this idea. Copyright ========= This document has been placed in the public domain.