+This "handler" encodes the search's having finished, and delivering a final
+answer to whatever else you wanted your program to do with the result of the
+search. If you like, at any stage in the search you might just give an argument
+to *this* handler, instead of giving an argument to the handler that continues
+the list traversal leftwards. Semantically, this would amount to *aborting* the
+list traversal! (As we've said before, whether the rest of the list traversal
+really gets evaluated will depend on what evaluation order is in place. But
+semantically we'll have avoided it. Our larger computation won't depend on the
+rest of the list traversal having been computed.)
+
+Do you have the basic idea? Think about how you'd implement it. A good
+understanding of the v2 lists will give you a helpful model.
+
+In broad outline, a single stage of the search would look like before, except
+now `f` would receive two extra, "handler" arguments. We'll reserve the name `f` for the original fold function, and use `f2` for the function that accepts two additional handler arguments. To get the general idea, you can regard these as interchangeable. If the extra precision might help, then you can pay attention to when we're talking about the handler-taking `f2` or the original `f`. You'll only be *supplying* the `f2` function; the idea will be that the behavior of the original `f` will be implicitly encoded in `f2`'s behavior.
+
+ f2 3 <sofar value that would have resulted from folding f and z over [2; 1]> <handler to continue folding leftwards> <handler to abort the traversal>
+
+`f2`'s job would be to check whether `3` matches the element we're searching for
+(here also `3`), and if it does, just evaluate to the result of passing `true` to
+the abort handler. If it doesn't, then evaluate to the result of passing
+`false` to the continue-leftwards handler.
+
+In this case, `f2` wouldn't need to consult the result of folding `f` and `z`
+over `[2; 1]`, since if we had found the element `3` in more rightward
+positions of the list, we'd have called the abort handler and this application
+of `f2` to `3` etc would never be needed. However, in other applications the
+result of folding `f` and `z` over the more rightward parts of the list would
+be needed. Consider if you were trying to multiply all the elements of the
+list, and were going to abort (with the result `0`) if you came across any
+element in the list that was zero. If you didn't abort, you'd need to know what
+the more rightward elements of the list multiplied to, because that would
+affect the answer you passed along to the continue-leftwards handler.
+
+A **version 5** list encodes the kind of fold operation we're envisaging here,
+in the same way that v3 (and [v4](/advanced_lambda/#index1h1)) lists encoded
+the simpler fold operation. Roughly, the list `[5;4;3;2;1]` would look like
+this:
+
+
+ \f2 z continue_leftwards_handler abort_handler.
+ <fold f2 and z over [4;3;2;1]>
+ (\result_of_folding_over_4321. f2 5 result_of_folding_over_4321 continue_leftwards_handler abort_handler)
+ abort_handler
+
+ ; or, expanding the fold over [4;3;2;1]:
+
+ \f2 z continue_leftwards_handler abort_handler.
+ (\continue_leftwards_handler abort_handler.
+ <fold f2 and z over [3;2;1]>
+ (\result_of_folding_over_321. f2 4 result_of_folding_over_321 continue_leftwards_handler abort_handler)
+ abort_handler
+ )
+ (\result_of_folding_over_4321. f2 5 result_of_folding_over_4321 continue_leftwards_handler abort_handler)
+ abort_handler
+
+ ; and so on
+
+Remarks: the `larger_computation` handler should be supplied as both the
+`continue_leftwards_handler` and the `abort_handler` for the leftmost
+application, where the head `5` is supplied to `f2`; because the result of this
+application should be passed to the larger computation, whether it's a "fall
+off the left end of the list" result or it's a "I'm finished, possibly early"
+result. The `larger_computation` handler also then gets passed to the next
+rightmost stage, where the head `4` is supplied to `f2`, as the `abort_handler` to
+use if that stage decides it has an early answer.
+
+Finally, notice that we're not supplying the application of `f2` to `4` etc as an argument to the application of `f2` to `5` etc---at least, not directly. Instead, we pass
+
+ (\result_of_folding_over_4321. f2 5 result_of_folding_over_4321 <one_handler> <another_handler>)
+
+*to* the application of `f2` to `4` as its "continue" handler. The application of `f2`
+to `4` can decide whether this handler, or the other, "abort" handler, should be
+given an argument and constitute its result.
+
+
+I'll say once again: we're using temporally-loaded vocabulary throughout this,
+but really all we're in a position to mean by that are claims about the result
+of the complex expression semantically depending only on this, not on that. A
+demon evaluator who custom-picked the evaluation order to make things maximally
+bad for you could ensure that all the semantically unnecessary computations got
+evaluated anyway. We don't yet know any way to prevent that. Later, we'll see
+ways to *guarantee* one evaluation order rather than another. Of
+course, in any real computing environment you'll know in advance that you're
+dealing with a fixed evaluation order and you'll be able to program efficiently
+around that.
+
+In detail, then, here's what our v5 lists will look like:
+
+ let empty = \f2 z continue_handler abort_handler. continue_handler z in
+ let make_list = \h t. \f2 z continue_handler abort_handler.
+ t f2 z (\sofar. f2 h sofar continue_handler abort_handler) abort_handler in
+ let isempty = \lst larger_computation. lst
+ ; here's our f2
+ (\hd sofar continue_handler abort_handler. abort_handler false)
+ ; here's our z
+ true
+ ; here's the continue_handler for the leftmost application of f2
+ larger_computation
+ ; here's the abort_handler
+ larger_computation in
+ let extract_head = \lst larger_computation. lst
+ ; here's our f2
+ (\hd sofar continue_handler abort_handler. continue_handler hd)
+ ; here's our z
+ junk
+ ; here's the continue_handler for the leftmost application of f2
+ larger_computation
+ ; here's the abort_handler
+ larger_computation in
+ let extract_tail = ; left as exercise
+
+These functions are used like this:
+
+ let my_list = make_list a (make_list b (make_list c empty) in
+ extract_head my_list larger_computation
+
+If you just want to see `my_list`'s head, the use `I` as the
+`larger_computation`.
+
+What we've done here does take some work to follow. But it should be within
+your reach. And once you have followed it, you'll be well on your way to
+appreciating the full terrible power of continuations.
+
+<!-- (Silly [cultural reference](http://www.newgrounds.com/portal/view/33440).) -->
+
+Of course, like everything elegant and exciting in this seminar, [Oleg
+discusses it in much more
+detail](http://okmij.org/ftp/Streams.html#enumerator-stream).
+
+> *Comments*:
+
+> 1. The technique deployed here, and in the v2 lists, and in our
+> implementations of pairs and booleans, is known as
+> **continuation-passing style** programming.
+
+> 2. We're still building the list as a right fold, so in a sense the
+> application of `f2` to the leftmost element `5` is "outermost". However,
+> this "outermost" application is getting lifted, and passed as a *handler*
+> to the next right application. Which is in turn getting lifted, and
+> passed to its next right application, and so on. So if you
+> trace the evaluation of the `extract_head` function to the list `[5;4;3;2;1]`,
+> you'll see `1` gets passed as a "this is the head sofar" answer to its
+> `continue_handler`; then that answer is discarded and `2` is
+> passed as a "this is the head sofar" answer to *its* `continue_handler`,
+> and so on. All those steps have to be evaluated to finally get the result
+> that `5` is the outer/leftmost head of the list. That's not an efficient way
+> to get the leftmost head.
+>
+> We could improve this by building lists as **left folds**. What's that?
+>
+> Well, the right fold of `f` over a list `[a;b;c;d;e]`, using starting value z, is:
+>
+> f a (f b (f c (f d (f e z))))
+>
+> The left fold on the other hand starts combining `z` with elements from the left. `f z a` is then combined with `b`, and so on:
+>
+> f (f (f (f (f z a) b) c) d) e
+>
+> or, if we preferred the arguments to each `f` flipped:
+>
+> f e (f d (f c (f b (f a z))))
+>
+> Recall we implemented v3 lists as their own right-fold functions. We could
+> instead implement lists as their own left-fold functions. To do that with our
+> v5 lists, we'd replace above:
+>
+> let make_list = \h t. \f2 z continue_handler abort_handler.
+> f2 h z (\z. t f2 z continue_handler abort_handler) abort_handler
+>
+> Having done that, now `extract_head` can return the leftmost head
+> directly, using its `abort_handler`:
+>
+> let extract_head = \lst larger_computation. lst
+> (\hd sofar continue_handler abort_handler. abort_handler hd)
+> junk
+> larger_computation
+> larger_computation
+>
+> 3. To extract tails efficiently, too, it'd be nice to fuse the apparatus
+> developed in these v5 lists with the ideas from
+> [v4](/advanced_lambda/#index1h1) lists. But that is left as an exercise.