4 To build a data-structure, you begin by deciding what the data-structure needs to do. When we built booleans, what they needed to do was select between two choices. When we built ordered pairs, what we needed was a way to wrap two elements into the pair, and ways to operate on the wrapped elements, especially a way to extract a specified one of them.
6 Now we're going to try to build lists. First, let's explain what is the difference bwteen a list and a pair.
8 A list can two elements, but it can also have more elements, or fewer. A list can even have zero elements: this is called the empty list. Sometimes this is written `nil`. In Scheme it's also written `'()` and `(list)`, and in OCaml it's written `[]`. Those languages are nice and have list structures pre-built into them. But we're going to build lists ourselves, from scratch.
10 OK, so a list doesn't have to have two elements, but still, what's the difference between a two-element list and a pair? And the difference between a three-element list and a triple?
12 The difference has to do with types. In the untyped lambda calculus we don't explicitly refer to or manipulate types, however even here we'll need to pay attention to which types of arguments we're giving to which functions. For instance, if you wrote:
14 and get-third-of-triple make-pair
16 This would evaluate to something, but it's not immediately obvious what it would be, and it's not likely to be especially useful. Even in the untyped lambda calculus, if we want computations that are easy to work with and reason about, we're going to want to pay some attention to the types we're treating formulas to have. For example, we're only going to want to pass formulas that represent boolean values as arguments to `and`. The computation may well terminate even when we don't. But the result won't be one we're in a position to make any good use of.
18 Moreover, lists and pairs (and triples, and so on), are data structures we'll find in many typed languages as well. Thinking about types is the way to understand what makes these data structures different.
22 * A list is type-homegeneous, that is, all its elements must be (or be treatable as being) of the same type.
24 * Not so a pair: its elements need not be of the same type.
26 We regard two pairs as being of the same type when their corresponding members are of the same type.
28 Another difference between lists and pairs:
30 * The length of a list is not essential to its type. A two-element list can be of the same type as a three-element list (whose members are of the right type).
32 * Not so a pair: no pair is of the same type as any triple, no matter what the types of their elements.
34 Q: Sometimes mathematicians <em>identify</em> the triple (1,2,3) with the pair (1,(2,3)), whose first member is 1 and whose second member is the pair (2,3). Wouldn't then a triple have the same type as pair, namely the pair it's identical to?
36 A: This is not an identity but an implementation. The claim that the triple (1,2,3) is not the same type as any pair is akin to the claim that the integer 1 is not the same type as any set. It allows you to go on and build set-theoretic constructions whose structure matches the desired behavior of the integers, and so is a reasonable implementation of them. This is exactly what we did when building higher-order functions to implement pairs, in the first place.
38 It is true, there are interesting and difficult questions here in the philosophical foundations of mathematics. But I hope we can proceed nonetheless.
44 OK, then, what sort of behavior do we want out of lists?
46 Well we need something to serve as the empty list. And we need a way to take an arbitrary object `hd`, and a list (possibly the empty list) `tl`, and return the list whose **head** is `hd` and whose **tail** is `tl`. And we need a way to tell, of an arbitrary list, whether it's the empty list. And we need a way to extract, from a non-empty list, that list's head and its tail. That's it. Given these basic operations, we should be able to do whatever else we want to do with lists: count their length, reverse them, or whatever. We'll explore how to do those more complex operations in the assignment.
48 It's natural to help ourselves to pairs as bricks to use in constructing lists. The way we described building a new list out of a head `hd` and a tail `tl`, for instance, sounds a lot like building a new pair `(hd, tl)`. But we also have to think about what the empty list will be---and more importantly, how to tell the empty list apart from other lists.
50 We'll work through a couple of different ways to do this. They'll get more principled as we proceed.
52 First, one way to handle the issue of "Is this the empty list or not?" is to have every list contain a boolean flag set to `true` if it's the empty list, and `false` if it isn't. Perhaps the lists could be, not pairs, but triples. The conventional implementation of this idea makes them instead a pair whose first member is the boolean flag, and whose second member, when the list is non-empty, is the list's head and tail. So a (non-empty) list whose head is `hd` and whose tail is `tl` would be represented by:
56 What about the empty list? Well we know it will be:
60 for some `N`. What should stand in for `N`?
62 No particular choice seems forced here. One strategy would be to go ahead and build your family of list operations, and see whether any particular choice for `N` made some of the other operations easier to define, or more elegant. Heres's an example. We shouldn't expect the result of extracting the head of the empty list to be meaningful. But what about the result of extracting its tail? You could argue that this operation should also be meaningless. Or you could argue that the empty list should be its own tail. If we went the latter way, it would be nice to let `N` in our construction of the empty list be some value, such that, when we tried to extract the empty list's tail in the same way we try to extract other lists' tails, we got back the empty list itself. And in fact it's possible to do this. (However, it requires a fixed-point combinator, which we won't discuss until next week.)
65 would be nice if nil tail = nil
66 nil tail = (t, N) get-second get-second = N get-second
67 so N get-second should be (t,N)
71 a fixed point g_ of g : g_ = (g g_)
73 N = (\n. K (t,n)) N is of the form N = g N
74 So if we just set N = Y g, we'll have what we want. N = (Y g) = g (Y g) = g N
76 i.e. N = Y g = Y (\n. K (t,n))
77 and nil = (t, N) = (t, Y (\n. K (t, n)))
79 nil get-second get-second ~~>
80 Y (\n. K (t, n)) get-second ~~>
81 (\n. K (t, n)) (Y (\n. K (t,n))) get-second ~~>
82 K (t, Y (\n. K (t,n))) get-second ~~>
86 For the time being, though, let's not worry about what stands in for `N` in our construction of the empty list. What should our other primitive list operations look like?
88 Well, building a list from a new element `hd` and an existing list `tl` isn't hard: we just build a pair whose first value is false and whose second value is a pair of `hd` and `tl`:
90 make-pair false (make-pair hd tl)
92 Determining whether a list is the empty list is just a matter of extracting the first element of the (outer) pair:
96 Given a non-empty list, extracting its head is just a matter of extracting the pair that is its second element, and extracting the first element of that pair:
98 some-non-empty-list get-second get-first
106 OK, version 1 works. But it might look ad hoc. Plus there's that matter of the `N` in the construction of the empty list that we don't know what should be.
108 If we do things just a bit differently, it will be easier to see some systematic rationale for them.
110 We've already seen some **enumerations**, These are data-structures that consist of several discrete values: such as true and false, or black and white, or red and green and blue. (Sometimes these groups are understood to have an order, but that's not important for our purposes.)
112 We've already seen how to build up data structures like this. For instance, red could be:
116 that is a function that waits to be supplied with three choices: the "if-you-are-red" choice, the "if-you-are-blue" choice, and the "if-you-are-green" choice, and then yields the "if-you-are-red" choice.
118 Now what if we wanted one of the enumerated possibilities to be associated with some further parameter. For instance, we wanted a structure that represented things as being (just) blue, or (just) green, or as *having some specific degree* of redness. How could we do that?
120 We might do it by adopting the convention that the "if-you-are-red" selection be not just an arbitrary value, but specifically a function that expected to be supplied with the degree of redness we're dealing with.
122 In other words, given an member `colored` of this new data structure, we'd use it like this:
124 colored (\deg. colored-is-red-to-degree-deg) colored-is-instead-green colored-is-instead-blue
126 and then if `colored` had the blue value from our data structure, this would evaluate to `colored-is-instead-blue`. If `colored` had a red-to-degree-deg value, it would evaluate to the result of supplying the relevant degree `deg` to the function `(\deg. colored-is-red-to-degree-deg)`.
128 To build a value of red-to-degree-deg, we'd replace our original:
138 If so, then you should be able to understand the underlying rationale of a list. We've just considered a data-structure that models an exclusive choice from among being-red-to-a-given-degree, just being green, or just being blue. Eliminate the blue choice. And let's associate a second parameter with being red, so that we have being-red-to-a-given-degree-and-illuminated-to-a-different-degree. Our red value would then look like this:
142 And our green value would look like this:
146 Why are we talking about this? Can you anticipate?
148 Answer: For "green", substitute "empty list". For "red", substitute "non-empty list". For "degree-of-redness", substitute "head of the non-empty list." for "degree-of-illumination," substitute "tail of the non-eempty list." And voilà!
150 Spelling it out explicitly, we say:
152 > **nil** is defined to be `\if-non-empty if-empty. if-empty`
154 > **make-list** is defined to be `\hd tl. \if-non-empty if-empty. if-non-empty hd tl`
156 Defining **isnil** and the head- and tail- extractors takes some more thought. When operating with any list implemented as we're proposing, we have to pass the list an "if-you're-non-empty" handler and a "if-you're-empty handler." If the list is non-empty, this will evaluate to the result of supplying the list's head and tail as arguments to the handler. If the list is empty, it will instead give us back the empty-handler, unprocessed.
158 So to check whether the list is empty, we could pass it an empty-handler of `true`, and a non-empty handler which accepts a head and tail argument, and then just returns the constant false:
160 some-list (\hd tl. false) true
162 What about extracting the head of a list? It only makes sense to do this when the list is known to be non-empty. In that case, the empty-handler can be anything since we know it's going to be discarded. We'll designate this dummy handler `H`. On the other hand, it's easy to see what our non-empty handler should be:
164 some-list (\hd tl. hd) H
166 Similarly for extracting the tail of a list.
168 When we get to discussing types, you'll see that the strategy deployed here has great generality. (Moreover, you can see the version 1 strategy as an approximate implementation of it.)
171 There are other reasonable choices you could make for how to implement lists. We'll come back later and discuss a third. If you're creative, you'll be able to design more yourself. The hard part is making the design principled and minimizing extraneous cruft, like the `N` and the `H` in our above discussion.
177 Now how might we go about building numbers? We'll just try to build the natural numbers: 0, 1, 2, ...
179 If you think about lists and numbers, you should start to see some interesting similarities between them. In each case there's a base value (the empty list, 0). And then further values are always the result of some operation (appending a new head to, taking the successor of) on an existing value.
181 Because of this underlying similarity, we could in fact use either of the strategies described above to implement numbers.
183 Following the version 1 strategy for lists, we could let 0 be:
187 for some useful---or, if need be, arbitrary---value `Z`. And given a number `n`, we could let the successor of `n` be:
191 Given this implementation of the numbers, it would be an easy matter to determine whether a given number was zero. (How would you do it?) And it would also be an easy matter to determine the predecessor of any number that wasn't zero. (How would you do it?) Other arithmetic operations, however, would be more complicated. We haven't yet learned the tools that would be needed to determine whether two numbers were equal, or to add two numbers.
193 Following the version 2 strategy for lists, we could adopt the convention that we'd operate on numbers by passing them an "if-you're-non-zero" handler and a "if-you're-zero" handler. If the number is non-zero, it will evaluate to the result of supplying the number's predecessor as an argument to the handler. If the number is zero, it will instead give us back the zero handler, unprocessed.
195 With that convention, we could let 0 be:
197 \if-non-zero if-zero. if-zero
199 and we could let the successor function be:
201 \n. \if-non-zero if-zero. if-non-zero n
203 This is a more principled implementation, and would again make some arithmetic operations easy to implement. But as before, others would be more difficult.
209 We're going now to describe a third strategy, which goes in a different direction.
211 The **composition** of two functions is the operation that first applies one of them, and then applies the second. For instance, the arithmetic operation that maps a real number *r* to *r<sup>2</sup>+1* is the composition of the squaring function and the successor function. This complex function is standardly written:
213 <pre><code>successor ∘ square</code></pre>
217 <pre><code>(s ∘ f) z</code></pre>
219 should be understood as:
223 Now consider the following series:
231 Remembering that I is the identity combinator, this could also be written:
236 (s ∘ s ∘ s) z
239 And we might adopt the following natural shorthand for this:
241 <pre><code>s<sup>0</sup> z
247 We haven't introduced any new constants 0, 1, 2 into the object language, nor any new form of syntactic combination. This is all just a metalanguage abbreviation for:
255 Church had the idea to implement the number *n* by an operation that accepted an arbitrary function `s` and base value `z` as arguments, and returned <code>s<sup><em>n</em></sup> z</code> as a result. In other words:
257 <pre><code>zero ≡ \s z. s<sup>0</sup> z ≡ \s z. z
258 one ≡ \s z. s<sup>1</sup> z ≡ \s z. s z
259 two ≡ \s z. s<sup>2</sup> z ≡ \s z. s (s z)
260 three ≡ \s z. s<sup>3</sup> z ≡ \s z. s (s (s z))
263 This is a very elegant idea. Implementing numbers this way, we'd let the successor function be:
265 <pre><code>succ ≡ \n. \s z. s (n s z)</code><pre>
270 ≡ (\n. \s z. s (n s z)) (\s z. s (s z))
271 ~~> \s z. s ((\s z, s (s z)) s z)
272 ~~> \s z. s (s (s z))</code></pre>
274 Adding *m* to *n* is a matter of applying the successor function to *n* *m* times. And we know how to apply an arbitrary function s to *n* *m* times: we just give that function s, and the base-value *n*, to *m* as arguments. Because that's what the function we're using to implement *m* *does*. Hence **add** can be defined to be, simply:
280 How would we tell whether a number was 0? Well, look again at the implementations of the first few numbers:
282 <pre><code>zero ≡ \s z. s<sup>0</sup> z ≡ \s z. z
283 one ≡ \s z. s<sup>1</sup> z ≡ \s z. s z
284 two ≡ \s z. s<sup>2</sup> z ≡ \s z. s (s z)
285 three ≡ \s z. s<sup>3</sup> z ≡ \s z. s (s (s z))
288 We can see that with the non-zero numbers, the function s is always applied to an argument at least once. With zero, on the other hand, we just get back the base-value. Hence we can determine whether a number is zero as follows:
290 some-number (\x. false) true
292 If some-number is zero, this will evaluate to the base value true. If some-number is non-zero, then it will evaluate to the result of applying (\x. false) to the result of applying ... to the result of applying (\x. false) to the base value true. But the result of applying (\x. false) to any argument is always false. So when some-number is non-zero, this expressions evaluates to false.
294 Perhaps not as elegant as addition, but still decently principled.
296 Multiplication is even more elegant. Consider that applying an arbitrary function s to a base value z *m × n* times is a matter of applying s to z *n* times, and then doing that again, and again, and so on...for *m* repetitions. In other words, it's a matter of applying the function (\z. n s z) to z *m* times. In other words, *m × n* can be represented as:
298 \s z. m (\z. n s z) z
301 which eta-reduces to:
307 However, at this point the elegance gives out. The predecessor function is substantially more difficult to construct on this implementation. As with all of these operations, there are several ways to do it, but they all take at least a bit of ingenuity. If you're only first learning programming right now, it would be unreasonable to expect you to be able to figure out how to do it.
309 However, if on the other hand you do have some experience programming, consider how you might construct a predecessor function for numbers implemented in this way. Using only the resources we've so far discussed. (So you have no general facility for performing recursion, for instance.)
324 nil = (t,N) = \f. f true N
329 head = L get-second get-first
330 tail = L get-second get-second
338 L (\h\t.K deal_with_h_and_t) if-nil
340 We've already seen enumerations: true | false, red | green | blue
341 What if you want one or more of the elements to have associated data? e.g. red | green | blue <how-blue>
343 could handle like this:
344 the-value if-red if-green (\n. handler-if-blue-to-degree-n)
346 then red = \r \g \b-handler. r
347 green = \r \g \b-handler. g
348 make-blue = \degree. \r \g \b-handler. b-handler degree
350 A list is basically: empty | non-empty <head,tail>
352 empty = \non-empty-handler \if-empty. if-empty = false
353 cons = \h \t. \non-empty-handler \if-empty. non-empty-handler h t
355 so [a] = cons a empty = \non-empty-handler \_. non-empty-handler a empty
360 [a; tl] isnil == (\f. f a tl) (\h \t.false) a b ~~> false a b
362 nil isnil == (\f. M) (\h \t. false) a b ~~> M[f:=isnil] a b == a
364 so M could be \a \b. a, i.e. true
365 so nil = \f. true == K true == K K = \_ K
373 nil tail = K true tail = true = \x\y.x = \f.f? such that f? = Kx. there is no such.
374 nil head = K true head = true. could mislead.
378 Church figured out how to encode integers and arithmetic operations
379 using lambda terms. Here are the basics:
387 Adding two integers involves applying a special function + such that
388 (+ 1) 2 = 3. Here is a term that works for +:
390 + = \m\n\f\x.m(f((n f) x))
393 (((\m\n\f\x.m(f((n f) x))) ;+
397 ~~>_beta targeting m for beta conversion
399 ((\n\f\x.[\f\x.fx](f((n f) x)))
402 \f\x.[\f\x.fx](f(([\f\x.fx] f) x))
404 \f\x.[\f\x.fx](f(fx))
418 let t = <<fun y _ -> y>>
419 let f = <<fun _ n -> n>>
420 let b = <<fun f g x -> f (g x)>>
424 let id = <<fun x -> x>>
425 let w = <<fun f -> f f>>
426 let w' = <<fun f n -> f f n>>
427 let pair = <<fun x y theta -> theta x y>>
429 let zero = <<fun s z -> z>>
430 let succ = <<fun n s z -> s (n s z)>>
431 let one = << $succ$ $zero$ >>
432 let two = << $succ$ $one$ >>
433 let three = << $succ$ $two$ >>
434 let four = << $succ$ $three$ >>
435 let five = << $succ$ $four$ >>
436 let six = << $succ$ $five$ >>
437 let seven = << $succ$ $six$ >>
438 let eight = << $succ$ $seven$ >>
439 let nine = << $succ$ $eight$ >>
442 let pred = <<fun n -> n (fun u v -> v (u $succ$)) ($k$ $zero$) $id$ >>
444 let pred = <<fun n s z -> n (fun u v -> v (u s)) ($k$ z) $id$ >>
445 (* ifzero n withp whenz *)
446 let ifzero = <<fun n -> n (fun u v -> v (u $succ$)) ($k$ $zero$) (fun n' withp
447 whenz -> withp n') >>
450 let iszero = << fun n -> n (fun _ -> $f$) $t$ >> in
451 << fun n -> n ( fun f z' -> $iszero$ (f $one$) z' ($succ$ (f z')) ) ($k$ $zero$) $zero$ >>
454 so n = zero ==> (k zero) zero
455 n = one ==> f=(k zero) z'=zero ==> z' i.e. zero
456 n = two ==> g(g (k zero)) zero
457 f = g(k zero) z'=zero
458 f = fun z'->z' z'=zero ==> succ (f z') = succ(zero)
459 n = three ==> g(g(g (k zero))) zero
460 f = g(g(k zero)) z'=zero
461 f = fun z' -> succ(i z') z'=zero
462 ==> succ (f z') ==> succ(succ(z'))
466 let shift = (* <n,_> -> <n+1,n> *)
467 <<fun d -> d (fun d1 _ -> $pair$ ($succ$ d1) d1) >> in
468 <<fun n -> n $shift$ ($pair$ $zero$ $zero$) $get2$ >>
472 let add = <<fun m n -> n $succ$ m>>
473 (* let add = <<fun m n -> fun s z -> m s (n s z) >> *)
474 let mul = << fun m n -> n (fun z' -> $add$ m z') $zero$ >>
477 (* we create a pairs-list of the numbers up to m, and take the
478 * head of the nth tail. the tails are in the form (k tail), with
479 * the tail of mzero being instead (id). We unwrap the content by:
480 * (k tail) tail_of_mzero
483 * we let tail_of_mzero be the mzero itself, so the nth predecessor of
484 * zero will still be zero.
488 let mzero = << $pair$ $zero$ $id$ >> in
489 let msucc = << fun d -> d (fun d1 _ -> $pair$ ($succ$ d1) ($k$ d)) >> in
490 let mtail = << fun d -> d $get2$ d >> in (* or could use d $get2$ $mzero$ *)
491 <<fun m n -> n $mtail$ (m $msucc$ $mzero$) $get1$ >>
493 let min' = <<fun m n -> $sub$ m ($sub$ m n) >>
494 let max' = <<fun m n -> $add$ n ($sub$ m n) >>
497 let mzero = << $pair$ $zero$ $id$ >> in
498 let msucc = << fun d -> d (fun d1 _ -> $pair$ ($succ$ d1) ($k$ d)) >> in
499 let mtail = << fun d -> d $get2$ d >> in (* or could use d $get2$ $mzero$ *)
500 <<fun n m -> n $mtail$ (m $msucc$ $mzero$) $get1$ (fun _ -> $t$) $f$ >>
503 let mzero = << $pair$ $zero$ $id$ >> in
504 let msucc = << fun d -> d (fun d1 _ -> $pair$ ($succ$ d1) ($k$ d)) >> in
505 let mtail = << fun d -> d $get2$ d >> in (* or could use d $get2$ $mzero$ *)
506 <<fun m n -> n $mtail$ (m $msucc$ $mzero$) $get1$ (fun _ -> $f$) $t$ >>
509 (* like leq, but now we make mzero have a self-referential tail *)
510 let mzero = << $pair$ $zero$ ($k$ ($pair$ $one$ $id$)) >> in
511 let msucc = << fun d -> d (fun d1 _ -> $pair$ ($succ$ d1) ($k$ d)) >> in
512 let mtail = << fun d -> d $get2$ d >> in (* or could use d $get2$ $mzero$ *)
513 <<fun m n -> n $mtail$ (m $msucc$ $mzero$) $get1$ (fun _ -> $f$) $t$ >>
516 let divmod' = << fun n d -> n
517 (fun f' -> f' (fun d' m' ->
518 $lt'$ ($succ$ m') d ($pair$ d' ($succ$ m')) ($pair$ ($succ$ d') $zero$)
520 ($pair$ $zero$ $zero$) >>
521 let div' = <<fun n d -> $divmod'$ n d $get1$ >>
522 let mod' = <<fun n d -> $divmod'$ n d $get2$ >>
526 let triple = << fun d1 d2 d3 -> fun sel -> sel d1 d2 d3 >> in
527 let mzero = << $triple$ $succ$ ($k$ $zero$) $id$ >> in
528 let msucc = << fun d -> $triple$ $id$ $succ$ ($k$ d) >> in
529 let mtail = (* open in dhead *)
530 << fun d -> d (fun dz mz df mf drest ->
531 fun sel -> (drest dhead) (sel (df dz) (mf mz))) >> in
533 ( fun dhead -> n $mtail$ (fun sel -> dhead (sel $zero$ $zero$)) )
534 (divisor $msucc$ $mzero$ (fun _ _ d3 -> d3 _))
535 (fun dz mz _ _ _ -> $pair$ dz mz) >>
537 let div' = <<fun n d -> $divmod'$ n d $get1$ >>
538 let mod' = <<fun n d -> $divmod'$ n d $get2$ >>
541 ISZERO = lambda n. n (lambda x. false) true,
543 LE = lambda x. lambda y. ISZERO (MONUS x y),
546 MONUS = lambda a. lambda b. b PRED a,
547 {NB. assumes a >= b >= 0}
549 DIVMOD = lambda x. lambda y.
550 let rec dm = lambda q. lambda x.
551 if LE y x then {y <= x}
552 dm (SUCC q) (MONUS x y)
557 (* f n =def. phi n_prev f_prev *)
558 let bernays = <<fun phi z n -> n (fun d -> $pair$ (d (fun n_prev f_prev -> $succ$ n_prev)) (d phi)) ($pair$ $zero$ z) (fun n f -> f)>>
562 let pred_b = << $bernays$ $k$ $zero$ >>
563 let fact_b = << $bernays$ (fun x p -> $mul$ ($succ$ x) p) $one$ >>
565 (* if m is zero, returns z; else returns withp (pred m) *)
566 let ifzero = <<fun z withp m -> $bernays$ (fun x p -> withp x) z m>>
567 let ifzero = <<fun z withp m -> m ($k$ (withp ($pred$ m))) z>>
570 let y = <<fun f -> (fun u -> f (u u)) (fun u -> f (u u))>>
571 (* strict y-combinator from The Little Schemer, Crockford's http://www.crockford.com/javascript/little.html *)
572 let y' = <<fun f -> (fun u -> f (fun n -> u u n)) (fun u -> f (fun n -> u u n))>>
574 let y'' = <<fun f -> (fun u n -> f (u u n)) (fun u n -> f (u u n))>>
577 let turing = <<(fun u f -> f (u u f)) (fun u f -> f (u u f))>>
578 let turing' = <<(fun u f -> f (fun n -> u u f n)) (fun u f -> f (fun n -> u u f n))>>
581 let fact_w = << $w$ (fun f n -> $ifzero$ n (fun p -> $mul$ n (f f
583 let fact_w' = <<(fun f n -> f f n) (fun f n -> $ifzero$ n (fun p ->
584 $mul$ n (f f p)) $one$)>>
585 let fact_w'' = let u = <<(fun f n -> $ifzero$ n (fun p -> $mul$ n (f f
586 p)) $one$)>> in <<fun n -> $u$ $u$ n>>
588 let fact_y = << $y$ (fun f n -> $ifzero$ n (fun p -> $mul$ n (f p)) $one$)>>
589 let fact_y' = << $y'$ (fun f n -> $ifzero$ n (fun p -> $mul$ n (f p)) $one$)>>
591 let fact_turing = << $turing$ (fun f n -> $ifzero$ n (fun p -> $mul$ n (f p)) $one$)>>
592 let fact_turing' = << $turing'$ (fun f n -> $ifzero$ n (fun p -> $mul$ n (f p)) $one$)>>
596 let zero_ = <<fun s z -> z>>;;
597 let succ_ = <<fun m s z -> s m (m s z)>>;;
598 let one_ = << $succ_$ $zero_$ >>;;
599 let two_ = << $succ_$ $one_$ >>;;
600 let three_ = << $succ_$ $two_$ >>;;
601 let four_ = << $succ_$ $three_$ >>;;
602 let five_ = << $succ_$ $four_$ >>;;
603 let six_ = << $succ_$ $five_$ >>;;
604 let seven_ = << $succ_$ $six_$ >>;;
605 let eight_ = << $succ_$ $seven_$ >>;;
606 let nine_ = << $succ_$ $eight_$ >>;;
608 let pred_ = <<fun n -> n (fun n' _ -> n') $zero_$ >>
609 let add_ = <<fun m n -> n (fun _ f' -> $succ_$ f') m>>
610 let mul_ = <<fun m n -> n (fun _ f' -> $add_$ m f') $zero_$ >>
613 let sub_ = <<fun m n -> n (fun _ f' -> $pred_$ f') m>>
614 let min_ = <<fun m n -> $sub_$ m ($sub_$ m n)>> (* m-max(m-n,0) = m+min(n-m,0) = min(n,m) *)
615 let max_ = <<fun m n -> $add_$ n ($sub_$ m n)>> (* n+max(m-n,0) = max(m,n) *)
617 let eq_ = <<fun m -> m (fun _ fm' n -> n (fun n' _ -> fm' n') $f$) (fun n -> n (fun _ _ -> $f$) $t$)>>
618 let lt_ = <<fun m n -> ($sub_$ n m) (fun _ _ -> $t$) $f$ >>
619 let leq_ = << fun m n -> ($sub_$ m n) (fun _ _ -> $f$) $t$ >>
621 let divmod_ = << fun n d -> n
622 (fun _ f' -> f' (fun d' m' ->
623 $lt_$ ($succ_$ m') d ($pair$ d' ($succ_$ m')) ($pair$ ($succ_$ d') $zero_$)
625 ($pair$ $zero_$ $zero_$) >>
626 let div_ = <<fun n d -> $divmod_$ n d $get1$ >>
627 let mod_ = <<fun n d -> $divmod_$ n d $get2$ >>