132:
398:, so the final result will be the same regardless of parenthesization, although the specific way in which it is calculated will be different. In the general case of non-associative binary functions, the order in which the elements are combined may influence the final result's value. On lists, there are two obvious ways to carry this out: either by combining the first element with the result of recursively combining the rest (called a
1868:
library (one needs to be aware of the fact though that forcing a value built with a lazy data constructor won't force its constituents automatically by itself). Combined with tail recursion, such folds approach the efficiency of loops, ensuring constant space operation, when lazy evaluation of the
5415:
Scala's symbolic fold syntax was intended to resemble the left- or right-leaning tree commonly used to explain the fold operation, but has since been reinterpreted as an illustration of a toppling domino. The colon comes from a general Scala syntax mechanism whereby the apparent infix operator is
1859:
Another technical point is that, in the case of left folds using lazy evaluation, the new initial parameter is not being evaluated before the recursive call is made. This can lead to stack overflows when one reaches the end of the list and tries to evaluate the resulting potentially gigantic
429:
In practice, it is convenient and natural to have an initial value which in the case of a right fold is used when one reaches the end of the list, and in the case of a left fold is what is initially combined with the first element of the list. In the example above, the value 0 (the
368:
and structures, like various sorts of trees. One writes a function which recursively replaces the constructors of the datatype with provided functions, and any constant values of the type with provided values. Such a function is generally referred to as a
478:
chain of applications to be possible. Whether it will be left- or right-oriented will be determined by the types expected of its arguments by the combining function. If it is the second argument that must be of the same type as the result, then
581:
which transforms its last element, by applying an additional argument function to it, into a value of the result type before starting the folding itself, and is thus able to use type-asymmetrical binary operation like the regular
557:. When no initial value seems appropriate, for example, when one wants to fold the function which computes the maximum of its two parameters over a non-empty list to get the maximum element of the list, there are variants of
510: is associative this value will be well-defined, i.e., same for any parenthesization, although the operational details of how it is calculated will be different. This can have significant impact on efficiency if
308:. Note that the parameters to cons must be flipped, because the element to add is now the right hand parameter of the combining function. Another easy result to see from this vantage-point is to write the higher-order
105:
to decide how to progressively construct a corecursive data structure, whereas a fold recursively breaks that structure down, replacing it with the results of applying a combining function at each node on its
1860:
expression. For this reason, such languages often provide a stricter variant of left folding which forces the evaluation of the initial parameter before making the recursive call. In
Haskell this is the
576:
These folds use type-symmetrical binary operation: the types of both its arguments, and its result, must be the same. Richard Bird in his 2010 book proposes "a general fold function on non-empty lists"
249:
259:
722:
If the list is empty, the result is the initial value. If not, fold the tail of the list using as new initial value the result of applying f to the old initial value and the first element.
255:
There's another way to perform the structural transformation in a consistent manner, with the order of the two links of each node flipped when fed into the combining function:
5416:
invoked as a method on the left operand with the right operand passed as an argument, or vice versa if the operator's last character is a colon, here applied symmetrically.
1711:), tail-recursively building a representation of expression that right-fold would build. The extraneous intermediate list structure can be eliminated with the
4800:
6067:
522:
573:, the 1 making reference to the automatic provision of an initial element, and the fact that the lists they are applied to must have at least one element.
470:), i.e. when the type of its result is different from the type of the list's elements. Then an initial value must be used, with the same type as that of
17:
1472:
to refer to its second argument first here, and be able to produce some part of its result without reference to the recursive case (here, on its
6338:
2254:
A finite prefix of primes is concisely defined as a folding of set difference operation over the lists of enumerated multiples of integers, as
565:
which use the last and first element of the list respectively as the initial value. In
Haskell and several other languages, these are called
153:
86:, and possibly some default values to be used under certain conditions. The fold then proceeds to combine elements of the data structure's
6470:
6282:
837:
If the list is empty, the result is the initial value z. If not, apply f to the first element and the result of folding the rest.
498:), and the result type is the same as the list elements' type, the parentheses may be placed in arbitrary fashion thus creating a
6317:
358:
6128:
196:, for example, are built up in many functional languages from two primitives: any list is either an empty list, commonly called
6115:
1587:
lists, that means that left-fold and reverse can be composed to perform a right fold in a tail-recursive way (cf.
192:
Folds can be regarded as consistently replacing the structural components of a data structure with functions and values.
387:. To a rough approximation, one can think of this fold as replacing the commas in the list with the + operation, giving
1881:
interpreter, the structural transformations which fold functions perform can be illustrated by constructing a string:
402:), or by combining the result of recursively combining all elements but the last one, with the last element (called a
6475:
3107:
2248:
179:
161:
107:
6460:
1449:
can be efficiently compiled as a loop, but can't deal with infinite lists at all — it will recurse forever in an
3740:
3413:
2569:
2085:
1878:
1392:
is able to produce some part of its result without reference to the recursive case on its "right" i.e., in its
411:
226:
157:
94:
6179:
74:
processing its constituent parts, building up a return value. Typically, a fold is presented with a combining
6256:
5428:
5049:
4888:
4816:
4797:
4084:
3667:
3364:
3258:
1496:, it allows for a lazy combining function to inspect list's elements from the right, if it so chooses (e.g.,
6153:
6445:
5723:
5320:
4021:
3506:
2081:
1842:
is only needed in languages like
Haskell with its flipped order of arguments to the combining function of
1488:, it allows for a lazy combining function to inspect list's elements from the left; and conversely, while
6440:
6205:
6047:
5236:
5090:
3920:
3607:
3033:
value on the first iteration); the current item; the current item's index or key; and a reference to the
309:
285:
6087:
4180:
3312:
2506:
1712:
6430:
3358:
6031:
4586:
845:
Lists can be folded over in a tree-like fashion, both for finite and for indefinitely defined lists:
4965:
3853:
3181:
2364:
142:
2089:
1846:
unlike e.g., in Scheme where the same order of arguments is used for combining functions to both
526:
210:
203:), or is constructed by prefixing an element in front of another list, creating what is called a
146:
79:
5032:
R supports right folding and left or right folding with or without an initial value through the
364:
This way of looking at things provides a simple route to designing fold-like functions on other
5965:
31:
1388:
to the head of the list and the recursive case of folding over the rest of the list. Thus, if
395:
59:
4808:
1445:
will immediately call itself with new parameters until it reaches the end of the list. This
1396:
argument, and the rest of the result is never demanded, then the recursion will stop (e.g.,
1377:
515:
304:
will not change the result. The left fold diagram suggests an easy way to reverse a list,
8:
6057:
2242:
2232:
365:
67:
6393:
6294:
6110:
Richard Bird, "Pearls of
Functional Algorithm Design", Cambridge University Press 2010,
1468:-applications, which is then presented to the caller to be evaluated. Were the function
70:
data structure and through use of a given combining operation, recombine the results of
6465:
6062:
6026:
2238:
6425:
5225:
Please notice that these implementations of right folds are wrong for non-commutative
6111:
5872:
4446:
3655:
2836:
431:
3913:
1084:
demand its second argument's value, at least not all of it, or not immediately (see
533:, tree-like folds are whole-list oriented and operate in a consistent manner across
6286:
4957:
4263:
2755:
2645:
546:
530:
447:
384:
with the addition operator would result in 15, the sum of the elements of the list
193:
5216:
In Ruby 1.8.7+, can also pass a symbol representing a function instead of a block.
4804:
2759:
1373:
491:
1446:
83:
5968:
can be implemented via fold, proving that iterations can be reduced to folds:
3304:
6454:
6290:
5961:
4784:
is not supplied, NULL is used, so this is not a true foldl1. Before PHP 5.3,
1450:
442:
for the left fold. For multiplication, an initial choice of 0 wouldn't work:
248:
6042:
6036:
370:
258:
111:
6363:
5645:
4334:
3810:, the folding function takes arguments in the opposite order as that for
3114:
2572:
2237:
operates on ordered lists in a local manner to efficiently produce their
499:
245:
with a specific function. These replacements can be viewed as a diagram:
115:
102:
6052:
3973:
3968:
2289:
75:
1441:). This allows right folds to operate on infinite lists. By contrast,
5599:
2075:"(1+((2+3)+(((4+5)+(6+7))+((((8+9)+(10+11))+(12+13))+0))))"
2027:"(((((1+2)+(3+4))+((5+6)+(7+8)))+(((9+10)+(11+12))+13))+0)"
1979:"(((((((((((((0+1)+2)+3)+4)+5)+6)+7)+8)+9)+10)+11)+12)+13)"
1931:"(1+(2+(3+(4+(5+(6+(7+(8+(9+(10+(11+(12+(13+0)))))))))))))"
1544:
Reversing a list is also tail-recursive (it can be implemented using
586:
to produce a result of type different from the list's elements type.
462:
The use of an initial value is necessary when the combining function
418:'s terminology. With a right fold, the sum would be parenthesized as
87:
71:
131:
4367:
without an initial value is supported in versions 10.0 and higher.
3027:
receives as arguments the result of the previous operation (or the
63:
6088:"Haskell unit 6: The higher-order fold functions | Antoni Diller"
4631:
4225:
3043:
2892:
2766:
5714:, the folding function takes arguments in the same order as for
1480:
argument), then the recursion would stop. This means that while
4372:
415:
5830:
5788:
4517:
2664:
6435:
6364:"A tutorial on the universality and expressiveness of fold"
4661:
3819:
2982:
205:
6441:"Constructing List Homomorphism from Left and Right Folds"
5710:
The supplied function takes its arguments in a tuple. For
1864:(note the apostrophe, pronounced 'prime') function in the
5418:
Scala also features the tree-like folds using the method
4707:
450:
for multiplication is 1. This would give us the outcome
422:, whereas with a left fold it would be parenthesized as
316:, by composing the function to act on the elements with
5817:
Two functions exist for each case because XPath offers
4441:
Requires
Symbolic Math Toolbox, supported from R2016b.
410:
being either right-associative or left-associative, in
6318:"An intuitive feel for Scala's /: operator (foldLeft)"
2296:) could be easily defined using tree-like folding as
2080:
Infinite tree-like folding is demonstrated e.g., in
540:
4167:(either success or failure) to the return type of
1367:
237:at the end of the list with a specific value, and
1657:so it reverses the order of its arguments (i.e.,
483: could be seen as a binary operation that
6452:
121:
6426:"Higher order functions — map, fold and filter"
6280:
3912:u/y applies the dyad u between the items of y.
434:) would be chosen as an initial value, giving
6129:"Array.prototype.reduce() - JavaScript | MDN"
1072:function, to avoid its runaway evaluation on
457:
357:where the period (.) is an operator denoting
6283:"Re: Blog: My verdict on the Scala language"
1091:
525:and operate in a consistent manner for each
276:visually. They also highlight the fact that
5807:array:fold-right($ input, $ zero, $ action)
5231:(also initial value is put on wrong side).
4250:standard library object. The abbreviations
2384:could have been defined through folding as
1869:final result is impossible or undesirable.
1384:will immediately return the application of
466: is asymmetrical in its types (e.g.
160:. Unsourced material may be challenged and
5798:array:fold-left($ input, $ zero, $ action)
4236:fold_right(Closure, Initial, List, Result)
90:, using the function in a systematic way.
6431:"Unit 6: The Higher-order fold Functions"
4231:fold_left(Closure, Initial, List, Result)
217: ), resulting from application of a
215:Cons(X1,Cons(X2,Cons(...(Cons(Xn,nil)))))
180:Learn how and when to remove this message
2475:
1653:), with a modification to the function
1456:Having reached the end of the list, an
14:
6453:
6206:"Result - Kotlin Programming Language"
6180:"reduce - Kotlin Programming Language"
2292:(and its duplicates-removing variety,
607:can be formulated in a few equations.
6104:
5803:fold-right($ input, $ zero, $ action)
3662:interface, arrays are also supported
280:is the identity function on lists (a
6315:
6154:"fold - Kotlin Programming Language"
5794:fold-left($ input, $ zero, $ action)
158:adding citations to reliable sources
125:
6394:"Getting a Fix from the Right Fold"
6339:"Fold API - Scala Standard Library"
2369:a duplicates-preserving variant of
229:). One can view a fold on lists as
24:
6361:
840:
257:
247:
221:function (written down as a colon
110:values and the recursive results (
25:
18:Accumulate (higher-order function)
6487:
6419:
6371:Journal of Functional Programming
4161:Result.fold(onSuccess, onFailure)
2887:is an unexpanded parameter pack.
2495:Right fold without initial value
589:
541:Special folds for non-empty lists
502:of nested sub-expressions, e.g.,
494:, i.e. symmetrical in its types (
6471:Programming language comparisons
6391:
4246:Meta-predicates provided by the
2659:Similarly in all .NET languages
2492:Left fold without initial value
452:1 * 1 * 2 * 3 * 4 * 5 = 120 = 5!
406:). This corresponds to a binary
130:
27:Family of higher-order functions
6385:
6355:
6331:
6309:
5866:
4151:Other collections also support
2852:
2086:unbounded sieve of Eratosthenes
1851:
1776:
1716:
1658:
1588:
1545:
1497:
1397:
1368:Evaluation order considerations
594:
6281:Odersky, Martin (2008-01-05).
6274:
6249:
6224:
6204:
6198:
6178:
6172:
6146:
6121:
6080:
5636:ANSI Smalltalk doesn't define
4942:functools.reduce(lambda x, y:
4912:functools.reduce(lambda x, y:
545:One often wants to choose the
394:In the example above, + is an
13:
1:
6073:
5640:but many implementations do.
2971:considering the emptiness of
2945:std::ranges::fold_right_last(
2932:std::ranges::fold_left_first(
1085:
599:Using Haskell as an example,
440:((((0 + 1) + 2) + 3) + 4) + 5
436:1 + (2 + (3 + (4 + (5 + 0))))
266:
122:As structural transformations
93:Folds are in a sense dual to
6245:from functools import reduce
6152:
2965:std::ranges::fold_right_last
2961:std::ranges::fold_left_first
2467:"last: Empty list"
2425:"head: Empty list"
7:
6048:Map (higher-order function)
6020:
3143::from-end t :initial-value
2845:is a binary operator (both
1872:
1076:defined lists the function
376:
101:value and apply a function
10:
6492:
6039:, a generalization of fold
3108:clojure.core.reducers/fold
2851:s must be the same, e.g.,
2835:Fold expression (only for
1713:continuation-passing style
506:. If the binary operation
458:Linear vs. tree-like folds
265:These pictures illustrate
6032:Iterated binary operation
2500:
2497:
2494:
2491:
2488:
2485:
2482:
1464:of nested left-deepening
1092:Folds for non-empty lists
521:Whereas linear folds are
444:0 * 1 * 2 * 3 * 4 * 5 = 0
6476:Iteration in programming
6257:"Iterator in core::iter"
5970:
5940:
5881:
2915:std::ranges::fold_right(
2386:
2298:
2288:For finite lists, e.g.,
2256:
2226:-- = g . g . g . g . ...
2094:
1883:
1095:
847:
724:
609:
474: 's result, for a
438:for the right fold, and
381:The folding of the list
322:
288:parlance), as replacing
58:) refers to a family of
5964:with infinite lists, a
2898:std::ranges::fold_left(
504:((1 + 2) + (3 + 4)) + 5
490:When the function is a
485:associates on the right
424:(((1 + 2) + 3) + 4) + 5
420:1 + (2 + (3 + (4 + 5)))
6461:Higher-order functions
5966:fixed point combinator
3914:"J Dictionary: Insert"
1535:"empty list"
1460:is in effect built by
1435:"empty list"
262:
252:
60:higher-order functions
32:functional programming
6133:developer.mozilla.org
5825:for structured data.
5821:for unstructured and
5577:(vector-unfold-right
4790:can only be integer.
2084:primes production by
553:as the initial value
396:associative operation
261:
251:
6316:Sterling, Nicholas.
6068:Structural recursion
5938:can be expressed as
5879:having a definition
4870:reduce(lambda x, y:
4840:reduce(lambda x, y:
2656:IEnumerable<T>
2476:In various languages
366:algebraic data types
359:function composition
154:improve this section
6058:Recursive data type
5478:(vector-fold-right
5313:DoubleEndedIterator
2231:where the function
1372:In the presence of
6343:www.scala-lang.org
6063:Reduction operator
6027:Aggregate function
5875:function. For any
4803:2020-11-28 at the
4258:may also be used.
4163:, which reduces a
3357:See also List API
3307:for example usage
2837:variadic templates
2363:with the function
487:, and vice versa.
263:
253:
6446:"The magic foldr"
6116:978-0-521-51338-8
5864:
5863:
5223:is an Enumeration
5040:arguments to the
4946:(y, x), reversed(
4929:functools.reduce(
4916:(y, x), reversed(
4895:functools.reduce(
4874:(y, x), reversed(
4844:(y, x), reversed(
4563:Array.fold_right
3656:extension methods
549:of the operation
432:additive identity
389:1 + 2 + 3 + 4 + 5
306:foldl (flip (:))
190:
189:
182:
16:(Redirected from
6483:
6413:
6412:
6410:
6408:
6401:The Monad.Reader
6398:
6389:
6383:
6382:
6380:
6378:
6373:(9 (4)): 355–372
6368:
6362:Hutton, Graham.
6359:
6353:
6352:
6350:
6349:
6335:
6329:
6328:
6326:
6324:
6313:
6307:
6306:
6304:
6302:
6293:. Archived from
6278:
6272:
6271:
6269:
6268:
6253:
6247:
6246:
6242:
6237:
6236:import functools
6233:
6232:functools.reduce
6228:
6222:
6221:
6219:
6217:
6202:
6196:
6195:
6193:
6191:
6176:
6170:
6169:
6167:
6165:
6150:
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6141:
6140:
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6101:
6099:
6098:
6084:
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5998:
5995:
5992:
5989:
5986:
5983:
5980:
5977:
5974:
5956:
5953:
5950:
5947:
5944:
5930:
5927:
5924:
5921:
5918:
5915:
5912:
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5903:
5900:
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5894:
5891:
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5885:
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5804:
5799:
5795:
5775:
5765:
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5741:
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5688:
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5629:
5616:
5591:
5574:
5557:
5539:
5519:
5504:
5489:
5475:
5460:
5446:
5421:
5420:list.fold(z)(op)
5410:
5398:
5386:
5370:
5354:
5338:
5314:
5310:
5304:
5294:
5282:
5270:
5254:
5230:
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5211:
5205:
5198:
5192:
5184:
5173:
5167:
5160:
5150:
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5133:
5125:
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5109:
5099:
5077:
5064:
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5035:
5027:
5014:
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4951:
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4866:
4853:
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4763:
4758:
4745:
4731:
4726:
4701:
4691:
4678:
4646:
4618:
4603:
4573:
4560:
4550:List.fold_right
4546:
4536:Array.fold_left
4533:
4508:
4495:
4482:
4465:
4438:
4434:
4421:
4408:
4391:
4366:
4361:
4356:
4351:
4346:
4341:
4325:
4312:
4299:
4282:
4257:
4253:
4249:
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4232:
4212:
4197:
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4170:
4166:
4162:
4159:. There is also
4158:
4154:
4146:
4134:
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4112:
4104:
4094:
4075:
4062:
4049:
4036:
4010:
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3957:
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3809:
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3676:
3661:
3643:
3632:
3621:
3597:
3588:iterator.unfold(
3582:
3573:iterator.reduce(
3570:
3557:
3544:list.fold_right(
3540:
3525:
3499:
3488:
3481:List.reduceBack
3477:
3467:
3456:
3442:
3429:
3400:
3383:
3348:
3331:
3294:
3277:
3253:
3245:
3240:
3231:
3218:
3216:
3200:
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3088:
3076:
3060:
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3026:
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3000:
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2806:
2787:
2756:function pointer
2753:
2746:
2740:
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2693:
2692:std::accumulate(
2688:
2671:
2670:std::accumulate(
2657:
2653:
2646:extension method
2643:
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2417:
2414:
2411:
2408:
2405:
2402:
2399:
2396:
2393:
2390:
2383:
2379:
2372:
2367:
2359:
2356:
2353:
2350:
2347:
2344:
2341:
2338:
2335:
2332:
2329:
2326:
2323:
2320:
2317:
2314:
2311:
2308:
2305:
2302:
2295:
2284:
2281:
2278:
2275:
2272:
2269:
2266:
2263:
2260:
2245:
2235:
2227:
2224:
2221:
2218:
2215:
2212:
2209:
2206:
2203:
2200:
2197:
2194:
2191:
2188:
2185:
2182:
2179:
2176:
2173:
2170:
2167:
2164:
2161:
2158:
2155:
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2149:
2146:
2143:
2140:
2137:
2134:
2131:
2128:
2125:
2122:
2119:
2116:
2113:
2110:
2107:
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2101:
2098:
2076:
2073:
2070:
2067:
2064:
2061:
2058:
2055:
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2049:
2046:
2043:
2040:
2037:
2034:
2031:
2028:
2025:
2022:
2019:
2016:
2013:
2010:
2007:
2004:
2001:
1998:
1995:
1992:
1989:
1986:
1983:
1980:
1977:
1974:
1971:
1968:
1965:
1962:
1959:
1956:
1953:
1950:
1947:
1944:
1941:
1938:
1935:
1932:
1929:
1926:
1923:
1920:
1917:
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1911:
1908:
1905:
1902:
1899:
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1887:
1867:
1863:
1855:
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1833:
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1827:
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1710:
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1679:
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1609:
1606:
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1600:
1597:
1594:
1591:
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1578:
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1569:
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1563:
1560:
1557:
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1540:
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1536:
1533:
1530:
1527:
1524:
1521:
1518:
1515:
1512:
1509:
1506:
1503:
1500:
1491:
1483:
1471:
1467:
1463:
1444:
1440:
1439:
1436:
1433:
1430:
1427:
1424:
1421:
1418:
1415:
1412:
1409:
1406:
1403:
1400:
1383:
1363:
1360:
1357:
1354:
1351:
1348:
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1342:
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1336:
1333:
1330:
1327:
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1318:
1315:
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1309:
1306:
1303:
1300:
1297:
1294:
1291:
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1285:
1282:
1279:
1276:
1273:
1270:
1267:
1264:
1261:
1258:
1255:
1252:
1249:
1246:
1243:
1240:
1237:
1234:
1231:
1228:
1225:
1222:
1219:
1216:
1213:
1210:
1207:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1183:
1180:
1177:
1174:
1171:
1168:
1165:
1162:
1159:
1156:
1153:
1150:
1147:
1144:
1141:
1138:
1135:
1132:
1129:
1126:
1123:
1120:
1117:
1114:
1111:
1108:
1105:
1102:
1099:
1079:
1071:
1064:
1061:
1058:
1055:
1052:
1049:
1046:
1043:
1040:
1037:
1034:
1031:
1028:
1025:
1022:
1019:
1016:
1013:
1010:
1007:
1004:
1001:
998:
995:
992:
989:
986:
983:
980:
977:
974:
971:
968:
965:
962:
959:
956:
953:
950:
947:
944:
941:
938:
935:
932:
929:
926:
923:
920:
917:
914:
911:
908:
905:
902:
899:
896:
893:
890:
887:
884:
881:
878:
875:
872:
869:
866:
863:
860:
857:
854:
851:
833:
830:
827:
824:
821:
818:
815:
812:
809:
806:
803:
800:
797:
794:
791:
788:
785:
782:
779:
776:
773:
770:
767:
764:
761:
758:
755:
752:
749:
746:
743:
740:
737:
734:
731:
728:
718:
715:
712:
709:
706:
703:
700:
697:
694:
691:
688:
685:
682:
679:
676:
673:
670:
667:
664:
661:
658:
655:
652:
649:
646:
643:
640:
637:
634:
631:
628:
625:
622:
619:
616:
613:
606:
602:
585:
580:
572:
568:
564:
560:
547:identity element
505:
497:
469:
453:
448:identity element
445:
441:
437:
425:
421:
390:
386:
383:
353:
350:
347:
344:
341:
338:
335:
332:
329:
326:
319:
315:
307:
303:
295:
279:
224:
220:
216:
202:
185:
178:
174:
171:
165:
134:
126:
21:
6491:
6490:
6486:
6485:
6484:
6482:
6481:
6480:
6451:
6450:
6422:
6417:
6416:
6406:
6404:
6396:
6390:
6386:
6376:
6374:
6366:
6360:
6356:
6347:
6345:
6337:
6336:
6332:
6322:
6320:
6314:
6310:
6300:
6298:
6291:comp.scala.lang
6279:
6275:
6266:
6264:
6255:
6254:
6250:
6244:
6240:
6238:
6235:
6231:
6229:
6225:
6215:
6213:
6203:
6199:
6189:
6187:
6177:
6173:
6163:
6161:
6151:
6147:
6138:
6136:
6127:
6126:
6122:
6109:
6105:
6096:
6094:
6086:
6085:
6081:
6076:
6023:
6018:
6017:
6014:
6011:
6008:
6005:
6002:
5999:
5996:
5993:
5990:
5987:
5984:
5981:
5978:
5975:
5972:
5958:
5957:
5954:
5951:
5948:
5945:
5942:
5932:
5931:
5928:
5925:
5922:
5919:
5916:
5913:
5910:
5907:
5904:
5901:
5898:
5895:
5892:
5889:
5886:
5883:
5869:
5849:
5835:
5806:
5805:
5802:
5797:
5796:
5793:
5760:
5743:
5742:
5728:
5715:
5711:
5690:
5689:
5677:
5663:
5662:
5650:
5637:
5621:
5604:
5594:srfi/1 srfi/43
5576:
5575:
5560:(vector-unfold
5559:
5558:
5541:
5540:
5522:
5507:
5492:
5477:
5476:
5463:
5448:
5447:
5434:
5419:
5401:
5389:
5372:
5371:
5357:
5340:
5339:
5325:
5312:
5306:
5299:
5285:
5273:
5257:
5241:
5226:
5224:
5218:
5217:
5200:
5199:
5187:
5175:
5174:
5163:
5145:
5144:
5128:
5111:
5110:
5095:
5067:
5054:
5041:
5037:
5033:
5017:
5004:
4987:
4970:
4941:
4928:
4911:
4894:
4869:
4856:
4839:
4822:
4805:Wayback Machine
4791:
4785:
4779:
4761:
4748:
4729:
4712:
4699:
4683:
4666:
4636:
4606:
4591:
4562:
4561:
4549:
4535:
4534:
4523:List.fold_left
4522:
4498:
4485:
4468:
4451:
4437:
4424:
4411:
4394:
4377:
4364:
4359:
4354:
4349:
4344:
4339:
4315:
4302:
4285:
4268:
4255:
4251:
4247:
4235:
4230:
4200:
4185:
4172:
4168:
4165:Result<T>
4164:
4160:
4156:
4152:
4137:
4125:
4107:
4089:
4065:
4052:
4039:
4026:
3999:
3979:
3972:
3946:
3926:
3899:
3888:
3873:
3858:
3824:
3811:
3807:
3795:
3784:
3773:
3759:
3745:
3720:
3707:
3689:
3672:
3659:
3634:
3633:
3623:
3622:
3612:
3601:
3587:
3572:
3571:
3560:
3543:
3527:
3526:
3511:
3491:
3480:
3469:
3468:
3459:
3445:
3431:
3430:
3418:
3386:
3369:
3334:
3317:
3280:
3263:
3251:
3234:
3221:
3203:
3186:
3163:
3151:
3135:
3127::initial-value
3119:
3091:
3079:
3063:
3048:
3034:
3028:
3022:
3005:
2987:
2972:
2968:
2964:
2960:
2944:
2931:
2914:
2897:
2882:
2877:
2874:
2871:
2868:
2865:
2862:
2859:
2856:
2853:
2846:
2840:
2821:
2809:
2790:
2771:
2760:function object
2749:
2748:
2742:
2736:
2730:
2724:
2723:
2721:<numeric>
2720:
2691:
2669:
2658:
2655:
2649:
2648:
2641:
2625:
2612:
2594:
2577:
2552:
2541:
2526:
2511:
2478:
2473:
2472:
2469:
2466:
2463:
2460:
2457:
2454:
2451:
2448:
2445:
2442:
2439:
2436:
2433:
2430:
2427:
2424:
2421:
2418:
2415:
2412:
2409:
2406:
2403:
2400:
2397:
2394:
2391:
2388:
2381:
2377:
2370:
2365:
2361:
2360:
2357:
2354:
2351:
2348:
2345:
2342:
2339:
2336:
2333:
2330:
2327:
2324:
2321:
2318:
2315:
2312:
2309:
2306:
2303:
2300:
2293:
2286:
2285:
2282:
2279:
2276:
2273:
2270:
2267:
2264:
2261:
2258:
2243:
2233:
2229:
2228:
2225:
2222:
2219:
2216:
2213:
2210:
2207:
2204:
2201:
2198:
2195:
2192:
2189:
2186:
2183:
2180:
2177:
2174:
2171:
2168:
2165:
2162:
2159:
2156:
2153:
2150:
2147:
2144:
2141:
2138:
2135:
2132:
2129:
2126:
2123:
2120:
2117:
2114:
2111:
2108:
2105:
2102:
2099:
2096:
2078:
2077:
2074:
2071:
2068:
2065:
2062:
2059:
2056:
2053:
2050:
2047:
2044:
2041:
2038:
2035:
2032:
2029:
2026:
2023:
2020:
2017:
2014:
2011:
2008:
2005:
2002:
1999:
1996:
1993:
1990:
1987:
1984:
1981:
1978:
1975:
1972:
1969:
1966:
1963:
1960:
1957:
1954:
1951:
1948:
1945:
1942:
1939:
1936:
1933:
1930:
1927:
1924:
1921:
1918:
1915:
1912:
1909:
1906:
1903:
1900:
1897:
1894:
1891:
1888:
1885:
1875:
1865:
1861:
1852:
1847:
1843:
1839:
1834:
1831:
1828:
1825:
1822:
1819:
1816:
1813:
1810:
1807:
1804:
1801:
1798:
1795:
1792:
1789:
1786:
1783:
1780:
1777:
1771:
1768:
1765:
1762:
1759:
1756:
1753:
1750:
1747:
1744:
1741:
1738:
1735:
1732:
1729:
1726:
1723:
1720:
1717:
1707:
1704:
1701:
1698:
1695:
1692:
1689:
1686:
1683:
1680:
1677:
1674:
1671:
1668:
1665:
1662:
1659:
1654:
1649:
1646:
1643:
1640:
1637:
1634:
1631:
1628:
1625:
1622:
1619:
1616:
1613:
1610:
1607:
1604:
1601:
1598:
1595:
1592:
1589:
1579:
1576:
1573:
1570:
1567:
1564:
1561:
1558:
1555:
1552:
1549:
1546:
1537:
1534:
1531:
1528:
1525:
1522:
1519:
1516:
1513:
1510:
1507:
1504:
1501:
1498:
1489:
1481:
1469:
1465:
1461:
1442:
1437:
1434:
1431:
1428:
1425:
1422:
1419:
1416:
1413:
1410:
1407:
1404:
1401:
1398:
1381:
1370:
1365:
1364:
1361:
1358:
1355:
1352:
1349:
1346:
1343:
1340:
1337:
1334:
1331:
1328:
1325:
1322:
1319:
1316:
1313:
1310:
1307:
1304:
1301:
1298:
1295:
1292:
1289:
1286:
1283:
1280:
1277:
1274:
1271:
1268:
1265:
1262:
1259:
1256:
1253:
1250:
1247:
1244:
1241:
1238:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1211:
1208:
1205:
1202:
1199:
1196:
1193:
1190:
1187:
1184:
1181:
1178:
1175:
1172:
1169:
1166:
1163:
1160:
1157:
1154:
1151:
1148:
1145:
1142:
1139:
1136:
1133:
1130:
1127:
1124:
1121:
1118:
1115:
1112:
1109:
1106:
1103:
1100:
1097:
1094:
1077:
1069:
1068:In the case of
1066:
1065:
1062:
1059:
1056:
1053:
1050:
1047:
1044:
1041:
1038:
1035:
1032:
1029:
1026:
1023:
1020:
1017:
1014:
1011:
1008:
1005:
1002:
999:
996:
993:
990:
987:
984:
981:
978:
975:
972:
969:
966:
963:
960:
957:
954:
951:
948:
945:
942:
939:
936:
933:
930:
927:
924:
921:
918:
915:
912:
909:
906:
903:
900:
897:
894:
891:
888:
885:
882:
879:
876:
873:
870:
867:
864:
861:
858:
855:
852:
849:
843:
841:Tree-like folds
835:
834:
831:
828:
825:
822:
819:
816:
813:
810:
807:
804:
801:
798:
795:
792:
789:
786:
783:
780:
777:
774:
771:
768:
765:
762:
759:
756:
753:
750:
747:
744:
741:
738:
735:
732:
729:
726:
720:
719:
716:
713:
710:
707:
704:
701:
698:
695:
692:
689:
686:
683:
680:
677:
674:
671:
668:
665:
662:
659:
656:
653:
650:
647:
644:
641:
638:
635:
632:
629:
626:
623:
620:
617:
614:
611:
604:
600:
597:
592:
583:
578:
570:
566:
562:
558:
543:
503:
495:
467:
460:
451:
443:
439:
435:
423:
419:
388:
385:
382:
379:
355:
354:
351:
348:
345:
342:
339:
336:
333:
330:
327:
324:
317:
313:
305:
301:
293:
277:
222:
218:
214:
201:
186:
175:
169:
166:
151:
135:
124:
97:, which take a
28:
23:
22:
15:
12:
11:
5:
6489:
6479:
6478:
6473:
6468:
6463:
6449:
6448:
6443:
6438:
6433:
6428:
6421:
6420:External links
6418:
6415:
6414:
6392:Pope, Bernie.
6384:
6354:
6330:
6308:
6297:on 14 May 2015
6273:
6248:
6239:For reference
6230:For reference
6223:
6197:
6171:
6145:
6120:
6103:
6092:www.cantab.net
6078:
6077:
6075:
6072:
6071:
6070:
6065:
6060:
6055:
6050:
6045:
6040:
6034:
6029:
6022:
6019:
5971:
5941:
5882:
5868:
5865:
5862:
5861:
5859:
5857:
5855:
5847:
5845:
5833:
5827:
5826:
5815:
5813:
5811:
5809:
5800:
5791:
5785:
5784:
5782:
5780:
5778:
5776:
5758:
5726:
5720:
5719:
5708:
5706:
5704:
5702:
5675:
5648:
5642:
5641:
5634:
5632:
5630:
5619:
5617:
5602:
5596:
5595:
5592:
5520:
5508:(reduce-right
5505:
5490:
5461:
5432:
5425:
5424:
5413:
5411:
5399:
5387:
5355:
5323:
5317:
5316:
5297:
5295:
5289:.rev().reduce(
5283:
5271:
5255:
5239:
5233:
5232:
5214:
5212:
5185:
5161:
5126:
5093:
5087:
5086:
5084:
5082:
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5065:
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5030:
5028:
5015:
5002:
4985:
4968:
4962:
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4954:
4952:
4939:
4926:
4909:
4892:
4885:
4884:
4882:
4880:
4867:
4854:
4837:
4820:
4813:
4812:
4776:
4774:
4764:array_reverse(
4759:
4746:
4732:array_reverse(
4727:
4710:
4704:
4703:
4696:
4694:
4692:
4681:
4679:
4664:
4658:
4657:
4655:
4653:
4651:
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4647:
4634:
4628:
4627:
4625:
4623:
4621:
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4604:
4589:
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4582:
4580:
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4547:
4520:
4514:
4513:
4511:
4509:
4496:
4483:
4466:
4449:
4443:
4442:
4439:
4435:
4422:
4409:
4392:
4375:
4369:
4368:
4362:
4357:
4352:
4347:
4342:
4337:
4331:
4330:
4328:
4326:
4313:
4300:
4283:
4266:
4260:
4259:
4244:
4242:
4240:
4238:
4233:
4228:
4222:
4221:
4219:
4217:
4215:
4213:
4198:
4183:
4177:
4176:
4149:
4147:
4135:
4123:
4105:
4087:
4081:
4080:
4078:
4076:
4063:
4050:
4037:
4024:
4018:
4017:
4015:
4013:
4011:
3997:
3995:
3977:
3965:
3964:
3962:
3960:
3958:
3944:
3942:
3924:
3917:
3916:
3910:
3908:
3897:
3886:
3871:
3856:
3850:
3849:
3847:
3845:
3843:
3841:
3839:
3822:
3816:
3815:
3804:
3793:
3782:
3771:
3757:
3743:
3737:
3736:
3734:
3732:
3718:
3705:
3687:
3670:
3664:
3663:
3652:
3650:
3648:
3646:
3644:
3627:.reduce(init,
3610:
3604:
3603:
3598:
3585:
3583:
3558:
3541:
3528:iterator.fold(
3509:
3503:
3502:
3500:
3489:
3478:
3457:
3446:List.foldBack
3443:
3416:
3410:
3409:
3407:
3405:
3403:
3401:
3384:
3367:
3361:
3360:
3355:
3353:
3351:
3349:
3332:
3315:
3309:
3308:
3301:
3299:
3297:
3295:
3278:
3261:
3255:
3254:
3248:
3246:
3232:
3219:
3201:
3184:
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3177:
3175:
3173:
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3149:
3133:
3117:
3111:
3110:
3104:
3102:
3089:
3077:
3061:
3046:
3040:
3039:
3019:
3017:
3015:
3003:
3001:
2985:
2979:
2978:
2957:
2955:
2942:
2929:
2912:
2895:
2889:
2888:
2833:
2831:
2819:
2807:
2788:
2769:
2763:
2762:
2717:
2715:
2713:
2711:
2689:
2667:
2661:
2660:
2639:
2637:
2623:
2610:
2592:
2575:
2566:
2565:
2563:
2561:
2550:
2539:
2524:
2509:
2503:
2502:
2499:
2496:
2493:
2490:
2487:
2484:
2477:
2474:
2387:
2299:
2257:
2249:set difference
2095:
1884:
1874:
1871:
1447:tail recursion
1369:
1366:
1096:
1093:
1090:
848:
842:
839:
725:
610:
596:
593:
591:
590:Implementation
588:
542:
539:
459:
456:
378:
375:
323:
274:fold of a list
188:
187:
138:
136:
129:
123:
120:
84:data structure
26:
9:
6:
4:
3:
2:
6488:
6477:
6474:
6472:
6469:
6467:
6464:
6462:
6459:
6458:
6456:
6447:
6444:
6442:
6439:
6437:
6436:"Fold in Tcl"
6434:
6432:
6429:
6427:
6424:
6423:
6402:
6395:
6388:
6372:
6365:
6358:
6344:
6340:
6334:
6319:
6312:
6296:
6292:
6288:
6284:
6277:
6262:
6258:
6252:
6227:
6211:
6207:
6201:
6185:
6181:
6175:
6159:
6155:
6149:
6134:
6130:
6124:
6117:
6113:
6107:
6093:
6089:
6083:
6079:
6069:
6066:
6064:
6061:
6059:
6056:
6054:
6051:
6049:
6046:
6044:
6041:
6038:
6035:
6033:
6030:
6028:
6025:
6024:
5969:
5967:
5963:
5962:lazy language
5939:
5937:
5880:
5878:
5874:
5860:
5858:
5856:
5852:
5848:
5846:
5842:
5838:
5834:
5832:
5829:
5828:
5824:
5820:
5816:
5814:
5812:
5810:
5801:
5792:
5790:
5787:
5786:
5783:
5781:
5779:
5777:
5773:
5769:
5763:
5759:
5755:
5751:
5747:
5739:
5735:
5731:
5727:
5725:
5722:
5721:
5709:
5707:
5705:
5703:
5700:
5697:
5694:
5687:
5684:
5681:
5676:
5673:
5670:
5667:
5660:
5657:
5654:
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5647:
5644:
5643:
5635:
5633:
5631:
5628:
5624:
5620:
5618:
5615:
5611:
5607:
5603:
5601:
5598:
5597:
5593:
5589:
5586:
5583:
5580:
5572:
5569:
5566:
5563:
5556:
5554:
5551:
5548:
5545:
5542:unfold-right
5537:
5535:
5532:
5529:
5526:
5521:
5517:
5514:
5511:
5506:
5502:
5499:
5496:
5493:(reduce-left
5491:
5487:
5484:
5481:
5473:
5470:
5467:
5462:
5458:
5455:
5452:
5449:(vector-fold
5444:
5441:
5438:
5433:
5430:
5427:
5426:
5423:
5414:
5412:
5408:
5405:.reduceRight(
5404:
5400:
5396:
5392:
5388:
5384:
5380:
5376:
5368:
5364:
5360:
5356:
5352:
5348:
5344:
5336:
5332:
5328:
5324:
5322:
5319:
5318:
5309:
5302:
5298:
5296:
5292:
5288:
5284:
5280:
5276:
5272:
5268:
5264:
5260:
5256:
5252:
5248:
5244:
5240:
5238:
5235:
5234:
5229:
5221:
5215:
5213:
5209:
5204:.reverse_each
5203:
5196:
5191:.reverse_each
5190:
5186:
5182:
5178:
5171:
5166:
5162:
5158:
5154:
5149:.reverse_each
5148:
5141:
5137:
5132:.reverse_each
5131:
5127:
5123:
5119:
5114:
5107:
5103:
5098:
5094:
5092:
5089:
5088:
5085:
5083:
5081:
5079:
5075:
5071:
5066:
5062:
5058:
5053:
5051:
5048:
5047:
5031:
5029:
5026:, right=TRUE)
5025:
5021:
5016:
5012:
5008:
5003:
5000:, right=TRUE)
4999:
4995:
4991:
4986:
4982:
4978:
4974:
4969:
4967:
4964:
4963:
4959:
4955:
4953:
4949:
4945:
4940:
4936:
4932:
4927:
4923:
4919:
4915:
4910:
4906:
4902:
4898:
4893:
4890:
4887:
4886:
4883:
4881:
4877:
4873:
4868:
4864:
4860:
4855:
4851:
4847:
4843:
4838:
4834:
4830:
4826:
4821:
4818:
4815:
4814:
4810:
4806:
4802:
4799:
4794:
4788:
4782:
4777:
4775:
4771:
4767:
4762:array_reduce(
4760:
4756:
4752:
4749:array_reduce(
4747:
4743:
4739:
4735:
4730:array_reduce(
4728:
4724:
4720:
4716:
4713:array_reduce(
4711:
4709:
4706:
4705:
4697:
4695:
4693:
4690:
4687:
4682:
4680:
4677:
4673:
4670:
4665:
4663:
4660:
4659:
4656:
4654:
4652:
4650:
4648:
4644:
4640:
4635:
4633:
4630:
4629:
4626:
4624:
4622:
4620:
4616:
4613:
4610:
4605:
4601:
4598:
4595:
4590:
4588:
4585:
4584:
4581:
4579:
4577:
4575:
4572:
4569:
4566:
4559:
4556:
4553:
4548:
4545:
4542:
4539:
4532:
4529:
4526:
4521:
4519:
4516:
4515:
4512:
4510:
4506:
4502:
4497:
4493:
4489:
4484:
4480:
4476:
4472:
4467:
4463:
4459:
4455:
4450:
4448:
4445:
4444:
4440:
4436:
4432:
4428:
4423:
4419:
4415:
4410:
4406:
4402:
4398:
4393:
4389:
4385:
4381:
4376:
4374:
4371:
4370:
4363:
4360:NestWhileList
4358:
4353:
4348:
4343:
4338:
4336:
4333:
4332:
4329:
4327:
4323:
4319:
4314:
4310:
4306:
4301:
4297:
4293:
4289:
4284:
4280:
4276:
4272:
4267:
4265:
4262:
4261:
4245:
4243:
4241:
4239:
4234:
4229:
4227:
4224:
4223:
4220:
4218:
4216:
4214:
4210:
4207:
4204:
4201:(lists:foldr
4199:
4195:
4192:
4189:
4186:(lists:foldl
4184:
4182:
4179:
4178:
4150:
4148:
4144:
4141:.reduceRight(
4140:
4136:
4132:
4128:
4124:
4120:
4116:
4110:
4106:
4102:
4098:
4092:
4088:
4086:
4083:
4082:
4079:
4077:
4073:
4069:
4064:
4060:
4056:
4051:
4047:
4043:
4038:
4034:
4030:
4025:
4023:
4020:
4019:
4016:
4014:
4012:
4008:
4002:
3998:
3996:
3992:
3988:
3982:
3978:
3975:
3970:
3967:
3966:
3963:
3961:
3959:
3955:
3949:
3945:
3943:
3939:
3935:
3929:
3925:
3922:
3919:
3918:
3915:
3911:
3909:
3906:
3902:
3898:
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3891:
3887:
3884:
3880:
3876:
3872:
3869:
3865:
3861:
3857:
3855:
3852:
3851:
3848:
3846:
3844:
3842:
3840:
3836:
3832:
3828:
3823:
3821:
3818:
3817:
3805:
3802:
3799:
3794:
3791:
3788:
3783:
3780:
3777:
3772:
3769:
3766:
3763:
3758:
3755:
3752:
3749:
3744:
3742:
3739:
3738:
3735:
3733:
3729:
3723:
3719:
3715:
3710:
3706:
3702:
3698:
3692:
3688:
3684:
3680:
3675:
3671:
3669:
3666:
3665:
3657:
3653:
3651:
3649:
3647:
3645:
3641:
3637:
3630:
3626:
3619:
3615:
3611:
3609:
3606:
3605:
3602:
3599:
3595:
3591:
3586:
3584:
3580:
3576:
3568:
3564:
3559:
3555:
3551:
3547:
3542:
3539:
3535:
3531:
3523:
3519:
3515:
3510:
3508:
3505:
3504:
3501:
3498:
3495:
3490:
3487:
3484:
3479:
3476:
3473:
3466:
3463:
3458:
3455:
3452:
3449:
3444:
3441:
3438:
3435:
3428:
3425:
3422:
3417:
3415:
3412:
3411:
3408:
3406:
3404:
3402:
3398:
3394:
3390:
3385:
3381:
3377:
3373:
3368:
3366:
3363:
3362:
3359:
3356:
3354:
3352:
3350:
3346:
3342:
3338:
3333:
3329:
3325:
3321:
3316:
3314:
3311:
3310:
3306:
3305:documentation
3302:
3300:
3298:
3296:
3292:
3288:
3284:
3279:
3275:
3271:
3267:
3262:
3260:
3257:
3256:
3252:std.algorithm
3249:
3247:
3243:
3238:
3233:
3229:
3225:
3220:
3215:
3211:
3207:
3202:
3198:
3194:
3190:
3185:
3183:
3180:
3179:
3176:
3174:
3170:
3167:
3162:
3158:
3155:
3150:
3146:
3142:
3139:
3134:
3130:
3126:
3123:
3118:
3116:
3113:
3112:
3109:
3105:
3103:
3099:
3095:
3090:
3086:
3083:
3078:
3074:
3070:
3067:
3062:
3058:
3055:
3052:
3047:
3045:
3042:
3041:
3037:
3031:
3025:
3020:
3018:
3016:
3012:
3008:
3004:
3002:
2998:
2994:
2990:
2986:
2984:
2981:
2980:
2975:
2969:std::optional
2958:
2956:
2952:
2948:
2943:
2939:
2935:
2930:
2926:
2922:
2918:
2913:
2909:
2905:
2901:
2896:
2894:
2891:
2890:
2885:
2849:
2843:
2838:
2834:
2832:
2828:
2825:
2820:
2816:
2813:
2808:
2804:
2801:
2797:
2794:
2789:
2785:
2782:
2778:
2775:
2770:
2768:
2765:
2764:
2761:
2757:
2752:
2747:are iterators
2745:
2739:
2733:
2727:
2718:
2716:
2714:
2712:
2708:
2704:
2700:
2696:
2690:
2686:
2682:
2678:
2674:
2668:
2666:
2663:
2662:
2652:
2647:
2640:
2638:
2634:
2628:
2624:
2620:
2615:
2611:
2607:
2603:
2597:
2593:
2589:
2585:
2580:
2576:
2574:
2571:
2568:
2567:
2564:
2562:
2559:
2555:
2551:
2548:
2544:
2540:
2537:
2533:
2529:
2525:
2522:
2518:
2514:
2510:
2508:
2505:
2504:
2481:
2385:
2374:
2368:
2297:
2291:
2255:
2252:
2250:
2246:
2240:
2236:
2093:
2091:
2087:
2083:
2060:"0"
2012:"0"
1964:"0"
1916:"0"
1882:
1880:
1870:
1857:
1775:; similarly,
1714:
1586:
1542:
1495:
1487:
1479:
1476:i.e., in its
1475:
1459:
1454:
1452:
1451:infinite loop
1448:
1395:
1391:
1387:
1379:
1375:
1089:
1087:
1083:
1075:
846:
838:
723:
608:
587:
574:
556:
552:
548:
538:
536:
532:
528:
524:
523:node-oriented
519:
517:
513:
509:
501:
493:
488:
486:
482:
477:
473:
465:
455:
449:
433:
427:
417:
413:
409:
405:
401:
397:
392:
374:
372:
367:
362:
360:
321:
311:
299:
291:
287:
283:
275:
273:
269:
260:
256:
250:
246:
244:
240:
236:
232:
228:
212:
208:
207:
199:
195:
184:
181:
173:
163:
159:
155:
149:
148:
144:
139:This section
137:
133:
128:
127:
119:
118:of unfolds).
117:
113:
109:
104:
103:corecursively
100:
96:
91:
89:
85:
81:
77:
73:
69:
65:
61:
57:
53:
49:
45:
41:
38:(also termed
37:
33:
19:
6405:. Retrieved
6400:
6387:
6375:. Retrieved
6370:
6357:
6346:. Retrieved
6342:
6333:
6321:. Retrieved
6311:
6299:. Retrieved
6295:the original
6276:
6265:. Retrieved
6260:
6251:
6226:
6214:. Retrieved
6209:
6200:
6188:. Retrieved
6183:
6174:
6162:. Retrieved
6157:
6148:
6137:. Retrieved
6135:. 2023-12-11
6132:
6123:
6106:
6095:. Retrieved
6091:
6082:
6043:Homomorphism
6037:Catamorphism
5959:
5935:
5933:
5876:
5870:
5867:Universality
5850:
5840:
5836:
5822:
5818:
5771:
5767:
5761:
5753:
5749:
5745:
5737:
5733:
5729:
5698:
5695:
5692:
5691:Array.foldr
5685:
5682:
5679:
5671:
5668:
5665:
5664:Array.foldl
5658:
5655:
5652:
5626:
5622:
5613:
5609:
5605:
5587:
5585:initial-seed
5584:
5581:
5578:
5570:
5568:initial-seed
5567:
5564:
5561:
5555:
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363:
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6263:. Rust Team
6212:. Jetbrains
6186:. Jetbrains
6160:. Jetbrains
5960:Also, in a
5873:polymorphic
5646:Standard ML
5623:aCollection
5606:aCollection
5435:(fold-left
5361:.foldRight(
4335:Mathematica
3638:.partition(
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3341:Accumulator
3335:List.foldr(
3324:Accumulator
3318:List.foldl(
3281:List.foldr(
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3115:Common Lisp
2631:.Aggregate(
2617:.Aggregate(
2600:.Aggregate(
2582:.Aggregate(
2489:Right fold
1715:technique,
1494:on the left
514: is
500:binary tree
116:anamorphism
72:recursively
6455:Categories
6348:2018-04-10
6301:14 October
6267:2021-06-22
6139:2024-01-16
6097:2023-04-04
6074:References
6053:Prefix sum
5871:Fold is a
5764:.reverse()
5513:defaultval
5498:defaultval
5329:.foldLeft(
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5195:&block
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4956:In module
4700:List::Util
4405:defaultVal
4388:defaultVal
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3969:JavaScript
3724:.reverse()
3693:.reverse()
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3512:list.fold(
3419:List.fold
3250:in module
2719:in header
2629:.Reverse()
2598:.Reverse()
2486:Left fold
2376:Functions
2290:merge sort
1458:expression
1378:non-strict
1082:not always
537:of nodes.
516:non-strict
400:right fold
278:foldr (:)
44:accumulate
6466:Recursion
6403:(6): 5–16
6377:March 26,
6287:Newsgroup
6012:undefined
6003:undefined
5819:sequences
5600:Smalltalk
5377: :\
5305:requires
4958:functools
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4169:onSuccess
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3244:.reverse)
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3106:See also
3096:(reverse
3071:(reverse
2754:can be a
2642:Aggregate
2483:Language
2301:mergesort
2239:set union
2082:recursive
1866:Data.List
1492:recurses
1484:recurses
496:a → a → a
468:a → b → b
404:left fold
239:replacing
231:replacing
200: (
170:June 2013
141:does not
114:, versus
88:hierarchy
68:recursive
48:aggregate
6216:29 March
6190:29 March
6164:29 March
6021:See also
5851:iterable
5837:iterable
5766:.reduce(
5746:sequence
5732:.reduce(
5638:#reduce:
5625:reduce:
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5523:(unfold
5311:to be a
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4801:Archived
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4499:rreduce(
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4322:sequence
4309:sequence
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4091:Iterable
3827:iterable
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3712:.inject(
3695:.inject(
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3660:Iterable
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3440:sequence
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3009:.reduce(
2991:.reduce(
2872:<<
2866:<<
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1877:Using a
1873:Examples
1838:(
1088:below).
408:operator
377:On lists
209:
108:terminal
78:, a top
76:function
52:compress
6323:24 June
6289::
6118:, p. 42
5853:.reduce
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5768:initval
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2498:Unfold
2331:nubsort
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2090:Haskell
1879:Haskell
1086:example
412:Haskell
227:Haskell
213:(
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147:sources
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6184:Kotlin
6158:Kotlin
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5429:Scheme
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4889:Python
4817:Python
4807:. Try
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2247:their
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2006:concat
1958:concat
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1394:second
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571:foldl1
567:foldr1
535:groups
476:linear
446:. The
416:Prolog
414:'s or
320:, as:
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4264:Maple
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4022:Julia
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3507:Gleam
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2900:range
2893:C++23
2810:(...
2767:C++17
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