I am trying to learn functional programming and Scala, so I'm reading the "Functional Programming in Scala" by Chiusano and Bjarnason. I' m having trouble understanding what fold left and fold right methods do in case of a list. I've looked around here but I haven't find something beginner friendly. So the code provided by the book is:
def foldRight[A,B](as: List[A], z: B)(f: (A, B) => B): B = as match {
case Nil => z
case Cons(h, t) => f(h, foldRight(t, z)(f))
}
def foldLeft[A,B](l: List[A], z: B)(f: (B, A) => B): B = l match {
case Nil => z
case Cons(h,t) => foldLeft(t, f(z,h))(f)
}
Where Cons and Nil are:
case class Cons[+A](head: A, tail: List[A]) extends List[A]
case object Nil extends List[Nothing]
So what do actually fold left and right do? Why are needed as "utility" methods? There are many other methods that use them and I have trouble to understand them as well, since I don't get those two.
According to my experience, one of the best ways to workout the intuition is to see how it works on the very simple examples:
List(1, 3, 8).foldLeft(100)(_ - _) == ((100 - 1) - 3) - 8 == 88
List(1, 3, 8).foldRight(100)(_ - _) == 1 - (3 - (8 - 100)) == -94
As you can see, foldLeft/Right just passes the element of the list and the result of the previous application to the the operation in second parentheses.
It should be also mentioned that if you apply these methods to the same list, they will return equal results only if the applied operation is associative.
Say you have a list of numbers, and you want to add them all up. How would you do that?
You add the first and the second, then take the result of that, add that to the third, take the result of that, add it to the fourth.. and so on.
That's what fold let's you do.
List(1,2,3,4,5).foldLeft(0)(_ + _)
The "+" is the function you want to apply, with the first operand being the result of its application to the elements so far, and the second operand being the next element.
As you don't have a "result so far" for the first application, you provide a start value - in this case 0, as it is the identity element for addition.
Say you want to multiply all of your list elements, with fold, that'd be
List(1,2,3,4,5).foldLeft(1)(_ * _)
Fold has it's own Wikipedia page you might want to check.
Of course there are also ScalaDoc entries for foldLeft and foldRight.
Another way of visualisation of leftFold and rightFold in Scala is through string concatenation, its clearly show how leftFold and rightFold worked, let's see the below example:
val listString = List("a", "b", "c") // : List[String] = List(a,b,c)
val leftFoldValue = listString.foldLeft("z")((el, acc) => el + acc) // : String = zabc
val rightFoldValue = listString.foldRight("z")((el, acc) => el + acc) // : abcz
OR in shorthand ways
val leftFoldValue = listString.foldLeft("z")(_ + _) // : String = zabc
val rightFoldValue = listString.foldRight("z")(_ + _) // : String = abcz
Explanation:
leftFold is worked as ( ( ('z' + 'a') + 'b') + 'c') = ( ('za' + 'b') + 'c') = ('zab' + 'c') = 'zabc'
and rightFold as ('a' + ('b' + ('c' + 'z'))) = ('a' + ('b' + 'cz')) = ('a' + 'bcz') = 'abcz'
Related
Write a function to divide the input list into three sublists.
The first sub-list is to include all the elements whose indexes satisfy the equation i mod 3 = 1.
The second sub-list is to include all the elements whose indexes satisfy the equation and mod 3 = 2.
The third sub-list is to contain the remaining elements.
The order of the elements must be maintained. Return the result as three lists.
Write a function using tail and non-tail recursion.
My attempt: I’m very confused in how to increase index so it can go through the list, any recommendation about how to make it recursive with increasing index each time?
def divide(list: List[Int]): (List[Int], List[Int], List[Int]) = {
var index:Int =0
def splitList(remaining: List[Int], firstSubList: List[Int], secondSubList: List[Int], thirdSubList: List[Int], index:Int): (List[Int], List[Int], List[Int]) = {
if(remaining.isEmpty) {
return (List[Int](), List[Int](), List[Int]())
}
val splitted = splitList(remaining.tail, firstSubList, secondSubList, thirdSubList, index)
val firstList = if (index % 3 == 1) List() ::: splitted._1 else splitted._1
val secondList = if (index % 3 == 2) List() ::: splitted._2 else splitted._2
val thirdList = if((index% 3 != 1) && (index % 3 != 2)) List() ::: splitted._3 else splitted._3
index +1
(firstSubList ::: firstList, secondSubList ::: secondList, thirdSubList ::: thirdList)
}
splitList(list, List(), List(), List(), index+1)
}
println(divide(List(0,11,22,33)))
Generalizing the requirement a little, here's one approach using a simple recursive function to compose a Map of Lists by modulo n of the original list indexes:
def splitList[T](list: List[T], n: Int): Map[Int, List[T]] = {
#scala.annotation.tailrec
def loop(zls: List[(T, Int)], lsMap: Map[Int, List[T]]): Map[Int, List[T]] =
zls match {
case Nil =>
lsMap.map{ case (i, ls) => (i, ls.reverse) }
case (x, i) :: rest =>
val j = i % n
loop(rest, lsMap + (j -> (x :: lsMap.getOrElse(j, Nil))))
}
loop(list.zipWithIndex, Map.empty[Int, List[T]])
}
splitList(List(0, 11, 22, 33, 44, 55, 66), 3)
// Map(0 -> List(0, 33, 66), 1 -> List(11, 44), 2 -> List(22, 55))
splitList(List('a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i'), 4)
// Map(0 -> List(a, e, i), 1 -> List(b, f), 2 -> List(c, g), 3 -> List(d, h))
To do this in real life, just label each value with its index and then group by that index modulo 3:
def divide[T](list: List[T]) = {
val g = list.zipWithIndex.groupMap(_._2 % 3)(_._1)
(g.getOrElse(1, Nil), g.getOrElse(2, Nil), g.getOrElse(0, Nil))
}
If you insist on a recursive version, it might look like this:
def divide[T](list: List[T]) = {
def loop(rem: List[T]): (List[T], List[T], List[T]) =
rem match {
case a::b::c::tail =>
val rem = loop(tail)
(b +: rem._1, c +: rem._2, a +: rem._3)
case a::b::Nil =>
(List(b), Nil, List(a))
case a::Nil =>
(Nil, Nil, List(a))
case Nil =>
(Nil, Nil, Nil)
}
loop(list)
}
Tail recursion would look like this:
def divide[T](list: List[T]) = {
#annotation.tailrec
def loop(rem: List[T], res: (List[T], List[T], List[T])): (List[T], List[T], List[T]) =
rem match {
case a::b::c::tail =>
loop(tail, (res._1 :+ b, res._2 :+ c, res._3 :+ a))
case a::b::Nil =>
(res._1 :+ b, res._2, res._3 :+ a)
case a::Nil =>
(res._1, res._2, res._3 :+ a)
case Nil =>
res
}
loop(list, (Nil, Nil, Nil))
}
And if you care about efficiency, this version would build the lists in the other order and reverse them when returning the result.
Your problem is that you put index+1 into a wrong place. Try swapping it around: put index+1 into the call where you have index, and index into the other one. Also remove the "standalone" index+1 statement in the middle, it doesn't do anything anyway.
That should make your code work ... but it is still not very good. A couple of problems with it (besides it being badly structured, non-idiomatic, and hard to read, which is kinda subjective):
It it is not tail-recursive, and effectively, creates another copy of the entire list on stack. This may be problematic when the list is long.
It concatenates (potentially long) lists. This is a bad idea. List in scala is a singly linked list, you have it's head readily available, but to get to the end, you need to spend O(N) cycles, iterating through each node. Thus things like foo:::bar in a iterative function instantly make any algorithm (at least) quadratic.
The usual "trick" to avoid the last problem is prepending elements to output one-by-one, and then reversing the result in the end. The first one can be avoided with tail-recursion. The "non-idiomatic" and "hard to read" problems are mostly addressed by using match statement in this case:
def split3(
in: List[Int],
one: List[Int],
two: List[Int],
three: List[Int],
index: Int = 0
} = (in, index % 3) match {
case (Nil, _) => (one.reverse, two.reverse, three.reverse)
case (head::tail, 1) => split3(tail, head::one, two, three, index+1)
case (head::tail, 2) => split3(tail, one, head::two, three, index+1)
case (head::tail, _) => split3(tail, one, two, head::three, index+1)
}
Now, this is a fine solution, albeit a little repetitive to my demanding eye ... But if want to be clever and really unleash the full power of scala standard library, forget recursion, you don't really need it in this case.
If you knew that number of elements in the list was always divisible by 3,
you could just do list.grouped(3).toSeq.transpose: break the list into groups of three (each group will have index%3=0 as first element, index%3=1 as second, index%3=2 as the third), and then transpose will turn a list of lists of 3 into a list of three lists where the first one contains all the first elements, the second - all the seconds etc. (I know, you wanted them in a different order, but that's trivial). If you are having trouble understanding what I am talking about, just try running it on some lists, and look at the results.
This would be a really elegant solution ... if it worked :/ The problem is, that it only does when you have 3*n elements in the original list. If not, transpose will fail on the last element if it doesn't have 3 elements like all others. Can we fix it? Well ... that's where the cleverness comes in.
val (triplets, tails) = list.grouped(3).toSeq.partition(_.size == 3)
triplets
.transpose
.padTo(3, Nil)
.zip(tails.flatten.map(Seq(_)).padTo(3, Nil))
.map { case (head, tail) => head ++ tail }
Basically, it is doing the same thing as the one-liner I described above (break into groups of 3 and transpose), but adds special handling for the case when the last group has less than three elements - it splits it out and pads with required number of empty lists, then just appends the result to transposed triplets.
This question already has answers here:
Scala - Combine two lists in an alternating fashion
(4 answers)
Closed 3 years ago.
The elements of the resulting list should alternate between the elements of the arguments. Assume that the two arguments have the same length.
USE RECURSION
My code as follows
val finalString = new ListBuffer[Int]
val buff2= new ListBuffer[Int]
def alternate(xs:List[Int], ys:List[Int]):List[Int] = {
while (xs.nonEmpty) {
finalString += xs.head + ys.head
alternate(xs.tail,ys.tail)
}
return finalString.toList
}
EXPECTED RESULT:
alternate ( List (1 , 3, 5) , List (2 , 4, 6)) = List (1 , 2, 3, 4, 6)
As far for the output, I don't get any output. The program is still running and cannot be executed.
Are there any Scala experts?
There are a few problems with the recursive solutions suggested so far (including yours, which would actually work, if you replace while with if): appending to end of list is a linear operation, making the whole thing quadratic (taking a .length of a list too, as well ас accessing elements by index), don't do that; also, if the lists are long, a recursion may require a lot of space on the stack, you should be using tail-recursion whenever possible.
Here is a solution that is free of both those problems: it builds the output backwards, by prepending elements to the list (constant time operation) rather than appending them, and reverses the result at the end. It is also tail-recursive: the recursive call is the last operation in the function, which allows the compiler to convert it into a loop, so that it will only use a single stack frame for execution regardless of the size of the lists.
#tailrec
def alternate(
a: List[Int],
b: List[Int],
result: List[Int] = Nil
): List[Int] = (a,b) match {
case (Nil, _) | (_, Nil) => result.reversed
case (ah :: at, bh :: bt) => alternate(at, bt, bh :: ah :: result)
}
(if the lists are of different lengths, the whole thing stops when the shortest one ends, and whatever is left in the longer one is thrown out. You may want to modify the first case (split it into two, perhaps) if you desire a different behavior).
BTW, your own solution is actually better than most suggested here: it is actually tail recursive (or rather can be made one if you add else after your if, which is now while), and appending to ListBuffer isn't actually as bad as to a List. But using mutable state is generally considered "code smell" in scala, and should be avoided (that's one of the main ideas behind using recursion instead of loops in the first place).
Condition xs.nonEmpty is true always so you have infinite while loop.
Maybe you meant if instead of while.
A more Scala-ish approach would be something like:
def alternate(xs: List[Int], ys: List[Int]): List[Int] = {
xs.zip(ys).flatMap{case (x, y) => List(x, y)}
}
alternate(List(1,3,5), List(2,4,6))
// List(1, 2, 3, 4, 5, 6)
A recursive solution using match
def alternate[T](a: List[T], b: List[T]): List[T] =
(a, b) match {
case (h1::t1, h2::t2) =>
h1 +: h2 +: alternate(t1, t2)
case _ =>
a ++ b
}
This could be more efficient at the cost of clarity.
Update
This is the more efficient solution:
def alternate[T](a: List[T], b: List[T]): List[T] = {
#annotation.tailrec
def loop(a: List[T], b: List[T], res: List[T]): List[T] =
(a, b) match {
case (h1 :: t1, h2 :: t2) =>
loop(t1, t2, h2 +: h1 +: res)
case _ =>
a ++ b ++ res.reverse
}
loop(a, b, Nil)
}
This retains the original function signature but uses an inner function that is an efficient, tail-recursive implementation of the algorithm.
You're accessing variables from outside the method, which is bad. I would suggest something like the following:
object Main extends App {
val l1 = List(1, 3, 5)
val l2 = List(2, 4, 6)
def alternate[A](l1: List[A], l2: List[A]): List[A] = {
if (l1.isEmpty || l2.isEmpty) List()
else List(l1.head,l2.head) ++ alternate(l1.tail, l2.tail)
}
println(alternate(l1, l2))
}
Still recursive but without accessing state from outside the method.
Assuming both lists are of the same length, you can use a ListBuffer to build up the alternating list. alternate is a pure function:
import scala.collection.mutable.ListBuffer
object Alternate extends App {
def alternate[T](xs: List[T], ys: List[T]): List[T] = {
val buffer = new ListBuffer[T]
for ((x, y) <- xs.zip(ys)) {
buffer += x
buffer += y
}
buffer.toList
}
alternate(List(1, 3, 5), List(2, 4, 6)).foreach(println)
}
Suppose that I use a sequence of various maps and/or flatMaps to generate a sequence of collections. Is it possible to access information about the "current" collection from within any of those methods? For example, without knowing anything specific about the functions used in the previous maps or flatMaps, and without using any intermediate declarations, how can I get the maximum value (or length, or first element, etc.) of the collection upon which the last map acts?
List(1, 2, 3)
.flatMap(x => f(x) /* some unknown function */)
.map(x => x + ??? /* what is the max element of the collection? */)
Edit for clarification:
In the example, I'm not looking for the max (or whatever) of the initial List. I'm looking for the max of the collection after the flatMap has been applied.
By "without using any intermediate declarations" I mean that I do not want to use any temporary collections en route to the final result. So, the example by Steve Waldman below, while giving the desired result, is not what I am seeking. (I include this condition is mostly for aesthetic reasons.)
Edit for clarification, part 2:
The ideal solution would be some magic keyword or syntactic sugar that lets me reference the current collection:
List(1, 2, 3)
.flatMap(x => f(x))
.map(x => x + theCurrentList.max)
I'm prepared to accept the fact, however, that this simply is not possible.
Maybe just define the list as a val, so you can name it? I don't know of any facility built into map(...) or flatMap(...) that would help.
val myList = List(1, 2, 3)
myList
.flatMap(x => f(x) /* some unknown function */)
.map(x => x + myList.max /* what is the max element of the List? */)
Update: By this approach at least, if you have multiple transformations and want to see the transformed version, you'd have to name that. You could get away with
val myList = List(1, 2, 3).flatMap(x => f(x) /* some unknown function */)
myList.map(x => x + myList.max /* what is the max element of the List? */)
Or, if there will be multiple transformations, get in the habit of naming the stages.
val rawList = List(1, 2, 3)
val smordified = rawList.flatMap(x => f(x) /* some unknown function */)
val maxified = smordified.map(x => x + smordified.max /* what is the max element of the List? */)
maxified
Update 2: Watch it work in the REPL even with heterogenous types:
scala> def f( x : Int ) : Vector[Double] = Vector(x * math.random, x * math.random )
f: (x: Int)Vector[Double]
scala> val rawList = List(1, 2, 3)
rawList: List[Int] = List(1, 2, 3)
scala> val smordified = rawList.flatMap(x => f(x) /* some unknown function */)
smordified: List[Double] = List(0.40730853571901315, 0.15151641399798665, 1.5305929709857609, 0.35211231420067435, 0.644241939254793, 0.15530230501048903)
scala> val maxified = smordified.map(x => x + smordified.max /* what is the max element of the List? */)
maxified: List[Double] = List(1.937901506704774, 1.6821093849837476, 3.0611859419715217, 1.8827052851864352, 2.1748349102405538, 1.6858952759962498)
scala> maxified
res3: List[Double] = List(1.937901506704774, 1.6821093849837476, 3.0611859419715217, 1.8827052851864352, 2.1748349102405538, 1.6858952759962498)
It is possible, but not pretty, and not likely something you want if you are doing it for "aesthetic reasons."
import scala.math.max
def f(x: Int): Seq[Int] = ???
List(1, 2, 3).
flatMap(x => f(x) /* some unknown function */).
foldRight((List[Int](),List[Int]())) {
case (x, (xs, Nil)) => ((x :: xs), List.fill(xs.size + 1)(x))
case (x, (xs, xMax :: _)) => ((x :: xs), List.fill(xs.size + 1)(max(x, xMax)))
}.
zipped.
map {
case (x, xMax) => x + xMax
}
// Or alternately, a slightly more efficient version using Streams.
List(1, 2, 3).
flatMap(x => f(x) /* some unknown function */).
foldRight((List[Int](),Stream[Int]())) {
case (x, (xs, Stream())) =>
((x :: xs), Stream.continually(x))
case (x, (xs, curXMax #:: _)) =>
val newXMax = max(x, curXMax)
((x :: xs), Stream.continually(newXMax))
}.
zipped.
map {
case (x, xMax) => x + xMax
}
Seriously though, I just took this on to see if I could do it. While the code didn't turn out as bad as I expected, I still don't think it's particularly readable. I'd discourage using this over something similar to Steve Waldman's answer. Sometimes, it's simply better to just introduce a val, rather than being dogmatic about it.
You could define a mapWithSelf (resp. flatMapWithSelf) operation along these lines and add it as an implicit enrichment to the collection. For List it might look like:
// Scala 2.13 APIs
object Enrichments {
implicit class WithSelfOps[A](val lst: List[A]) extends AnyVal {
def mapWithSelf[B](f: (A, List[A]) => B): List[B] =
lst.map(f(_, lst))
def flatMapWithSelf[B](f: (A, List[A]) => IterableOnce[B]): List[B] =
lst.flatMap(f(_, lst))
}
}
The enrichment basically fixes the value of the collection before the operation and threads it through. It should be possible to generify this (at least for the strict collections), though it would look a little different in 2.12 vs. 2.13+.
Usage would look like
import Enrichments._
val someF: Int => IterableOnce[Int] = ???
List(1, 2, 3)
.flatMap(someF)
.mapWithSelf { (x, lst) =>
x + lst.max
}
So at the usage site, it's aesthetically pleasant. Note that if you're computing something which traverses the list, you'll be traversing the list every time (leading to a quadratic runtime). You can get around that with some mutability or by just saving the intermediate list after the flatMap.
One somewhat-simple way of referencing prior output within the current map/collect operation is to use a named reference outside the map, then reference it from within the map block:
var prevOutput = ... // starting value of whatever is referenced within the map
myValues.map {
prevOutput = ... // expression that references prior `prevOutput`
prevOutput // return above computed value for the map to collect
}
This draws attention to the fact that we're referencing prior elements while building the new sequence.
This would be more messy, though, if you wanted to reference arbitrarily previous values, not just the previous one.
Is there a way to manipulate multiple values of a tuple without using a temporary variable and starting a new statement?
Say I have a method that returns a tuple and I want to do something with those values inline.
e.g. if I want to split a string at a certain point and reverse the pieces
def backToFront(s: String, n:Int) = s.splitAt(n)...
I can do
val (a, b) = s.splitAt(n)
b + a
(requires temporary variables and new statement) or
List(s.splitAt(n)).map(i => i._2 + i._1).head
(works, but seems a bit dirty, creating a single element List just for this) or
s.splitAt(n).swap.productIterator.mkString
(works for this particular example, but only because there happens to be a swap method that does what I want, so it's not very general).
The zipped method on tuples seems just to be for tuples of Lists.
As another example, how could you turn the tuple ('a, 'b, 'c) into ('b, 'a, 'c) in one statement?
Tuples are just convenient return type, and it is not assumed that you will make complicated manipulations with it. Also there was similar discussion on scala forums.
About the last example, couldn't find anything better than pattern-matching.
('a, 'b, 'c) match { case (a, b, c) => (b, a ,c) }
Unfortunately, the built-in methods on tuples are pretty limited.
Maybe you want something like these in your personal library,
def fold2[A, B, C](x: (A, B))(f: (A, B) => C): C = f(x._1, x._2)
def fold3[A, B, C, D](x: (A, B, C))(f: (A, B, C) => D): D = f(x._1, x._2, x._3)
With the appropriate implicit conversions, you could do,
scala> "hello world".splitAt(5).swap.fold(_ + _)
res1: java.lang.String = " worldhello"
scala> (1, 2, 3).fold((a, b, c) => (b, c, a))
res2: (Int, Int, Int) = (2,3,1)
An alternative to the last expression would be the "pipe" operator |> (get it from Scalaz or here),
scala> ('a, 'b, 'c) |> (t => (t._2, t._3, t._1))
res3: (Symbol, Symbol, Symbol) = ('b,'c,'a)
This would be nice, if not for the required annotations,
scala> ("hello ", "world") |> (((_: String) + (_: String)).tupled)
res4: java.lang.String = hello world
How about this?
s.splitAt(n) |> Function.tupled(_ + _)
[ Edit: Just noticed your arguments to function are reversed. In that case, you will have to give up placeholder syntax and instead go for: s.splitAt(n) |> Function.tupled((a, b) => b + a) ]
For your last example, can't think of anything better than a pattern match (as shown by #4e6.)
With the current development version of shapeless, you can achieve this without unpacking the tuple:
import shapeless.syntax.std.tuple._
val s = "abcdefgh"
val n = 3
s.splitAt(n).rotateRight[shapeless.Nat._1].mkString("", "", "") // "defghabc"
I think you shouldn't have to wait too long (matter of days I'd say) before the syntax of the methods of the last line get cleaned, and you can simply write
s.splitAt(n).rotateRight(1).mkString
I know that you can do matching on lists in a way like
val list = List(1,2,3)
list match {
case head::tail => head
case _ => //whatever
}
so I started to wonder how this works. If I understand correctly, :: is just an operator, so what's to stop me from doing something like
4 match {
case x + 2 => x //I would expect x=2 here
}
If there is a way to create this kind of functionality, how is it done; if not, then why?
Pattern matching takes the input and decomposes it with an unapply function. So in your case, unapply(4) would have to return the two numbers that sum to 4. However, there are many pairs that sum to 4, so the function wouldn't know what to do.
What you need is for the 2 to be accessible to the unapply function somehow. A special case class that stores the 2 would work for this:
case class Sum(addto: Int) {
def unapply(i: Int) = Some(i - addto)
}
val Sum2 = Sum(2)
val Sum2(x) = 5 // x = 3
(It would be nice to be able to do something like val Sum(2)(y) = 5 for compactness, but Scala doesn't allow parameterized extractors; see here.)
[EDIT: This is a little silly, but you could actually do the following too:
val `2 +` = Sum(2)
val `2 +`(y) = 5 // y = 3
]
EDIT: The reason the head::tail thing works is that there is exactly one way to split the head from the tail of a list.
There's nothing inherently special about :: versus +: you could use + if you had a predetermined idea of how you wanted it to break a number. For example, if you wanted + to mean "split in half", then you could do something like:
object + {
def unapply(i: Int) = Some(i-i/2, i/2)
}
and use it like:
scala> val a + b = 4
a: Int = 2
b: Int = 2
scala> val c + d = 5
c: Int = 3
d: Int = 2
EDIT: Finally, this explains that, when pattern matching, A op B means the same thing as op(A,B), which makes the syntax look nice.
Matching with case head :: tail uses an infix operation pattern of the form p1 op p2 which gets translated to op(p1, p2) before doing the actual matching. (See API for ::)
The problem with + is the following:
While it is easy to add an
object + {
def unapply(value: Int): Option[(Int, Int)] = // ...
}
object which would do the matching, you may only supply one result per value. E.g.
object + {
def unapply(value: Int): Option[(Int, Int)] = value match {
case 0 => Some(0, 0)
case 4 => Some(3, 1)
case _ => None
}
Now this works:
0 match { case x + 0 => x } // returns 0
also this
4 match { case x + 1 => x } // returns 3
But this won’t and you cannot change it:
4 match { case x + 2 => x } // does not match
No problem for ::, though, because it is always defined what is head and what is tail of a list.
There are two ::s (pronounced "cons") in Scala. One is the operator on Lists and the other is a class, which represents a non empty list characterized by a head and a tail. So head :: tail is a constructor pattern, which has nothing to do with the operator.