Haskell FAQ

Please note: this document was originally compiled for my work at De Montfort University teaching undergrads functional programming in Haskell. It hasn't been updated in a few years.


How do I open GHCi?

Just open a terminal window and simply type ghci to open the Glasgow Haskell Compiler in interactive mode.

In the computer labs in the Gateway building you need to log into Linux where Haskell is already installed. If you want to install Haskell on your laptop or home computer you can download it here. Alternatively, if you just quickly want to try an expression but do not have Haskell installed on your computer at home, you can use the interactive prompt on haskell.org or tryhaskell.org. There is even an app for iOS devices (Raskell). Let me know if you find a good app for Android!

What is the point of :set +t?

Setting this will show the type of any value returned. It is optional but useful. You will have to set this every time you restart GHCi. Do not forget to type the : at the start of the expression. In some of the examples throughout the lab sessions you will encounter instructions telling you to type :s +t which might not work on your machine. Use :set +t instead.

Setting up your GHCi config file

You may want to setup a config file on your profile which can save you typing out certain commands every time you start a new session. For example you can put the :set +t command into this file and you will never have to remember it again.

To do this, open your home folder on Linux and enable the viewing of hidden files. Find the .ghc folder. You need to create a new file called ghci.conf within that folder. Now, open this file and simply type in :set +t and save.

There are various other things you can do with this config file. For example you can change the prompt from Prelude> to your name or the Haskell “logo” lambda (λ) using this command: :set prompt "λ>".

If you have installed GHC on your laptop or home computer you may have noticed that :set +t produces slightly different results than on the Gateway Linux lab machines. This is due to the version you are using.

In the labs (which use version 7.4.1) you would see something like this:

> 4 + 6
it :: Integer

If you installed the newest version of GHC (7.10.3) then you will see this instead:

> 4 + 6
it :: Num a => a

Right now this might not mean much to you, so you can change this setting in your ghci.conf file (see above) by adding this command: :set -XMonomorphismRestriction. To change it back to what it was before change the command to: :set -XNoMonomorphismRestriction.

An example ghci.conf file can look like this:

:set +t
:set -XMonomorphismRestriction
:set prompt "λ> "

Some other useful commands for GHCi


Do not forget you can check what functions do in Hoogle: floor. For example, you can also search for functions by their type declarations: String -> Int Hoogle: String -> Int.

How do I load scripts into GHCi?

Once you have typed a few expressions into a text editor of your choice and saved the file with an .hs extension, you can load it into GHCi.

First you need to make sure your terminal prompt is in the right directory. Use the cd command to enter the folder where you saved the Haskell script. So, if you called your file Script101.hs and saved it in a folder called CTEC1901 within your home directory, you need to enter the following commands into your terminal prompt: cd CTEC1901, then ghci and then :load Script101.hs.

~$ cd CTEC1901
~/CTEC1901$ ghci
Prelude> :load Script101.hs

How do I use script files and functions?

Let us assume you have a script file which contains the following code:

add x y = x + y

You can call this function in GHCi by loading the script and then typing:

add 4 6

which should then return:

it :: Integer

Similarly if your file contained the following code:

add :: (Int, Int) -> Int
add (x,y) = x + y

Then you need to type the following to call the function:

add (4,6)

Note that the first line in your script add :: (Int, Int) -> Int tells you which types of values you need to pass to the function as parameters. In this case the left hand side (before the ::) tells you that the function is called add and the right hand side tells you that the function takes a tuple of two Integers (Int, Int) as input (before the ->) and that the output of the function is an Int (after the ->).

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Div and mod are not working?

The expressions x `div` y and x `mod` y need backwards quotes (left of the ‘1’ key on the keyboard). Note that the above notation is just syntactic sugar for div x y and mod x y - you can interchange them as you like.

What do floor and ceiling do?

floor x returns the greatest integer not greater than x and similarly, ceiling x returns the least integer not less than x. So floor 1.4 will return 1 and ceiling 1.4 will return 2. This is different from rounding!

How does 1 + if 2==2 then 3 else 4 work?

The if statement automatically takes precedence over the addition:

1 + if 2 == 2 then 3 else 4
1 + ( if 2 == 2 then 3 else 4 )
1 + ( if True then 3 else 4 ) 
-- 2 == 2 is True so we enter the `then` statement
1 + ( 3 )
1 + 3

How does (if 2/=2 then reverse else tail) "abc" work?

Here, the if statement determines which function to apply to the string "abc". Since 2 is not not equal to 2, it is False and the function performs tail "abc" and returns "bc".

Why does two * length name produce an error?

In this particular example, the variable two is defined as 2 in your script which Haskell interprets as having type Integer. You can test this by using the :type or :t command:

:t two
two :: Integer

name is defined as a String "Haskell" (of type[Char]). The function length returns an Int (see here), so length name will return an 7 of type Int.
Now, the multiplication function * requires you to call it with two values of the same numeric type (see here). But we have seen that GHCi interprets the value two as an Integer rather than an Int. Therefore trying to multiply it with length name (which returns an Int) GHCi will return an error:

Could not match expected type `Integer’ with actual type `Int’
In the return type of a call of `lengthIn the second argument of `(*)’, namely `length name’
In the expression: two * length name

On the other hand, if you type 2 * length name into your terminal it will return 14 of type Int. That is because manually typing a number into GHCi will default to an Int. One way around this would be to convert the two parts of the evaluation to the same type. For example you can convert an Int to an Integer by using the fromIntegral function (see here).

two * fromIntegral (length name)
it :: Integer

Or using the fromInteger function (see here) similarly:

fromInteger two * length name
it:: Int

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Curry, Lambda and Sections

See page 64 of David Smallwoods tutorial file. The following examples are taken from wunki.org

Curried and uncurried functions

Uncurried function:

fooU :: (Int, Int) -> Int
fooU (x, y) = x * y

Curried function:

fooC :: Int -> Int -> Int
fooC x y = x * y

The main difference in very simple terms is that the function parameters x and y are given separately in curried functions compared to a tuple in uncurried functions (x,y).

Anonymous / Lambda functions

Lambda functions are sometimes called anonymous functions because they are not given a name.

fooA :: Int -> Int -> Int
fooA x = \y -> x * y

Note that the type signature indicates that the function takes 2 input parameters of type Int, even though in the function expression it only lists 1 of them as x.

> let foo x = \y -> x * y
> foo 2 3

Sections / Partial application

“Partial function application is the ability to take a function of many parameters and apply arguments to some of the parameters to create a new function that needs only the application of the remaining arguments to produce the equivalent of applying all arguments to the original function.” (Source & Code examples)

Consider the following example:

sumTwo x y = x + y
addSeven = sumTwo 7

The second line is equivalent to:

addSeven = \y -> 7 + y


addSeven y = sumTwo 7 y

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Tuples and Lists

What is the difference between tuples and lists?

Lists are sequences of elements of the same type. They are enclosed by square brackets and separated by commas. E.g. [1,2,3] or [’a’,’c’,’b’].

It is possible to have infinite lists, for example [1,3..] will produce the list of all odd numbers from 1 to infinity. If you type that into your terminal, remember you can interrupt a process by pressing ctrl c.

Tuples are finite sequences of elements of possibly different types. They are enclosed by round brackets and separated by commas. E.g. (1, "Hello") or (True, False).


Why do Strings have the type [Char]?

Because in Haskell Strings are interpreted as a list of characters. This allows you to perform list functions like head and tail on Strings:

head "Haskell"Hit :: Char
tail "Haskell"
it :: [Char]

Also, this means that if you type in lists of characters into GHCi it will return them as strings of type [Char].

it :: [Char]

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Types, Polymorphism and Order of Execution

What is the meaning of :t?

:t is short for :type and returns the type definition of a function. E.g.:

:t sum
sum :: Num a => [a] -> a

This is different from the :set +t command which sets GHCi to automatically return the type of an output from an expression (for the current session only though - so you have to set that every time you restart GHCi).

What are function types?

In general a function type definition has the following structure.

functionName :: InputType -> OutputType

What is the difference between add :: Int and add :: Num a => a?

In short, the first is more specific than the second.

add :: Int -> Int -> Int -> Int
add x y z = x + y + z

Calling this function with Ints will work fine:

add 1 2 3
it :: Int

But trying to call it with any other kinds of numbers will not work:

add 1.4
No instance for (Fractional Int) arising from the literal `1.4In the first argument of `add’, namely `1.4In the expression: add 1.4 2.6 3.0
In an equation for `it’: it = add 1.4 2.6 3.0

If your add function has the specific kind of type definition then trying to run add 1.4 returns an error for example. This is because you have told Haskell that your function only works with Ints. If you use the more generic (polymorphic and overloaded) type definition however, then you would get the result.

add :: Num a => a -> a -> a -> a
add x y z = x + y + z
add 1 2 3
it :: Num a => a
add 1.4
it :: Fractional a => a

What are polymorphic and overloaded functions again?

A function is polymorphic if it takes values that can have more than one type. See the tutorial by David Smallwood here. For example, length is defined polymorphically. In this case a is a type variable and represents any type. The function therefore takes a list of any type of value ([Int], [[Char]], [(Double, Bool)], etc) but then specifically returns an Int.

length :: [a] -> Int

A polymorphic function is called overloaded if its type contains one or more class constraints. For example, the sum function takes a list [a] as input and returns a value of type a but this time a has to be of a class of numeric types. The numeric class (Num) contains all sorts of types: (Int, Integer, Float, Double, Fractional, etc) (See Prelude#t:Num).

sum :: Num a => [a] -> a
sum [1,2,3]
it :: Num a => a
sum [2.5, 3, 4.5]
it :: Fractional a => a

functionName :: ClassConstraint => InputType -> OutputType.

Other type classes include Eq for equality (see Prelude#t:Eq) and Ord for orderable datatypes (see Prelude#t:Ord). E.g.:

(+)  :: Num a => a -> a -> a
(==) :: Eq a => a -> a -> Bool
(<)  :: Ord a => a -> a -> Bool

See also the LearnYouAHasekll chapter on types here.

What is the meaning of in function names?

Often in the examples you will have encountered functions with a at the end like add’ vs. add. There is no specific meaning to this other than to give the function a slightly different name. We could have equally called it add1 or anotherAdd to set it apart from the first add function we defined.
You might want to review Haskell naming conventions.

In which order is mult x y z evaulated?

“the -> operator is right associative, and function application is left associative” (Source)

Because function application is left associative, mult x y z gets evaluated as ((mult x) y) z.

mult :: Int -> Int -> Int -> Int
mult x y z

But remeber that the -> operator is right associative, hence the function definition really looks something like this.

mult :: Int -> (Int -> (Int -> Int))
((mult x) y) z

You do not have to add these parenthesis, but you should know that this is the default order of evaluation!

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How do I write a function from scratch?

In the first exercise we are asked to write a function that splits a list into two halves. We are given the type signature as a starting point.

halve :: [a] -> ([a],[a])

Let us start filling in the bits we know, for example we know the name of the function (halve) and that the output is structured as a tuple.

halve :: [a] -> ([a],[a])
halve = (   ,   )

Note that this will not run - GHCi will return an error because it is an incomplete function!

Next, we know the input to the function is a list of polymorphic type, so we know that we will be able to call this function in GHCi like this:

halve [1,2,3,4,5,6]

and it should return a result like this:


Because the function is polymorphic we can also call it with Char types for example:

halve [’a’,’b’,’c’,’d’]

which would return:


Also, because “lists of Chars” are synonymous to Strings we can call the function with a String like this:

halve "Haskell"

which would return:


Now, let us get back to writing our function by filling in the gaps of halve = ( , ).

We know from the type signature halve :: [a] -> ([a],[a]) that there is one input parameter of type [a] and one output tuple of type ([a],[a]).

$$ \definecolor{blue}{RGB}{181,204,227} \text{halve} \ {\underbrace{\qquad\qquad\qquad} \atop \color{blue}\small\text{input parameter}} \ = \ ( {\underbrace{\qquad\qquad\qquad} \atop \color{blue}\small\text{first half}} \ , \ {\underbrace{\qquad\qquad\qquad} \atop \color{blue}\small\text{second half}} \ ) $$

Let us call the input parameter xs: remember the convention to put an s on variables that represent lists!

halve xs = (   ,   )

Next, let us think of how to get the first half of xs. How do we know what half of the input list is? Well, it is the total length of xs divided by 2, right?

length xs `div` 2

Let us give this value a name (n) so that we can use it in our function by writing it into a where statement in a new line and slightly indented (this is important).

halve :: [a] -> ([a],[a])
halve xs = (   ,   )
      where n = length xs `div` 2

Now we can define the two halves in our output tuple using the take and drop functions from the Standard Haskell Prelude (which is loaded into GHCi by default and we do not have to manually import the library).

Since n now represents the length of half the input list xs we can take the first n elements in the list xs which translates into Haskell like this: take n xs. Similarly we can drop the first n elements from the list xs to get the second half: drop n xs.

The complete function then looks like this:

halve :: [a] -> ([a],[a])
halve xs = ( take n xs , drop n xs )
      where n = length xs `div` 2

How do I split a list into three parts?

We can use a similar approach to the halve function explained above.

split3 :: [a] -> ([a],[a],[a])
split3 xs = (   ,   ,   )
      where n = length xs `div` 3

Using take and drop and a variable n which represents a third of the length of xs we can create the various parts of our output tuple as such:

take drop example
take drop example

So our final solution could be this (note from the table above that there are various options on how to extract the three thirds of the input list):

split3 :: [a] -> ([a],[a],[a])
split3 xs = ( take n xs , take n (drop n xs) , drop (n*2) xs )
      where n = length xs `div` 3

What are the different kinds of techniques for writing functions?

There are various techniques in Haskell for writing functions and expressions. Some of them are fairly similar in style to other languages and others are quite different. Deciding when to use one technique over another depends on the purpose of the function.


Functions have a name and usually one or more input parameters. There is only ever one output. The type signature of a function is optional when writing functions in a script - Haskell can derive it by itself - but it is good practice to include the type signature as a form of documentation.

add :: Num a => a -> a > a -> a
add x y z = x + y + z
add 1 2 3


$$ \definecolor{blue}{RGB}{181,204,227} \underbrace{ \ \text{add} \ }_{\color{blue}\small\text{function name}} :: \underbrace{\ \text{Num a} \Rightarrow \ }_{\color{blue}\small\text{class constraint}} \underbrace{\ \text{a} \rightarrow \ }_{\color{blue}\small\text{input 1}} \underbrace{ \ \text{a} \rightarrow \ }_{\color{blue}\small\text{input 2}} \underbrace{ \ \text{a} \rightarrow \ }_{\color{blue}\small\text{input 3}} \underbrace{ \ \text{a} \ }_{\color{blue}\small\text{output}} $$

$$ \definecolor{blue}{RGB}{181,204,227} \underbrace{ \ \text{add} \ }_{\color{blue}\small\text{function name}} \ \underbrace{ \quad \text{x} \quad }_{\color{blue}\small\text{input 1}} \ \underbrace{ \quad \text{y} \quad }_{\color{blue}\small\text{input 2}} \ \underbrace{ \quad \text{z} \quad }_{\color{blue}\small\text{input 3}} \ \underbrace{ \qquad\quad \text{=} \qquad\quad }_{\color{blue}\small\text{assignment operator}} \ \underbrace{ \ \text{x+y+z} \ }_{\color{blue}\small\text{output}} $$

The class constraint is optional, and there can be 0 or more input parameters but always one result.

Anonymous functions

Anonymous functions or lambda functions are nameless functions. This allows us to declare them on the fly. The structure is as follows.

$$ \definecolor{blue}{RGB}{181,204,227} \underbrace{ \ \text{\\} \ }_{\color{blue}\small\text{lambda}} \ \underbrace{ \quad \text{x} \quad }_{\color{blue}\small\text{input}} \ \underbrace{ \qquad \rightarrow \qquad }_{\color{blue}\small\text{assignment operator}} \ \underbrace{ \ \text{x*2} \ }_{\color{blue}\small\text{output}} $$

Instead of a function name we now have the lambda symbol (a backslash in Haskell) \ and instead of the normal assignment operator = we use the -> symbol.

Similar to normal functions, we can use various techniques within a lambda expression and have more than one parameter. For example:

We can use lambda expressions as inputs for the map function:

map (\x -> if x == 1 then 0 else 1) [1,0,1,0,0,0,1,1]
map (\(x,y) -> x + y) [(1,2),(2,3),(4,5)]

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List comprehensions

List comprehensions definition

List comprehensions are defined in the official documentation as: [ e | q1, ..., qn ]
where n>=1 and the qi qualifiers are either

See also the tutorial by David Smallwood on page 84.

List comprehensions vs loops

In pseudo C using for-loops:

for ( int x = 1; x <= 4; x++ ) {
      for ( int y = 1; y <= 2; y++ ) {
            //add (x,y) to myList

In Haskell:

myList = [(x,y) | x <- [1..4], y <- [1,2]]
ghci> mylist

Infinite lists

Example taken from haskell.org:

ghci> take 5 [ (i,j) | i <- [1,2], j <- [1..]] 

“each successive generator refines the results of the previous generator. Thus, if the second list is infinite, one will never reach the second element of the first list.” Source

So, because the second generator (the inner loop) [1..] produces an infinite list, this list comprehension theoretically creates an infinite list of tuples: [(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),.. etc. The first value of the tuple (i) is never incremented because we infinitely increment the second one (j) first.

The whole expression however asks only for the first 5 elements (take 5) of the infinite list that is being generated through the list comprehension.

Set Notation

Examples taken from Wikipedia: In mathematics, set notation looks like this:

$$ \definecolor{blue}{RGB}{181,204,227} S = \{ \ 2 \cdot x \ | \ x \in \mathbb{N}, \ x^2 > 3 \ \} $$

This means:

$$ \definecolor{blue}{RGB}{181,204,227} S = \{ \underbrace{ \ 2 \cdot x \ }_{\color{blue}\text{output expr.}} \ | \ \underbrace{ \ x \ }_{\color{blue}\text{variable}} \in \underbrace{ \ \mathbb{N} \ }_{\color{blue}\text{input set}}, \ \underbrace{ \ x^2 > 3 \ }_{\color{blue}\text{predicate}} \} $$

In Haskell this translates directly into:

s = [ 2*x | x <- [0..], x^2 > 3]

This reads something like:
“for each x from 0 to infinity: if x^2 is larger than 3 then return 2*x”

More List comprehension examples

if-then-else statements within the list comprehension:

Taken from: learnyouahaskell.com

boomBangs xs = [ if x < 10 then "BOOM!" else "BANG!" | x <- xs, odd x]
ghci> boomBangs [7..13]  

Nested List comprehensions:

ghci> let xxs = [[1,3,5,2,3,1,2,4,5],[1,2,3,4,5,6,7,8,9],[1,2,4,2,1,6,3,1,3,2,3,6]]  
ghci> [ [ x | x <- xs, even x ] | xs <- xxs]  

Local bindings using let expressions:

Taken from: learnyouahaskell.com

calcBmis :: (RealFloat a) => [(a, a)] -> [a]  
calcBmis xs = [bmi | (w, h) <- xs, let bmi = w / h ^ 2]
calcBmis :: (RealFloat a) => [(a, a)] -> [a]  
calcBmis xs = [bmi | (w, h) <- xs, let bmi = w / h ^ 2, bmi >= 25.0]

Lecture example

positions :: Eq a => a -> [a] -> [Int]
positions x xs =
      [i | (x’,i) <- zip xs [0..n], x == x’]
      where n = length xs - 1

We can run this function as follows:

ghci> positions 0 [1,0,0,1,0,1,1,0]

To understand this function, let us look at the list comprehension. Remember the expression around the left pointing arrow <- is called a generator.

The generator of this list comprehension ((x’,i) <- zip xs [0..n]) creates a list of type [(x’,i)] by zipping xs and [0..n] together. We are calling the function positions with x as 0 and xs as [1,0,0,1,0,1,1,0] and n is locally defined using the where notation as the length of our list xs minus 1. The length of xs is 8, therefore n must be 7. This means that the generator will zip these two lists together:

zip [1,0,0,1,0,1,1,0] [0,1,2,3,4,5,6,7]

which returns:


Each of those tuples in this list is of type (x",i). Now, the list comprehension states that we are only interested in those elements of the list where x is equal to x". This is called the guard expression and is separated from the generator by a single comma. Remember that x is 0. This means we can delete any elements of the list where x" is not equal to 0. We are then left with:


The list comprehension also states that the type of our output list should be [i] (the bit before | in the list comprehension), that means we can discard all x" in our list of tuples [(x",i)] so that we are left with only the i:


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Guards and map

Guarded equations

Here is an example of a guarded equation:

sign :: (Ord a, Num a) => a -> Int
sign x 
      | x &lt; 0      = -1
      | x == 0     =  0
      | otherwise  =  1

This is very similar to mathematical notation:

$$ \text{sign}(x) = \begin{cases} -1 & \quad \text{if } x < 0\\ 0 & \quad \text{if } x == 0\\ 1 & \quad \text{otherwise} \end{cases} $$

To write the zoe function we need to put the three specified cases into this guarded notation. If the input parameter x is 0 then we want to return 0; if x is odd we want to return 1 and if x is even we return 2. Putting what we know so far into code is easy:

zoe :: (Integral a, Num a) => a -> Int
zoe x
      | "if x is equal to 0 return 0"
      | "if x is odd then return 1"
      | "if x is even then return 2"

These statements obviously need to be replaced with the appropriate conditional statements in Haskell.

"if x is equal to 0 return 0" can be written as x == 0 = 0. Checking if a number is even or odd can be done in two ways: we can either use the Standard Haskell Prelude functions odd and even, or we can use mod (for example, if the remainder of a number divided by 2 is 0 then the number must be even; if the remainder is 1 then it must be odd).

zoe :: (Integral a, Num a) => a -> Int
zoe x
      | x == 0 = 0
      | odd x  = 1
      | even x = 2


zoe :: (Integral a, Num a) => a -> Int
zoe x
      | x == 0    = 0
      | odd x     = 1
      | otherwise = 2


zoe :: (Integral a, Num a) => a -> Int
zoe x
      | x == 0         = 0
      | x `mod` 2 == 1 = 1
      | x `mod` 2 == 0 = 2

How do I use the map function?

map f xs is the list obtained by applying f to each element of xsSource

The map function has the following type signature:

map :: (a -> b) -> [a] -> [b]

which means it takes a function of type (a -> b) as the first input parameter and a list of type [a] as the second. The function returns a list of type [b].

Now that you have seen recursive functions you should be able to understand the actual function definition of map.

map :: (a -> b) -> [a] -> [b]
map _ []     = []
map f (x:xs) = f x : map f xs

We can run this function like this:

map (+1) [1,2,3,4,5,6]

It evaluates recursively like this:

map (+1) [1,2,3,4,5,6]
= (+1) 1 : map (+1) [2,3,4,5,6]
= 2 : (+1) 2 : map (+1) [3,4,5,6]
= 2 : 3 : (+1) 3 : map (+1) [4,5,6]
= 2 : 3 : 4 : (+1) 4 : map (+1) [5,6]
= 2 : 3 : 4 : 5 : (+1) 5 : map (+1) [6]
= 2 : 3 : 4 : 5 : 6 : (+1) 6 : map (+1) []
= 2 : 3 : 4 : 5 : 6 : 7 : []
= [2,3,4,5,6,7]

Note also that when providing the map function with an anonymous function as the first input parameter, then we can shorten that as such:

$$ \definecolor{blue}{RGB}{181,204,227} \definecolor{salmon}{RGB}{250, 128, 114} \definecolor{normal}{RGB}{162,153,161} \text{map} \ (\underbrace{\color{salmon} \backslash x \rightarrow x}_{\color{blue}\text{not needed}} \color{normal} + 7) \ [1..10] $$

Why does const flip id const 3 id 4 return 4?

Because the expression is evaluated from left to right and functions can be input to functions. The three functions are defined as follows:

id    = (\x -> x)
const = (\x y -> x)
flip  = (\f x y -> f y x)

The expression const flip id const 3 id 4 evaluates as such:

$$ \definecolor{blue}{RGB}{181,204,227} \definecolor{normal}{RGB}{162,153,161} \begin{matrix} & const & flip & id & const & 3 & id & 4 \\ = \ & ( \color{blue}const\color{normal} & \underbrace{flip}_{\color{blue}x} & \underbrace{id}_{\color{blue}y}) & const & 3 & id & 4 \\ = \ & & ( \color{blue}flip\color{normal} & & \underbrace{const}_{\color{blue}f} & \underbrace{3}_{\color{blue}x} & \underbrace{id}_{\color{blue}y}) & 4 \\ = \ & & & & ( \color{blue}const\color{normal} & \underbrace{id}_{\color{blue}x} & \underbrace{3}_{\color{blue}y}) & 4 \\ = \ & & & & & & ( \color{blue}id\color{normal} & \underbrace{4}_{\color{blue}x}) \\ = & & & & & & & 4 \end{matrix} $$

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