In this lab we will look at algebraic data types, pattern matching with the case
command, and guarded expressions.
 Prelab Checklist
 Getting Started
 Case Expressions
 Guards
 Nesting guards and cases
 Testing your code with Doctest
 Area of a triangle done properly (Optional)
 References
Prelab Checklist
 You should have completed Lab01 and Lab02.
 You need to somewhat comfortable using Git with VSCodium.
 You should have read through lectures up to Case Expressions in Week 2.
Getting Started

Go to the project’s dashboard under the Project tab and click on the Fork button. The gitlab url for this lab is https://gitlab.cecs.anu.edu.au/comp1100/2024s2/2024s2studentfiles/comp11002024s2lab03

You will be asked where to fork the repository. Click on the user or group to where you’d like to add the forked project. You should see your own name here.
Once you have successfully forked a project, you will be redirected to a new webpage where you will notice your name before
> projectname
. This means you are now the owner of this repository. The url in your browser should reflect the change: https://gitlab.cecs.anu.edu.au/uXXXXXXX/2024s2/2024s2studentfiles/comp11002024s2lab03 
Finally, clone your forked repository to your computer according to the instructions in the Week 2 Lab.
Case Expressions
Quick Recap: In Haskell, you can create your own Algebraic Data Types (ADTs) using sum types, product types, and combinations of the two.
Sum types
data Season = Spring  Summer  Autumn  Winter
deriving Show
The deriving Show
gets GHC to automatically generate default code
for converting a Season
to a String
. GHCi uses this when printing
the value of an expression of type Season
, which will make testing
your code easier.
We will see later in the course precisely what the deriving
keyword
does in general, so don’t worry too much about it for now.
Product types
data Student = Person Name Age Gender
deriving Show
Here, Person
is called a data constructor (it holds everything together)
and Name, Age, Gender
are the types of the data inside this product type.
So, for example,
johnSmith :: Student
johnSmith = Person "John Smith" 21 Male
You can also think of Person
as a function of type
Person :: Name > Age > Gender > Student
That is, it takes three inputs, the name, the age, and the gender,
and uses this information to construct a Student
.
Combining sum and product types
data Shape = Circle Double  Rectangle Double Double
Case expressions can be used for pattern matching. Patterns can be
literal values (2
, 'a'
, True
), variables, wildcard (_
),
tuples, other constructors etc. Case expressions allow you to match
against the “shape” of the input, and extract out useful information
from within.
Remember that case expressions have general syntax
case <expression> of
<pattern1> > <result1>
<pattern2> > <result2>
...
The expression <expression>
is matched against each of the patterns <pattern1>
, <pattern2>
, … during evaluation in sequence from top to bottom. Once a pattern is matched, the function returns the corresponding result.
Exercise 1
Open Season.hs
. You will find a predefined function isCold
.
isCold :: Season > Bool
isCold season = case season of
Spring > False
Summer > False
Autumn > False
Winter > False
Winter > True
We can type ghci Wall Season.hs
in Terminal to start the ghci interpreter
(with all warnings turned on) and test how this function performs.
ghci> isCold Summer
False
ghci> isCold Winter
False
That doesn’t seem right, Winter is a cold season.
Task 1
Fix the function so that isCold Winter
returns True
. Hint: What
do you think Woverlappingpatterns
in the following warning could
mean?
[1 of 1] Compiling Season ( Season.hs, interpreted )
Season.hs:12:5: warning: [Woverlappingpatterns]
Pattern match is redundant
In a case alternative: Winter > ...

12  Winter > True
 ^^^^^^^^^^^^^^
Ok, one module loaded.
Task 2
Since only one of the seasons returns True
, do we really need to
pattern match against all four seasons? Consider rewriting accordingly.
Hint: You should only need two patterns.
Submission required: Exercise 1
Each exercise should be committed using Git integration in
VSCodium as shown in Lab 02. Use an
appropriate commit message, e.g. “Lab03: completed Ex1”. Feel free to
write your own, as long as it is meaningful.
Exercise 2
Create a new file and name it Mensuration.hs
.
On the first line, write:
module Mensuration where
VSCodium will probably automatically indent the next line, press backspace to return the cursor to the far left of the file.
Now, consider the algebraic data type Shape
:
data Shape
= Circle Double  ^ Radius
 Square Double  ^ Side length
 Rectangle Double Double  ^ Width, Height
Write a function area
that calculates the area of a given shape.
You will need to use case expressions to check which kind of shape the
input is, and then compute the area as appropriate. Feel free to look
online if you are not sure of the mathematical formula required.
Thinking back to Lab 01:
 What do you think should be the type signature of this function?
 What do you think should be on the second line of this function?
 What could you include as a comment for the function that is useful to the reader?
Example outputs:
ghci> area (Circle 10)
314.1592653589793
ghci> area (Square 3)
9.0
ghci> area (Rectangle 3 5)
15.0
Why area (Circle 10)
and not area Circle 10
? Try both, and see
what happens.
Submission required: Exercise 2
Commit your work, as you did for Exercise 1.
Guards
There are several ways of writing conditional functions in Haskell – two important ones are case, as we have seen above, and guarded expressions, also known as guards. Let’s explore guards through an example:
 check whether an integer is even
isEven :: Int > String
isEven num
 mod num 2 == 0 = show num ++ " is even."
 if number is even
 otherwise = show num ++ " is not even."
 if the previous guard(s) is not True
As the name suggests, a guarded expression is an expression which will
only be reached if the associated guard (the Boolean expression right
in front of it) evaluates to True
. Each line  <guard> = <expression>
of the function isEven
can thus be read as: if <guard>
then <expression>
is returned. You may find the operators &&
(Boolean AND), the 
operator (Boolean OR) and not
(Boolean
negation) useful for constructing Boolean expressions.
Exercise 3
Finding your grade at the ANU from your raw mark isn’t a smooth, continuous function, but it has discrete output values. Depending on your mark, your grade will be calculated as:
data Grade = Fail  Pass  Credit  Distinction  HighDistinction
deriving Show
type Mark = Int
markToGrade :: Mark > Grade
markToGrade mark
 mark >= 80 && mark <= 100 = HighDistinction
 mark >= 70 && mark < 80 = Distinction
 mark >= 60 && mark < 70 = Credit
 mark >= 50 && mark < 60 = Pass
 mark >= 0 && mark < 50 = Fail
Task 1
Open GPACalculator.hs
and then load the module into GHCi by
executing the command ghci Wall GPACalculator.hs
in the
Terminal. You will get the following warning:
GPACalculator.hs:14:1: warning: [Wincompletepatterns]
Pattern match(es) are nonexhaustive
In an equation for ‘markToGrade’: Patterns not matched: _
Try out:
ghci> markToGrade 116
...
ghci> markToGrade (16)
...
Since the mark is an Int
, numbers below 0 and above 100 are also
valid inputs, but not valid as a mark. Your task is to add a guard
that returns an error message “markToGrade: Not a valid mark” if the
input mark is strictly less than 0 OR strictly greater than 100.
Hint: Have a look at the isEven
function above for the guards
syntax, and for returning an error, see the lecture material on Case Expressions.
Once the task is completed, reload the module, and try evaluating
markToGrade 116
again. You should see the following:
ghci> markToGrade 116
*** Exception: markToGrade: Not a valid mark
CallStack (from HasCallStack):
error, called at GPACalculator.hs:20:32 in main:GPACalculator
Task 2
If you reload the module with the flag Wall
, you may get the
following warning (if not, you can just read through this section,
since you’ve already fixed it):
GPACalculator.hs:14:1: warning: [Wincompletepatterns]
Pattern match(es) are nonexhaustive
In an equation for ‘markToGrade’: Patterns not matched: _
What does this mean? Can we modify our guarded expression to remove this warning?
Haskell is a language that takes code correctness and reliability very
seriously. As such, it will give warnings for functions which it
believes do not have a definition for every possible valid input
(i.e., every possible value of the input(s) type). Even though it is
likely that you have matched every possible Integer input to this
function, Haskell isn’t sure, and so it gives the above warning. To
satisfy Haskell, we typically will end a set of guards with an
otherwise
— upon reaching an otherwise
case as the last line
(when evaluating guard conditions from top to bottom) Haskell will
always evaluate the function as the definition on the right hand
side of the otherwise
.
This is not a test. Ask your tutor or your classmate if you’re confused.
Submission required: Exercise 3
.
Commit your work.
Exercise 4
A tuple is a pair of elements, possibly of different types. A 3tuple is a triple of elements, a 4tuple is a grouping of 4 elements, in general an Ntuple is a grouping of n elements for some n.
Tuples are useful when we want to group information together. In Haskell we represent tuple types with parentheses.
As an example, consider your overall grade for a particular course, we can
represent this as a pair of a Course
and a positive Integer like so:
(Comp 1100, 85)
or (Busn 1001, 97)
.
The tuple can be thought of as the Haskell representation of the product type.
(Math 1005,92) :: (Course, Integer)  A product of a Course and Integer
Your task is to write a function markToGrade'
that takes an input
value of type (Course, Mark)
and returns a Grade
.
You can define Course
as
data Course = Art Int  Int denoted the course code
 Comp Int
 Busn Int
 Math Int
As before, Mark
is another name for Int
, and Grade
is as defined
in Exercise 3.
Example outputs:
ghci> markToGrade' (Comp 1100, 45)
Fail
ghci> markToGrade' (Busn 1001, 95)
HighDistinction
Hint: This function can be written in a single line.
What happens when we enter an invalid mark as input?
Submission required: Exercise 4
Commit your work.
Exercise 5
There are several other ways to handle exceptional values or situations which you will learn about. One way is to use types that can also hold some exceptional values “on top of” their “normal” values. The simplest of these is the Maybe type, which can hold values of a specific type or one special value (which is defined as the value Nothing here). This way we can write function which will return (or accept) values that are of a specific type, but also have the option to say they cannot return anything and actually returns the value Nothing instead.
For example, supposing you write a function to divide numbers. What
should the function return if we divide by zero? It doesn’t seem
sensible to return any number at all (except perhaps the value
infinity), and throwing an error and killing the program also may not
be ideal (imagine if this software is running on an aeroplane, and
division by zero occurs, so shutting down the plane’s control computer
in midflight — not ideal!) Returning Nothing
instead is a more
graceful way to handle the situation where we would not otherwise be
able to return a value.
After you have read the above you will probably notice that the name “Nothing” is perhaps not the most sensible choice of name, as it is not literally “nothing”, but in fact a special value. But, planet Haskell has defined it thus, so we are stuck with it here.
Consequently, the Maybe Type as already predefined in Haskell looks like this:
data Maybe a = Nothing  Just a
where a
stands for “any type”. We can now write another version of
our grading program.
Your task is to use the Maybe
type to stop the markToGrade
function throwing an error when the input mark is out of bounds. Write
function markToGradeSafe
with the type signature:
markToGradeSafe :: Mark > Maybe Grade
Note that when the mark is within bounds, instead of a value X
of
the type Grade
, we now return Just X
, which is of type Maybe Grade
.
When the mark is out of bounds, we now return Nothing
.
Test your implementation of markToGradeSafe
and see what GHCi
displays as return values. Other numerical types use a similar concept.
For instance, try to divide 1 by 0 in GHCi, or take the square root of
a negative number, and see if the output is what you expect:
ghci> 1 / 0
...
ghci> sqrt (1)
See the following example for how you might write a safe division
function using a Maybe type (maybe use something like this to write
markToGradeSafe
):
safeDiv :: Double > Double > Maybe Double
safeDiv x y
 y == 0 = Nothing
 otherwise = Just (x / y)
Submission required: Exercise 5
.
Commit your work.
Exercise 6
Here’s another problem. At some point, you will want to figure out what GPA corresponds to which Grade. To simplify things, here’s a table modified from ANU GPA reference.
Grade  Grade Point Value 

HighDistinction  7 
Distinction  6 
Credit  5 
Pass  4 
Fail  0 
It would be handy if you could enter your mark, and it automatically
calculated the GPA associated with that mark.
For example, markToGPA 80
would return 7 as the result.
Let’s break this down to two steps. We have already written the part
where you convert your mark to a grade. The next step is to write
another function to convert grades to a GPA. Therefore the new
function should have following type signature (where GPA
is another
name for Double
): Maybe Grade > GPA
You have already learnt that the Maybe
type is defined as follows:
data Maybe a = Just a  Nothing
How do you actually use this productively? Let’s start with an example:
maybeToInt :: Maybe Int > Int
maybeToInt m = case m of
Just i > i
Nothing > 0
The example above takes in a Maybe Int
which is either Just Int
or
Nothing
. If it follows the pattern of Just <some Int value i>
,
return i
, if it’s Nothing
, return 0
.
You can see we’ve used pattern matching here to “look inside” the
input, and extract out the number contained inside the Maybe
type
(if such a number exists).
Have a closer look at the third line of this example — what is the
type of i
?
Your task is to add a new function to GPACalculator.hs:
maybeGradeToGPA :: Maybe Grade > GPA
Note, if the Maybe Grade is Nothing
, return 0
.
Submission required: Exercise 6
Commit your work.
Exercise 7
Let’s link all of these things together! First, let’s recall the functions you’ve already written and are going to use:
markToGradeSafe :: Mark > Maybe Grade
: This takes a rawMark
and outputs aMaybe Grade
.maybeGradeToGPA :: Maybe Grade > GPA
: This takes aMaybe Grade
and outputs aGPA
associated with that grade.
What’s the next step?
Write a function: markToGPA :: Mark > GPA
, (using the functions
already written to help you) which takes in a raw mark as an Int
in
the range 0 to 100 and outputs a GPA (0 to 7) associated with that
mark.
Submission required: Exercise 7
.
Commit your work.
Nesting guards and cases
A useful property of cases and guards is being able to nest them (i.e. use cases within cases).
Remember how we defined the data type Shape
in section combining sum and product types.
Let us define a valid circle as a Circle
with nonnegative radius, and a valid rectangle as a Rectangle
with nonnegative length and width.
data Validity = Valid  Invalid
Using this let us define a function that takes a Shape
and returns the Validity
of that shape.
validShape :: Shape > Validity
validShape shape = case shape of
Circle rad
 rad < 0 > Invalid
 otherwise > Valid
Rectangle h w
 h < 0  w < 0 > Invalid
 otherwise > Valid
In this example we have used a guard within a case. The case checks what the shape is and the guard considers the criteria for the shape to be valid.
Note how the nested guard uses an >
and not a =
Exercise 8 (Tricky)
Using the above concept, we will be trying to remake the function markToGrade'
. This time we are given the additional information that 1000 level Busn
courses will be scaled down by 10% (i.e. multiplied by 0.90) and all other Busn
courses will be scaled down by 5% (i.e. multiplied by 95%). All the other courses calculate the grade without scaling.
Submission required: Exercise 8
.
Commit your work.
Testing your code with Doctest
Exercise 9
Just because your code compiles, it doesn’t mean it will do what it is supposed to. Consider the following:
addOne :: Int > Int
addOne x = x + 50
Does this compile? Of course it does! Does this do what it’s supposed to? Deducing from the name, it should take a number, and add one to it. However, what it actually does is to take a number and add 50 to it. For a simple functions like this, it’s relatively easy to figure out that it’s wrong before even compiling it. However, for a more complicated function it is often necessary to compile it and run the code with a predetermined input and expected output, to see if it does what you want it to.
Loading the above function into GHCI, and typing: addOne 5
, you
might expect 6
to be the answer, but it actually gives the result
55
. There is a useful tool in Haskell which takes away the manual
labour of testing everything yourself. You simply tell it what to do,
what to expect, and it will check the functions for you. This is
called Doctest, which we will extensively use throughout the
semester.
What does a Doctest statement look like?
  Adding one.

 >>> addOne 5
 6

 >>> addOne 100
 101
addOne :: Int > Int
addOne x = x + 50
Please remember to place the doctests above the function
definition. In this case, the doctests for the function addOne
is
written above the function. Note that doctests can be very picky about
formatting, so you can use the above example to check against.
If you add the above code to a file called Adding.hs
and run it with
the command doctest Adding.hs
, you should get the following output:
> doctest Adding.hs
### Failure in Adding.hs:9: expression `addOne 5'
expected: 6
but got: 55
Examples: 2 Tried: 1 Errors: 0 Failures: 1
This means that two tests were present, one ran (and failed). Doctest will stop running tests as soon as the first one fails. Here, doctest is reporting back that the output was 55, but the test says it should have been 6. This indicates that the behaviour of the code, and the tests do not agree. (Of course, it could be possible that the test itself is broken.)
Your task is to add doctests to at least one function in every module of this
lab, namely Season.hs
, GPACalculator.hs
, and Mensuration.hs
.
Submission required: Exercise 9
.
Commit your work.
You are required to submit your attempts at the exercises above in order to receive full participation marks for this lab. You have until the start of your next lab to do this. You should submit by committing and pushing all your changes to your Gitlab repository.
Area of a triangle done properly
[Optional, not required for participation marks]
Exercise 10: Is the triangle even valid?
If you look back at last week’s lab, you will now see that the
function areaOfTriangle
could not have been the last word. This
function should not return an area value if the provided edge lengths
cannot form a triangle. So as an intermediate step we will first write
a function isValidTriangle
that will tell you whether the sum of any
two sides of the triangle is larger than the remaining side.
Copy the Area.hs
file from Lab 2 to the
current working directory. You can reuse some of that code in the next
task and fix the definition of areaOfTriangle
using isValidTriangle
.
The first challenge is to work out what the type of the function
should be (Hint: remember the output of the function can be either
True
or False
). Write a type signature for the function.
Next write the actual function definition, which can well be a single line, or alternatively a sequence of guarded expressions. Test your function with some values in GHCi before you proceed to the final exercise. \(\{1, 2 ,4\}\) is for instance a set of edge lengths which cannot form a triangle. What does your function make of it?
Area of a valid triangle
You now have all the tools at hand to do the job properly. Your new
function areaOfTriangleSafe
has the type:
areaOfTriangleSafe :: Double > Double > Double > Maybe Double
It will return a numerical value as Just <Double value>
for all
valid triangles and Nothing
for invalid triangles. Put all the bits
from this lab together for this function.
Write your function and a few doctests for both valid and invalid triangles to justify your function.
Submission: Exercise 10: Area of valid triangle
.
References
[1] ANU Grade Point Average, http://www.anu.edu.au/students/programadministration/assessmentsexams/gradepointaveragegpa
[2] Haskell Doctest, http://hackage.haskell.org/package/doctest