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Induction Data Types Type Classes Functors Homework
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Software System Design and Implementation
Induction, Data Types and Type Classes
Dr. Christine Rizkallah
UNSW Sydney Term 2 2021

Induction Data Types Type Classes Functors Homework
Recap: Induction
Suppose we want to prove that a property P(n) holds for all natural numbers n. Remember that the set of natural numbers N can be defined as follows:
Definition of Natural Numbers
1 0 is a natural number.
2 For any natural number n, n + 1 is also a natural number.
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Induction Data Types Type Classes Functors Homework
Recap: Induction
Therefore, to show P(n) for all n, it suffices to show:
1 P(0) (the base case), and
2 assuming P(k) (the inductive hypothesis), ⇒ P(k + 1) (the inductive case).
Example
Show that f(n) = n2 for all n ∈ N, where:
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(done on iPad)
􏰀0 if n = 0
f (n) =
2n−1+f(n−1) ifn>0

Induction Data Types Type Classes Functors Homework
Induction on Lists
Haskell lists can be defined similarly to natural numbers.
Definition of Haskell Lists
1 [] is a list.
2 For any list xs, x:xs is also a list (for any item x).
This means, if we want to prove that a property P(ls) holds for all lists ls, it suffices to show:
1 P([]) (the base case)
2 P(x:xs) for all items x, assuming the inductive hypothesis P(xs).
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Induction Data Types Type Classes Functors Homework
Example
Prove for all ls: (done on iPad)
sum ls == foldr (+) 0 ls
Induction on Lists: Example
sum :: [Int] -> Int
sum [] = 0 — 1 sum (x:xs) = x + sum xs — 2
foldr :: (a -> b -> b) -> b -> [a] -> b
foldr f z [] = z
foldr f z (x:xs) = x `f` foldr f z xs
— A — B
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Induction Data Types Type Classes Functors Homework
Custom Data Types
So far, we have seen type synonyms using the type keyword. For a graphics library, we might define:
type Point = (Float, Float)
type Vector = (Float, Float)
type Line = (Point, Point)
type Colour = (Int, Int, Int, Int) — RGBA
movePoint :: Point -> Vector -> Point
movePoint (x,y) (dx,dy) = (x + dx, y + dy)
But these definitions allow Points and Vectors to be used interchangeably, increasing the likelihood of errors.

Induction Data Types Type Classes Functors Homework
Product Types
We can define our own compound types using the data keyword:
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Constructor name
data Point = Point Float Float
deriving (Show, Eq)
data Vector = Vector Float Float
deriving (Show, Eq)
movePoint :: Point -> Vector -> Point
movePoint (Point x y) (Vector dx dy)
= Point (x + dx) (y + dy)
Constructor argument types
Type name

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Induction Data Types Type Classes Functors Homework
We could define Colour similarly:
data Colour = Colour Int Int Int Int
But this has so many parameters, it’s hard to tell which is which.
Records

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Induction Data Types Type Classes Functors Homework
We could define Colour similarly:
Records
data Colour = Colour Int Int Int Int
But this has so many parameters, it’s hard to tell which is which.
Haskell lets us declare these types as records, which is identical to the declaration style on the previous slide, but also gives us projection functions and record syntax:
data Colour = Colour { redC
, greenC
:: Int
:: Int
:: Int
, blueC
, opacityC :: Int
} deriving (Show, Eq)
Here, the code redC (Colour 255 128 0 255) gives 255.

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Induction Data Types Type Classes Functors Homework
Enumeration Types
Similar to enums in C and Java, we can define types to have one of a set of predefined values:
data LineStyle = Solid
| Dashed
| Dotted
deriving (Show, Eq)
data FillStyle = SolidFill | NoFill
deriving (Show, Eq)
Types with more than one constructor are called sum types.

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Induction Data Types Type Classes Functors Homework
Algebraic Data Types
Just as the Point constructor took two Float arguments, constructors for sum types can take parameters too, allowing us to model different kinds of shape:
data PictureObject
= Path [Point] Colour LineStyle
| Circle Point Float Colour LineStyle FillStyle
| Polygon [Point] Colour LineStyle FillStyle
| Ellipse Point Float Float Float
Colour LineStyle FillStyle
deriving (Show, Eq)
type Picture = [PictureObject]

Induction Data Types Type Classes Functors Homework
Live Coding: Cool Graphics
Example (Ellipses and Curves)
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Induction Data Types Type Classes Functors Homework
Recursive and Parametric Types
Data types can also be defined with parameters, such as the well known Maybe type, defined in the standard library:
data Maybe a = Just a | Nothing

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Induction Data Types Type Classes Functors Homework
Recursive and Parametric Types
Data types can also be defined with parameters, such as the well known Maybe type, defined in the standard library:
data Maybe a = Just a | Nothing
Types can also be recursive. If lists weren’t already defined in the standard library, we
could define them ourselves:
data List a = Nil | Cons a (List a)

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Induction Data Types Type Classes Functors Homework
Recursive and Parametric Types
Data types can also be defined with parameters, such as the well known Maybe type, defined in the standard library:
data Maybe a = Just a | Nothing
Types can also be recursive. If lists weren’t already defined in the standard library, we
could define them ourselves:
data List a = Nil | Cons a (List a)
We can even define natural numbers, where 2 is encoded as Succ(Succ Zero): data Natural = Zero | Succ Natural

Induction Data Types Type Classes Functors Homework
Types in Design
Sage Advice
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An old adage due to Yaron Minsky (of Jane Street) is:
Make illegal states unrepresentable.
Choose types that constrain your implementation as much as possible. Then failure scenarios are eliminated automatically.

Induction Data Types Type Classes Functors Homework
Sage Advice
Types in Design
An old adage due to Yaron Minsky (of Jane Street) is:
Make illegal states unrepresentable.
Choose types that constrain your implementation as much as possible. Then failure scenarios are eliminated automatically.
Example (Contact Details)
data Contact = C Name (Maybe Address) (Maybe Email)
is changed to:
data ContactDetails = EmailOnly Email | PostOnly Address
| Both Address Email
data Contact = C Name ContactDetails What failure state is eliminated here?
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Induction Data Types Type Classes Functors Homework
Partial Functions
Failure to follow Yaron’s excellent advice leads to partial functions.
Definition
A partial function is a function not defined for all possible inputs. Examples: head, tail, (!!), division
Partial functions are to be avoided, because they cause your program to crash if undefined cases are encountered.
To eliminate partiality, we must either:
enlarge the codomain, usually with a Maybe type:
safeHead :: [a] -> Maybe a — Q: How is this safer?
safeHead (x:xs) = Just x
safeHead [] = Nothing
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Induction Data Types Type Classes Functors Homework
Type Classes
You have already seen functions such as:
compare
(==)
(+)
show
that work on multiple types, and their corresponding constraints on type variables Ord, Eq, Num and Show.

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Induction Data Types Type Classes Functors Homework
Type Classes
You have already seen functions such as:
compare
(==)
(+)
show
that work on multiple types, and their corresponding constraints on type variables Ord, Eq, Num and Show.
These constraints are called type classes, and can be thought of as a set of types for which certain operations are implemented.

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Induction Data Types Type Classes Functors Homework
Show
The Show type class is a set of types that can be converted to strings. It is defined like:
class Show a where — nothing to do with OOP show :: a -> String

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Induction Data Types Type Classes Functors Homework
Show
The Show type class is a set of types that can be converted to strings. It is defined like: class Show a where — nothing to do with OOP
show :: a -> String
Types are added to the type class as an instance like so:
instance Show Bool where
show True = “True”
show False = “False”

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Induction Data Types Type Classes Functors Homework
Show
The Show type class is a set of types that can be converted to strings. It is defined like: class Show a where — nothing to do with OOP
show :: a -> String
Types are added to the type class as an instance like so:
instance Show Bool where
show True = “True”
show False = “False”
We can also define instances that depend on other instances:
instance Show a => Show (Maybe a) where
show (Just x) = “Just ” ++ show x
show Nothing = “Nothing”
Fortunately for us, Haskell supports automatically deriving instances for some classes, including Show.

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Induction Data Types Type Classes Functors Homework
Read
Type classes can also overload based on the type returned, unlike similar features like Java’s interfaces:
class Read a where
read :: String -> a
Some examples:
read “34” :: Int

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Induction Data Types Type Classes Functors Homework
Read
Type classes can also overload based on the type returned, unlike similar features like Java’s interfaces:
class Read a where
read :: String -> a
Some examples:
read “34” :: Int
read “22” :: Char

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Induction Data Types Type Classes Functors Homework
Read
Type classes can also overload based on the type returned, unlike similar features like Java’s interfaces:
class Read a where
read :: String -> a
Some examples:
read “34” :: Int
read “22” :: Char Runtime error! show (read “34”) :: String

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Induction Data Types Type Classes Functors Homework
Read
Type classes can also overload based on the type returned, unlike similar features like Java’s interfaces:
class Read a where
read :: String -> a
Some examples:
read “34” :: Int
read “22” :: Char Runtime error!
show (read “34”) :: String Type error!

Induction Data Types Type Classes Functors Homework
Semigroup
Semigroups
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A semigroup is a pair of a set S and an operation • : S → S → S where the operation • is associative.

Induction Data Types Type Classes Functors Homework
Semigroup
Semigroups
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A semigroup is a pair of a set S and an operation • : S → S → S where the operation • is associative.
Associativity is defined as, for all a, b, c:
(a • (b • c)) = ((a • b) • c)
Haskell has a type class for semigroups! The associativity law is enforced only by programmer discipline:
class Semigroup s where
(<>) :: s -> s -> s
— Law: (<>) must be associative.
What instances can you think of?

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Induction Data Types Type Classes Functors Homework
Semigroup
Lets implement additive colour mixing:
instance Semigroup Colour where
Colour r1 g1 b1 a1 <> Colour r2 g2 b2 a2
= Colour (mix r1 r2)
(mix g1 g2)
(mix b1 b2)
(mix a1 a2)
where
mix x1 x2 = min 255 (x1 + x2) Observe that associativity is satisfied.

Induction Data Types Type Classes Functors Homework
Monoid
Monoids
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A monoid is a semigroup (S,•) equipped with a special identity element z : S such that x • z = x and z • y = y for all x , y .

Induction Data Types Type Classes Functors Homework
Monoid
Monoids
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A monoid is a semigroup (S,•) equipped with a special identity element z : S such that x • z = x and z • y = y for all x , y .
class (Semigroup a) => Monoid a where
mempty :: a

Induction Data Types Type Classes Functors Homework
Monoid
Monoids
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A monoid is a semigroup (S,•) equipped with a special identity element z : S such that x • z = x and z • y = y for all x , y .
class (Semigroup a) => Monoid a where
mempty :: a
For colours, the identity element is transparent black:
instance Monoid Colour where
mempty = Colour 0 0 0 0
For each of the semigroups discussed previously: Are they monoids?
If so, what is the identity element?

Induction Data Types Type Classes Functors Homework
Monoid
Monoids
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A monoid is a semigroup (S,•) equipped with a special identity element z : S such that x • z = x and z • y = y for all x , y .
class (Semigroup a) => Monoid a where
mempty :: a
For colours, the identity element is transparent black:
instance Monoid Colour where
mempty = Colour 0 0 0 0
For each of the semigroups discussed previously: Are they monoids?
If so, what is the identity element?
Are there any semigroups that are not monoids?

Induction Data Types Type Classes Functors Homework
Monoid
Monoids
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A monoid is a semigroup (S,•) equipped with a special identity element z : S such that x • z = x and z • y = y for all x , y .
class (Semigroup a) => Monoid a where
mempty :: a
For colours, the identity element is transparent black:
instance Monoid Colour where
mempty = Colour 0 0 0 0
For each of the semigroups discussed previously: Are they monoids?
If so, what is the identity element?
Are there any semigroups that are not monoids? Non-empty lists, maximum

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Induction Data Types Type Classes Functors Homework
Newtypes
There are multiple possible monoid instances for numeric types like Integer: The operation (+) is associative, with identity element 0
The operation (*) is associative, with identity element 1

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Induction Data Types Type Classes Functors Homework
Newtypes
There are multiple possible monoid instances for numeric types like Integer: The operation (+) is associative, with identity element 0
The operation (*) is associative, with identity element 1
Haskell doesn’t use any of these, because there can be only one instance per type per class in the entire program (including all dependencies and libraries used).
A common technique is to define a separate type that is represented identically to the original type, but can have its own, different type class instances.

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Induction Data Types Type Classes Functors Homework
Newtypes
There are multiple possible monoid instances for numeric types like Integer: The operation (+) is associative, with identity element 0
The operation (*) is associative, with identity element 1
Haskell doesn’t use any of these, because there can be only one instance per type per class in the entire program (including all dependencies and libraries used).
A common technique is to define a separate type that is represented identically to the original type, but can have its own, different type class instances.
In Haskell, this is done with the newtype keyword.

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Induction Data Types Type Classes Functors Homework
Newtypes
A newtype declaration is much like a data declaration except that there can be only one constructor and it must take exactly one argument:
newtype Score = S Integer
instance Semigroup Score where
S x <> S y = S (x + y)
instance Monoid Score where
mempty = S 0
Here, Score is represented identically to Integer, and thus no performance penalty is
incurred to convert between them.
In general, newtypes are a great way to prevent mistakes. Use them frequently!

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Induction Data Types Type Classes Functors Homework
Ord
Ord is a type class for inequality comparison: class Ord a where
(<=) :: a -> a -> Bool What laws should instances satisfy?

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Induction Data Types Type Classes Functors Homework
Ord
Ord is a type class for inequality comparison: class Ord a where
(<=) :: a -> a -> Bool What laws should instances satisfy?
For all x, y, and z:
1 Reflexivity: x <= x. 45 Induction Data Types Type Classes Functors Homework Ord Ord is a type class for inequality comparison: class Ord a where (<=) :: a -> a -> Bool What laws should instances satisfy?
For all x, y, and z:
1 Reflexivity: x <= x. 2 Transitivity: If x <= yandy <= zthenx <= z. 46 Induction Data Types Type Classes Functors Homework Ord Ord is a type class for inequality comparison: class Ord a where (<=) :: a -> a -> Bool What laws should instances satisfy?
For all x, y, and z:
1 Reflexivity: x <= x. 2 Transitivity: If x <= yandy <= zthenx <= z. 3 Antisymmetry: If x <= yandy <= xthenx == y. 47 Induction Data Types Type Classes Functors Homework Ord Ord is a type class for inequality comparison: class Ord a where (<=) :: a -> a -> Bool What laws should instances satisfy?
For all x, y, and z:
1 Reflexivity: x <= x. 2 Transitivity: If x <= yandy <= zthenx <= z. 3 Antisymmetry: If x <= yandy <= xthenx == y. 4 Totality: Eitherx <= yory <= x 48 Induction Data Types Type Classes Functors Homework Ord Ord is a type class for inequality comparison: class Ord a where (<=) :: a -> a -> Bool What laws should instances satisfy?
For all x, y, and z:
1 Reflexivity: x <= x. 2 Transitivity: If x <= yandy <= zthenx <= z. 3 Antisymmetry: If x <= yandy <= xthenx == y. 4 Totality: Eitherx <= yory <= x Relations that satisfy these four properties are called total orders. Without the fourth (totality), they are called partial orders. 49 Induction Data Types Type Classes Functors Homework Eq Eq is a type class for equality or equivalence: class Eq a where (==) :: a -> a -> Bool What laws should instances satisfy?

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Induction Data Types Type Classes Functors Homework
Eq
Eq is a type class for equality or equivalence: class Eq a where
(==) :: a -> a -> Bool What laws should instances satisfy?
For all x, y, and z:
1 Reflexivity: x == x.

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Induction Data Types Type Classes Functors Homework
Eq
Eq is a type class for equality or equivalence: class Eq a where
(==) :: a -> a -> Bool What laws should instances satisfy?
For all x, y, and z:
1 Reflexivity: x == x.
2 Transitivity: If x == yandy == zthenx == z.

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Induction Data Types Type Classes Functors Homework
Eq
Eq is a type class for equality or equivalence: class Eq a where
(==) :: a -> a -> Bool What laws should instances satisfy?
For all x, y, and z:
1 Reflexivity: x == x.
2 Transitivity: If x == yandy == zthenx == z.
3 Symmetry: If x == ytheny == x.

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Induction Data Types Type Classes Functors Homework
Eq
Eq is a type class for equality or equivalence: class Eq a where
(==) :: a -> a -> Bool What laws should instances satisfy?
For all x, y, and z:
1 Reflexivity: x == x.
2 Transitivity: If x == yandy == zthenx == z.
3 Symmetry: If x == ytheny == x.
Relations that satisfy these are called equivalence relations.

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Induction Data Types Type Classes Functors Homework
Eq
Eq is a type class for equality or equivalence: class Eq a where
(==) :: a -> a -> Bool What laws should instances satisfy?
For all x, y, and z:
1 Reflexivity: x == x.
2 Transitivity: If x == yandy == zthenx == z.
3 Symmetry: If x == ytheny == x.
Relations that satisfy these are called equivalence relations.
Some argue that the Eq class should be only for equality, requiring stricter laws like:
Ifx == ythenf x == f yforallfunctionsf
But this is debated.

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Induction Data Types Type Classes Functors Homework
Types and Values
Haskell is actually comprised of two languages.
The value-level language, consisting of expressions such as if, let, 3 etc.

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Induction Data Types Type Classes Functors Homework
Types and Values
Haskell is actually comprised of two languages.
The value-level language, consisting of expressions such as if, let, 3 etc.
The type-level language, consisting of types Int, Bool, synonyms like String, and type constructors like Maybe, (->), [ ] etc.

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Induction Data Types Type Classes Functors Homework
Types and Values
Haskell is actually comprised of two languages.
The value-level language, consisting of expressions such as if, let, 3 etc.
The type-level language, consisting of types Int, Bool, synonyms like String, and type constructors like Maybe, (->), [ ] etc.
This type level language itself has a type system!

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Induction Data Types Type Classes Functors Homework
Kinds
Just as terms in the value level language are given types, terms in the type level language are given kinds.
The most basic kind is written as *.
Types such as Int and Bool have kind *.
Seeing as Maybe is parameterised by one argument, Maybe has kind * -> *: given a type (e.g. Int), it will return a type (Maybe Int).

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Induction Data Types Type Classes Functors Homework
Suppose we have a function:
Lists
toString :: Int -> String
And we also have a function to give us some numbers: getNumbers :: Seed -> [Int]
How can I compose toString with getNumbers to get a function f of type Seed -> [String]?

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Induction Data Types Type Classes Functors Homework
Suppose we have a function:
toString :: Int -> String
Lists
And we also have a function to give us some numbers:
getNumbers :: Seed -> [Int]
How can I compose toString with getNumbers to get a function f of type Seed -> [String]?
Answer: we use map:
f = map toString . getNumbers

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Induction Data Types Type Classes Functors Homework
Suppose we have a function:
Maybe
toString :: Int -> String
And we also have a function that may give us a number: tryNumber :: Seed -> Maybe Int
How can I compose toString with tryNumber to get a function f of type Seed -> Maybe String?

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Induction Data Types Type Classes Functors Homework
Suppose we have a function:
toString :: Int -> String
Maybe
And we also have a function that may give us a number:
tryNumber :: Seed -> Maybe Int
How can I compose toString with tryNumber to get a function f of type Seed -> Maybe String?
We want a map function but for the Maybe type: f = maybeMap toString . tryNumber
Let’s implement it.

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Induction Data Types Type Classes Functors Homework
Functor
All of these functions are in the interface of a single type class, called Functor.
class Functor f where
fmap :: (a -> b) -> f a -> f b

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Induction Data Types Type Classes Functors Homework
Functor
All of these functions are in the interface of a single type class, called Functor. class Functor f where
fmap :: (a -> b) -> f a -> f b
Unlike previous type classes we’ve seen like Ord and Semigroup, Functor is over types of kind * -> *.

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Induction Data Types Type Classes Functors Homework
Functor
All of these functions are in the interface of a single type class, called Functor. class Functor f where
fmap :: (a -> b) -> f a -> f b
Unlike previous type classes we’ve seen like Ord and Semigroup, Functor is over types
of kind * -> *.
Instances for: Lists
Maybe
Tuples (how?)
Functions (how?)
Demonstrate in live-coding

Induction Data Types Type Classes Functors Homework
Functor Laws
The functor type class must obey two laws:
Functor Laws
1 fmap id == id
2 fmap f . fmap g == fmap (f . g)
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Induction Data Types Type Classes Functors Homework
Functor Laws
The functor type class must obey two laws:
Functor Laws
1 fmap id == id
2 fmap f . fmap g == fmap (f . g)
In Haskell’s type system it’s impossible to make a total fmap function that satisfies the first law but violates the second.
This is due to parametricity, a property we will return to in Week 8 or 9
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Induction
Data Types Type Classes Functors Homework
Homework
1 Do the first programming exercise, and ask us on Piazza if you get stuck. It will be due in exactly 1 week from the start of the Friday lecture.
2 Last week’s quiz is due this Friday. Make sure you submit your answers.
3 This week’s quiz is also up, due next Friday (the Friday after this one).

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