# Notes for ‘Thinking with Types: Type-level Programming in Haskell’, Chapters 1–5

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Haskell—with its powerful type system—has a great support for type-level programming and it has gotten much better in the recent times with the new releases of the GHC compiler. But type-level programming remains a daunting topic even with seasoned haskellers. *Thinking with Types: Type-level Programming in Haskell* by Sandy Maguire is a book which attempts to fix that. I’ve taken some notes to summarize my understanding of the same.

## Introduction

- Type-level Programming (TLP) is writing programs that run at compile-time, unlike term-level programming which is writing programs that run at run-time.
- TLP should be used in moderation.
- TLP should be mostly used
- for programs that are catastrophic to get wrong (finance, healthcare, etc).
- when it simplifies the program API massively.
- when power-to-weight ratio of adding TLP is high.

- Types are not a silver bullet for fixing all errors:
- Correct programs can be not well-typed.
- It can be hard to assign type for useful programs. e.g.
`printf`

from C.

- Types can turn possible runtime errors into compile-time errors.

## Chapter 1. The Algebra Behind Types

### Isomorphisms and Cardinalities

*Cardinality*of a type is the number of values it can have ignoring bottoms. The values of a type are also called the*inhabitants*of the type.

```
data Void
-- no possible values. cardinality: 0
data Unit = Unit
-- only one possible value. cardinality: 1
data Bool = True | False
-- only two possible values. cardinality: 2
```

- Cardinality is written using notation:
`|Void| = 0`

- Two types are said to be
*Isomorphic*if they have same cardinality. - An
*isomorphism*between types`a`

and`b`

is a pair of functions`to`

and`from`

such that:

```
to :: a -> b
from :: b -> a
. from = id
to . to = id from
```

### Sum, Product and Exponential Types

`Either a b`

is a*Sum*type. Its number of inhabitants is sum of the number of inhabitants of type`a`

and`b`

like so:`|a|`

possible values with the`Left`

constructor and`|b|`

possible values with the`Right`

constructor. Formally:

`|Either a b| = |a| + |b|`

`(a, b)`

is a*Product*type. Its number of inhabitant is the product of the number of inhabitants of types`a`

and`b`

. Formally:

`|(a, b)| = |a| * |b|`

- Some more examples:

```
|Maybe a| = |Nothing| + |Just a| = 1 + |a|
|[a]| = 1 + |a| + |a|^2 + |a|^3 + ...
|Either a Void| = |a| + 0 = |a|
|Either Void a| = 0 + |a| = |a|
|(a, Unit)| = |a| * 1 = |a|
|(Unit, a)| = 1 * |a| = |a|
```

- Function types are exponentiation types.

`|a -> b| = |b|^|a|`

For every value in domain `a`

there can be `|b|`

possible values in the range `b`

. And there are `|a|`

possible values in domain `a`

. So:

```
|a -> b|
= |b| * |b| * ... * |b| -- (|a| times)
= |b|^|a|
```

- Data can be represented in many possible isomorphic types. Some of them are more useful than others. Example:

```
data TicTacToe1 a = TicTacToe1
topLeft :: a
{ topCenter :: a
, topRight :: a
, middleLeft :: a
, middleCenter :: a
, middleRight :: a
, bottomLeft :: a
, bottomCenter :: a
, bottomRight :: a
,
}
|TicTacToe1 a|
= |a| * |a| * ... * |a| -- 9 times
= |a|^9
emptyBoard1 :: TicTacToe1 (Maybe Bool)
=
emptyBoard1 TicTacToe1 Nothing Nothing Nothing
Nothing Nothing Nothing
Nothing Nothing Nothing
-- Alternatively
data Three = One | Two | Three
data TicTacToe2 a =
TicTacToe2 (Three -> Three -> a)
|TicTacToe2 a| = |a|^(|Three| * |Three|)
= |a|^(3*3)
= |a|^9
emptyBoard2 :: TicTacToe2 (Maybe Bool)
=
emptyBoard2 TicTacToe2 $ const $ const Nothing
```

### The Curry-Howard Isomorphism

- Every logic statement can be expressed as an equivalent computer program.
- Helps us analyze mathematical theorems through programming.

### Canonical Representations

- Since multiple equivalent representations of a type are possible, the representation in form of sum of products is considered the canonical representation of the type. Example:

```
Either a (Either b (c, d)) -- canonical
Bool) -- not canonical
(a, Either a a
-- same cardinality as above but canonical
```

## Chapter 2. Terms, Types and Kinds

### The Kind System

*Terms*are things manipulated at runtime.*Types*of terms are used by compiler to prove “things” about the terms.- Similarly,
*Types*are things manipulated at compile-time.*Kinds*of types are used by the compiler to prove “things” about the types. - Kinds are “the types of the Types”.
- Kind of things that can exist at runtime (terms) is
`*`

. That is, kind of`Int`

,`String`

etc is`*`

.

```
> :type True
True :: Bool> :kind Bool
Bool :: *
```

- There are kinds other than
`*`

. For example:

```
> :kind Show Int
Show Int :: Constraint
```

- Higher-kinded types have
`(->)`

in their kind signature:

```
> :kind Maybe
Maybe :: * -> *> :kind Maybe Int
Maybe Int :: *
> :type Control.Monad.Trans.Maybe.MaybeT
Control.Monad.Trans.Maybe.MaybeT
:: m (Maybe a) -> Control.Monad.Trans.Maybe.MaybeT m a> :kind Control.Monad.Trans.Maybe.MaybeT
Control.Monad.Trans.Maybe.MaybeT :: (* -> *) -> * -> *> :kind Control.Monad.Trans.Maybe.MaybeT IO Int
Control.Monad.Trans.Maybe.MaybeT IO Int :: *
```

### Data Kinds

`DataKinds`

extension lets us create new kinds.- It lifts data constructors into type constructors and types into kinds.

```
> :set -XDataKinds
> data Allow = Yes | No
> :type Yes -- Yes is data constructor
Yes :: Allow> :kind Allow -- Allow is a type
Allow :: *> :kind 'Yes -- 'Yes is a type too. Its kind is 'Allow.
'Yes :: Allow
```

- Lifted constructors and types are written with a preceding
`'`

(called*tick*).

### Promotion of Built-In Types

`DataKinds`

extension promotes built-in types too.- Strings are promoted to the kind
`Symbol`

. - Natural numbers are promoted to the kind
`Nat`

.

```
> :kind "hi" -- "hi" is a type-level string
"hi" :: GHC.Types.Symbol> :kind 123 -- 123 is a type-level natural number
123 :: GHC.Types.Nat
```

- We can do type level operations on
`Symbol`

s and`Nat`

s.

```
> :m +GHC.TypeLits
> :kind AppendSymbol
AppendSymbol :: Symbol -> Symbol -> Symbol> :kind! AppendSymbol "hello " "there"
AppendSymbol "hello " "there" :: Symbol
= "hello there"> :set -XTypeOperators
> :kind! (1 + 2) ^ 7
(1 + 2) ^ 7 :: Nat = 2187
```

`TypeOperators`

extension is needed for applying type-level functions with symbolic identifiers.- There are type-level lists and tuples:

```
> :kind '[ 'True ]
'[ 'True ] :: [Bool]> :kind '[1,2,3]
'[1,2,3] :: [Nat]> :kind '["abc"]
'["abc"] :: [Symbol]> :kind 'False ': 'True ': '[]
'False ': 'True ': '[] :: [Bool]> :kind '(6, "x", 'False)
'(6, "x", 'False) :: (Nat, Symbol, Bool)
```

### Type-level Functions

- With the
`TypeFamilies`

extension, it’s possible to write new type-level functions as closed type families:

```
> :set -XDataKinds
> :set -XTypeFamilies
> :{
> type family And (x :: Bool) (y :: Bool) :: Bool where
> And 'True 'True = 'True
> And _ _ = 'False
> :}
> :kind And
And :: Bool -> Bool -> Bool> :kind! And 'True 'False
And 'True 'False :: Bool
= 'False> :kind! And 'True 'True
And 'True 'True :: Bool
= 'True> :kind! And 'False 'True
And 'False 'True :: Bool = 'False
```

## Chapter 3. Variance

- There are three types of
*Variance*(`T`

here a type of kind`* -> *`

):- Covariant: any function of type
`a -> b`

can be lifted into a function of type`T a -> T b`

. Covariant types are instances of the`Functor`

typeclass:

`class Functor f where fmap :: (a -> b) -> f a -> f b`

- Contravariant: any function of type
`a -> b`

can be lifted into a function of type`T b -> T a`

. Contravariant functions are instances of the`Contravariant`

typeclass:

`class Contravariant f where contramap :: (a -> b) -> f b -> f a`

- Invariant: no function of type
`a -> b`

can be lifted into a function of type`T a`

. Invariant functions are instances of the`Invariant`

typeclass:

`class Invariant f where invmap :: (a -> b) -> (b -> a) -> f a -> f b`

- Covariant: any function of type
- Variance of a type
`T`

is specified with respect to a particular type parameter. A type`T`

with two parameters`a`

and`b`

could be covariant wrt.`a`

and contravariant wrt.`b`

. - Variance of a type
`T`

wrt. a particular type parameter is determined by whether the parameter appears in positive or negative*position*s.- If a type parameter appears on the left-hand side of a function, it is said to be in a negative position. Else it is said to be in a positive position.
- If a type parameter appears only in positive positions then the type is covariant wrt. that parameter.
- If a type parameter appears only in negative positions then the type is contravariant wrt. that parameter.
- If a type parameter appears in both positive and negative positions then the type is invariant wrt. that parameter.
- positions follow the laws of multiplication for their
*signs*.

a | b | a * b |
---|---|---|

+ | + | + |

+ | - | - |

- | + | - |

- | - | + |

- Examples:

```
newtype T1 a = T1 (Int -> a)
-- a is in +ve position, T1 is covariant wrt. a.
newtype T2 a = T2 (a -> Int)
-- a is in -ve position, T2 is contravariant wrt. a.
newtype T3 a = T3 (a -> a)
-- a is in both -ve and +ve position. T3 is invariant wrt. a.
newtype T4 a = T4 ((Int -> a) -> Int)
-- a is in +ve position but (Int -> a) is in -ve position.
-- So a is in -ve position overall. T4 is contravariant wrt. a.
newtype T5 a = T5 ((a -> Int) -> Int)
-- a is in -ve position but (a -> Int) is in -ve position.
-- So a is in +ve position overall. T5 is covariant wrt. a.
```

- Covariant parameters are said to be
*produced*or*owned*by the type. - Contravariant parameters are said to be
*consumed*by the type. - A type that has two parameters and is covariant in both of them is an instance of
`BiFunctor`

. - A type that has two parameters and is contravariant in first parameter and covariant in second parameter is an instance of
`Profunctor`

.

## Chapter 4. Working with Types

- Standard Haskell has no notion of scopes for types.
`ScopedTypeVariables`

extension lets us bind type variables to a scope. It requires an explicitly`forall`

quantifier in type signatures.

```
> -- This does not compile.
> :{
> comp :: (a -> b) -> (b -> c) -> a -> c
> comp f g a = go f
> where
> go :: (a -> b) -> c
> go f' = g (f' a)
> :}
<interactive>:11:11: error:
• Couldn't match expected type ‘c1’ with actual type ‘c’
‘c1’ is a rigid type variable bound by
the type signature for:
go :: forall a1 b1 c1. (a1 -> b1) -> c1
at <interactive>:10:3-21
‘c’ is a rigid type variable bound by
the type signature for:
comp :: forall a b c. (a -> b) -> (b -> c) -> a -> c
at <interactive>:7:1-38
• In the expression: g (f' a)
<interactive>:11:14: error:
• Couldn't match expected type ‘b’ with actual type ‘b1’
‘b1’ is a rigid type variable bound by
the type signature for:
go :: forall a1 b1 c1. (a1 -> b1) -> c1
at <interactive>:10:3-21
‘b’ is a rigid type variable bound by
the type signature for:
comp :: forall a b c. (a -> b) -> (b -> c) -> a -> c
at <interactive>:7:1-38
• In the first argument of ‘g’, namely ‘(f' a)’
<interactive>:11:17: error:
• Couldn't match expected type ‘a1’ with actual type ‘a’
‘a1’ is a rigid type variable bound by
the type signature for:
go :: forall a1 b1 c1. (a1 -> b1) -> c1
at <interactive>:10:3-21
‘a’ is a rigid type variable bound by
the type signature for:
comp :: forall a b c. (a -> b) -> (b -> c) -> a -> c
at <interactive>:7:1-38
• In the first argument of ‘f'’, namely ‘a’
> -- But this does.
> :set -XScopedTypeVariables
> :{
> comp :: forall a b c. (a -> b) -> (b -> c) -> a -> c
> comp f g a = go f
> where
> go :: (a -> b) -> c
> go f' = g (f' a)
> :}
```

`TypeApplications`

extension lets us directly apply types to expressions:

```
> :set -XTypeApplications
> :type traverse
traverse
:: (Traversable t, Applicative f) =>
(a -> f b) -> t a -> f (t b)> :type traverse @Maybe
traverse @Maybe
:: Applicative f =>
(a -> f b) -> Maybe a -> f (Maybe b)> :type traverse @Maybe @[]
traverse @Maybe @[]
:: (a -> [b]) -> Maybe a -> [Maybe b]> :type traverse @Maybe @[] @Int
traverse @Maybe @[] @Int
:: (Int -> [b]) -> Maybe Int -> [Maybe b]> :type traverse @Maybe @[] @Int @String
traverse @Maybe @[] @Int @String :: (Int -> [String]) -> Maybe Int -> [Maybe String]
```

- Types are applied in the order they appear in the type signature. It is possible to avoid applying types by using a type with an underscore:
`@_`

```
> :type traverse @Maybe @_ @_ @String
traverse @Maybe @_ @_ @String
:: Applicative w1 => (w2 -> w1 String) -> Maybe w2 -> w1 (Maybe String)
```

- Sometimes the compiler cannot infer the type of an expression.
`AllowAmbiguousTypes`

extension allow such programs to compile.

```
> :set -XScopedTypeVariables
> :{
> f :: forall a. Show a => Bool
> f = True
> :}
<interactive>:7:6: error:
• Could not deduce (Show a0)
from the context: Show a
bound by the type signature for:
f :: forall a. Show a => Bool
at <interactive>:7:6-29
The type variable ‘a0’ is ambiguous
• In the ambiguity check for ‘f’
To defer the ambiguity check to use sites, enable AllowAmbiguousTypes In the type signature: f :: forall a. Show a => Bool
```

`Proxy`

is a type isomorphic to`()`

except with a phantom type parameter:

`data Proxy a = Proxy`

- With all the three extensions enabled, it is possible to get a term-level representation of types using the
`Data.Typeable`

module:

```
> :set -XScopedTypeVariables
> :set -XTypeApplications
> :set -XAllowAmbiguousTypes
> :m +Data.Typeable
> :{
> typeName :: forall a. Typeable a => String
> typeName = show . typeRep $ Proxy @a
> :}
> typeName @String
"[Char]"> typeName @(IO Int)
"IO Int"
```

## Chapter 5. Constraints and GADTs

### Constraints

*Constraints*are a kind different than the types (`*`

).- Constraints are what appear on the left-hand side on the fat context arrow
`=>`

, like`Show a`

.

```
> :k Show
Show :: * -> Constraint> :k Show Int
Show Int :: Constraint> :k (Show Int, Eq String)
(Show Int, Eq String) :: Constraint
```

- Type equalities
`(Int ~ a)`

are another way of creating Constraints.`(Int ~ a)`

says`a`

is same as`Int`

. - Type equalities are
- reflexive:
`a ~ a`

always - symmetrical:
`a ~ b`

implies`b ~ a`

- transitive:
`a ~ b`

and`b ~ c`

implies`a ~ c`

- reflexive:

### GADTs

*GADTs*are Generalized Algebraic DataTypes. They allow writing explicit type signatures for data constructors. Here is the code for a length-typed list using GADTs:

```
> :set -XGADTs
> :set -XKindSignatures
> :set -XTypeOperators
> :set -XDataKinds
> :m +GHC.TypeLits
> :{
> data List (a :: *) (n :: Nat) where
> Nil :: List a 0
> (:~) :: a -> List a n -> List a (n + 1)
> infixr 5 :~
> :}
> :type Nil
Nil :: List a 0> :type 'a' :~ Nil
'a' :~ Nil :: List Char 1> :type 'b' :~ 'a' :~ Nil
'b' :~ 'a' :~ Nil :: List Char 2> :type True :~ 'a' :~ Nil
<interactive>:1:9: error:
• Couldn't match type ‘Char’ with ‘Bool’
Expected type: List Bool 1
Actual type: List Char (0 + 1)
• In the second argument of ‘(:~)’, namely ‘'a' :~ Nil’ In the expression: True :~ 'a' :~ Nil
```

- GADTs are just syntactic sugar for ADTs with type equalities. The above definition is equivalent to:

```
> :set -XGADTs
> :set -XKindSignatures
> :set -XTypeOperators
> :set -XDataKinds
> :m +GHC.TypeLits
> :{
> data List (a :: *) (n :: Nat)
> = (n ~ 0) => Nil
> | a :~ List a (n - 1)
> infixr 5 :~
> :}
> :type 'a' :~ Nil
'a' :~ Nil :: List Char 1> :type 'b' :~ 'a' :~ Nil
'b' :~ 'a' :~ Nil :: List Char 2
```

- Type-safety of this list can be used to write a safe
`head`

function which does not compile for an empty list:

```
> :{
> safeHead :: List a (n + 1) -> a
> safeHead (x :~ _) = x
> :}
> safeHead ('a' :~ 'b' :~ Nil)
'a'> safeHead Nil
<interactive>:21:10: error:
• Couldn't match type ‘1’ with ‘0’
Expected type: List a (0 + 1)
Actual type: List a 0
• In the first argument of ‘safeHead’, namely ‘Nil’
In the expression: safeHead Nil In an equation for ‘it’: it = safeHead Nil
```

### Heterogeneous Lists

We can use GADTs to build heterogeneous lists which can store values of different types and are type-safe to use.

First, the required extensions and imports:

```
{-# LANGUAGE KindSignatures #-}
{-# LANGUAGE DataKinds #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE GADTs #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE TypeApplications #-}
{-# LANGUAGE ScopedTypeVariables #-}
module HList where
import Data.Typeable
```

`HList`

is defined as a GADT:

```
data HList (ts :: [*]) where
HNil :: HList '[]
(:#) :: t -> HList ts -> HList (t ': ts)
infixr 5 :#
```

Example usage:

```
> :type HNil
HNil :: HList '[]> :type 'a' :# HNil
'a' :# HNil :: HList '[Char]> :type True :# 'a' :# HNil
True :# 'a' :# HNil :: HList '[Bool, Char]
```

We can write operations on `HList`

:

```
hLength :: HList ts -> Int
HNil = 0
hLength :# xs) = 1 + hLength xs
hLength (x
hHead :: HList (t ': ts) -> t
:# _) = t hHead (t
```

Example usage:

```
> hLength $ True :# 'a' :# HNil
2> hHead $ True :# 'a' :# HNil
True> hHead HNil
<interactive>:7:7: error:
• Couldn't match type ‘'[]’ with ‘t : ts0’
Expected type: HList (t : ts0)
Actual type: HList '[]
• In the first argument of ‘hHead’, namely ‘HNil’
In the expression: hHead HNil
In an equation for ‘it’: it = hHead HNil • Relevant bindings include it :: t (bound at <interactive>:7:1)
```

We need to define instances of typeclasses like `Eq`

, `Ord`

etc. for `HList`

because GHC cannot derive them automatically yet:

```
instance Eq (HList '[]) where
HNil == HNil = True
instance (Eq t, Eq (HList ts))
=> Eq (HList (t ': ts)) where
:# xs) == (y :# ys) =
(x == y && xs == ys
x
instance Ord (HList '[]) where
HNil `compare` HNil = EQ
instance (Ord t, Ord (HList ts))
=> Ord (HList (t ': ts)) where
:# xs) `compare` (y :# ys) =
(x `compare` y <> xs `compare` ys
x
instance Show (HList '[]) where
show HNil = "[]"
instance (Typeable t, Show t, Show (HList ts))
=> Show (HList (t ': ts)) where
show (x :# xs) =
show x
++ "@" ++ show (typeRep (Proxy @t))
++ " :# " ++ show xs
```

The instances are defined recursively: one for the base case and one for the inductive case.

Example usage:

```
> True :# 'a' :# HNil == True :# 'a' :# HNil
True> True :# 'a' :# HNil == True :# 'b' :# HNil
False> True :# 'a' :# HNil == True :# HNil
<interactive>:17:24: error:
• Couldn't match type ‘'[]’ with ‘'[Char]’
Expected type: HList '[Bool, Char]
Actual type: HList '[Bool]
• In the second argument of ‘(==)’, namely ‘True :# HNil’
In the expression: True :# 'a' :# HNil == True :# HNil
In an equation for ‘it’: it = True :# 'a' :# HNil == True :# HNil> show $ True :# 'a' :# HNil
"True@Bool :# 'a'@Char :# []"
```

### Creating New Constraints

- Type families can be used to create new Constraints:

```
> :set -XKindSignatures
> :set -XDataKinds
> :set -XTypeOperators
> :set -XTypeFamilies
> :m +Data.Constraint
> :{
> type family AllEq (ts :: [*]) :: Constraint where
> AllEq '[] = ()
> AllEq (t ': ts) = (Eq t, AllEq ts)
> :}
> :kind! AllEq '[Bool, Char]
AllEq '[Bool, Char] :: Constraint = (Eq Bool, (Eq Char, () :: Constraint))
```

`AllEq`

is a type-level function from a list of types to a constraint.- With the
`ConstraintKinds`

extension,`AllEq`

can be made polymorphic over all constraints instead of just`Eq`

:

```
> :set -XConstraintKinds
> :{
> type family All (c :: * -> Constraint)
> (ts :: [*]) :: Constraint where
> All c '[] = ()
> All c (t ': ts) = (c t, All c ts)
> :}
```

- With
`All`

, instances for`HList`

can be written non-recursively:

```
instance All Eq ts => Eq (HList ts) where
HNil == HNil = True
:# as) == (b :# bs) = a == b && as == bs (a
```

## Conclusion

I’m still in the process of reading the book and I’ll post the notes for the rest of the chapters in a later post. The complete code for `HList`

can be found here.

## Got suggestions, corrections, or thoughts? Post a comment!

## 5 comments

## sharno

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## Abhinav Sarkar