Type Families with Class, Type Classes with Family · Haskell Symposium 2015. Motivation Type-level...

Type Families with Class,Type Classes with Family

Alejandro Serrano1 Jurriaan Hage1 Patrick Bahr2

1Utrecht University

2IT University of Copenhagen

Haskell Symposium 2015

Motivation

Type-level programming in Haskell/GHC

I functional dependencies

I type families

I data type promotion, kind polymorphism

I closed type families

I use type families to simulate type classes

I gain flexibility and expressiveness

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Motivation

Type-level programming in Haskell/GHC

I functional dependencies

I type families

I data type promotion, kind polymorphism

I closed type families

I use type families to simulate type classes

I gain flexibility and expressiveness

2 / 18

Overview

I Simulate type classes using type families(without class methods)

I What can we do with this encoding?

I Proposal: type families with elaboration

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Encoding type classes

Simplification (for now): type class = predicate on types

class C t1 ... tn IsC :: κ1 → ...→ κn → Bool

Instead of Bool we use:

data Defined = Yes | No

Type family declaration

type family IsC t1 ... tn :: Defined

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class C t1 ... tn IsC :: κ1 → ...→ κn → Defined

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class C t1 ... tn IsC :: κ1 → ...→ κn → Defined

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Example: Eq

Type class declaration

class Eq a where(≡) :: a→ a→ Bool

Translation into type family declaration

type family IsEq (a :: ∗) :: Defined

type Eq a = IsEq a ∼ Yes

f :: Eq a ⇒ a→ a

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Example: Eq

class Eq a where

(≡) :: a→ a→ Bool

f :: Eq a ⇒ a→ a

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Example: Eq

class Eq a where

(≡) :: a→ a→ Bool

f :: Eq a ⇒ a→ a

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Example: Eq

class Eq a where

(≡) :: a→ a→ Bool

f :: Eq a ⇒ a→ a

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Example: Eq

class Eq a where

(≡) :: a→ a→ Bool

f :: IsEq a ∼ Yes ⇒ a→ a

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Example: Eq

class Eq a where

(≡) :: a→ a→ Bool

f :: IsEq a ∼ Yes ⇒ a→ a

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Example: Eq

class Eq a where

(≡) :: a→ a→ Bool

f :: Eq a ⇒ a→ a

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Example: Eq

Type class instance declarations

instance Eq Int where ...

instance Eq a⇒ Eq [a ] where ...

instance (Eq a,Eq b)⇒ Eq (a, b) where ...

Type family instance declarations

type instance IsEq Int = Yes

type instance IsEq [a ] = IsEq a

type instance IsEq (a, b) = And (IsEq a) (IsEq b)

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Example: Eq

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Example: Eq

type family And (x :: Defined) (y :: Defined) :: Defined whereAnd Yes Yes = YesAnd x y = No

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What Can We Express

I super classes

I limited forms functional dependencies(requires injective type families)

I not supported: overlapping instances

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What else can we do with this encoding?

I type class directives

I custom error messages

I instance chains

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What else can we do with this encoding?

I type class directives

I custom error messages

I instance chains

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Non-membership

type instance IsShow (a→ b) = No

Couldn’t match type ’No with ’Yes

Expected type: ’Yes

Actual type: IsShow (t -> t)

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Non-membership

type instance IsShow (a→ b) = No

Couldn’t match type ’No with ’Yes

Actual type: IsShow (t -> t)

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Closed set of instances

type family IsIntegral t whereIsIntegral Int = YesIsIntegral Integer = YesIsIntegral t = No

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Disjointness

data IntegralOrFractional = Integral | Fractional | None

type family IsIntegralOrFractional t :: IntegralOrFractional

type instance IsIntegralOrFractional Int = Integraltype instance IsIntegralOrFractional Integer = Integraltype instance IsIntegralOrFractional Double = Fractional

type IsIntegral t = IsIntegral ′ (IsIntegralOrFractional t)

type family IsIntegral ′ c :: Defined whereIsIntegral ′ Integral = YesIsIntegral ′ c = No

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Disjointness

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Disjointness

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Custom Error Messages

data Defined

= Yes | No

Example

type family IsIntegral ′ c :: Defined Symbol whereIsIntegral ′ Integral = YesIsIntegral ′ Fractional = No "is instance of Factional"

IsIntegral ′ c = No ""

Couldn’t match type ’No "is instance of Fractional"

with ’Yes

Actual type: IsIntegral Double

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data Defined e = Yes | No e

Example

with ’Yes

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Example

with ’Yes

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Example

with ’Yes

Actual type: IsIntegral Double12 / 18

Instance Chains [Morris & Jones]

I explicit failure

I provide alternatives (else clause)

Example

instance Show (a→ b) if (Enum a, Show a,Show b) whereshow = ...

else instance Show (a→ b) fails

As type family

type instance IsShow (a→ b) = IsShowFn a btype family IsShowFn a b

= And3 (IsEnum a) (IsShow a) (IsShow b)

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I explicit failure

Example

instance Show (a→ b) if (Enum a, Show a, Show b) whereshow = ...

As type family

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I explicit failure

Example

instance Show (a→ b) if (Enum a, Show a, Show b) whereshow = ...

As type family

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What about class methods?

Two possibilities:

I Use type classes

I Extend type families with elaboration

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Two possibilities:

I Use type classes

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Two possibilities:

I Use type classes

data Defined e

| No e

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Two possibilities:

I Use type classes

data Defined e p = Yes p | No e

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Two possibilities:

I Use type classes

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Two possibilities:

I Use type classes

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Elaboration at Rewriting

assign a dictionary to type families

Example

type family IsEq (t :: ∗) :: Defineddictionary eq :: t → t → Bool

type instance IsEq Int = Yesdictionary eq = primEqInt -- the primitive Int comparison

type instance IsEq [a ] = e@(IsEq a)dictionary eq [ ] [ ] = True

eq (x : xs) (y : ys) = e@eq x y ∧ eq xs yseq = False

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Example

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Example

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Example

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Example: Data Types a la Carte

type family f :<: g :: Defineddictionary inj :: f a→ g awhere e :<: e = Yes dictionary inj = id

f :<: (x :+: y) = d@(Choose f x y l@(f :<: x) r@(f :<: y))dictionary inj = d@choice l@inj r@inj

f :<: g = No

type family Choose f x y fx fy :: Defineddictionary choice :: (f a→ x a)→ (f a→ y a)→ f a→ (x :+: y) awhere Choose f x y Yes fy = Yes

dictionary choice x y = Inl ◦ xChoose f x y fx Yes = Yes

dictionary choice x y = Inr ◦ yChoose f x y fx fy = No

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Conclusions

Future work

I study elaboration further

I use encoding as implementation of type classes?

In the paper

I encoding of functional dependencies

I encoding of superclasses

I more examples

I proof of soundness & completeness (using OutsideIn(X))

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Conclusions

Future work

I study elaboration further

I use encoding as implementation of type classes?

In the paper

I encoding of functional dependencies

I encoding of superclasses

I more examples

I proof of soundness & completeness (using OutsideIn(X))

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Type Families with Class,Type Classes with Family

Alejandro Serrano1 Jurriaan Hage1 Patrick Bahr2

1Utrecht University

2IT University of Copenhagen

Haskell Symposium 2015

Type Families with Class, Type Classes with Family · Haskell Symposium 2015. Motivation Type-level...

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