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u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Faculty of Science

Composing and Decomposing Data Types

A Closed Type Families Implementation

• f Data Types `

a la Carte Patrick Bahr

University of Copenhagen, Department of Computer Science paba@di.ku.dk

10th ACM SIGPLAN Workshop on Generic Programming, 31st August, 2014 Slide 1

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Introduction

Experimenting with Closed Type Families

• What can we do with them?
• How do they compare to type classes?
• How do they interact with type classes?

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 2

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Introduction

Experimenting with Closed Type Families

• What can we do with them?
• How do they compare to type classes?
• How do they interact with type classes?

Application: Data Types ` a la Carte

Specifically: the subtyping constraint : ≺:

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 2

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Introduction

Experimenting with Closed Type Families

• What can we do with them?
• How do they compare to type classes?
• How do they interact with type classes?

Application: Data Types ` a la Carte

Specifically: the subtyping constraint : ≺:

• Can we get rid of some of the restrictions?
• Can we improve error messages?
• What price do we have to pay?

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 2

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Data Types ` a la Carte [Swierstra 2008]

Idea: Decompose data types into two-level types:

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 3

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Data Types ` a la Carte [Swierstra 2008]

Idea: Decompose data types into two-level types:

Recursive data type

data Exp = Val Int | Add Exp Exp

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 3

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Data Types ` a la Carte [Swierstra 2008]

Idea: Decompose data types into two-level types:

Recursive data type

data Exp = Val Int | Add Exp Exp = ⇒

Fixpoint of functor

data Arith a = Val Int | Add a a type Exp = Fix Arith

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 3

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Data Types ` a la Carte [Swierstra 2008]

Idea: Decompose data types into two-level types:

Recursive data type

data Exp = Val Int | Add Exp Exp = ⇒

Fixpoint of functor

data Arith a = Val Int | Add a a type Exp = Fix Arith data Fix f = In (f (Fix f ))

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 3

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Data Types ` a la Carte [Swierstra 2008]

Idea: Decompose data types into two-level types:

Recursive data type

data Exp = Val Int | Add Exp Exp = ⇒

Fixpoint of functor

data Arith a = Val Int | Add a a type Exp = Fix Arith

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 3

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Data Types ` a la Carte [Swierstra 2008]

Idea: Decompose data types into two-level types:

Recursive data type

data Exp = Val Int | Add Exp Exp = ⇒

Fixpoint of functor

data Arith a = Val Int | Add a a type Exp = Fix Arith Functors can be combined by coproduct construction :+:

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 3

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Data Types ` a la Carte [Swierstra 2008]

Idea: Decompose data types into two-level types:

Recursive data type

data Exp = Val Int | Add Exp Exp = ⇒

Fixpoint of functor

data Arith a = Val Int | Add a a type Exp = Fix Arith Functors can be combined by coproduct construction :+: data Mul a = Mul a a type Exp′ = Fix (Arith :+: Mul)

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 3

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Data Types ` a la Carte [Swierstra 2008]

Idea: Decompose data types into two-level types:

Recursive data type

data Exp = Val Int | Add Exp Exp = ⇒

Fixpoint of functor

data Arith a = Val Int | Add a a type Exp = Fix Arith Functors can be combined by coproduct construction :+: data (f :+: g) a = Inl (f a) | Inr (g a) data Mul a = Mul a a type Exp′ = Fix (Arith :+: Mul)

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 3

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Data Types ` a la Carte (cont.)

Subtyping constraint : ≺:

class f : ≺: g where inj :: f a → g a prj :: g a → Maybe (f a)

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 4

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Data Types ` a la Carte (cont.)

Subtyping constraint : ≺:

class f : ≺: g where inj :: f a → g a prj :: g a → Maybe (f a) e.g. Mul : ≺: Arith :+: Mul

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 4

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Data Types ` a la Carte (cont.)

Subtyping constraint : ≺:

class f : ≺: g where inj :: f a → g a prj :: g a → Maybe (f a) e.g. Mul : ≺: Arith :+: Mul

Example: smart constructors

add :: (Arith : ≺: f ) ⇒ Fix f → Fix f → Fix f add x y = In (inj (Add x y))

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 4

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Data Types ` a la Carte (cont.)

Subtyping constraint : ≺:

class f : ≺: g where inj :: f a → g a prj :: g a → Maybe (f a) e.g. Mul : ≺: Arith :+: Mul

Example: smart constructors

add :: (Arith : ≺: f ) ⇒ Fix f → Fix f → Fix f add x y = In (inj (Add x y)) exp :: Fix (Arith :+: Mul) exp = val 1 ‘add‘ (val 2 ‘mul‘ val 3)

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 4

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Limitations of : ≺:

Definition of : ≺:

instance f : ≺: f where . . . instance (f : ≺: f1) ⇒ f : ≺: (f1 :+: f2) where . . . instance (f : ≺: f2) ⇒ f : ≺: (f1 :+: f2) where . . .

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 5

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Limitations of : ≺:

Definition of : ≺:

instance f : ≺: f where . . . instance f : ≺: (f :+: f2) where . . . instance (f : ≺: f2) ⇒ f : ≺: (f1 :+: f2) where . . .

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 5

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Limitations of : ≺:

Definition of : ≺:

instance f : ≺: f where . . . instance f : ≺: (f :+: f2) where . . . instance (f : ≺: f2) ⇒ f : ≺: (f1 :+: f2) where . . .

• Asymmetric treatment of :+:
• Left-hand side is not inspected
• Ambiguity

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 5

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Limitations of : ≺:

Definition of : ≺:

instance f : ≺: f where . . . instance f : ≺: (f :+: f2) where . . . instance (f : ≺: f2) ⇒ f : ≺: (f1 :+: f2) where . . .

• Asymmetric treatment of :+:

A : ≺: A :+: (B :+: C)

• Left-hand side is not inspected
• Ambiguity

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 5

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Limitations of : ≺:

Definition of : ≺:

instance f : ≺: f where . . . instance f : ≺: (f :+: f2) where . . . instance (f : ≺: f2) ⇒ f : ≺: (f1 :+: f2) where . . .

• Asymmetric treatment of :+:

A : ≺: (A :+: B) :+: C

• Left-hand side is not inspected
• Ambiguity

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 5

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Limitations of : ≺:

Definition of : ≺:

instance f : ≺: f where . . . instance f : ≺: (f :+: f2) where . . . instance (f : ≺: f2) ⇒ f : ≺: (f1 :+: f2) where . . .

• Asymmetric treatment of :+:

A : ≺: (A :+: B) :+: C

• Left-hand side is not inspected

A :+: B : ≺: (A :+: B) :+: C

• Ambiguity

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 5

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Limitations of : ≺:

Definition of : ≺:

instance f : ≺: f where . . . instance f : ≺: (f :+: f2) where . . . instance (f : ≺: f2) ⇒ f : ≺: (f1 :+: f2) where . . .

• Asymmetric treatment of :+:

A : ≺: (A :+: B) :+: C

• Left-hand side is not inspected

A :+: B : ≺: A :+: (B :+: C)

• Ambiguity

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 5

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Limitations of : ≺:

Definition of : ≺:

instance f : ≺: f where . . . instance f : ≺: (f :+: f2) where . . . instance (f : ≺: f2) ⇒ f : ≺: (f1 :+: f2) where . . .

• Asymmetric treatment of :+:

A : ≺: (A :+: B) :+: C

• Left-hand side is not inspected

A :+: B : ≺: A :+: (B :+: C)

• Ambiguity

A : ≺: A :+: (A :+: B)

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 5

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Contributions We re-implemented : ≺: such that:

• Subtyping behaves as intuitively expected*
• Ambiguous subtyping is avoided
• We can express isomorphism :≃:

*terms and conditions may apply

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 6

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Improved subtyping constraint : ≺:

Subtyping : ≺: behaves as intuitively expected

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 7

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Improved subtyping constraint : ≺:

Subtyping : ≺: behaves as intuitively expected

f : ≺: g ⇐ ⇒ ∃ unique injection from f to g

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 7

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Improved subtyping constraint : ≺:

Subtyping : ≺: behaves as intuitively expected

f : ≺: g ⇐ ⇒ ∃ unique injection from f to g C :+: A : ≺: A :+: B :+: C

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 7

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Improved subtyping constraint : ≺:

Subtyping : ≺: behaves as intuitively expected

f : ≺: g ⇐ ⇒ ∃ unique injection from f to g C :+: A : ≺: A :+: B :+: C

Avoid ambiguous subtyping

Multiple occurrences of signatures are rejected:

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 7

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Improved subtyping constraint : ≺:

Subtyping : ≺: behaves as intuitively expected

f : ≺: g ⇐ ⇒ ∃ unique injection from f to g C :+: A : ≺: A :+: B :+: C

Avoid ambiguous subtyping

Multiple occurrences of signatures are rejected: A : ≺: A :+: A :+: C A :+: A : ≺: A :+: B

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 7

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Improved subtyping constraint : ≺:

Subtyping : ≺: behaves as intuitively expected

f : ≺: g ⇐ ⇒ ∃ unique injection from f to g C :+: A : ≺: A :+: B :+: C

Avoid ambiguous subtyping

Multiple occurrences of signatures are rejected: A : ≺: A :+: A :+: C A :+: A : ≺: A :+: B injection not unique!

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 7

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Improved subtyping constraint : ≺:

Subtyping : ≺: behaves as intuitively expected

f : ≺: g ⇐ ⇒ ∃ unique injection from f to g C :+: A : ≺: A :+: B :+: C

Avoid ambiguous subtyping

Multiple occurrences of signatures are rejected: A : ≺: A :+: A :+: C A :+: A : ≺: A :+: B injection not unique!

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 7

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Improved subtyping constraint : ≺:

Subtyping : ≺: behaves as intuitively expected

f : ≺: g ⇐ ⇒ ∃ unique injection from f to g C :+: A : ≺: A :+: B :+: C

Avoid ambiguous subtyping

Multiple occurrences of signatures are rejected: A : ≺: A :+: A :+: C A :+: A : ≺: A :+: B injection not unique! “injection” not injective!

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 7

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Improved subtyping constraint : ≺:

Subtyping : ≺: behaves as intuitively expected

f : ≺: g ⇐ ⇒ ∃ unique injection from f to g C :+: A : ≺: A :+: B :+: C

Avoid ambiguous subtyping

Multiple occurrences of signatures are rejected: A : ≺: A :+: A :+: C A :+: A : ≺: A :+: B injection not unique! “injection” not injective!

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 7

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Type isomorphism constraint :≃:

We can express isomorphism :≃:

f :≃: g ⇐ ⇒ ∃ unique bijection from f to g

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 8

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Type isomorphism constraint :≃:

We can express isomorphism :≃:

f :≃: g ⇐ ⇒ ∃ unique bijection from f to g Easy to implement: f :≃: g = (f : ≺: g, g : ≺: f )

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 8

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Type isomorphism constraint :≃:

We can express isomorphism :≃:

f :≃: g ⇐ ⇒ ∃ unique bijection from f to g Easy to implement: f :≃: g = (f : ≺: g, g : ≺: f )

Use case: improved projection function

The type of the projection function is unsatisfying: prj :: (f : ≺: g) ⇒ g a → Maybe (f a)

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 8

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Type isomorphism constraint :≃:

We can express isomorphism :≃:

f :≃: g ⇐ ⇒ ∃ unique bijection from f to g Easy to implement: f :≃: g = (f : ≺: g, g : ≺: f )

Use case: improved projection function

The type of the projection function is unsatisfying: prj :: (f : ≺: g) ⇒ g a → Maybe (f a) With :≃: we can do better: split :: (g :≃: f :+: r) ⇒ g a → Either (f a) (r a)

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 8

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Type isomorphism constraint :≃:

We can express isomorphism :≃:

f :≃: g ⇐ ⇒ ∃ unique bijection from f to g Easy to implement: f :≃: g = (f : ≺: g, g : ≺: f )

Use case: improved projection function

The type of the projection function is unsatisfying: prj :: (f : ≺: g) ⇒ g a → Maybe (f a) With :≃: we can do better: split :: (g :≃: f :+: r) ⇒ g a → (f a → b) → (r a → b) → b

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 8

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Implementation of : ≺:

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 9

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Idea

Type-level function Embed:

• take two signatures f , g as arguments
• check whether f :

≺: g

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 10

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Idea

Type-level function Embed:

• take two signatures f , g as arguments
• check whether f :

≺: g Derive implementation of inj and prj: ???

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 10

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Idea

Type-level function Embed:

• take two signatures f , g as arguments
• produce proof object p for f :

≺: g Derive implementation of inj and prj:

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 10

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Idea

Type-level function Embed:

• take two signatures f , g as arguments
• produce proof object p for f :

≺: g Derive implementation of inj and prj:

• also use a type class
• But: use proof object as oracle in instance declarations

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 10

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Idea

Type-level function Embed:

• take two signatures f , g as arguments
• produce proof object p for f :

≺: g Derive implementation of inj and prj:

• also use a type class
• But: use proof object as oracle in instance declarations

No singleton types. This all happens at compile time!

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 10

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Idea

Type-level function Embed:

• take two signatures f , g as arguments
• produce proof object p for f :

≺ ≺: g Derive implementation of inj and prj:

• also use a type class
• But: use proof object as oracle in instance declarations

No singleton types. This all happens at compile time!

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 10

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Idea

Type-level function Embed:

• take two signatures f , g as arguments
• produce proof object p for f :

≺ ≺: g

• check whether p also proves f :

≺: g Derive implementation of inj and prj:

• also use a type class
• But: use proof object as oracle in instance declarations

No singleton types. This all happens at compile time!

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 10

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Proof Objects

Definition

data Prf = Refl | Left Prf | Right Prf | Sum Prf Prf

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 11

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Proof Objects

Definition

data Prf = Refl | Left Prf | Right Prf | Sum Prf Prf Refl : f : ≺ ≺: f

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 11

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Proof Objects

Definition

data Prf = Refl | Left Prf | Right Prf | Sum Prf Prf Refl : f : ≺ ≺: f p : f : ≺ ≺: g1 Left p : f : ≺ ≺: g1 :+: g2 p : f : ≺ ≺: g2 Right p : f : ≺ ≺: g1 :+: g2

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Proof Objects

Definition

data Prf = Refl | Left Prf | Right Prf | Sum Prf Prf Refl : f : ≺ ≺: f p : f : ≺ ≺: g1 Left p : f : ≺ ≺: g1 :+: g2 p : f : ≺ ≺: g2 Right p : f : ≺ ≺: g1 :+: g2 p1 : f1 : ≺ ≺: g p2 : f2 : ≺ ≺: g Sum p1 p2 : f1 :+: f2 : ≺ ≺: g

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 11

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Proof Objects

Definition

data Prf = Refl | Left Prf | Right Prf | Sum Prf Prf Refl : f : ≺ ≺: f p : f : ≺ ≺: g1 Left p : f : ≺ ≺: g1 :+: g2 p : f : ≺ ≺: g2 Right p : f : ≺ ≺: g1 :+: g2 p1 : f1 : ≺ ≺: g p2 : f2 : ≺ ≺: g Sum p1 p2 : f1 :+: f2 : ≺ ≺: g kind

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Proof Objects

Definition

data Prf = Refl | Left Prf | Right Prf | Sum Prf Prf Refl : f : ≺ ≺: f p : f : ≺ ≺: g1 Left p : f : ≺ ≺: g1 :+: g2 p : f : ≺ ≺: g2 Right p : f : ≺ ≺: g1 :+: g2 p1 : f1 : ≺ ≺: g p2 : f2 : ≺ ≺: g Sum p1 p2 : f1 :+: f2 : ≺ ≺: g kind type

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Proof Objects

Definition

data Prf = Refl | Left Prf | Right Prf | Sum Prf Prf Refl : f : ≺ ≺: f p : f : ≺ ≺: g1 Left p : f : ≺ ≺: g1 :+: g2 p : f : ≺ ≺: g2 Right p : f : ≺ ≺: g1 :+: g2 p1 : f1 : ≺ ≺: g p2 : f2 : ≺ ≺: g Sum p1 p2 : f1 :+: f2 : ≺ ≺: g kind type type constructor

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Construct Proof Objects

data Emb = Found Prf | NotFound | Ambiguous

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Construct Proof Objects

data Emb = Found Prf | NotFound | Ambiguous type family Embed (f :: ∗ → ∗) (g :: ∗ → ∗) :: Emb where

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Construct Proof Objects

data Emb = Found Prf | NotFound | Ambiguous type family Embed (f :: ∗ → ∗) (g :: ∗ → ∗) :: Emb where Embed f f = Found Refl Embed (f1 :+: f2) g = Sum′ (Embed f1 g) (Embed f2 g) Embed f (g1 :+: g2) = Choose (Embed f g1) (Embed f g2) Embed f g = NotFound

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u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Construct Proof Objects

data Emb = Found Prf | NotFound | Ambiguous type family Embed (f :: ∗ → ∗) (g :: ∗ → ∗) :: Emb where Embed f f = Found Refl Embed (f1 :+: f2) g = Sum′ (Embed f1 g) (Embed f2 g) Embed f (g1 :+: g2) = Choose (Embed f g1) (Embed f g2) Embed f g = NotFound type family Choose (e1 :: Emb) (e2 :: Emb) :: Emb where Choose (Found p1) (Found p2) = Ambiguous Choose Ambiguous e2 = Ambiguous Choose e1 Ambiguous = Ambiguous Choose (Found p1) e2 = Found (Left p1) Choose e1 (Found p2) = Found (Right p2) Choose NotFound NotFound = NotFound

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Post-Processing

This is almost what we want.

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Post-Processing

This is almost what we want.

• We avoid ambiguity on the right-hand side:

A : ≺: A :+: A :+: C

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 13

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Post-Processing

This is almost what we want.

• We avoid ambiguity on the right-hand side:

A : ≺: A :+: A :+: C

• We still have ambiguity on the left-hand side:

A :+: A : ≺: A :+: B

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 13

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Post-Processing

This is almost what we want.

• We avoid ambiguity on the right-hand side:

A : ≺: A :+: A :+: C

• We still have ambiguity on the left-hand side:

A :+: A : ≺: A :+: B

Solution: check for duplicates in Prf

type family Dupl (p :: Prf ) :: Bool where . . .

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 13

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Post-Processing

This is almost what we want.

• We avoid ambiguity on the right-hand side:

A : ≺: A :+: A :+: C

• We still have ambiguity on the left-hand side:

A :+: A : ≺: A :+: B

Solution: check for duplicates in Prf

type family Dupl (p :: Prf ) :: Bool where . . . Sum (Left Refl) (Left Refl)

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Are we there yet?

• Construct proof p for f :

≺ ≺: g

• Check whether p proves f :

≺: g

• Derive inj and prj

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 14

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Are we there yet?

• Construct proof p for f :

≺ ≺: g

• Check whether p proves f :

≺: g

• Derive inj and prj

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 14

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Are we there yet?

• Construct proof p for f :

≺ ≺: g

• Check whether p proves f :

≺: g

• Derive inj and prj

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 14

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Are we there yet?

• Construct proof p for f :

≺ ≺: g

• Check whether p proves f :

≺: g

• Derive inj and prj

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 14

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Derive inj and prj

class f : ≺: g where inj :: f a → g a prj :: g a → Maybe (f a) instance f : ≺: f where . . . instance f : ≺: (f :+: g2) where . . . instance f : ≺: g2 ⇒ f : ≺: (g1 :+: g2) where . . .

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Derive inj and prj

class f : ≺: g where inj :: f a → g a prj :: g a → Maybe (f a) instance f : ≺: f where . . . instance f : ≺: g1 ⇒ f : ≺: (g1 :+: g2) where . . . instance f : ≺: g2 ⇒ f : ≺: (g1 :+: g2) where . . .

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Derive inj and prj

class f : ≺: g where inj :: f a → g a prj :: g a → Maybe (f a) instance f : ≺: f where . . . instance f : ≺: g1 ⇒ f : ≺: (g1 :+: g2) where . . . instance f : ≺: g2 ⇒ f : ≺: (g1 :+: g2) where . . . instance ( f1 : ≺: g, f2 : ≺: g) ⇒ (f1 :+: f2) : ≺: g where . . .

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 15

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Derive inj and prj

class Sub f g where inj :: f a → g a prj :: g a → Maybe (f a) instance Sub f f where . . . instance Sub f g1 ⇒ Sub f (g1 :+: g2) where . . . instance Sub f g2 ⇒ Sub f (g1 :+: g2) where . . . instance (Sub f1 g, Sub f2 g) ⇒ Sub (f1 :+: f2) g where . . .

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 15

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Derive inj and prj

class Sub (e :: Emb) f g where inj :: f a → g a prj :: g a → Maybe (f a) instance Sub f f where . . . instance Sub f g1 ⇒ Sub f (g1 :+: g2) where . . . instance Sub f g2 ⇒ Sub f (g1 :+: g2) where . . . instance (Sub f1 g, Sub f2 g) ⇒ Sub (f1 :+: f2) g where . . .

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 15

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Derive inj and prj

class Sub (e :: Emb) f g where inj :: f a → g a prj :: g a → Maybe (f a) instance Sub (Found Refl) f f where . . . instance Sub (Found p) f g1 ⇒ Sub (Found (Left p)) f (g1 :+: g2) where . . . instance Sub (Found p) f g2 ⇒ Sub (Found (Right p)) f (g1 :+: g2) where . . . instance (Sub (Found p1) f1 g, Sub (Found p2) f2 g) ⇒ Sub (Found (Sum p1 p2)) (f1 :+: f2) g where . . .

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 15

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Derive inj and prj

class Sub (e :: Emb) f g where inj :: f a → g a prj :: g a → Maybe (f a) type f : ≺: g = Sub (Embed f g) f g instance Sub (Found Refl) f f where . . . instance Sub (Found p) f g1 ⇒ Sub (Found (Left p)) f (g1 :+: g2) where . . . instance Sub (Found p) f g2 ⇒ Sub (Found (Right p)) f (g1 :+: g2) where . . . instance (Sub (Found p1) f1 g, Sub (Found p2) f2 g) ⇒ Sub (Found (Sum p1 p2)) (f1 :+: f2) g where . . .

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 15

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Conclusion

• This approach generalises to similar applications
• Improves type class-based implementation in many

aspects

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 16

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Conclusion

• This approach generalises to similar applications
• Improves type class-based implementation in many

aspects

• But:
• We need a way to customise error messages.
• Compile time performance unpredictable.

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 16

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Conclusion

• This approach generalises to similar applications
• Improves type class-based implementation in many

aspects

• But:
• We need a way to customise error messages.
• Compile time performance unpredictable.
• Implemented in the compdata package

> cabal install compdata

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Discussion

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 17

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Error Messages

• A :

≺: B :+: C ?

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Error Messages

• A :

≺: B :+: C ? No instance for (Sub NotFound A (B :+: C))

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 18

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Error Messages

• A :

≺: B :+: C ? No instance for (Sub NotFound A (B :+: C)) The original implementation would give: No instance for (A :<: C)

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 18

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Error Messages

• A :

≺: B :+: C ? No instance for (Sub NotFound A (B :+: C))

• A :+: A :

≺: A :+: B ? No instance for (Sub Ambiguous (A :+: A) (A :+: B))

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 18

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Error Messages

• A :

≺: B :+: C ? No instance for (Sub NotFound A (B :+: C))

• A :+: A :

≺: A :+: B ? No instance for (Sub Ambiguous (A :+: A) (A :+: B))

• A :

≺: A :+: B ?

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 18

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Error Messages

• A :

≺: B :+: C ? No instance for (Sub NotFound A (B :+: C))

• A :+: A :

≺: A :+: B ? No instance for (Sub Ambiguous (A :+: A) (A :+: B))

• a :

≺: a :+: B ?

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 18

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Error Messages

• A :

≺: B :+: C ? No instance for (Sub NotFound A (B :+: C))

• A :+: A :

≺: A :+: B ? No instance for (Sub Ambiguous (A :+: A) (A :+: B))

• a :

≺: a :+: B ? No instance for (Sub (Post (Embed a (a :+: B))) a (a :+: B))

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 18

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Compile Time Performance

• If done “wrong”, this implementation can be very slow!

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 19

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Compile Time Performance

• If done “wrong”, this implementation can be very slow!
• Implementation presented here: O(n2)

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 19

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Compile Time Performance

• If done “wrong”, this implementation can be very slow!
• Implementation presented here: O(n2)
• Slightly different implementation: O(2n)

(but essentially the same)

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 19

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Compile Time Performance

• If done “wrong”, this implementation can be very slow!
• Implementation presented here: O(n2)
• Slightly different implementation: O(2n)

(but essentially the same)

• micro benchmark:
• derive F :

≺: G

• 9 summands in F and G

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 19

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Compile Time Performance

• If done “wrong”, this implementation can be very slow!
• Implementation presented here: O(n2)
• Slightly different implementation: O(2n)

(but essentially the same)

• micro benchmark:
• derive F :

≺: G

• 9 summands in F and G
• Implementation presented here: 0.5s

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 19

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Compile Time Performance

• If done “wrong”, this implementation can be very slow!
• Implementation presented here: O(n2)
• Slightly different implementation: O(2n)

(but essentially the same)

• micro benchmark:
• derive F :

≺: G

• 9 summands in F and G
• Implementation presented here: 0.5s
• Naive implementation: 45s

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 19

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f c o m p u t e r s c i e n c e

Compile Time Performance

• If done “wrong”, this implementation can be very slow!
• Implementation presented here: O(n2)
• Slightly different implementation: O(2n)

(but essentially the same)

• micro benchmark:
• derive F :

≺: G

• 9 summands in F and G
• Implementation presented here: 0.5s
• Naive implementation: 45s
• Type families on kind ∗ are expensive!

Patrick Bahr — Composing and Decomposing Data Types — WGP ’14, 31st August, 2014 Slide 19