[PPT] - Constructing Strictly Positive Types A Joint venture of the PowerPoint Presentation

SLIDE 1

Constructing Strictly Positive Types

A Joint venture of the Nottingham Container Consortium and the Epigram Team

Peter Morris w/ Thorsten Altenkirch

pwm@cs.nott.ac.uk

University of Nottingham

Constructing Strictly Positive Types – p. 1/10

SLIDE 2

First up

This talk is about dependently typed programming,

specifically the datatypes we use to do it

Constructing Strictly Positive Types – p. 2/10

SLIDE 3

First up

This talk is about dependently typed programming,

specifically the datatypes we use to do it

What we have:

data A : ⋆ n : Nat Vec A n : ⋆ where ε : Vec A 0 ; a : A as : Vec A n a: :as : Vec A (1+ n)

Constructing Strictly Positive Types – p. 2/10

SLIDE 4

First up

This talk is about dependently typed programming,

specifically the datatypes we use to do it

What we have:

data A : ⋆ n : Nat Vec A n : ⋆ where ε : Vec A 0 ; a : A as : Vec A n a: :as : Vec A (1+ n)

+ Pattern matching, Structural Recursion. . .

Constructing Strictly Positive Types – p. 2/10

SLIDE 5

What else?

We really want to add more support for the

programmer

Reusable libraries of code to use with a range of

datatypes

Datatype Generics (Vectors, Lists, Telescopes. . . )
Remove Boilerplate

Constructing Strictly Positive Types – p. 3/10

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What else?

We really want to add more support for the

programmer

Reusable libraries of code to use with a range of

datatypes

Datatype Generics (Vectors, Lists, Telescopes. . . )
Remove Boilerplate
We’d also like to be able to express the theory of our

datatypes in the language

To better define what datatypes ARE
To explain the generation of ⇐ rec and ⇐ case

gadgets

To build Epigram in Epigram?

Constructing Strictly Positive Types – p. 3/10

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data currently

Data declarations must conform to Luo’s syntactic test for strict positivity

Constructing Strictly Positive Types – p. 4/10

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data currently

Data declarations must conform to Luo’s syntactic test for strict positivity · · · a : A f : B a → WA B sup a f : WA B Is OK

Constructing Strictly Positive Types – p. 4/10

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data currently

Data declarations must conform to Luo’s syntactic test for strict positivity · · · a : A f : B a → WA B sup a f : WA B Is OK · · · f : (X → Bool) → Bool c f : X Is not (Negative)

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data currently

Data declarations must conform to Luo’s syntactic test for strict positivity · · · a : A f : B a → WA B sup a f : WA B Is OK · · · f : (X → Bool) → Bool c f : X Is not (Negative) · · · ts : List (RoseTree A) node ts : RoseTree a Is also rejected. . .

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What are containers?

Containers have a set of Shapes. S : ⋆. . .

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What are containers?

Containers have a set of Shapes. S : ⋆. . . For example: Lists have a shape very similar to the natural numbers. . . ε

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What are containers?

Containers have a set of Shapes. S : ⋆. . . and for each shape, some set of Positions where data goes P : ∀s : S ⇒ ⋆ For example: Lists have a shape very similar to the natural numbers. . . and given a shape the set of positions has exactly that many elements ε

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What are containers?

Containers have a set of Shapes. S : ⋆. . . and for each shape, some set of Positions where data goes P : ∀s : S ⇒ ⋆ For example: Lists have a shape very similar to the natural numbers. . . and given a shape the set of positions has exactly that many elements ε

Closed under: µ, +, ×, K →. . .

Constructing Strictly Positive Types – p. 5/10

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Indexed Containers

Are given by:

A Type of Shapes

I , O : ⋆ IC I O : ⋆ where S : ⋆ ( |S⊳ | ) : IC I O

Constructing Strictly Positive Types – p. 6/10

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Indexed Containers

Are given by:

A Type of Shapes
For each shape an Output index. . .

I , O : ⋆ IC I O : ⋆ where S : ⋆ q : S → O (q|S⊳ | ) : IC I O

Constructing Strictly Positive Types – p. 6/10

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Indexed Containers

Are given by:

A Type of Shapes
For each shape an Output index. . .
and a Type of Positions

I , O : ⋆ IC I O : ⋆ where S : ⋆ q : S → O P : S → ⋆ (q|S⊳P| ) : IC I O

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SLIDE 18

Indexed Containers

Are given by:

A Type of Shapes
For each shape an Output index. . .
and a Type of Positions
And for each position an Input index

I , O : ⋆ IC I O : ⋆ where S : ⋆ q : S → O P : S → ⋆ r : ∀s : S ⇒ P s → I (q|S⊳P|r) : IC I O

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Extension

The Extension of an Indexed Container (q|S⊳P|r) : IC I O gives rise to a functor: q|S⊳P|r : (I → ⋆) → (O → ⋆)

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Extension

The Extension of an Indexed Container (q|S⊳P|r) : IC I O gives rise to a functor: q|S⊳P|r : (I → ⋆) → (O → ⋆) which is given by: q|S⊳P|r X o ⇒ ∃s : S ⇒ (o = q s) × (∀p : P s ⇒ X (r s p))

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What we have done

We have a Universe of Indexed Containers which

contains codes for all Strictly Positive Families.

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What we have done

We have a Universe of Indexed Containers which

contains codes for all Strictly Positive Families.

The codes and interpretation are both Epigram

datatypes.

Constructing Strictly Positive Types – p. 8/10

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What we have done

We have a Universe of Indexed Containers which

contains codes for all Strictly Positive Families.

The codes and interpretation are both Epigram

datatypes.

This gives us a semantic, compositional notion of SPFs

in Epigram

Constructing Strictly Positive Types – p. 8/10

SLIDE 24

What we have done

We have a Universe of Indexed Containers which

contains codes for all Strictly Positive Families.

The codes and interpretation are both Epigram

datatypes.

This gives us a semantic, compositional notion of SPFs

in Epigram

Functions in this universe are generic programs.

Constructing Strictly Positive Types – p. 8/10

SLIDE 25

What we have done

We have a Universe of Indexed Containers which

contains codes for all Strictly Positive Families.

The codes and interpretation are both Epigram

datatypes.

This gives us a semantic, compositional notion of SPFs

in Epigram

Functions in this universe are generic programs.
Conclusion? Defining what we really mean by

‘datatype’ can give us Generic programming for all Epigram datatypes

Constructing Strictly Positive Types – p. 8/10

SLIDE 26

SPF

data I : Vec ⋆ n O : ⋆ SPF I O : ⋆ where ‘Z’ : SPF ( I : :O) O T : SPF I O ‘wk’ T : SPF ( I : :I ) O f : ∀t : Fin n ⇒ SPF I O ‘Tag’ f : SPF I (O × Fin n) ‘0’, ‘1’ : SPF I O f : O → O′ T : SPF I O ‘Σ’O f T : SPF I O′ f : O′ → O T : SPF I O ‘∆’O f T : SPF I O′ f : O → O′ T : SPF I O ‘Π’O f T : SPF I O′ T : SPF ( I : :O) O ‘ µ’ T : SPF I O

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SLIDE 27

El

data T : F I O

T : Tel

I

: O

T T o : ⋆ where v : T T X o top v : ‘Z’ ( T: :T) o v : T T X o pop v : ‘wk’ T ( T: :T) o v : f t T o tag v : ‘Tag’f T (o; t) void : ‘1’ T o v : T T o σo v : ‘Σ’f T T (f o) v : T T (f o) δ v : ‘∆’f T T o

v : ∀o : O; p : (f o) = o′T

T o π v : ‘Π’f T T o′ v : T ( T: :(‘ µ’ T)) o in v : ‘ µ’ T T o

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