Зависимые типы в GHC 8. Максим Талдыкин

Зависимые типы в GHC 8
Максим Талдыкин
max@formalmethods.ru

Just 1 : Maybe Int : * : □
Maybe : * -> * : □
Vec Int Z : * : □
Vec : * -> Nat -> * : □

map : forall a b. (a -> b) -> [a] -> [b]
map f (x:xs) = f x : map f xs
map f [] = []

map :: forall a b. (a -> b) -> [a] -> [b]
map =
(@a)
(@b)
(f :: a -> b)
(ds :: [a]) ->
case ds of
[] -> GHC.Types.[] @b
: x xs ->
GHC.Types.: @b (f x) (map @a @b f xs)

forall
(a :: *)
(b :: *)
(f :: a -> b)
(xs :: [a])
. [b]

replicate :: pi (n : Nat) -> a -> Vec a n
replicate Z _ = Nil
replicate (S m) x = x :> replicate m a

vis dep rel
forall (a : *). a - + -
a -> a + - +
Monad m => m a - - +
pi (a : Bool). f a - + +
pi (a : Bool) -> f a + + +

strncpy
:: forall (p q :: Nat)
=> (LE n p ~ True, LE n q ~ True)
. pi (n :: Nat)
-> Ptr Char p
-> Ptr Char q
-> IO ()

Π-Ware: Hardware Description
Swierstra et al., 2015
Correct-by-Construction Concurrency
Brady & Hammond, 2009
Security-Typed Programming
Morgenstern & Licata, 2010

data Nat = Z | S Nat
type family
(m :: Nat) :+ (n :: Nat) :: Nat
where
Z :+ n = n
(S k) :+ n = S (k :+ n)
-- Vec a m -> Vec a n -> Vec a (m :+ n)

GADTs
Generalized Algebraic Data Types

data Vec :: Nat -> * -> *
Nil :: Vec Z a
(:>) :: a -> Vec n a -> Vec (S n) a

data Vec (n :: Nat) (a :: *) :: * where
Nil :: n ~ Z => Nil n a
(:>) :: forall (m :: Nat)
. n ~ S m
=> a -> Vec m a -> Vec n a

tail :: Vec (S k) a -> Vec k a
tail (_ :> xs) = xs
-- (m :: Nat) (S k ~ S m)

type family
(v :: Vec a m) :++ (w :: Vec a n) :: Vec a k
where
Nil :++ w = w
(x :> xs) :++ w = x :> (xs :++ w)

type family
(v :: Vec a m) :++ (w :: Vec a n) :: Vec a k
where
Nil :++ w = w
(x :> xs) :++ w = x :> (xs :++ w)
‘Vec’ of kind ‘* -> Nat -> *’ is not promotable
In the kind ‘Vec a m’

replicate :: pi (n :: Nat) -> a -> Vec a n
replicate Z a = Nil
replicate (S m) a = a :> replicate m a

data Nat = Z | S Nat
data Nat's :: Nat -> * where
Z's :: Nat's Z
S's :: Nat's n -> Nat's (S n)
-- S's Z's :: Nat's (S Z)

replicate :: Nat's n -> a -> Vec a n
replicate Z's a = Nil
replicate (S's m) a = a :> replicate m a

Singleton types here
Singleton types there
Singleton types everywhere
Monnier & Haguenauer, 2009

Hasochism
Lindley & mcBride, 2013

data Fin :: Nat -> * where
FZ :: Fin (S n)
FS :: Fin n -> Fin (S n)

lookup :: pi (f :: Fin n) -> Vec a n -> a
lookup FZ (x :> _) = x
lookup (FS i) (_ :> xs) = lookup i xs

data Fin's (n :: Nat) (f :: Fin n) :: * where
FZ's :: Fin's (S m) FZ
FS's :: Fin's m g -> Fin's (S m) (FS g)
-- FS's FZ's :: Fin's 4 (FS FZ :: Fin 4)

data Fin's (n :: Nat) (f :: Fin n) :: * where
FZ's :: Fin's (S m) FZ
FS's :: Fin's m g -> Fin's (S m) (FS g)
Kind variable also used as type variable: ‘n’
In the data type declaration for ‘Fin's’

get "/Contract/:id" $ do
intParam "id" >>= queryDb "Contract" >>= json
type API
= "Contract"
:> Capture "id" Int
:> Get '[JSON] Contract

type Api
= "Contract"
:> Capture "id" Int
:> RoleFilter "Contract" "owner" '[43, 265]
:> Get '[JSON] (Obj Contract)

data User (fieldName :: Symbol) where
Id :: Int -> User "id"
Name :: Text -> User "name"
Roles :: [Vector (Ref Role)] -> User "roles"
data Contract (fieldName :: Symbol) where
Id :: Int -> Contract "id"
Owner :: Ref Role -> Contract "owner"
type family TableName (m :: Symbol -> *) :: Symbol
type instance TableName User = "User"
type instance TableName Contract = "Contract"

type Api
= "Contract"
:> Capture "id" Int
:> RoleFilter "Contract" "owner" '[43, 265]
:> Get '[JSON] (Obj Contract)
type Api
= MkFilter Contract
(RoleFilter C.Owner '[Role1, Role2])

type family MkFilter
(f :: Symbol)
(m :: Symbol -> *)
(filter :: *)
where
MkFilter f m (RoleFilter (own :: m g) rs)
= TableName m
:> Capture
(FieldName (TableId m))
(FieldType (TableId m))
:> RoleFilter' (TableName m) g rs
:> Get '[JSON] (Obj m)

Termination
Equality
Consistency

System FC with Explicit Kind Equality, 2013
Dependent Types in Haskell, draft
Type Inference, Haskell and Dependent Types, 2013

Per Martin-Löf, Intuitionistic type theory (1984)

Зависимые типы в GHC 8. Максим Талдыкин

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