{- This module provides type-level finite maps.
The implementation is similar to that shown in the paper.
"Embedding effect systems in Haskell" Orchard, Petricek 2014 -}
{-# LANGUAGE TypeOperators, PolyKinds, DataKinds, KindSignatures,
TypeFamilies, UndecidableInstances, MultiParamTypeClasses,
FlexibleInstances, GADTs, FlexibleContexts, ScopedTypeVariables, ConstraintKinds #-}
module Data.Type.Map (Mapping(..), Union, Unionable, union, Var(..), Map(..),
Combine, Combinable(..), Cmp,
Lookup, Member, (:\)) where
import GHC.TypeLits
import Data.Type.Bool
import Data.Type.Equality
import Data.Type.Set hiding (Set(..), Nub,Union,Nubable,Sortable,Unionable,append,union,quicksort,nub)
{- Throughout, type variables
'k' ranges over "keys"
'v' ranges over "values"
'kvp' ranges over "key-value-pairs"
'm', 'n' range over "maps" -}
-- Mappings
infixr 4 :->
{-| A key-value pair -}
data Mapping k v = k :-> v
{-| Union of two finite maps -}
type Union m n = Nub (Sort (m :++ n))
{-| Apply 'Combine' to values with matching key (removes duplicate keys) -}
type family Nub t where
Nub '[] = '[]
Nub '[kvp] = '[kvp]
Nub ((k :-> v1) ': (k :-> v2) ': m) = Nub ((k :-> Combine v1 v2) ': m)
Nub (kvp1 ': kvp2 ': s) = kvp1 ': Nub (kvp2 ': s)
{-| Open type family for combining values in a map (that have the same key) -}
type family Combine (a :: v) (b :: v) :: v
{-| Delete elements from a map by key -}
type family (m :: [Mapping k v]) :\ (c :: k) :: [Mapping k v] where
'[] :\ k = '[]
((k :-> v) ': m) :\ k = m :\ k
(kvp ': m) :\ k = kvp ': (m :\ k)
{-| Lookup elements from a map -}
type family Lookup (m :: [Mapping k v]) (c :: k) :: Maybe v where
Lookup '[] k = Nothing
Lookup ((k :-> v) ': m) k = Just v
Lookup (kvp ': m) k = Lookup m k
{-| Membership test -}
type family Member (c :: k) (m :: [Mapping k v]) :: Bool where
Member k '[] = False
Member k ((k :-> v) ': m) = True
Member k (kvp ': m) = Member k m
-----------------------------------------------------------------
-- Value-level map with a type-level representation
{-| Pair a symbol (representing a variable) with a type -}
data Var (k :: Symbol) = Var
instance KnownSymbol k => Show (Var k) where
show = symbolVal
{-| A value-level heterogenously-typed Map (with type-level representation in terms of lists) -}
data Map (n :: [Mapping Symbol *]) where
Empty :: Map '[]
Ext :: Var k -> v -> Map m -> Map ((k :-> v) ': m)
instance Show (Map '[]) where
show Empty = "{}"
instance (KnownSymbol k, Show v, Show' (Map s)) => Show (Map ((k :-> v) ': s)) where
show (Ext k v s) = "{" ++ show k ++ " :-> " ++ show v ++ (show' s) ++ "}"
class Show' t where
show' :: t -> String
instance Show' (Map '[]) where
show' Empty = ""
instance (KnownSymbol k, Show v, Show' (Map s)) => Show' (Map ((k :-> v) ': s)) where
show' (Ext k v s) = ", " ++ show k ++ " :-> " ++ show v ++ (show' s)
{-| Union of two finite maps -}
union :: (Unionable s t) => Map s -> Map t -> Map (Union s t)
union s t = nub (quicksort (append s t))
type Unionable s t = (Nubable (Sort (s :++ t)), Sortable (s :++ t))
append :: Map s -> Map t -> Map (s :++ t)
append Empty x = x
append (Ext k v xs) ys = Ext k v (append xs ys)
type instance Cmp (k :: Symbol) (k' :: Symbol) = CmpSymbol k k'
type instance Cmp (k :-> v) (k' :-> v) = CmpSymbol k k'
{-| Value-level quick sort that respects the type-level ordering -}
class Sortable xs where
quicksort :: Map xs -> Map (Sort xs)
instance Sortable '[] where
quicksort Empty = Empty
instance (Sortable (Filter FMin (k :-> v) xs),
Sortable (Filter FMax (k :-> v) xs), FilterV FMin k v xs, FilterV FMax k v xs) => Sortable ((k :-> v) ': xs) where
quicksort (Ext k v xs) = ((quicksort (less k v xs)) `append` (Ext k v Empty)) `append` (quicksort (more k v xs))
where less = filterV (Proxy::(Proxy FMin))
more = filterV (Proxy::(Proxy FMax))
{- Filter out the elements less-than or greater-than-or-equal to the pivot -}
class FilterV (f::Flag) k v xs where
filterV :: Proxy f -> Var k -> v -> Map xs -> Map (Filter f (k :-> v) xs)
instance FilterV f k v '[] where
filterV _ k v Empty = Empty
instance (Conder ((Cmp x (k :-> v)) == LT), FilterV FMin k v xs) => FilterV FMin k v (x ': xs) where
filterV f@Proxy k v (Ext k' v' xs) = cond (Proxy::(Proxy ((Cmp x (k :-> v)) == LT)))
(Ext k' v' (filterV f k v xs)) (filterV f k v xs)
instance (Conder (((Cmp x (k :-> v)) == GT) || ((Cmp x (k :-> v)) == EQ)), FilterV FMax k v xs) => FilterV FMax k v (x ': xs) where
filterV f@Proxy k v (Ext k' v' xs) = cond (Proxy::(Proxy (((Cmp x (k :-> v)) == GT) || ((Cmp x (k :-> v)) == EQ))))
(Ext k' v' (filterV f k v xs)) (filterV f k v xs)
class Combinable t t' where
combine :: t -> t' -> Combine t t'
class Nubable t where
nub :: Map t -> Map (Nub t)
instance Nubable '[] where
nub Empty = Empty
instance Nubable '[e] where
nub (Ext k v Empty) = Ext k v Empty
instance {-# OVERLAPPING #-}
(Combinable v v', Nubable ((k :-> Combine v v') ': s)) => Nubable ((k :-> v) ': (k :-> v') ': s) where
nub (Ext k v (Ext k' v' s)) = nub (Ext k (combine v v') s)
instance {-# OVERLAPPING #-}
(Nub (e ': f ': s) ~ (e ': Nub (f ': s)),
Nubable (f ': s)) => Nubable (e ': f ': s) where
nub (Ext k v (Ext k' v' s)) = Ext k v (nub (Ext k' v' s))
class Conder g where
cond :: Proxy g -> Map s -> Map t -> Map (If g s t)
instance Conder True where
cond _ s t = s
instance Conder False where
cond _ s t = t