Bound.Scope

This is the work-horse of the bound library.

Scope provides a single generalized de Bruijn level and is often used inside of the definition of binders.

Documentation

newtype Scope b f a

Scope b f a is an f expression with bound variables in b, and free variables in a

We store bound variables as their generalized de Bruijn representation in that we're allowed to lift (using F) an entire tree rather than only succ individual variables, but we're still only allowed to do so once per Scope. Weakening trees permits O(1) weakening and permits more sharing opportunities. Here the deBruijn 0 is represented by the B constructor of Var, while the de Bruijn succ (which may be applied to an entire tree!) is handled by F.

NB: equality and comparison quotient out the distinct F placements allowed by the generalized de Bruijn representation and return the same result as a traditional de Bruijn representation would.

Logically you can think of this as if the shape were the traditional f (Var b a), but the extra f a inside permits us a cheaper lift.

Constructors

Scope

Fields

• unscope :: f (Var b (f a))

Instances

MFunctor (Scope b)

Methods

hoist :: Monad m => (forall a. m a -> n a) -> Scope b m b -> Scope b n b

MonadTrans (Scope b)

Methods

lift :: Monad m => m a -> Scope b m a

Bound (Scope b)

Methods

(>>>=) :: Monad f => Scope b f a -> (a -> f c) -> Scope b f c

Monad f => Monad (Scope b f)

The monad permits substitution on free variables, while preserving bound variables

Methods

(>>=) :: Scope b f a -> (a -> Scope b f b) -> Scope b f b

(>>) :: Scope b f a -> Scope b f b -> Scope b f b

return :: a -> Scope b f a

fail :: String -> Scope b f a

Functor f => Functor (Scope b f)

Methods

fmap :: (a -> b) -> Scope b f a -> Scope b f b

(<$) :: a -> Scope b f b -> Scope b f a

(Functor f, Monad f) => Applicative (Scope b f)

Methods

pure :: a -> Scope b f a

(<*>) :: Scope b f (a -> b) -> Scope b f a -> Scope b f b

(*>) :: Scope b f a -> Scope b f b -> Scope b f b

(<*) :: Scope b f a -> Scope b f b -> Scope b f a

Foldable f => Foldable (Scope b f)

toList is provides a list (with duplicates) of the free variables

Methods

fold :: Monoid m => Scope b f m -> m

foldMap :: Monoid m => (a -> m) -> Scope b f a -> m

foldr :: (a -> b -> b) -> b -> Scope b f a -> b

foldr' :: (a -> b -> b) -> b -> Scope b f a -> b

foldl :: (b -> a -> b) -> b -> Scope b f a -> b

foldl' :: (b -> a -> b) -> b -> Scope b f a -> b

foldr1 :: (a -> a -> a) -> Scope b f a -> a

foldl1 :: (a -> a -> a) -> Scope b f a -> a

toList :: Scope b f a -> [a]

null :: Scope b f a -> Bool

length :: Scope b f a -> Int

elem :: Eq a => a -> Scope b f a -> Bool

maximum :: Ord a => Scope b f a -> a

minimum :: Ord a => Scope b f a -> a

sum :: Num a => Scope b f a -> a

product :: Num a => Scope b f a -> a

Traversable f => Traversable (Scope b f)

Methods

traverse :: Applicative f => (a -> f b) -> Scope b f a -> f (Scope b f b)

sequenceA :: Applicative f => Scope b f (f a) -> f (Scope b f a)

mapM :: Monad m => (a -> m b) -> Scope b f a -> m (Scope b f b)

sequence :: Monad m => Scope b f (m a) -> m (Scope b f a)

(Monad f, Eq b, Eq1 f) => Eq1 (Scope b f)

Methods

liftEq :: (a -> b -> Bool) -> Scope b f a -> Scope b f b -> Bool

(Monad f, Ord b, Ord1 f) => Ord1 (Scope b f)

Methods

liftCompare :: (a -> b -> Ordering) -> Scope b f a -> Scope b f b -> Ordering

(Read b, Read1 f) => Read1 (Scope b f)

Methods

liftReadsPrec :: (Int -> ReadS a) -> ReadS [a] -> Int -> ReadS (Scope b f a)

liftReadList :: (Int -> ReadS a) -> ReadS [a] -> ReadS [Scope b f a]

(Show b, Show1 f) => Show1 (Scope b f)

Methods

liftShowsPrec :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> Int -> Scope b f a -> ShowS

liftShowList :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> [Scope b f a] -> ShowS

(Serial b, Serial1 f) => Serial1 (Scope b f)

Methods

serializeWith :: MonadPut m => (a -> m ()) -> Scope b f a -> m ()

deserializeWith :: MonadGet m => m a -> m (Scope b f a)

(Hashable b, Monad f, Hashable1 f) => Hashable1 (Scope b f)

Methods

liftHashWithSalt :: (Int -> a -> Int) -> Int -> Scope b f a -> Int

(Monad f, Eq b, Eq1 f, Eq a) => Eq (Scope b f a)

Methods

(==) :: Scope b f a -> Scope b f a -> Bool

(/=) :: Scope b f a -> Scope b f a -> Bool

(Typeable * b, Typeable (* -> *) f, Data a, Data (f (Var b (f a)))) => Data (Scope b f a)

Methods

gfoldl :: (forall d c. Data d => c (d -> c) -> d -> c c) -> (forall g. g -> c g) -> Scope b f a -> c (Scope b f a)

gunfold :: (forall c r. Data c => c (c -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (Scope b f a)

toConstr :: Scope b f a -> Constr

dataTypeOf :: Scope b f a -> DataType

dataCast1 :: Typeable (* -> *) t => (forall d. Data d => c (t d)) -> Maybe (c (Scope b f a))

dataCast2 :: Typeable (* -> * -> *) t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (Scope b f a))

gmapT :: (forall c. Data c => c -> c) -> Scope b f a -> Scope b f a

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Scope b f a -> r

gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Scope b f a -> r

gmapQ :: (forall d. Data d => d -> u) -> Scope b f a -> [u]

gmapQi :: Int -> (forall d. Data d => d -> u) -> Scope b f a -> u

gmapM :: Monad m => (forall d. Data d => d -> m d) -> Scope b f a -> m (Scope b f a)

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Scope b f a -> m (Scope b f a)

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Scope b f a -> m (Scope b f a)

(Monad f, Ord b, Ord1 f, Ord a) => Ord (Scope b f a)

Methods

compare :: Scope b f a -> Scope b f a -> Ordering

(<) :: Scope b f a -> Scope b f a -> Bool

(<=) :: Scope b f a -> Scope b f a -> Bool

(>) :: Scope b f a -> Scope b f a -> Bool

(>=) :: Scope b f a -> Scope b f a -> Bool

max :: Scope b f a -> Scope b f a -> Scope b f a

min :: Scope b f a -> Scope b f a -> Scope b f a

(Read b, Read1 f, Read a) => Read (Scope b f a)

Methods

readsPrec :: Int -> ReadS (Scope b f a)

readList :: ReadS [Scope b f a]

readPrec :: ReadPrec (Scope b f a)

readListPrec :: ReadPrec [Scope b f a]

(Show b, Show1 f, Show a) => Show (Scope b f a)

Methods

showsPrec :: Int -> Scope b f a -> ShowS

show :: Scope b f a -> String

showList :: [Scope b f a] -> ShowS

(Binary b, Serial1 f, Binary a) => Binary (Scope b f a)

Methods

put :: Scope b f a -> Put

get :: Get (Scope b f a)

putList :: [Scope b f a] -> Put

(Serial b, Serial1 f, Serial a) => Serial (Scope b f a)

Methods

serialize :: MonadPut m => Scope b f a -> m ()

deserialize :: MonadGet m => m (Scope b f a)

(Serialize b, Serial1 f, Serialize a) => Serialize (Scope b f a)

Methods

put :: Putter (Scope b f a)

get :: Get (Scope b f a)

NFData (f (Var b (f a))) => NFData (Scope b f a)

Methods

rnf :: Scope b f a -> ()

(Hashable b, Monad f, Hashable1 f, Hashable a) => Hashable (Scope b f a)

Methods

hashWithSalt :: Int -> Scope b f a -> Int

hash :: Scope b f a -> Int

Abstraction

abstract :: Monad f => (a -> Maybe b) -> f a -> Scope b f a

Capture some free variables in an expression to yield a Scope with bound variables in b

>>> :m + Data.List
>>> abstract (`elemIndex` "bar") "barry"
Scope [B 0,B 1,B 2,B 2,F "y"]

abstract1 :: (Monad f, Eq a) => a -> f a -> Scope () f a

Abstract over a single variable

>>> abstract1 'x' "xyz"
Scope [B (),F "y",F "z"]

Instantiation

instantiate :: Monad f => (b -> f a) -> Scope b f a -> f a

Enter a scope, instantiating all bound variables

>>> :m + Data.List
>>> instantiate (\x -> [toEnum (97 + x)]) $ abstract (`elemIndex` "bar") "barry"
"abccy"

instantiate1 :: Monad f => f a -> Scope n f a -> f a

Enter a Scope that binds one variable, instantiating it

>>> instantiate1 "x" $ Scope [B (),F "y",F "z"]
"xyz"

Traditional de Bruijn

fromScope :: Monad f => Scope b f a -> f (Var b a)

fromScope quotients out the possible placements of F in Scope by distributing them all to the leaves. This yields a more traditional de Bruijn indexing scheme for bound variables.

Since,

fromScope . toScope ≡ id

we know that

fromScope . toScope . fromScope ≡ fromScope

and therefore (toScope . fromScope) is idempotent.

toScope :: Monad f => f (Var b a) -> Scope b f a

Convert from traditional de Bruijn to generalized de Bruijn indices.

This requires a full tree traversal

Bound variable manipulation

splat :: Monad f => (a -> f c) -> (b -> f c) -> Scope b f a -> f c

Perform substitution on both bound and free variables in a Scope.

bindings :: Foldable f => Scope b f a -> [b]

Return a list of occurences of the variables bound by this Scope.

mapBound :: Functor f => (b -> b') -> Scope b f a -> Scope b' f a

Perform a change of variables on bound variables.

mapScope :: Functor f => (b -> d) -> (a -> c) -> Scope b f a -> Scope d f c

Perform a change of variables, reassigning both bound and free variables.

liftMBound :: Monad m => (b -> b') -> Scope b m a -> Scope b' m a

Perform a change of variables on bound variables given only a Monad instance

liftMScope :: Monad m => (b -> d) -> (a -> c) -> Scope b m a -> Scope d m c

A version of mapScope that can be used when you only have the Monad instance

foldMapBound :: (Foldable f, Monoid r) => (b -> r) -> Scope b f a -> r

Obtain a result by collecting information from bound variables

foldMapScope :: (Foldable f, Monoid r) => (b -> r) -> (a -> r) -> Scope b f a -> r

Obtain a result by collecting information from both bound and free variables

traverseBound_ :: (Applicative g, Foldable f) => (b -> g d) -> Scope b f a -> g ()

traverse_ the bound variables in a Scope.

traverseScope_ :: (Applicative g, Foldable f) => (b -> g d) -> (a -> g c) -> Scope b f a -> g ()

traverse both the variables bound by this scope and any free variables.

mapMBound_ :: (Monad g, Foldable f) => (b -> g d) -> Scope b f a -> g ()

mapM_ over the variables bound by this scope

mapMScope_ :: (Monad m, Foldable f) => (b -> m d) -> (a -> m c) -> Scope b f a -> m ()

A traverseScope_ that can be used when you only have a Monad instance

traverseBound :: (Applicative g, Traversable f) => (b -> g c) -> Scope b f a -> g (Scope c f a)

traverse the bound variables in a Scope.

traverseScope :: (Applicative g, Traversable f) => (b -> g d) -> (a -> g c) -> Scope b f a -> g (Scope d f c)

Traverse both bound and free variables

mapMBound :: (Monad m, Traversable f) => (b -> m c) -> Scope b f a -> m (Scope c f a)

mapM over both bound and free variables

mapMScope :: (Monad m, Traversable f) => (b -> m d) -> (a -> m c) -> Scope b f a -> m (Scope d f c)

A traverseScope that can be used when you only have a Monad instance

serializeScope :: (Serial1 f, MonadPut m) => (b -> m ()) -> (v -> m ()) -> Scope b f v -> m ()

deserializeScope :: (Serial1 f, MonadGet m) => m b -> m v -> m (Scope b f v)

hoistScope :: Functor f => (forall x. f x -> g x) -> Scope b f a -> Scope b g a

Lift a natural transformation from f to g into one between scopes.

bitraverseScope :: (Bitraversable t, Applicative f) => (k -> f k') -> (a -> f a') -> Scope b (t k) a -> f (Scope b (t k') a')

This allows you to bitraverse a Scope.

bitransverseScope :: Applicative f => (forall a a'. (a -> f a') -> t a -> f (u a')) -> (c -> f c') -> Scope b t c -> f (Scope b u c')

transverseScope :: (Applicative f, Monad f, Traversable g) => (forall r. g r -> f (h r)) -> Scope b g a -> f (Scope b h a)

This is a higher-order analogue of traverse.

instantiateVars :: Monad t => [a] -> Scope Int t a -> t a

instantiate bound variables using a list of new variables