Copyright	(C) 2012-2013 Edward Kmett
License	BSD-style (see the file LICENSE)
Maintainer	Edward Kmett <ekmett@gmail.com>
Stability	experimental
Portability	portable
Safe Haskell	Trustworthy
Language	Haskell98

Bound.Scope

Contents

Abstraction
Instantiation
Traditional de Bruijn
Bound variable manipulation

Description

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.

Synopsis

Documentation

newtype Scope b f a Source

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

MonadTrans (Scope b)
Bound (Scope b)
Monad f => Monad (Scope b f)	The monad permits substitution on free variables, while preserving bound variables
Functor f => Functor (Scope b f)
(Functor f, Monad f) => Applicative (Scope b f)
Foldable f => Foldable (Scope b f)	`toList` is provides a list (with duplicates) of the free variables
Traversable f => Traversable (Scope b f)
(Serial b, Serial1 f) => Serial1 (Scope b f)
(Hashable b, Monad f, Hashable1 f) => Hashable1 (Scope b f)
(Monad f, Eq b, Eq1 f) => Eq1 (Scope b f)
(Monad f, Ord b, Ord1 f) => Ord1 (Scope b f)
(Functor f, Show b, Show1 f) => Show1 (Scope b f)
(Functor f, Read b, Read1 f) => Read1 (Scope b f)
Typeable (* -> (* -> ) -> -> *) Scope
(Monad f, Eq b, Eq1 f, Eq a) => Eq (Scope b f a)
(Typeable * b, Typeable (* -> *) f, Data a, Data (f (Var b (f a)))) => Data (Scope b f a)
(Monad f, Ord b, Ord1 f, Ord a) => Ord (Scope b f a)
(Functor f, Read b, Read1 f, Read a) => Read (Scope b f a)
(Functor f, Show b, Show1 f, Show a) => Show (Scope b f a)
(Binary b, Serial1 f, Binary a) => Binary (Scope b f a)
(Serial b, Serial1 f, Serial a) => Serial (Scope b f a)
(Serialize b, Serial1 f, Serialize a) => Serialize (Scope b f a)
(Hashable b, Monad f, Hashable1 f, Hashable a) => Hashable (Scope b f a)

Abstraction

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

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 Source

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 Source

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 Source

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) Source

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 Source

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 Source

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

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

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 Source

Perform a change of variables on bound variables.

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

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

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

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 Source

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 Source

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

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

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 () Source

traverse_ the bound variables in a Scope.

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

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 () Source

mapM_ over the variables bound by this scope

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

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) Source

Traverse both bound and free variables

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

Traverse both bound and free variables

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

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) Source

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 () Source

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

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

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') Source

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') Source

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

This is a higher-order analogue of traverse.

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

instantiate bound variables using a list of new variables