Extendable Lambda Calculus Interpreter

Lambda.hs

{- Copyright (C) Freddy Angel Cubas "Superstar64"
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{-# Language DeriveTraversable, TypeFamilies #-}
module Lambda where

import Data.List
import Data.Maybe
import Data.Foldable

import Data.Void
import Data.Functor.Const
import Data.Functor.Foldable

names prefix = prefix : (map (prefix ++) $ map show [0..])
inside list item = isJust $ find (==item) list


data Term e = TVariable String | TAbstraction String (Term e) | TApplication (Term e) (Term e) | TMisc (e (Term e))

data TermF e x = TVariableF String | TAbstractionF String x | TApplicationF x x | TMiscF (e x) deriving (Functor, Foldable, Traversable)

type instance Base (Term e) = TermF e

instance Functor e => Recursive (Term e) where
 project (TVariable x) = TVariableF x
 project (TAbstraction x e) = TAbstractionF x e
 project (TApplication e e') = TApplicationF e e'
 project (TMisc m) = TMiscF m

instance Functor e => Corecursive (Term e) where
 embed (TVariableF x) = TVariable x
 embed (TAbstractionF x e) = TAbstraction x e
 embed (TApplicationF e e') = TApplication e e'
 embed (TMiscF m) = TMisc m 

data Normal e = BAbstraction String (Normal e) | BApplication String [Normal e] | BMisc (e (Normal e))

data NormalF e x = BAbstractionF String x | BApplicationF String [x] | BMiscF (e x) deriving (Functor, Foldable, Traversable)

type instance Base (Normal e) = NormalF e

instance Functor e => Recursive (Normal e) where
 project (BAbstraction x e) = BAbstractionF x e
 project (BApplication x e) = BApplicationF x e
 project (BMisc m) = BMiscF m

instance Functor e => Corecursive (Normal e) where
 embed (BAbstractionF x e) = BAbstraction x e
 embed (BApplicationF x e) = BApplication x e
 embed (BMiscF m) = BMisc m

vanilla :: Functor e => Normal e -> Term e
vanilla (BAbstraction x e) = TAbstraction x (vanilla e)
vanilla (BApplication x e) = foldl TApplication (TVariable x) (map vanilla e)
vanilla (BMisc m) = TMisc (fmap vanilla m)

freeVariables :: Foldable t => Term t -> [String]
freeVariables (TVariable x) = [x]
freeVariables (TAbstraction x e) = filter (/= x) (freeVariables e)
freeVariables (TApplication e e') = freeVariables e ++ freeVariables e'
freeVariables (TMisc m) = foldMap freeVariables m


substitute :: (Foldable e, Functor e) => Term e -> String -> Term e -> Term e
substitute r x (TVariable y) | x == y = r
substitute r x (TVariable y) = TVariable y
substitute r x (TAbstraction y e) | x == y = TAbstraction y e
substitute r x (TAbstraction y e) = TAbstraction fresh (substitute r x e') where 
 fresh = fromJust $ find (not . inside (freeVariables r) ) (names y)
 e' = substitute (TVariable fresh) y e 
substitute r x (TApplication t s) = TApplication (substitute r x t) (substitute r x s)
substitute r x (TMisc m) = TMisc $ fmap (substitute r x) m

beta :: (Traversable e) => Term e -> Maybe (Normal e)
beta (TVariable x) = Just $ BApplication x []
beta (TAbstraction x e) = beta e >>= \e' -> Just $ BAbstraction x e'
beta (TApplication e1 e2) = beta e1 >>= \e1' -> case e1' of
 (BAbstraction x e) -> beta (substitute e2 x (vanilla e))
 (BApplication x e) -> beta e2 >>= \e2' -> Just $ BApplication x (e ++ [e2'])
 (BMisc m) -> Nothing
beta (TMisc m) = fmap BMisc $ traverse beta m


extract :: Traversable e => Normal e -> Maybe (Fix e)
extract (BMisc m) = fmap Fix (traverse extract m)
extract _ = Nothing

pretty :: (e (Term e) -> String) -> Term e -> String
pretty misc (TVariable x) = x
pretty misc (TAbstraction x e) = x ++ " => " ++ pretty misc e
pretty misc (TApplication e e') = "(" ++ pretty misc e ++ ")" ++ "(" ++ pretty misc e' ++ ")"
pretty misc (TMisc m) = misc m

test = pretty (absurd . getConst) . vanilla . fromJust . beta

twoLevel :: Term (TermF (Const Void))
twoLevel = TApplication (TAbstraction "x" (TMisc (TAbstractionF "y" (TVariable "x")) )) (TMisc (TVariableF "f"))

sample :: Term (Const Void)
sample = let (Just e) = beta twoLevel >>= extract in ana unfix e