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module HW (
fv,
subst,
normalstep,
pickFresh,
repeatedly,
printnormal,
) where
import AST
import qualified Data.Set as Set
import Parse (parse)
-- | Return the free variables in an expression
fv :: Expr -> Set.Set String
fv (Var x) = Set.singleton x
fv (Lam x m) = x `Set.delete` (fv m)
fv (App m1 m2) = (fv m1) `Set.union` (fv m2)
{- | Substitute n for x in e, avoiding name capture
subst n x e e[x := n]
-}
subst :: Expr -> String -> Expr -> Expr
subst n x (Var e)
| x == e = n
| otherwise = Var e
subst n x (App m1 m2) = App (subst n x m1) (subst n x m2)
subst n x (Lam y m)
| x == y = Lam y m
| y `Set.notMember` (fv n) = Lam y (subst n x m)
| otherwise = Lam y' (subst n x (subst (Var y') y m))
where
y' = pickFresh ((fv n) `Set.union` (fv m)) y
-- | Take a single step in normal order reduction or return Nothing
normalstep :: Expr -> Maybe Expr
-- beta
normalstep (App (Lam x m) n) = Just (subst n x m)
-- body
normalstep (Lam x m) = normalstep m >>= return . Lam x
-- arg
normalstep (App m n) | normalstep m == Nothing = normalstep n >>= return . App m
-- func
normalstep (App m n) = normalstep m >>= return . (`App` n)
-- No further reductions
normalstep _ = Nothing
{- | Return a "fresh" name not already in the set.
Tries x' then x'', etc.
-}
pickFresh :: Set.Set String -> String -> String
pickFresh s = pickFresh'
where
pickFresh' n | n `Set.notMember` s = n
pickFresh' n = pickFresh' $ n ++ "'"
{- | Repeatedly apply a function to transform a value, returning the list
of steps it took. The result list starts with the given initial value
-}
repeatedly :: (a -> Maybe a) -> a -> [a]
repeatedly f = repeatedly'
where
repeatedly' x =
x : case f x of
Nothing -> []
Just y -> repeatedly' y
-- | Print out the series of normal order reduction steps
printnormal :: String -> IO ()
printnormal = mapM_ print . repeatedly normalstep . parse
|
