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{-# LANGUAGE GADTs, RankNTypes #-}
module Main where
-- abstract syntax -------------------------------------------------------------
data Ty t where
Bool :: Ty Bool
Arr :: Ty a -> Ty b -> Ty (a -> b)
data Exp g t where
Var :: Var g t -> Exp g t
Lam :: Ty a -> Exp (g,a) b -> Exp g (a->b)
App :: Exp g (s -> t) -> Exp g s -> Exp g t
If :: Exp g Bool -> Exp g t -> Exp g t -> Exp g t
ETrue :: Exp g Bool
EFalse :: Exp g Bool
data Var g t where
ZVar :: Var (h,t) t
SVar :: Var h t -> Var (h,s) t
-- smart constructors ----------------------------------------------------------
lamE :: Ty s -> (Exp (g,s) s -> Exp (g,s) t) -> Exp g (s -> t)
lamE s f = Lam s (f (Var ZVar))
ifE :: Exp g Bool -> Exp g t -> Exp g t -> Exp g t
ifE t ETrue EFalse = t
ifE t e e' = if eqE e e' then e else If t e e'
-- boring equality tests -------------------------------------------------------
eqB :: BoxExp t -> BoxExp s -> Bool
eqB (Box e) (Box e_) = eqE e e_
eqE :: Exp g t -> Exp h s -> Bool
eqE (Var x) (Var y) = eqV x y
eqE (Lam s e) (Lam s_ e_) = eqT s s_ && eqE e e_
eqE (App e1 e2) (App e1_ e2_) = eqE e1 e1_ && eqE e2 e2_
eqE (If e1 e2 e3) (If e1_ e2_ e3_) = eqE e1 e1_ && (eqE e2 e2_ && eqE e3 e3_)
eqE (ETrue) (ETrue) = True
eqE (EFalse) (EFalse) = True
eqE _ _ = False
eqT :: Ty t -> Ty s -> Bool
eqT (Arr s t) (Arr s_ t_) = eqT s s_ && eqT t t_
eqT Bool Bool = True
eqT _ _ = False
eqV :: Var g t -> Var h s -> Bool
eqV (SVar x) (SVar y) = eqV x y
eqV ZVar ZVar = True
eqV _ _ = False
-- evaluation ------------------------------------------------------------------
var :: Var g t -> g -> t
var ZVar (_,t) = t
var (SVar x) (h,s) = var x h
eval :: Exp g t -> g -> t
eval (Var x) g = var x g
eval (Lam _ e) g = \a -> eval e (g,a)
eval (App e e') g = eval e g (eval e' g)
eval (ETrue) g = True
eval (EFalse) g = False
eval (If c t e) g = if eval c g then eval t g else eval e g
-- type inference --------------------------------------------------------------
data TyEnv g where
Nil :: TyEnv g
Cons :: Ty t -> TyEnv h -> TyEnv (h,t)
infer :: TyEnv g -> Exp g t -> Ty t
infer g (Var x) = inferVar g x
infer g (Lam t e) = Arr t (infer (Cons t g) e)
infer g (App e e') = case infer g e of Arr _ t -> t
infer g (ETrue) = Bool
infer g (EFalse) = Bool
infer g (If _ e _) = infer g e
inferVar :: TyEnv g -> Var g t -> Ty t
inferVar (Cons t h) (SVar x) = inferVar h x
inferVar (Cons t h) (ZVar) = t
-- tree monad ------------------------------------------------------------------
data Tree a = Val a | Choice (Tree a) (Tree a)
-- doesn't yet force trees to be fully balanced:
-- Val :: a -> Tree a Z
-- Choice :: Tree a n -> Tree a n -> Tree a (S n)
instance Monad Tree where
return x = Val x
(Val a) >>= f = f a
(Choice l r) >>= f = Choice (l >>= f) (r >>= f)
tmap :: Monad m => (a->b) -> m a -> m b
tmap f x = do { a <- x; return (f a) }
flatten t = flatten_ t []
where
flatten_ (Val a) k = a:k
flatten_ (Choice l r) k = flatten_ l (flatten_ r k)
-- quote & friends -------------------------------------------------------------
-- for values --------------------------
enumV :: Ty t -> Tree t
questionsV :: Ty t -> [t -> Bool]
enumV Bool = Choice (Val True) (Val False)
enumV (Arr s t) = mkEnum (questionsV s) (enumV t)
where
mkEnum [] t = tmap const t
mkEnum (q:qs) es = do
f1 <- mkEnum qs es
f2 <- mkEnum qs es
return (\d -> if q d then f1 d else f2 d)
questionsV Bool = return (\x -> x)
questionsV (Arr s t) = do
d <- flatten (enumV s)
q <- questionsV t
return (\f -> q (f d))
-- for expressions ---------------------
enumE :: Ty t -> Tree (Exp g t)
questionsE :: Ty t -> [Exp g t -> Exp g Bool]
enumE Bool = Choice (Val ETrue) (Val EFalse)
enumE (Arr s t) = tmap (lamE s) (mkEnumE (questionsE s) (enumE t))
where
mkEnumE [] t = tmap const t
mkEnumE (q:qs) es = do
f1 <- mkEnumE qs es
f2 <- mkEnumE qs es
return (\d -> ifE (q d) (f1 d) (f2 d))
questionsE Bool = return (\x -> x)
questionsE (Arr s t) = do
d <- flatten (enumE s)
q <- questionsE t
return (\f -> q (App f d))
-- should be
-- find (List (Exp g Bool) n) -> Tree (Exp g a) n -> Exp g a
find :: [Exp g Bool] -> Tree (Exp g a) -> Exp g a
find [] (Val a) = a
find (b:bs) (Choice l r) = ifE b (find bs l) (find bs r)
find _ _ = error "bad arguments to find"
quote :: Ty t -> t -> Exp g t
quote Bool t = case t of True -> ETrue; False -> EFalse
quote (Arr s t) f = lamE s (\e -> find (do q <- questionsE s; return (q e))
(tmap (quote t . f) (enumV s)))
-- normalization (by evaluation) -----------------------------------------------
data BoxExp t = Box (forall g. Exp g t)
normalize :: Ty t -> BoxExp t -> BoxExp t
normalize s (Box e) = Box (quote s (eval e ()))
-- examples --------------------------------------------------------------------
b2b = Arr Bool Bool
b22b = Arr b2b b2b
zero = Var ZVar
one = Var (SVar ZVar)
once = Box (Lam b2b (Lam Bool (App one zero)))
twice = Box (Lam b2b (Lam Bool (App one (App one zero))))
thrice = Box (Lam b2b (Lam Bool (App one (App one (App one zero)))))
test = [ eqB (nf b22b thrice) (nf b22b once)
, eqB (nf b22b twice) (nf b22b once)]
where nf = normalize
main = print test
|
