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TypingHaskellInHaskell_modernized.hs
-- I have made minor changes to the file below to make it compile
-- with GHC 8.0
-----------------------------------------------------------------------------
-- Thih: Typing Haskell in Haskell, main program
--
-- Part of `Typing Haskell in Haskell', version of November 23, 2000
-- Copyright (c) Mark P Jones and the Oregon Graduate Institute
-- of Science and Technology, 1999-2000
--
-- This program is distributed as Free Software under the terms
-- in the file "License" that is included in the distribution
-- of this software, copies of which may be obtained from:
-- http://www.cse.ogi.edu/~mpj/thih/
--
--
-- This file contains all the text from the paper in one single file
-- that can be loaded into Hugs for type checking. No additional code
-- is included.
--
-- Like other incarnations of this program, this version is generated
-- automatically from the very same source code as the others. Its
-- *sole* purpose in life is to provide a quick way to check that the
-- code in the paper is free from syntax and type errors. If you
-- actually want to run the program with real data, use the multiple
-- file version.
--
-----------------------------------------------------------------------------
module TypingHaskellInHaskell where
import Data.List(nub, (\\), intersect, union, partition)
import Control.Monad(msum, ap, liftM)
-----------------------------------------------------------------------------
-- Id: Identifiers
-----------------------------------------------------------------------------
type Id = String
enumId :: Int -> Id
enumId n = "v" ++ show n
-----------------------------------------------------------------------------
-- Kind: Kinds
-----------------------------------------------------------------------------
data Kind = Star | Kfun Kind Kind
deriving Eq
-----------------------------------------------------------------------------
-- Type: Types
-----------------------------------------------------------------------------
data Type = TVar Tyvar | TCon Tycon | TAp Type Type | TGen Int
deriving Eq
data Tyvar = Tyvar Id Kind
deriving Eq
data Tycon = Tycon Id Kind
deriving Eq
tUnit = TCon (Tycon "()" Star)
tChar = TCon (Tycon "Char" Star)
tInt = TCon (Tycon "Int" Star)
tInteger = TCon (Tycon "Integer" Star)
tFloat = TCon (Tycon "Float" Star)
tDouble = TCon (Tycon "Double" Star)
tList = TCon (Tycon "[]" (Kfun Star Star))
tArrow = TCon (Tycon "(->)" (Kfun Star (Kfun Star Star)))
tTuple2 = TCon (Tycon "(,)" (Kfun Star (Kfun Star Star)))
tString :: Type
tString = list tChar
infixr 4 `fn`
fn :: Type -> Type -> Type
a `fn` b = TAp (TAp tArrow a) b
list :: Type -> Type
list t = TAp tList t
pair :: Type -> Type -> Type
pair a b = TAp (TAp tTuple2 a) b
class HasKind t where
kind :: t -> Kind
instance HasKind Tyvar where
kind (Tyvar v k) = k
instance HasKind Tycon where
kind (Tycon v k) = k
instance HasKind Type where
kind (TCon tc) = kind tc
kind (TVar u) = kind u
kind (TAp t _) = case (kind t) of
(Kfun _ k) -> k
-----------------------------------------------------------------------------
-- Subst: Substitutions
-----------------------------------------------------------------------------
type Subst = [(Tyvar, Type)]
nullSubst :: Subst
nullSubst = []
(+->) :: Tyvar -> Type -> Subst
u +-> t = [(u, t)]
class Types t where
apply :: Subst -> t -> t
tv :: t -> [Tyvar]
instance Types Type where
apply s (TVar u) = case lookup u s of
Just t -> t
Nothing -> TVar u
apply s (TAp l r) = TAp (apply s l) (apply s r)
apply s t = t
tv (TVar u) = [u]
tv (TAp l r) = tv l `union` tv r
tv t = []
instance Types a => Types [a] where
apply s = map (apply s)
tv = nub . concat . map tv
infixr 4 @@
(@@) :: Subst -> Subst -> Subst
s1 @@ s2 = [ (u, apply s1 t) | (u,t) <- s2 ] ++ s1
merge :: Monad m => Subst -> Subst -> m Subst
merge s1 s2 = if agree then return (s1++s2) else fail "merge fails"
where agree = all (\v -> apply s1 (TVar v) == apply s2 (TVar v))
(map fst s1 `intersect` map fst s2)
-----------------------------------------------------------------------------
-- Unify: Unification
-----------------------------------------------------------------------------
mgu :: Monad m => Type -> Type -> m Subst
varBind :: Monad m => Tyvar -> Type -> m Subst
mgu (TAp l r) (TAp l' r') = do s1 <- mgu l l'
s2 <- mgu (apply s1 r) (apply s1 r')
return (s2 @@ s1)
mgu (TVar u) t = varBind u t
mgu t (TVar u) = varBind u t
mgu (TCon tc1) (TCon tc2)
| tc1==tc2 = return nullSubst
mgu t1 t2 = fail "types do not unify"
varBind u t | t == TVar u = return nullSubst
| u `elem` tv t = fail "occurs check fails"
| kind u /= kind t = fail "kinds do not match"
| otherwise = return (u +-> t)
match :: Monad m => Type -> Type -> m Subst
match (TAp l r) (TAp l' r') = do sl <- match l l'
sr <- match r r'
merge sl sr
match (TVar u) t | kind u == kind t = return (u +-> t)
match (TCon tc1) (TCon tc2)
| tc1==tc2 = return nullSubst
match t1 t2 = fail "types do not match"
-----------------------------------------------------------------------------
-- Pred: Predicates
-----------------------------------------------------------------------------
data Qual t = [Pred] :=> t
deriving Eq
data Pred = IsIn Id Type
deriving Eq
instance Types t => Types (Qual t) where
apply s (ps :=> t) = apply s ps :=> apply s t
tv (ps :=> t) = tv ps `union` tv t
instance Types Pred where
apply s (IsIn i t) = IsIn i (apply s t)
tv (IsIn i t) = tv t
mguPred, matchPred :: Pred -> Pred -> Maybe Subst
mguPred = lift mgu
matchPred = lift match
lift m (IsIn i t) (IsIn i' t')
| i == i' = m t t'
| otherwise = fail "classes differ"
type Class = ([Id], [Inst])
type Inst = Qual Pred
-----------------------------------------------------------------------------
data ClassEnv = ClassEnv { classes :: Id -> Maybe Class,
defaults :: [Type] }
super :: ClassEnv -> Id -> [Id]
super ce i = case classes ce i of Just (is, its) -> is
insts :: ClassEnv -> Id -> [Inst]
insts ce i = case classes ce i of Just (is, its) -> its
defined :: Maybe a -> Bool
defined (Just x) = True
defined Nothing = False
modify :: ClassEnv -> Id -> Class -> ClassEnv
modify ce i c = ce{classes = \j -> if i==j then Just c
else classes ce j}
initialEnv :: ClassEnv
initialEnv = ClassEnv { classes = \i -> fail "class not defined",
defaults = [tInteger, tDouble] }
type EnvTransformer = ClassEnv -> Maybe ClassEnv
infixr 5 <:>
(<:>) :: EnvTransformer -> EnvTransformer -> EnvTransformer
(f <:> g) ce = do ce' <- f ce
g ce'
addClass :: Id -> [Id] -> EnvTransformer
addClass i is ce
| defined (classes ce i) = fail "class already defined"
| any (not . defined . classes ce) is = fail "superclass not defined"
| otherwise = return (modify ce i (is, []))
addPreludeClasses :: EnvTransformer
addPreludeClasses = addCoreClasses <:> addNumClasses
addCoreClasses :: EnvTransformer
addCoreClasses = addClass "Eq" []
<:> addClass "Ord" ["Eq"]
<:> addClass "Show" []
<:> addClass "Read" []
<:> addClass "Bounded" []
<:> addClass "Enum" []
<:> addClass "Functor" []
<:> addClass "Monad" []
addNumClasses :: EnvTransformer
addNumClasses = addClass "Num" ["Eq", "Show"]
<:> addClass "Real" ["Num", "Ord"]
<:> addClass "Fractional" ["Num"]
<:> addClass "Integral" ["Real", "Enum"]
<:> addClass "RealFrac" ["Real", "Fractional"]
<:> addClass "Floating" ["Fractional"]
<:> addClass "RealFloat" ["RealFrac", "Floating"]
addInst :: [Pred] -> Pred -> EnvTransformer
addInst ps p@(IsIn i _) ce
| not (defined (classes ce i)) = fail "no class for instance"
| any (overlap p) qs = fail "overlapping instance"
| otherwise = return (modify ce i c)
where its = insts ce i
qs = [ q | (_ :=> q) <- its ]
c = (super ce i, (ps:=>p) : its)
overlap :: Pred -> Pred -> Bool
overlap p q = defined (mguPred p q)
exampleInsts :: EnvTransformer
exampleInsts = addPreludeClasses
<:> addInst [] (IsIn "Ord" tUnit)
<:> addInst [] (IsIn "Ord" tChar)
<:> addInst [] (IsIn "Ord" tInt)
<:> addInst [IsIn "Ord" (TVar (Tyvar "a" Star)),
IsIn "Ord" (TVar (Tyvar "b" Star))]
(IsIn "Ord" (pair (TVar (Tyvar "a" Star))
(TVar (Tyvar "b" Star))))
-----------------------------------------------------------------------------
bySuper :: ClassEnv -> Pred -> [Pred]
bySuper ce p@(IsIn i t)
= p : concat [ bySuper ce (IsIn i' t) | i' <- super ce i ]
byInst :: ClassEnv -> Pred -> Maybe [Pred]
byInst ce p@(IsIn i t) = msum [ tryInst it | it <- insts ce i ]
where tryInst (ps :=> h) = do u <- matchPred h p
Just (map (apply u) ps)
entail :: ClassEnv -> [Pred] -> Pred -> Bool
entail ce ps p = any (p `elem`) (map (bySuper ce) ps) ||
case byInst ce p of
Nothing -> False
Just qs -> all (entail ce ps) qs
-----------------------------------------------------------------------------
inHnf :: Pred -> Bool
inHnf (IsIn c t) = hnf t
where hnf (TVar v) = True
hnf (TCon tc) = False
hnf (TAp t _) = hnf t
toHnfs :: Monad m => ClassEnv -> [Pred] -> m [Pred]
toHnfs ce ps = do pss <- mapM (toHnf ce) ps
return (concat pss)
toHnf :: Monad m => ClassEnv -> Pred -> m [Pred]
toHnf ce p | inHnf p = return [p]
| otherwise = case byInst ce p of
Nothing -> fail "context reduction"
Just ps -> toHnfs ce ps
simplify :: ClassEnv -> [Pred] -> [Pred]
simplify ce = loop []
where loop rs [] = rs
loop rs (p:ps) | entail ce (rs++ps) p = loop rs ps
| otherwise = loop (p:rs) ps
reduce :: Monad m => ClassEnv -> [Pred] -> m [Pred]
reduce ce ps = do qs <- toHnfs ce ps
return (simplify ce qs)
scEntail :: ClassEnv -> [Pred] -> Pred -> Bool
scEntail ce ps p = any (p `elem`) (map (bySuper ce) ps)
-----------------------------------------------------------------------------
-- Scheme: Type schemes
-----------------------------------------------------------------------------
data Scheme = Forall [Kind] (Qual Type)
deriving Eq
instance Types Scheme where
apply s (Forall ks qt) = Forall ks (apply s qt)
tv (Forall ks qt) = tv qt
quantify :: [Tyvar] -> Qual Type -> Scheme
quantify vs qt = Forall ks (apply s qt)
where vs' = [ v | v <- tv qt, v `elem` vs ]
ks = map kind vs'
s = zip vs' (map TGen [0..])
toScheme :: Type -> Scheme
toScheme t = Forall [] ([] :=> t)
-----------------------------------------------------------------------------
-- Assump: Assumptions
-----------------------------------------------------------------------------
data Assump = Id :>: Scheme
instance Types Assump where
apply s (i :>: sc) = i :>: (apply s sc)
tv (i :>: sc) = tv sc
find :: Monad m => Id -> [Assump] -> m Scheme
find i [] = fail ("unbound identifier: " ++ i)
find i ((i':>:sc):as) = if i==i' then return sc else find i as
-----------------------------------------------------------------------------
-- TIMonad: Type inference monad
-----------------------------------------------------------------------------
newtype TI a = TI (Subst -> Int -> (Subst, Int, a))
instance Functor TI where
fmap = liftM
instance Applicative TI where
pure = return
(<*>) = ap
instance Monad TI where
return x = TI (\s n -> (s,n,x))
TI f >>= g = TI (\s n -> case f s n of
(s',m,x) -> let TI gx = g x
in gx s' m)
runTI :: TI a -> a
runTI (TI f) = x where (s,n,x) = f nullSubst 0
getSubst :: TI Subst
getSubst = TI (\s n -> (s,n,s))
unify :: Type -> Type -> TI ()
unify t1 t2 = do s <- getSubst
u <- mgu (apply s t1) (apply s t2)
extSubst u
extSubst :: Subst -> TI ()
extSubst s' = TI (\s n -> (s'@@s, n, ()))
newTVar :: Kind -> TI Type
newTVar k = TI (\s n -> let v = Tyvar (enumId n) k
in (s, n+1, TVar v))
freshInst :: Scheme -> TI (Qual Type)
freshInst (Forall ks qt) = do ts <- mapM newTVar ks
return (inst ts qt)
class Instantiate t where
inst :: [Type] -> t -> t
instance Instantiate Type where
inst ts (TAp l r) = TAp (inst ts l) (inst ts r)
inst ts (TGen n) = ts !! n
inst ts t = t
instance Instantiate a => Instantiate [a] where
inst ts = map (inst ts)
instance Instantiate t => Instantiate (Qual t) where
inst ts (ps :=> t) = inst ts ps :=> inst ts t
instance Instantiate Pred where
inst ts (IsIn c t) = IsIn c (inst ts t)
-----------------------------------------------------------------------------
-- TIMain: Type Inference Algorithm
-----------------------------------------------------------------------------
-- Infer: Basic definitions for type inference
-----------------------------------------------------------------------------
type Infer e t = ClassEnv -> [Assump] -> e -> TI ([Pred], t)
-----------------------------------------------------------------------------
-- Lit: Literals
-----------------------------------------------------------------------------
data Literal = LitInt Integer
| LitChar Char
| LitRat Rational
| LitStr String
tiLit :: Literal -> TI ([Pred],Type)
tiLit (LitChar _) = return ([], tChar)
tiLit (LitInt _) = do v <- newTVar Star
return ([IsIn "Num" v], v)
tiLit (LitStr _) = return ([], tString)
tiLit (LitRat _) = do v <- newTVar Star
return ([IsIn "Fractional" v], v)
-----------------------------------------------------------------------------
-- Pat: Patterns
-----------------------------------------------------------------------------
data Pat = PVar Id
| PWildcard
| PAs Id Pat
| PLit Literal
| PNpk Id Integer
| PCon Assump [Pat]
tiPat :: Pat -> TI ([Pred], [Assump], Type)
tiPat (PVar i) = do v <- newTVar Star
return ([], [i :>: toScheme v], v)
tiPat PWildcard = do v <- newTVar Star
return ([], [], v)
tiPat (PAs i pat) = do (ps, as, t) <- tiPat pat
return (ps, (i:>:toScheme t):as, t)
tiPat (PLit l) = do (ps, t) <- tiLit l
return (ps, [], t)
tiPat (PNpk i k) = do t <- newTVar Star
return ([IsIn "Integral" t], [i:>:toScheme t], t)
tiPat (PCon (i:>:sc) pats) = do (ps,as,ts) <- tiPats pats
t' <- newTVar Star
(qs :=> t) <- freshInst sc
unify t (foldr fn t' ts)
return (ps++qs, as, t')
tiPats :: [Pat] -> TI ([Pred], [Assump], [Type])
tiPats pats = do psasts <- mapM tiPat pats
let ps = concat [ ps' | (ps',_,_) <- psasts ]
as = concat [ as' | (_,as',_) <- psasts ]
ts = [ t | (_,_,t) <- psasts ]
return (ps, as, ts)
-----------------------------------------------------------------------------
data Expr = Var Id
| Lit Literal
| Const Assump
| Ap Expr Expr
| Let BindGroup Expr
tiExpr :: Infer Expr Type
tiExpr ce as (Var i) = do sc <- find i as
(ps :=> t) <- freshInst sc
return (ps, t)
tiExpr ce as (Const (i:>:sc)) = do (ps :=> t) <- freshInst sc
return (ps, t)
tiExpr ce as (Lit l) = do (ps,t) <- tiLit l
return (ps, t)
tiExpr ce as (Ap e f) = do (ps,te) <- tiExpr ce as e
(qs,tf) <- tiExpr ce as f
t <- newTVar Star
unify (tf `fn` t) te
return (ps++qs, t)
tiExpr ce as (Let bg e) = do (ps, as') <- tiBindGroup ce as bg
(qs, t) <- tiExpr ce (as' ++ as) e
return (ps ++ qs, t)
-----------------------------------------------------------------------------
type Alt = ([Pat], Expr)
tiAlt :: Infer Alt Type
tiAlt ce as (pats, e) = do (ps, as', ts) <- tiPats pats
(qs,t) <- tiExpr ce (as'++as) e
return (ps++qs, foldr fn t ts)
tiAlts :: ClassEnv -> [Assump] -> [Alt] -> Type -> TI [Pred]
tiAlts ce as alts t = do psts <- mapM (tiAlt ce as) alts
mapM (unify t) (map snd psts)
return (concat (map fst psts))
-----------------------------------------------------------------------------
split :: Monad m => ClassEnv -> [Tyvar] -> [Tyvar] -> [Pred]
-> m ([Pred], [Pred])
split ce fs gs ps = do ps' <- reduce ce ps
let (ds, rs) = partition (all (`elem` fs) . tv) ps'
rs' <- defaultedPreds ce (fs++gs) rs
return (ds, rs \\ rs')
type Ambiguity = (Tyvar, [Pred])
ambiguities :: ClassEnv -> [Tyvar] -> [Pred] -> [Ambiguity]
ambiguities ce vs ps = [ (v, filter (elem v . tv) ps) | v <- tv ps \\ vs ]
numClasses :: [Id]
numClasses = ["Num", "Integral", "Floating", "Fractional",
"Real", "RealFloat", "RealFrac"]
stdClasses :: [Id]
stdClasses = ["Eq", "Ord", "Show", "Read", "Bounded", "Enum", "Ix",
"Functor", "Monad", "MonadPlus"] ++ numClasses
candidates :: ClassEnv -> Ambiguity -> [Type]
candidates ce (v, qs) = [ t' | let is = [ i | IsIn i t <- qs ]
ts = [ t | IsIn i t <- qs ],
all ((TVar v)==) ts,
any (`elem` numClasses) is,
all (`elem` stdClasses) is,
t' <- defaults ce,
all (entail ce []) [ IsIn i t' | i <- is ] ]
withDefaults :: Monad m => ([Ambiguity] -> [Type] -> a)
-> ClassEnv -> [Tyvar] -> [Pred] -> m a
withDefaults f ce vs ps
| any null tss = fail "cannot resolve ambiguity"
| otherwise = return (f vps (map head tss))
where vps = ambiguities ce vs ps
tss = map (candidates ce) vps
defaultedPreds :: Monad m => ClassEnv -> [Tyvar] -> [Pred] -> m [Pred]
defaultedPreds = withDefaults (\vps ts -> concat (map snd vps))
defaultSubst :: Monad m => ClassEnv -> [Tyvar] -> [Pred] -> m Subst
defaultSubst = withDefaults (\vps ts -> zip (map fst vps) ts)
-----------------------------------------------------------------------------
type Expl = (Id, Scheme, [Alt])
tiExpl :: ClassEnv -> [Assump] -> Expl -> TI [Pred]
tiExpl ce as (i, sc, alts)
= do (qs :=> t) <- freshInst sc
ps <- tiAlts ce as alts t
s <- getSubst
let qs' = apply s qs
t' = apply s t
fs = tv (apply s as)
gs = tv t' \\ fs
sc' = quantify gs (qs':=>t')
ps' = filter (not . entail ce qs') (apply s ps)
(ds,rs) <- split ce fs gs ps'
if sc /= sc' then
fail "signature too general"
else if not (null rs) then
fail "context too weak"
else
return ds
-----------------------------------------------------------------------------
type Impl = (Id, [Alt])
restricted :: [Impl] -> Bool
restricted bs = any simple bs
where simple (i,alts) = any (null . fst) alts
tiImpls :: Infer [Impl] [Assump]
tiImpls ce as bs = do ts <- mapM (\_ -> newTVar Star) bs
let is = map fst bs
scs = map toScheme ts
as' = zipWith (:>:) is scs ++ as
altss = map snd bs
pss <- sequence (zipWith (tiAlts ce as') altss ts)
s <- getSubst
let ps' = apply s (concat pss)
ts' = apply s ts
fs = tv (apply s as)
vss = map tv ts'
gs = foldr1 union vss \\ fs
(ds,rs) <- split ce fs (foldr1 intersect vss) ps'
if restricted bs then
let gs' = gs \\ tv rs
scs' = map (quantify gs' . ([]:=>)) ts'
in return (ds++rs, zipWith (:>:) is scs')
else
let scs' = map (quantify gs . (rs:=>)) ts'
in return (ds, zipWith (:>:) is scs')
-----------------------------------------------------------------------------
type BindGroup = ([Expl], [[Impl]])
tiBindGroup :: Infer BindGroup [Assump]
tiBindGroup ce as (es,iss) =
do let as' = [ v:>:sc | (v,sc,alts) <- es ]
(ps, as'') <- tiSeq tiImpls ce (as'++as) iss
qss <- mapM (tiExpl ce (as''++as'++as)) es
return (ps++concat qss, as''++as')
tiSeq :: Infer bg [Assump] -> Infer [bg] [Assump]
tiSeq ti ce as [] = return ([],[])
tiSeq ti ce as (bs:bss) = do (ps,as') <- ti ce as bs
(qs,as'') <- tiSeq ti ce (as'++as) bss
return (ps++qs, as''++as')
-----------------------------------------------------------------------------
-- TIProg: Type Inference for Whole Programs
-----------------------------------------------------------------------------
type Program = [BindGroup]
tiProgram :: ClassEnv -> [Assump] -> Program -> [Assump]
tiProgram ce as bgs = runTI $
do (ps, as') <- tiSeq tiBindGroup ce as bgs
s <- getSubst
rs <- reduce ce (apply s ps)
s' <- defaultSubst ce [] rs
return (apply (s'@@s) as')
-----------------------------------------------------------------------------
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