{- |

Module : $Header$

Copyright : (c) Christian Maeder and Uni Bremen 2003

Licence : similar to LGPL, see HetCATS/LICENCE.txt or LIZENZ.txt

Maintainer : hets@tzi.de

Stability : experimental

Portability : portable

substitution and unification of types

-}

module HasCASL.Unify where

import HasCASL.As

import HasCASL.AsUtils

import Common.PrettyPrint

import Common.Id

import HasCASL.Le

import Common.Lib.State

import Common.Lib.Parsec

import qualified Common.Lib.Map as Map

import qualified Common.Lib.Set as Set

import Common.Result

import Data.List

import Data.Maybe

-- | vars

varsOf :: Type -> Set.Set TypeArg

varsOf = leaves (>0)

-- | vars or other ids

leaves :: (Int -> Bool) -> Type -> Set.Set TypeArg

leaves b t =

case t of

TypeName j k i -> if b(i)

then Set.single $ TypeArg j k Other []

else Set.empty

TypeAppl t1 t2 -> leaves b t1 `Set.union` leaves b t2

KindedType tk _ _ -> leaves b tk

LazyType tl _ -> leaves b tl

ProductType l _ -> Set.unions $ map (leaves b) l

FunType t1 _ t2 _ -> leaves b t1 `Set.union` leaves b t2

_ -> error ("leaves: " ++ show t)

generalize :: TypeScheme -> TypeScheme

generalize (TypeScheme _ q@(_ :=> ty) ps) =

TypeScheme (Set.toList $ varsOf ty) q ps

-- | composition (reversed: first substitution first!)

compSubst :: Subst -> Subst -> Subst

compSubst s1 s2 = Map.union (Map.map (subst s2) s1) s2

-- | unifiability of type schemes including instantiation with fresh variables

-- (and looking up type aliases)

isUnifiable :: TypeMap -> Int -> TypeScheme -> TypeScheme -> Bool

isUnifiable tm c = asSchemes c (unify tm)

-- | test if second scheme is a substitution instance

instScheme :: TypeMap -> Int -> TypeScheme -> TypeScheme -> Bool

instScheme tm c = asSchemes c (subsume tm)

-- | lift 'State' Int to 'State' Env

toEnvState :: State Int a -> State Env a

toEnvState p =

do s <- get

let (r, c) = runState p $ counter s

put s { counter = c }

return r

toSchemes :: (Type -> Type -> a) -> TypeScheme -> TypeScheme -> State Int a

toSchemes f sc1 sc2 =

do t1 <- freshInst sc1

t2 <- freshInst sc2

return $ f t1 t2

asSchemes :: Int -> (Type -> Type -> a) -> TypeScheme -> TypeScheme -> a

asSchemes c f sc1 sc2 = fst $ runState (toSchemes f sc1 sc2) c

-- -------------------------------------------------------------------------

freshInst :: TypeScheme -> State Int Type

freshInst (TypeScheme tArgs (_ :=> t) _) =

do m <- mkSubst tArgs

return $ subst m t

freshVar :: State Int Id

freshVar =

do c <- get

put (c + 1)

return $ simpleIdToId $ mkSimpleId ("_var_" ++ show c)

mkSingleSubst :: TypeArg -> State Int Subst

mkSingleSubst tv@(TypeArg _ k _ _) =

do ty <- freshVar

return $ Map.single tv $ TypeName ty k 1

mkSubst :: [TypeArg] -> State Int Subst

mkSubst tas = do ms <- mapM mkSingleSubst tas

return $ Map.unions ms

type Subst = Map.Map TypeArg Type

eps :: Subst

eps = Map.empty

class Unifiable a where

subst :: Subst -> a -> a

match :: TypeMap -> (Bool, a) -> (Bool, a) -> Result Subst

-- | most general unifier via 'match'

-- where both sides may contribute substitutions

mgu :: Unifiable a => TypeMap -> a -> a -> Result Subst

mgu tm a b = match tm (True, a) (True, b)

unify :: Unifiable a => TypeMap -> a -> a -> Bool

unify tm a b = isJust $ maybeResult $ mgu tm a b

subsume :: Unifiable a => TypeMap -> a -> a -> Bool

subsume tm a b = isJust $ maybeResult $ match tm (False, a) (True, b)

equalSubs :: Unifiable a => TypeMap -> a -> a -> Bool

equalSubs tm a b = subsume tm a b && subsume tm b a

idsOf :: (Int -> Bool) -> Type -> Set.Set TypeId

idsOf b = Set.image ( \ (TypeArg j _ _ _) -> j) . leaves b

occursIn :: TypeMap -> TypeId -> Type -> Bool

occursIn tm i = Set.any (relatedTypeIds tm i) . idsOf (const True)

relatedTypeIds :: TypeMap -> TypeId -> TypeId -> Bool

relatedTypeIds tm i1 i2 =

not $ Set.disjoint (allRelIds tm i1) $ allRelIds tm i2

allRelIds :: TypeMap -> TypeId -> Set.Set TypeId

allRelIds tm i = Set.union (superIds tm i) $ subIds tm i

rename :: (TypeId -> Kind -> Int -> Type) -> Type -> Type

rename m t = case t of

TypeName i k n -> m i k n

TypeAppl t1 t2 ->

TypeAppl (rename m t1) (rename m t2)

TypeToken _ -> t

BracketType b l ps ->

BracketType b (map (rename m) l) ps

KindedType tk k ps ->

KindedType (rename m tk) k ps

MixfixType l -> MixfixType $ map (rename m) l

LazyType tl ps -> LazyType (rename m tl) ps

ProductType l ps -> ProductType (map (rename m) l) ps

FunType t1 a t2 ps -> FunType (rename m t1) a (rename m t2) ps

instance Unifiable Type where

subst m = rename (\ i k n ->

case Map.lookup (TypeArg i k Other []) m of

Just s -> s

_ -> TypeName i k n)

match m (a, s) (b, t) = mm m (a, unalias m s) (b, unalias m t)

where

mm tm t1 (b2, LazyType t2 _) = mm tm t1 (b2, t2)

mm tm (b1, LazyType t1 _) t2 = mm tm (b1, t1) t2

mm tm t1 (b2, KindedType t2 _ _) = mm tm t1 (b2, t2)

mm tm (b1, KindedType t1 _ _) t2 = mm tm (b1, t1) t2

mm tm (b1, t1@(TypeName i1 k1 v1)) (b2, t2@(TypeName i2 k2 v2)) =

if relatedTypeIds tm i1 i2

then return eps

else if v1 > 0 && b1 then return $

Map.single (TypeArg i1 k1 Other []) t2

else if v2 > 0 && b2 then return $

Map.single (TypeArg i2 k2 Other []) t1

else uniResult "typename" i1

"is not unifiable with typename" i2

mm tm (b1, TypeName i1 k1 v1) (_, t2) =

if v1 > 0 && b1 then

if occursIn tm i1 t2 then

uniResult "var" i1 "occurs in" t2

else return $

Map.single (TypeArg i1 k1 Other []) t2

else uniResult "typename" i1

"is not unifiable with type" t2

mm tm t2 t1@(_, TypeName _ _ _) = mm tm t1 t2

mm tm (b1, TypeAppl t1 t2) (b2, TypeAppl t3 t4) =

match tm (b1, (t1, t2)) (b2, (t3, t4))

mm tm (b1, ProductType p1 _) (b2, ProductType p2 _) =

match tm (b1, p1) (b2, p2)

mm tm (b1, FunType t1 _ t2 _) (b2, FunType t3 _ t4 _) =

match tm (b1, (t1, t2)) (b2, (t3, t4))

mm _ (_,t1) (_,t2) = uniResult "type" t1

"is not unifiable with type" t2

showPrettyWithPos :: (PrettyPrint a, PosItem a) => a -> ShowS

showPrettyWithPos a = let p = getMyPos a

s = ("'" ++) . showPretty a . ("'" ++)

n = sourceName p in

if nullPos == p then s else s . (" (" ++) .

(if null n then id else (n ++) . (", " ++))

. shows (sourceLine p)

. ("." ++) . shows (sourceColumn p)

. (")" ++)

uniResult :: (PrettyPrint a, PosItem a, PrettyPrint b, PosItem b) =>

String -> a -> String -> b -> Result Subst

uniResult s1 a s2 b =

Result [Diag Hint ("in type\n" ++ " " ++ s1 ++ " " ++

showPrettyWithPos a "\n " ++ s2 ++ " " ++

showPrettyWithPos b "") nullPos] Nothing

instance (Unifiable a, Unifiable b) => Unifiable (a, b) where

subst s (t1, t2) = (subst s t1, subst s t2)

match tm (b1, (t1, t2)) (b2, (t3, t4)) =

let r1@(Result _ m1) = match tm (b1, t1) (b2, t3) in

case m1 of

Nothing -> r1

Just s1 -> let r2@(Result _ m2) =

match tm (b1, if b1 then subst s1 t2 else t2)

(b2, if b2 then subst s1 t4 else t4)

in case m2 of

Nothing -> r2

Just s2 -> return $ compSubst s1 s2

instance (PrettyPrint a, PosItem a, Unifiable a) => Unifiable [a] where

subst s = map (subst s)

match _ (_, []) (_, []) = return eps

match tm (b1, a1:r1) (b2, a2:r2) = match tm (b1, (a1, r1)) (b2, (a2, r2))

match tm (b1, []) l = match tm l (b1, [])

match _ (_, (a:_)) (_, []) = uniResult "type component" a

"is not unifiable with the empty list"

(mkSimpleId "[]")

instance (PrettyPrint a, PosItem a, Unifiable a) => Unifiable (Maybe a) where

subst s = fmap (subst s)

match _ (_, Nothing) _ = return eps

match _ _ (_, Nothing) = return eps

match tm (b1, Just a1) (b2, Just a2) = match tm (b1, a1) (b2, a2)

unalias :: TypeMap -> Type -> Type

unalias tm = fst . expandAlias tm

expandAlias :: TypeMap -> Type -> (Type, Bool)

expandAlias tm t =

let (ps, as, ta, b) = expandAliases tm t in

if b && length ps == length as then

(subst (Map.fromList (zip ps $ reverse as)) ta, b)

else (ta, b)

expandAliases :: TypeMap -> Type -> ([TypeArg], [Type], Type, Bool)

expandAliases tm t@(TypeName i _ _) =

case Map.lookup i tm of

Just (TypeInfo _ _ _

(AliasTypeDefn (TypeScheme l (_ :=> ts) _))) ->

(l, [], unalias tm ts, True)

Just (TypeInfo _ _ _

(Supertype _ (TypeScheme l (_ :=> ts) _) _)) ->

(l, [], unalias tm ts, True)

_ -> ([], [], t, False)

expandAliases tm (TypeAppl t1 t2) =

let (ps, as, ta, b) = expandAliases tm t1

(t3, b2) = expandAlias tm t2

in if b then

(ps, t3:as, ta, b) -- reverse later on

else ([], [], TypeAppl t1 t3, b2)

expandAliases tm (FunType t1 a t2 ps) =

let (t3, b1) = expandAlias tm t1

(t4, b2) = expandAlias tm t2

in ([], [], FunType t3 a t4 ps, b1 || b2)

expandAliases tm (ProductType ts ps) =

let tls = map (expandAlias tm) ts

in ([], [], ProductType (map fst tls) ps, any snd tls)

expandAliases tm (LazyType t ps) =

let (newT, b) = expandAlias tm t

in ([], [], LazyType newT ps, b)

expandAliases tm (KindedType t k ps) =

let (newT, b) = expandAlias tm t

in ([], [], KindedType newT k ps, b)

expandAliases _ t = ([], [], t, False)

-- | super type ids

superIds :: TypeMap -> Id -> Set.Set Id

superIds tm = supIds tm Set.empty . Set.single

subIds :: TypeMap -> Id -> Set.Set Id

subIds tm i = foldr ( \ j s ->

if Set.member i $ superIds tm j then

Set.insert j s else s) Set.empty $ Map.keys tm

supIds :: TypeMap -> Set.Set Id -> Set.Set Id -> Set.Set Id

supIds tm known new =

if Set.isEmpty new then known else

let more = Set.unions $ map superTypeToId $

concatMap ( \ i -> superTypes

$ Map.findWithDefault starTypeInfo i tm)

$ Set.toList new

newKnown = Set.union known new

in supIds tm newKnown (more Set.\\ newKnown)

starTypeInfo :: TypeInfo

starTypeInfo = TypeInfo star [] [] NoTypeDefn

superTypeToId :: Type -> Set.Set Id

superTypeToId t =

case t of

TypeName i _ _ -> Set.single i

_ -> Set.empty