{- |

Module : $Header$

Copyright : (c) Christian Maeder and Uni Bremen 2003

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

Maintainer : maeder@tzi.de

Stability : experimental

Portability : portable

substitution and unification of types

-}

module HasCASL.Unify where

import HasCASL.As

import HasCASL.AsUtils

import HasCASL.Le

import qualified Common.Lib.Map as Map

import qualified Common.Lib.Set as Set

import Common.PrettyPrint

import Common.Id

import Common.Lib.State

import Common.Result

import Data.List as List

import Data.Maybe

-- | bound vars

genVarsOf :: Type -> [TypeArg]

genVarsOf = map snd . leaves (<0)

-- | vars or other ids

leaves :: (Int -> Bool) -> Type -> [(Int, TypeArg)]

leaves b t =

case t of

TypeName j k i -> if b(i)

then [(i, TypeArg j k Other [])]

else []

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

ExpandedType _ t2 -> leaves b t2

KindedType tk _ _ -> leaves b tk

LazyType tl _ -> leaves b tl

ProductType l _ -> foldl List.union [] $ map (leaves b) l

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

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

-- | 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, _) <- freshInstList sc1

(t2, _) <- freshInstList 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

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

freshInstList :: TypeScheme -> State Int (Type, [Type])

freshInstList (TypeScheme tArgs t _) =

do let ls = leaves (< 0) t -- generic vars

vs = map snd ls

ts <- mkSubst vs

return (subst (Map.fromList $ zip (map fst ls) ts) t,

map (mapArg $ zip vs ts) tArgs)

mapArg :: [(TypeArg, a)] -> TypeArg -> a

mapArg ts (TypeArg i k _ _) =

maybe (error "mapArg") snd $

find (\ (TypeArg j l _ _, _) -> i == j && k == l) ts

freshVar :: Pos -> State Int (Id, Int)

freshVar p =

do c <- get

put (c + 1)

return (simpleIdToId $ Token ("_var_" ++ show c) p, c)

mkSingleSubst :: TypeArg -> State Int Type

mkSingleSubst tv@(TypeArg _ k _ _) =

do (ty, c) <- freshVar $ posOfTypeArg tv

return (TypeName ty k c)

mkSubst :: [TypeArg] -> State Int [Type]

mkSubst = mapM mkSingleSubst

type Subst = Map.Map Int Type

eps :: Subst

eps = Map.empty

class Unifiable a where

subst :: Subst -> a -> a

match :: TypeMap -> (TypeId -> TypeId -> Bool)

-> (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)

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

shapeMatch tm a b = match tm (const $ const True) (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

-- | get the type variable

getTypeVar :: TypeArg -> Id

getTypeVar(TypeArg v _ _ _) = v

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

idsOf b = Set.fromList . map (getTypeVar . snd) . leaves b

instance Unifiable Type where

subst m = rename (\ i k n ->

case Map.lookup n m of

Just s -> s

_ -> TypeName i k n)

match tm rel t1 (b2, ExpandedType _ t2) = match tm rel t1 (b2, t2)

match tm rel (b1, ExpandedType _ t1) t2 = match tm rel (b1, t1) t2

match tm rel t1 (b2, LazyType t2 _) = match tm rel t1 (b2, t2)

match tm rel (b1, LazyType t1 _) t2 = match tm rel (b1, t1) t2

match tm rel t1 (b2, KindedType t2 _ _) = match tm rel t1 (b2, t2)

match tm rel (b1, KindedType t1 _ _) t2 = match tm rel (b1, t1) t2

match _ rel (b1, t1@(TypeName i1 _k1 v1)) (b2, t2@(TypeName i2 _k2 v2)) =

if rel i1 i2 && v1 == v2

then return eps

else if v1 > 0 && b1 then return $

Map.single v1 t2

else if v2 > 0 && b2 then return $

Map.single v2 t1

else uniResult "typename" t1

"is not unifiable with typename" t2

match _tm _ (b1, TypeName i1 _ v1) (_, t2) =

if v1 > 0 && b1 then

if null $ leaves (==v1) t2 then

return $ Map.single v1 t2

else uniResult "var" i1 "occurs in" t2

else uniResult "typename" i1

"is not unifiable with type" t2

match tm rel t2 t1@(_, TypeName _ _ _) = match tm rel t1 t2

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

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

match tm rel (b1, ProductType p1 _) (b2, ProductType p2 _) =

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

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

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

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

"is not unifiable with type" t2

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

showPrettyWithPos a = let p = getMyPos a in

showChar '\'' . showPretty a . showChar '\''

. noShow (isNullPos p) (showChar ' ' .

showParen True (showPos 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 rel (b1, (t1, t2)) (b2, (t3, t4)) =

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

case m1 of

Nothing -> r1

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

(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 rel (b1, a1:r1) (b2, a2:r2) =

match tm rel (b1, (a1, r1)) (b2, (a2, r2))

match tm rel (b1, []) l = match tm rel 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 rel (b1, Just a1) (b2, Just a2) = match tm rel (b1, a1) (b2, a2)

repl :: Map.Map TypeArg Type -> Type -> Type

repl m = rename ( \ i k n ->

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

Just s -> s

Nothing -> TypeName i k n)

expand :: TypeMap -> TypeScheme -> TypeScheme

expand = mapTypeOfScheme . expandAlias

expandAlias :: TypeMap -> Type -> Type

expandAlias tm t =

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

if b && length ps == length as then

ExpandedType t $ repl (Map.fromList (zip ps $ reverse as)) ta

else ta

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

expandAliases tm t = case t of

TypeName i _ _ -> case Map.lookup i tm of

Just (TypeInfo _ _ _

(AliasTypeDefn (TypeScheme l ty _))) ->

(l, [], ty, True)

Just (TypeInfo _ _ _

(Supertype _ (TypeScheme l ty _) _)) ->

case ty of

TypeName _ _ _ -> wrap t

_ -> (l, [], ty, True)

_ -> wrap t

TypeAppl t1 t2 ->

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

t3 = expandAlias tm t2

in if b then

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

else wrap $ TypeAppl t1 t3

FunType t1 a t2 ps ->

wrap $ FunType (expandAlias tm t1) a (expandAlias tm t2) ps

ProductType ts ps -> wrap $ ProductType (map (expandAlias tm) ts) ps

LazyType ty ps -> wrap $ LazyType (expandAlias tm ty) ps

ExpandedType t1 t2 -> wrap $ ExpandedType t1 $ expandAlias tm t2

KindedType ty k ps -> wrap $ KindedType (expandAlias tm ty) k ps

_ -> wrap t

where wrap ty = ([], [], ty, False)