{- |

Module : $Header$

Description : generalized unification of types

Copyright : (c) Christian Maeder and Uni Bremen 2003-2005

License : similar to LGPL, see HetCATS/LICENSE.txt or LIZENZ.txt

Maintainer : Christian.Maeder@dfki.de

Stability : experimental

Portability : portable

substitution and unification of types

-}

module HasCASL.Unify where

import HasCASL.As

import HasCASL.AsUtils

import HasCASL.PrintAs ()

import HasCASL.TypeAna

import HasCASL.Le

import qualified Data.Map as Map

import qualified Data.Set as Set

import Common.DocUtils

import Common.Id

import Common.Lib.State

import Common.Result

import Data.List as List

import Data.Maybe

-- | bound vars

genVarsOf :: Type -> [(Id, RawKind)]

genVarsOf = map snd . leaves (<0)

-- | 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

substTypeArg :: Subst -> TypeArg -> VarKind

substTypeArg s (TypeArg _ _ vk _ _ _ _) = case vk of

Downset super -> Downset $ subst s super

_ -> vk

mapArgs :: Subst -> [(Id, Type)] -> [TypeArg] -> [(Type, VarKind)]

mapArgs s ts = foldr ( \ ta l ->

maybe l ( \ (_, t) -> (t, substTypeArg s ta) : l) $

find ( \ (j, _) -> getTypeVar ta == j) ts) []

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

freshInst (TypeScheme tArgs t _) =

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

vs = map snd ls

ts <- mkSubst vs

let s = Map.fromList $ zip (map fst ls) ts

return (subst s t, mapArgs s (zip (map fst vs) ts) tArgs)

inc :: State Int Int

inc = do

c <- get

put (c + 1)

return c

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

freshVar ps = do

c <- inc

return (simpleIdToId $ Token ("_v" ++ show c) ps, c)

mkSingleSubst :: (Id, RawKind) -> State Int Type

mkSingleSubst (i, rk) = do

(ty, c) <- freshVar $ posOfId i

return $ TypeName ty rk c

mkSubst :: [(Id, RawKind)] -> State Int [Type]

mkSubst = mapM mkSingleSubst

type Subst = Map.Map Int Type

eps :: Subst

eps = Map.empty

flatKind :: Type -> RawKind

flatKind = mapKindV (const InVar) id . rawKindOfType

noAbs :: Type -> Bool

noAbs t = case t of

TypeAbs _ _ _ -> False

_ -> True

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

-> (Bool, Type) -> (Bool, Type) -> Result Subst

match tm rel p1@(b1, ty1) p2@(b2, ty2) =

if flatKind ty1 == flatKind ty2 then case (ty1, ty2) of

(_, ExpandedType _ t2) | noAbs t2 -> match tm rel p1 (b2, t2)

(ExpandedType _ t1, _) | noAbs t1 -> match tm rel (b1, t1) p2

(_, TypeAppl (TypeName l _ _) t2) | l == lazyTypeId ->

match tm rel p1 (b2, t2)

(TypeAppl (TypeName l _ _) t1, _) | l == lazyTypeId ->

match tm rel (b1, t1) p2

(_, KindedType t2 _ _) -> match tm rel p1 (b2, t2)

(KindedType t1 _ _, _) -> match tm rel (b1, t1) p2

(TypeName i1 _k1 v1, TypeName i2 _k2 v2) ->

if rel i1 i2 && v1 == v2

then return eps

else if v1 > 0 && b1 then return $ Map.singleton v1 ty2

else if v2 > 0 && b2 then return $ Map.singleton v2 ty1

{- the following two conditions only guarantee that instScheme also matches for

a partial function that is mapped to a total one.

Maybe a subtype condition is better. -}

else if not b1 && b2 && v1 == 0 && v2 == 0 && Set.member i1

(superIds tm i2) then return eps

else if b1 && not b2 && v1 == 0 && v2 == 0 && Set.member i2

(superIds tm i1) then return eps

else uniResult "typename" ty1 "is not unifiable with typename" ty2

(TypeName _ _ v1, _) ->

if hasRedex ty2 then match tm rel p1 (b2, redStep ty2) else

if v1 > 0 && b1 then

if null $ leaves (==v1) ty2 then

return $ Map.singleton v1 ty2

else uniResult "var" ty1 "occurs in" ty2

else uniResult "typename" ty1

"is not unifiable with type" ty2

(_, TypeName _ _ _) -> match tm rel p2 p1

(TypeAppl f1 a1, TypeAppl f2 a2) ->

let res = do

s1 <- match tm rel (b1, f1) (b2, f2)

s2 <- match tm rel (b1, if b1 then subst s1 a1 else a1)

(b2, if b2 then subst s1 a2 else a2)

return $ compSubst s1 s2

res1@(Result _ ms1) = if hasRedex ty1 then

match tm rel (b1, redStep ty1) p2

else fail "match1"

res2@(Result _ ms2) = if hasRedex ty2 then

match tm rel p1 (b2, redStep ty2)

else fail "match2"

in case ms1 of

Nothing -> case ms2 of

Nothing -> res

Just _ -> res2

Just _ -> res1

_ -> if ty1 == ty2 then return eps else

uniResult "type" ty1 "is not unifiable with type" ty2

else uniResult "type" ty1 "is not unifiable with differently kinded type" ty2

-- | most general unifier via 'match'

-- where both sides may contribute substitutions

mgu :: TypeMap -> Type -> Type -> Result Subst

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

mguList :: TypeMap -> [Type] -> [Type] -> Result Subst

mguList tm l1 l2 = case (l1, l2) of

([], []) -> return eps

(h1 : t1, h2 : t2) -> do

s1 <- mgu tm h1 h2

s2 <- mguList tm (map (subst s1) t1) $ map (subst s1) t2

return $ compSubst s1 s2

_ -> mkError "no unification of differently long argument lists"

(head $ l1 ++ l2)

shapeMatch :: TypeMap -> Type -> Type -> Result Subst

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

unify :: TypeMap -> Type -> Type -> Bool

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

subsume :: TypeMap -> Type -> Type -> Bool

subsume tm a b =

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

subst :: Subst -> Type -> Type

subst m = rename (\ i k n ->

case Map.lookup n m of

Just s -> s

_ -> TypeName i k n)

showDocWithPos :: Type -> ShowS

showDocWithPos a = let p = getRange a in

showChar '\'' . showDoc a . showChar '\''

. noShow (isNullRange p) (showChar ' ' .

showParen True (showPos $ maximumBy comparePos (rangeToList p)))

uniResult :: String -> Type -> String -> Type -> Result Subst

uniResult s1 a s2 b =

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

showDocWithPos a "\n " ++ s2 ++ " " ++

showDocWithPos b "") nullRange] Nothing

-- | make representation of bound variables unique

generalize :: [TypeArg] -> Type -> Type

generalize tArgs =

subst $ Map.fromList $ zipWith

( \ (TypeArg i _ _ rk c _ _) n ->

(c, TypeName i rk n)) tArgs [-1, -2..]

genTypeArgs :: [TypeArg] -> [TypeArg]

genTypeArgs tArgs = snd $ mapAccumL ( \ n (TypeArg i v vk rk _ s ps) ->

(n-1, TypeArg i v (case vk of

Downset t -> Downset $ generalize tArgs t

_ -> vk) rk n s ps)) (-1) tArgs