types/Unify.lhs

% % (c) The University of Glasgow 2006 % \begin{code}

module Unify ( 
	-- Matching of types: 
	--	the "tc" prefix indicates that matching always
	--	respects newtypes (rather than looking through them)
	tcMatchTy, tcMatchTys, tcMatchTyX, 
	ruleMatchTyX, tcMatchPreds, 

	MatchEnv(..), matchList, 

	typesCantMatch,

        -- Side-effect free unification
        tcUnifyTys, BindFlag(..),
        niFixTvSubst, niSubstTvSet

   ) where

#include "HsVersions.h"

import Var
import VarEnv
import VarSet
import Kind
import Type
import TyCon
import TypeRep
import Outputable
import ErrUtils
import Util
import Maybes
import FastString

import Control.Monad (guard)

\end{code} %************************************************************************ %* * Matching %* * %************************************************************************ Matching is much tricker than you might think. 1. The substitution we generate binds the *template type variables* which are given to us explicitly. 2. We want to match in the presence of foralls; e.g (forall a. t1) ~ (forall b. t2) That is what the RnEnv2 is for; it does the alpha-renaming that makes it as if a and b were the same variable. Initialising the RnEnv2, so that it can generate a fresh binder when necessary, entails knowing the free variables of both types. 3. We must be careful not to bind a template type variable to a locally bound variable. E.g. (forall a. x) ~ (forall b. b) where x is the template type variable. Then we do not want to bind x to a/b! This is a kind of occurs check. The necessary locals accumulate in the RnEnv2. \begin{code}

data MatchEnv
  = ME	{ me_tmpls :: VarSet	-- Template variables
 	, me_env   :: RnEnv2	-- Renaming envt for nested foralls
	}			--   In-scope set includes template variables
    -- Nota Bene: MatchEnv isn't specific to Types.  It is used
    --            for matching terms and coercions as well as types

tcMatchTy :: TyVarSet		-- Template tyvars
	  -> Type		-- Template
	  -> Type		-- Target
	  -> Maybe TvSubst	-- One-shot; in principle the template
				-- variables could be free in the target

tcMatchTy tmpls ty1 ty2
  = case match menv emptyTvSubstEnv ty1 ty2 of
	Just subst_env -> Just (TvSubst in_scope subst_env)
	Nothing	       -> Nothing
  where
    menv     = ME { me_tmpls = tmpls, me_env = mkRnEnv2 in_scope }
    in_scope = mkInScopeSet (tmpls `unionVarSet` tyVarsOfType ty2)
	-- We're assuming that all the interesting 
	-- tyvars in tys1 are in tmpls

tcMatchTys :: TyVarSet		-- Template tyvars
	   -> [Type]		-- Template
	   -> [Type]		-- Target
	   -> Maybe TvSubst	-- One-shot; in principle the template
				-- variables could be free in the target

tcMatchTys tmpls tys1 tys2
  = case match_tys menv emptyTvSubstEnv tys1 tys2 of
	Just subst_env -> Just (TvSubst in_scope subst_env)
	Nothing	       -> Nothing
  where
    menv     = ME { me_tmpls = tmpls, me_env = mkRnEnv2 in_scope }
    in_scope = mkInScopeSet (tmpls `unionVarSet` tyVarsOfTypes tys2)
	-- We're assuming that all the interesting 
	-- tyvars in tys1 are in tmpls

-- This is similar, but extends a substitution
tcMatchTyX :: TyVarSet 		-- Template tyvars
	   -> TvSubst		-- Substitution to extend
	   -> Type		-- Template
	   -> Type		-- Target
	   -> Maybe TvSubst
tcMatchTyX tmpls (TvSubst in_scope subst_env) ty1 ty2
  = case match menv subst_env ty1 ty2 of
	Just subst_env -> Just (TvSubst in_scope subst_env)
	Nothing	       -> Nothing
  where
    menv = ME {me_tmpls = tmpls, me_env = mkRnEnv2 in_scope}

tcMatchPreds
	:: [TyVar]			-- Bind these
	-> [PredType] -> [PredType]
   	-> Maybe TvSubstEnv
tcMatchPreds tmpls ps1 ps2
  = matchList (match_pred menv) emptyTvSubstEnv ps1 ps2
  where
    menv = ME { me_tmpls = mkVarSet tmpls, me_env = mkRnEnv2 in_scope_tyvars }
    in_scope_tyvars = mkInScopeSet (tyVarsOfTheta ps1 `unionVarSet` tyVarsOfTheta ps2)

-- This one is called from the expression matcher, which already has a MatchEnv in hand
ruleMatchTyX :: MatchEnv 
	 -> TvSubstEnv		-- Substitution to extend
	 -> Type		-- Template
	 -> Type		-- Target
	 -> Maybe TvSubstEnv

ruleMatchTyX menv subst ty1 ty2 = match menv subst ty1 ty2	-- Rename for export

\end{code} Now the internals of matching \begin{code}

match :: MatchEnv	-- For the most part this is pushed downwards
      -> TvSubstEnv 	-- Substitution so far:
			--   Domain is subset of template tyvars
			--   Free vars of range is subset of 
			--	in-scope set of the RnEnv2
      -> Type -> Type	-- Template and target respectively
      -> Maybe TvSubstEnv
-- This matcher works on core types; that is, it ignores PredTypes
-- Watch out if newtypes become transparent agin!
-- 	this matcher must respect newtypes

match menv subst ty1 ty2 | Just ty1' <- coreView ty1 = match menv subst ty1' ty2
			 | Just ty2' <- coreView ty2 = match menv subst ty1 ty2'

match menv subst (TyVarTy tv1) ty2
  | Just ty1' <- lookupVarEnv subst tv1'	-- tv1' is already bound
  = if eqTypeX (nukeRnEnvL rn_env) ty1' ty2
	-- ty1 has no locally-bound variables, hence nukeRnEnvL
    then Just subst
    else Nothing	-- ty2 doesn't match

  | tv1' `elemVarSet` me_tmpls menv
  = if any (inRnEnvR rn_env) (varSetElems (tyVarsOfType ty2))
    then Nothing	-- Occurs check
    else do { subst1 <- match_kind menv subst tv1 ty2
			-- Note [Matching kinds]
	    ; return (extendVarEnv subst1 tv1' ty2) }

   | otherwise	-- tv1 is not a template tyvar
   = case ty2 of
	TyVarTy tv2 | tv1' == rnOccR rn_env tv2 -> Just subst
	_                                       -> Nothing
  where
    rn_env = me_env menv
    tv1' = rnOccL rn_env tv1

match menv subst (ForAllTy tv1 ty1) (ForAllTy tv2 ty2) 
  = match menv' subst ty1 ty2
  where		-- Use the magic of rnBndr2 to go under the binders
    menv' = menv { me_env = rnBndr2 (me_env menv) tv1 tv2 }

match menv subst (PredTy p1) (PredTy p2) 
  = match_pred menv subst p1 p2
match menv subst (TyConApp tc1 tys1) (TyConApp tc2 tys2) 
  | tc1 == tc2 = match_tys menv subst tys1 tys2
match menv subst (FunTy ty1a ty1b) (FunTy ty2a ty2b) 
  = do { subst' <- match menv subst ty1a ty2a
       ; match menv subst' ty1b ty2b }
match menv subst (AppTy ty1a ty1b) ty2
  | Just (ty2a, ty2b) <- repSplitAppTy_maybe ty2
	-- 'repSplit' used because the tcView stuff is done above
  = do { subst' <- match menv subst ty1a ty2a
       ; match menv subst' ty1b ty2b }

match _ _ _ _
  = Nothing

--------------
match_kind :: MatchEnv -> TvSubstEnv -> TyVar -> Type -> Maybe TvSubstEnv
-- Match the kind of the template tyvar with the kind of Type
-- Note [Matching kinds]
match_kind _ subst tv ty
  = guard (typeKind ty `isSubKind` tyVarKind tv) >> return subst

-- Note [Matching kinds]
-- ~~~~~~~~~~~~~~~~~~~~~
-- For ordinary type variables, we don't want (m a) to match (n b) 
-- if say (a::*) and (b::*->*).  This is just a yes/no issue. 
--
-- For coercion kinds matters are more complicated.  If we have a 
-- coercion template variable co::a~[b], where a,b are presumably also
-- template type variables, then we must match co's kind against the 
-- kind of the actual argument, so as to give bindings to a,b.  
--
-- In fact I have no example in mind that *requires* this kind-matching
-- to instantiate template type variables, but it seems like the right
-- thing to do.  C.f. Note [Matching variable types] in Rules.lhs

--------------
match_tys :: MatchEnv -> TvSubstEnv -> [Type] -> [Type] -> Maybe TvSubstEnv
match_tys menv subst tys1 tys2 = matchList (match menv) subst tys1 tys2

--------------
matchList :: (env -> a -> b -> Maybe env)
	   -> env -> [a] -> [b] -> Maybe env
matchList _  subst []     []     = Just subst
matchList fn subst (a:as) (b:bs) = do { subst' <- fn subst a b
				      ; matchList fn subst' as bs }
matchList _  _     _      _      = Nothing

--------------
match_pred :: MatchEnv -> TvSubstEnv -> PredType -> PredType -> Maybe TvSubstEnv
match_pred menv subst (ClassP c1 tys1) (ClassP c2 tys2)
  | c1 == c2 = match_tys menv subst tys1 tys2
match_pred menv subst (IParam n1 t1) (IParam n2 t2)
  | n1 == n2 = match menv subst t1 t2
match_pred _    _     _ _ = Nothing

\end{code} %************************************************************************ %* * GADTs %* * %************************************************************************ Note [Pruning dead case alternatives] ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Consider data T a where T1 :: T Int T2 :: T a newtype X = MkX Int newtype Y = MkY Char type family F a type instance F Bool = Int Now consider case x of { T1 -> e1; T2 -> e2 } The question before the house is this: if I know something about the type of x, can I prune away the T1 alternative? Suppose x::T Char. It's impossible to construct a (T Char) using T1, Answer = YES (clearly) Suppose x::T (F a), where 'a' is in scope. Then 'a' might be instantiated to 'Bool', in which case x::T Int, so ANSWER = NO (clearly) Suppose x::T X. Then *in Haskell* it's impossible to construct a (non-bottom) value of type (T X) using T1. But *in FC* it's quite possible. The newtype gives a coercion CoX :: X ~ Int So (T CoX) :: T X ~ T Int; hence (T1 `cast` sym (T CoX)) is a non-bottom value of type (T X) constructed with T1. Hence ANSWER = NO (surprisingly) Furthermore, this can even happen; see Trac #1251. GHC's newtype-deriving mechanism uses a cast, just as above, to move from one dictionary to another, in effect giving the programmer access to CoX. Finally, suppose x::T Y. Then *even in FC* we can't construct a non-bottom value of type (T Y) using T1. That's because we can get from Y to Char, but not to Int. Here's a related question. data Eq a b where EQ :: Eq a a Consider case x of { EQ -> ... } Suppose x::Eq Int Char. Is the alternative dead? Clearly yes. What about x::Eq Int a, in a context where we have evidence that a~Char. Then again the alternative is dead. Summary We are really doing a test for unsatisfiability of the type constraints implied by the match. And that is clearly, in general, a hard thing to do. However, since we are simply dropping dead code, a conservative test suffices. There is a continuum of tests, ranging from easy to hard, that drop more and more dead code. For now we implement a very simple test: type variables match anything, type functions (incl newtypes) match anything, and only distinct data types fail to match. We can elaborate later. \begin{code}

typesCantMatch :: [(Type,Type)] -> Bool
typesCantMatch prs = any (\(s,t) -> cant_match s t) prs
  where
    cant_match :: Type -> Type -> Bool
    cant_match t1 t2
	| Just t1' <- coreView t1 = cant_match t1' t2
	| Just t2' <- coreView t2 = cant_match t1 t2'

    cant_match (FunTy a1 r1) (FunTy a2 r2)
	= cant_match a1 a2 || cant_match r1 r2

    cant_match (TyConApp tc1 tys1) (TyConApp tc2 tys2)
	| isDataTyCon tc1 && isDataTyCon tc2
	= tc1 /= tc2 || typesCantMatch (zipEqual "typesCantMatch" tys1 tys2)

    cant_match (FunTy {}) (TyConApp tc _) = isDataTyCon tc
    cant_match (TyConApp tc _) (FunTy {}) = isDataTyCon tc
	-- tc can't be FunTyCon by invariant

    cant_match (AppTy f1 a1) ty2
	| Just (f2, a2) <- repSplitAppTy_maybe ty2
	= cant_match f1 f2 || cant_match a1 a2
    cant_match ty1 (AppTy f2 a2)
	| Just (f1, a1) <- repSplitAppTy_maybe ty1
	= cant_match f1 f2 || cant_match a1 a2

    cant_match _ _ = False      -- Safe!

-- Things we could add;
--	foralls
--	look through newtypes
--	take account of tyvar bindings (EQ example above)

\end{code} %************************************************************************ %* * Unification %* * %************************************************************************ \begin{code}

tcUnifyTys :: (TyVar -> BindFlag)
	   -> [Type] -> [Type]
	   -> Maybe TvSubst	-- A regular one-shot (idempotent) substitution
-- The two types may have common type variables, and indeed do so in the
-- second call to tcUnifyTys in FunDeps.checkClsFD
--
tcUnifyTys bind_fn tys1 tys2
  = maybeErrToMaybe $ initUM bind_fn $
    do { subst <- unifyList emptyTvSubstEnv tys1 tys2

	-- Find the fixed point of the resulting non-idempotent substitution
        ; return (niFixTvSubst subst) }

\end{code} %************************************************************************ %* * Non-idempotent substitution %* * %************************************************************************ Note [Non-idempotent substitution] ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ During unification we use a TvSubstEnv that is (a) non-idempotent (b) loop-free; ie repeatedly applying it yields a fixed point \begin{code}

niFixTvSubst :: TvSubstEnv -> TvSubst
-- Find the idempotent fixed point of the non-idempotent substitution
-- ToDo: use laziness instead of iteration?
niFixTvSubst env = f env
  where
    f e | not_fixpoint = f (mapVarEnv (substTy subst) e)
        | otherwise    = subst
        where
          range_tvs    = foldVarEnv (unionVarSet . tyVarsOfType) emptyVarSet e
          subst        = mkTvSubst (mkInScopeSet range_tvs) e 
          not_fixpoint = foldVarSet ((||) . in_domain) False range_tvs
          in_domain tv = tv `elemVarEnv` e

niSubstTvSet :: TvSubstEnv -> TyVarSet -> TyVarSet
-- Apply the non-idempotent substitution to a set of type variables,
-- remembering that the substitution isn't necessarily idempotent
-- This is used in the occurs check, before extending the substitution
niSubstTvSet subst tvs
  = foldVarSet (unionVarSet . get) emptyVarSet tvs
  where
    get tv = case lookupVarEnv subst tv of
	       Nothing -> unitVarSet tv
               Just ty -> niSubstTvSet subst (tyVarsOfType ty)

\end{code} %************************************************************************ %* * The workhorse %* * %************************************************************************ \begin{code}

unify :: TvSubstEnv	-- An existing substitution to extend
      -> Type -> Type 	-- Types to be unified, and witness of their equality
      -> UM TvSubstEnv		-- Just the extended substitution, 
				-- Nothing if unification failed
-- We do not require the incoming substitution to be idempotent,
-- nor guarantee that the outgoing one is.  That's fixed up by
-- the wrappers.

-- Respects newtypes, PredTypes

-- in unify, any NewTcApps/Preds should be taken at face value
unify subst (TyVarTy tv1) ty2  = uVar subst tv1 ty2
unify subst ty1 (TyVarTy tv2)  = uVar subst tv2 ty1

unify subst ty1 ty2 | Just ty1' <- tcView ty1 = unify subst ty1' ty2
unify subst ty1 ty2 | Just ty2' <- tcView ty2 = unify subst ty1 ty2'

unify subst (PredTy p1) (PredTy p2) = unify_pred subst p1 p2

unify subst (TyConApp tyc1 tys1) (TyConApp tyc2 tys2) 
  | tyc1 == tyc2 = unify_tys subst tys1 tys2

unify subst (FunTy ty1a ty1b) (FunTy ty2a ty2b) 
  = do	{ subst' <- unify subst ty1a ty2a
	; unify subst' ty1b ty2b }

	-- Applications need a bit of care!
	-- They can match FunTy and TyConApp, so use splitAppTy_maybe
	-- NB: we've already dealt with type variables and Notes,
	-- so if one type is an App the other one jolly well better be too
unify subst (AppTy ty1a ty1b) ty2
  | Just (ty2a, ty2b) <- repSplitAppTy_maybe ty2
  = do	{ subst' <- unify subst ty1a ty2a
        ; unify subst' ty1b ty2b }

unify subst ty1 (AppTy ty2a ty2b)
  | Just (ty1a, ty1b) <- repSplitAppTy_maybe ty1
  = do	{ subst' <- unify subst ty1a ty2a
        ; unify subst' ty1b ty2b }

unify _ ty1 ty2 = failWith (misMatch ty1 ty2)
	-- ForAlls??

------------------------------
unify_pred :: TvSubstEnv -> PredType -> PredType -> UM TvSubstEnv
unify_pred subst (ClassP c1 tys1) (ClassP c2 tys2)
  | c1 == c2 = unify_tys subst tys1 tys2
unify_pred subst (IParam n1 t1) (IParam n2 t2)
  | n1 == n2 = unify subst t1 t2
unify_pred _ p1 p2 = failWith (misMatch (PredTy p1) (PredTy p2))
 
------------------------------
unify_tys :: TvSubstEnv -> [Type] -> [Type] -> UM TvSubstEnv
unify_tys subst xs ys = unifyList subst xs ys

unifyList :: TvSubstEnv -> [Type] -> [Type] -> UM TvSubstEnv
unifyList subst orig_xs orig_ys
  = go subst orig_xs orig_ys
  where
    go subst []     []     = return subst
    go subst (x:xs) (y:ys) = do { subst' <- unify subst x y
				; go subst' xs ys }
    go _ _ _ = failWith (lengthMisMatch orig_xs orig_ys)

---------------------------------
uVar :: TvSubstEnv	-- An existing substitution to extend
     -> TyVar           -- Type variable to be unified
     -> Type            -- with this type
     -> UM TvSubstEnv

-- PRE-CONDITION: in the call (uVar swap r tv1 ty), we know that
--	if swap=False	(tv1~ty)
--	if swap=True	(ty~tv1)

uVar subst tv1 ty
 = -- Check to see whether tv1 is refined by the substitution
   case (lookupVarEnv subst tv1) of
     Just ty' -> unify subst ty' ty     -- Yes, call back into unify'
     Nothing  -> uUnrefined subst       -- No, continue
			    tv1 ty ty

uUnrefined :: TvSubstEnv          -- An existing substitution to extend
           -> TyVar               -- Type variable to be unified
           -> Type                -- with this type
           -> Type                -- (version w/ expanded synonyms)
           -> UM TvSubstEnv

-- We know that tv1 isn't refined

uUnrefined subst tv1 ty2 ty2'
  | Just ty2'' <- tcView ty2'
  = uUnrefined subst tv1 ty2 ty2''	-- Unwrap synonyms
		-- This is essential, in case we have
		--	type Foo a = a
		-- and then unify a ~ Foo a

uUnrefined subst tv1 ty2 (TyVarTy tv2)
  | tv1 == tv2		-- Same type variable
  = return subst

    -- Check to see whether tv2 is refined
  | Just ty' <- lookupVarEnv subst tv2
  = uUnrefined subst tv1 ty' ty'

  -- So both are unrefined; next, see if the kinds force the direction
  | eqKind k1 k2	-- Can update either; so check the bind-flags
  = do	{ b1 <- tvBindFlag tv1
	; b2 <- tvBindFlag tv2
	; case (b1,b2) of
	    (BindMe, _) 	 -> bind tv1 ty2
	    (Skolem, Skolem)	 -> failWith (misMatch ty1 ty2)
	    (Skolem, _)		 -> bind tv2 ty1
	}

  | k1 `isSubKind` k2 = bindTv subst tv2 ty1  -- Must update tv2
  | k2 `isSubKind` k1 = bindTv subst tv1 ty2  -- Must update tv1

  | otherwise = failWith (kindMisMatch tv1 ty2)
  where
    ty1 = TyVarTy tv1
    k1 = tyVarKind tv1
    k2 = tyVarKind tv2
    bind tv ty = return $ extendVarEnv subst tv ty

uUnrefined subst tv1 ty2 ty2'	-- ty2 is not a type variable
  | tv1 `elemVarSet` niSubstTvSet subst (tyVarsOfType ty2')
  = failWith (occursCheck tv1 ty2)	-- Occurs check
  | not (k2 `isSubKind` k1)
  = failWith (kindMisMatch tv1 ty2)	-- Kind check
  | otherwise
  = bindTv subst tv1 ty2		-- Bind tyvar to the synonym if poss
  where
    k1 = tyVarKind tv1
    k2 = typeKind ty2'

bindTv :: TvSubstEnv -> TyVar -> Type -> UM TvSubstEnv
bindTv subst tv ty	-- ty is not a type variable
  = do  { b <- tvBindFlag tv
	; case b of
	    Skolem -> failWith (misMatch (TyVarTy tv) ty)
	    BindMe -> return $ extendVarEnv subst tv ty
	}

\end{code} %************************************************************************ %* * Binding decisions %* * %************************************************************************ \begin{code}

data BindFlag 
  = BindMe	-- A regular type variable

  | Skolem	-- This type variable is a skolem constant
		-- Don't bind it; it only matches itself

\end{code} %************************************************************************ %* * Unification monad %* * %************************************************************************ \begin{code}

newtype UM a = UM { unUM :: (TyVar -> BindFlag)
		         -> MaybeErr Message a }

instance Monad UM where
  return a = UM (\_tvs -> Succeeded a)
  fail s   = UM (\_tvs -> Failed (text s))
  m >>= k  = UM (\tvs -> case unUM m tvs of
			   Failed err -> Failed err
			   Succeeded v  -> unUM (k v) tvs)

initUM :: (TyVar -> BindFlag) -> UM a -> MaybeErr Message a
initUM badtvs um = unUM um badtvs

tvBindFlag :: TyVar -> UM BindFlag
tvBindFlag tv = UM (\tv_fn -> Succeeded (tv_fn tv))

failWith :: Message -> UM a
failWith msg = UM (\_tv_fn -> Failed msg)

maybeErrToMaybe :: MaybeErr fail succ -> Maybe succ
maybeErrToMaybe (Succeeded a) = Just a
maybeErrToMaybe (Failed _)    = Nothing

\end{code} %************************************************************************ %* * Error reporting We go to a lot more trouble to tidy the types in TcUnify. Maybe we'll end up having to do that here too, but I'll leave it for now. %* * %************************************************************************ \begin{code}

misMatch :: Type -> Type -> SDoc
misMatch t1 t2
  = ptext (sLit "Can't match types") <+> quotes (ppr t1) <+> 
    ptext (sLit "and") <+> quotes (ppr t2)

lengthMisMatch :: [Type] -> [Type] -> SDoc
lengthMisMatch tys1 tys2
  = sep [ptext (sLit "Can't match unequal length lists"), 
	 nest 2 (ppr tys1), nest 2 (ppr tys2) ]

kindMisMatch :: TyVar -> Type -> SDoc
kindMisMatch tv1 t2
  = vcat [ptext (sLit "Can't match kinds") <+> quotes (ppr (tyVarKind tv1)) <+> 
	    ptext (sLit "and") <+> quotes (ppr (typeKind t2)),
	  ptext (sLit "when matching") <+> quotes (ppr tv1) <+> 
		ptext (sLit "with") <+> quotes (ppr t2)]

occursCheck :: TyVar -> Type -> SDoc
occursCheck tv ty
  = hang (ptext (sLit "Can't construct the infinite type"))
       2 (ppr tv <+> equals <+> ppr ty)

\end{code}