root/compiler/types/Unify.lhs

%
% (c) The University of Glasgow 2006
%

\begin{code}
{-# OPTIONS -fno-warn-tabs #-}
-- The above warning supression flag is a temporary kludge.
-- While working on this module you are encouraged to remove it and
-- detab the module (please do the detabbing in a separate patch). See
--     http://hackage.haskell.org/trac/ghc/wiki/Commentary/CodingStyle#TabsvsSpaces
-- for details

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
\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 menv) emptyTvSubstEnv ps1 ps2
  where
    menv = ME { me_tmpls = mkVarSet tmpls, me_env = mkRnEnv2 in_scope_tyvars }
    in_scope_tyvars = mkInScopeSet (tyVarsOfTypes ps1 `unionVarSet` tyVarsOfTypes 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

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 (tyVarKind tv1) (typeKind 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) 
  = do { subst' <- match_kind menv subst (tyVarKind tv1) (tyVarKind tv2)
       ; 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 (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 _ subst (LitTy x) (LitTy y) | x == y  = return subst

match _ _ _ _
  = Nothing

--------------
match_kind :: MatchEnv -> TvSubstEnv -> Kind -> Kind -> Maybe TvSubstEnv
-- Match the kind of the template tyvar with the kind of Type
-- Note [Matching kinds]
match_kind menv subst k1 k2
  | k2 `isSubKind` k1
  = return subst

  | otherwise
  = match menv subst k1 k2

-- 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
\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 we can prune the T1 branch (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 we can't prune the T1 branch (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)
        | isDistinctTyCon tc1 && isDistinctTyCon tc2
        = tc1 /= tc2 || typesCantMatch (zipEqual "typesCantMatch" tys1 tys2)

    cant_match (FunTy {}) (TyConApp tc _) = isDistinctTyCon tc
    cant_match (TyConApp tc _) (FunTy {}) = isDistinctTyCon 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 (LitTy x) (LitTy y) = x /= y

    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 (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 subst (LitTy x) (LitTy y) | x == y = return subst

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

------------------------------
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'

  | otherwise

  = do {   -- So both are unrefined; unify the kinds
       ; subst' <- unify subst (tyVarKind tv1) (tyVarKind tv2)

           -- And then bind one or the other, 
           -- depending on which is bindable
           -- NB: unlike TcUnify we do not have an elaborate sub-kinding 
           --     story.  That is relevant only during type inference, and
           --     (I very much hope) is not relevant here.
       ; b1 <- tvBindFlag tv1
       ; b2 <- tvBindFlag tv2
       ; let ty1 = TyVarTy tv1
       ; case (b1, b2) of
           (Skolem, Skolem) -> failWith (misMatch ty1 ty2)
           (BindMe, _)      -> return (extendVarEnv subst' tv1 ty2)
           (_, BindMe)      -> return (extendVarEnv subst' tv2 ty1) }

uUnrefined subst tv1 ty2 ty2'   -- ty2 is not a type variable
  | tv1 `elemVarSet` niSubstTvSet subst (tyVarsOfType ty2')
  = failWith (occursCheck tv1 ty2)      -- Occurs check
  | otherwise
  = do { subst' <- unify subst k1 k2
       ; 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 MsgDoc 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 MsgDoc a
initUM badtvs um = unUM um badtvs

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

failWith :: MsgDoc -> 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) ]

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