types/Unify.hs

    -- (c) The University of Glasgow 2006

    {-# LANGUAGE CPP #-}

    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
            tcUnifyTy, tcUnifyTys, BindFlag(..),

            UnifyResultM(..), UnifyResult, tcUnifyTysFG

       ) where

    #include "HsVersions.h"

    import Var
    import VarEnv
    import VarSet
    import Kind
    import Type
    import TyCon
    import TypeRep
    import Util

    import Control.Monad (liftM, ap)
    #if __GLASGOW_HASKELL__ < 709
    import Control.Applicative (Applicative(..))
    #endif

    {-
    ************************************************************************
    *                                                                      *
                    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.
    -}

    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 ty1 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

    -- Now the internals of matching

    -- | Workhorse matching function.  Our goal is to find a substitution
    -- on all of the template variables (specified by @me_tmpls menv@) such
    -- that @ty1@ and @ty2@ unify.  This substitution is accumulated in @subst@.
    -- If a variable is not a template variable, we don't attempt to find a
    -- substitution for it; it must match exactly on both sides.  Furthermore,
    -- only @ty1@ can have template variables.
    --
    -- This function handles binders, see 'RnEnv2' for more details on
    -- how that works.
    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
                            -- ezyang: Is this really an occurs check?  It seems
                            -- to just reject matching \x. A against \x. x (maintaining
                            -- the invariant that the free vars of the range of @subst@
                            -- are a subset of the in-scope set in @me_env menv@.)
        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

    {-
    ************************************************************************
    *                                                                      *
                    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.
    -}

    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)

    {-
    ************************************************************************
    *                                                                      *
                 Unification
    *                                                                      *
    ************************************************************************

    Note [Fine-grained unification]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Do the types (x, x) and ([y], y) unify? The answer is seemingly "no" --
    no substitution to finite types makes these match. But, a substitution to
    *infinite* types can unify these two types: [x |-> [[[...]]], y |-> [[[...]]] ].
    Why do we care? Consider these two type family instances:

    type instance F x x   = Int
    type instance F [y] y = Bool

    If we also have

    type instance Looper = [Looper]

    then the instances potentially overlap. The solution is to use unification
    over infinite terms. This is possible (see [1] for lots of gory details), but
    a full algorithm is a little more power than we need. Instead, we make a
    conservative approximation and just omit the occurs check.

    [1]: http://research.microsoft.com/en-us/um/people/simonpj/papers/ext-f/axioms-extended.pdf

    tcUnifyTys considers an occurs-check problem as the same as general unification
    failure.

    tcUnifyTysFG ("fine-grained") returns one of three results: success, occurs-check
    failure ("MaybeApart"), or general failure ("SurelyApart").

    See also Trac #8162.

    It's worth noting that unification in the presence of infinite types is not
    complete. This means that, sometimes, a closed type family does not reduce
    when it should. See test case indexed-types/should_fail/Overlap15 for an
    example.

    Note [The substitution in MaybeApart]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    The constructor MaybeApart carries data with it, typically a TvSubstEnv. Why?
    Because consider unifying these:

    (a, a, Int) ~ (b, [b], Bool)

    If we go left-to-right, we start with [a |-> b]. Then, on the middle terms, we
    apply the subst we have so far and discover that we need [b |-> [b]]. Because
    this fails the occurs check, we say that the types are MaybeApart (see above
    Note [Fine-grained unification]). But, we can't stop there! Because if we
    continue, we discover that Int is SurelyApart from Bool, and therefore the
    types are apart. This has practical consequences for the ability for closed
    type family applications to reduce. See test case
    indexed-types/should_compile/Overlap14.

    Note [Unifying with skolems]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    If we discover that two types unify if and only if a skolem variable is
    substituted, we can't properly unify the types. But, that skolem variable
    may later be instantiated with a unifyable type. So, we return maybeApart
    in these cases.

    Note [Lists of different lengths are MaybeApart]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    It is unusual to call tcUnifyTys or tcUnifyTysFG with lists of different
    lengths. The place where we know this can happen is from compatibleBranches in
    FamInstEnv, when checking data family instances. Data family instances may be
    eta-reduced; see Note [Eta reduction for data family axioms] in TcInstDcls.

    We wish to say that

      D :: * -> * -> *
      axDF1 :: D Int ~ DFInst1
      axDF2 :: D Int Bool ~ DFInst2

    overlap. If we conclude that lists of different lengths are SurelyApart, then
    it will look like these do *not* overlap, causing disaster. See Trac #9371.

    In usages of tcUnifyTys outside of family instances, we always use tcUnifyTys,
    which can't tell the difference between MaybeApart and SurelyApart, so those
    usages won't notice this design choice.
    -}

    tcUnifyTy :: Type -> Type       -- All tyvars are bindable
              -> Maybe TvSubst      -- A regular one-shot (idempotent) substitution
    -- Simple unification of two types; all type variables are bindable
    tcUnifyTy ty1 ty2
      = case initUM (const BindMe) (unify emptyTvSubstEnv ty1 ty2) of
          Unifiable subst_env -> Just (niFixTvSubst subst_env)
          _other              -> Nothing

    -----------------
    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
      = case tcUnifyTysFG bind_fn tys1 tys2 of
          Unifiable subst -> Just subst
          _               -> Nothing

    -- This type does double-duty. It is used in the UM (unifier monad) and to
    -- return the final result. See Note [Fine-grained unification]
    type UnifyResult = UnifyResultM TvSubst
    data UnifyResultM a = Unifiable a        -- the subst that unifies the types
                        | MaybeApart a       -- the subst has as much as we know
                                             -- it must be part of an most general unifier
                                             -- See Note [The substitution in MaybeApart]
                        | SurelyApart

    -- See Note [Fine-grained unification]
    tcUnifyTysFG :: (TyVar -> BindFlag)
                 -> [Type] -> [Type]
                 -> UnifyResult
    tcUnifyTysFG bind_fn tys1 tys2
      = initUM bind_fn $
        do { subst <- unifyList emptyTvSubstEnv tys1 tys2

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

    {-
    ************************************************************************
    *                                                                      *
                    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

    Note [Finding the substitution fixpoint]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Finding the fixpoint of a non-idempotent substitution arising from a
    unification is harder than it looks, because of kinds.  Consider
       T k (H k (f:k)) ~ T * (g:*)
    If we unify, we get the substitution
       [ k -> *
       , g -> H k (f:k) ]
    To make it idempotent we don't want to get just
       [ k -> *
       , g -> H * (f:k) ]
    We also want to substitute inside f's kind, to get
       [ k -> *
       , g -> H k (f:*) ]
    If we don't do this, we may apply the substitition to something,
    and get an ill-formed type, i.e. one where typeKind will fail.
    This happened, for example, in Trac #9106.

    This is the reason for extending env with [f:k -> f:*], in the
    definition of env' in niFixTvSubst
    -}

    niFixTvSubst :: TvSubstEnv -> TvSubst
    -- Find the idempotent fixed point of the non-idempotent substitution
    -- See Note [Finding the substitution fixpoint]
    -- ToDo: use laziness instead of iteration?
    niFixTvSubst env = f env
      where
        f env | not_fixpoint = f (mapVarEnv (substTy subst') env)
              | otherwise    = subst
            where
              not_fixpoint  = foldVarSet ((||) . in_domain) False all_range_tvs
              in_domain tv  = tv `elemVarEnv` env

              range_tvs     = foldVarEnv (unionVarSet . tyVarsOfType) emptyVarSet env
              all_range_tvs = closeOverKinds range_tvs
              subst         = mkTvSubst (mkInScopeSet all_range_tvs) env

                 -- env' extends env by replacing any free type with
                 -- that same tyvar with a substituted kind
                 -- See note [Finding the substitution fixpoint]
              env'          = extendVarEnvList env [ (rtv, mkTyVarTy $ setTyVarKind rtv $
                                                           substTy subst $ tyVarKind rtv)
                                                   | rtv <- varSetElems range_tvs
                                                   , not (in_domain rtv) ]
              subst'        = mkTvSubst (mkInScopeSet all_range_tvs) env'

    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)

    {-
    ************************************************************************
    *                                                                      *
                    The workhorse
    *                                                                      *
    ************************************************************************
    -}

    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 _ _ _ = surelyApart
            -- 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 subst _ _ = maybeApart subst  -- See Note [Lists of different lengths are MaybeApart]

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

    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) -> maybeApart subst' -- See Note [Unification with skolems]
               (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')
      = maybeApart subst                    -- Occurs check
                                            -- See Note [Fine-grained unification]
      | otherwise
      = do { subst' <- unify subst k1 k2
           -- Note [Kinds Containing Only Literals]
           ; 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 -> maybeApart subst -- See Note [Unification with skolems]
                BindMe -> return $ extendVarEnv subst tv ty
            }

    {-
    ************************************************************************
    *                                                                      *
                    Binding decisions
    *                                                                      *
    ************************************************************************
    -}

    data BindFlag
      = BindMe      -- A regular type variable

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

    {-
    ************************************************************************
    *                                                                      *
                    Unification monad
    *                                                                      *
    ************************************************************************
    -}

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

    instance Functor UM where
          fmap = liftM

    instance Applicative UM where
          pure = return
          (<*>) = ap

    instance Monad UM where
      return a = UM (\_tvs -> Unifiable a)
      fail _   = UM (\_tvs -> SurelyApart) -- failed pattern match
      m >>= k  = UM (\tvs -> case unUM m tvs of
                               Unifiable v -> unUM (k v) tvs
                               MaybeApart v ->
                                 case unUM (k v) tvs of
                                   Unifiable v' -> MaybeApart v'
                                   other        -> other
                               SurelyApart -> SurelyApart)

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

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

    maybeApart :: TvSubstEnv -> UM TvSubstEnv
    maybeApart subst = UM (\_tv_fn -> MaybeApart subst)

    surelyApart :: UM a
    surelyApart = UM (\_tv_fn -> SurelyApart)