    -- (c) The University of Glasgow 2006
    -- (c) The GRASP/AQUA Project, Glasgow University, 1998
    --
    -- Type - public interface

    {-# LANGUAGE CPP #-}
    {-# OPTIONS_GHC -fno-warn-orphans #-}

    -- | Main functions for manipulating types and type-related things
    module Type (
            -- Note some of this is just re-exports from TyCon..

            -- * Main data types representing Types
            -- $type_classification

            -- $representation_types
            TyThing(..), Type, KindOrType, PredType, ThetaType,
            Var, TyVar, isTyVar,

            -- ** Constructing and deconstructing types
            mkTyVarTy, mkTyVarTys, getTyVar, getTyVar_maybe,

            mkAppTy, mkAppTys, splitAppTy, splitAppTys,
            splitAppTy_maybe, repSplitAppTy_maybe,

            mkFunTy, mkFunTys, splitFunTy, splitFunTy_maybe,
            splitFunTys, splitFunTysN,
            funResultTy, funArgTy, zipFunTys,

            mkTyConApp, mkTyConTy,
            tyConAppTyCon_maybe, tyConAppArgs_maybe, tyConAppTyCon, tyConAppArgs,
            splitTyConApp_maybe, splitTyConApp, tyConAppArgN, nextRole,

            mkForAllTy, mkForAllTys, splitForAllTy_maybe, splitForAllTys,
            mkPiKinds, mkPiType, mkPiTypes,
            applyTy, applyTys, applyTysD, applyTysX, dropForAlls,

            mkNumLitTy, isNumLitTy,
            mkStrLitTy, isStrLitTy,

            coAxNthLHS,

            -- (Newtypes)
            newTyConInstRhs,

            -- Pred types
            mkFamilyTyConApp,
            isDictLikeTy,
            mkEqPred, mkCoerciblePred, mkPrimEqPred, mkReprPrimEqPred,
            mkClassPred,
            isClassPred, isEqPred,
            isIPPred, isIPPred_maybe, isIPTyCon, isIPClass,

            -- Deconstructing predicate types
            PredTree(..), EqRel(..), eqRelRole, classifyPredType,
            getClassPredTys, getClassPredTys_maybe,
            getEqPredTys, getEqPredTys_maybe, getEqPredRole,
            predTypeEqRel,

            -- ** Common type constructors
            funTyCon,

            -- ** Predicates on types
            isTypeVar, isKindVar, allDistinctTyVars, isForAllTy,
            isTyVarTy, isFunTy, isDictTy, isPredTy, isVoidTy,

            -- (Lifting and boxity)
            isUnLiftedType, isUnboxedTupleType, isAlgType, isClosedAlgType,
            isPrimitiveType, isStrictType,

            -- * Main data types representing Kinds
            -- $kind_subtyping
            Kind, SimpleKind, MetaKindVar,

            -- ** Finding the kind of a type
            typeKind,

            -- ** Common Kinds and SuperKinds
            anyKind, liftedTypeKind, unliftedTypeKind, openTypeKind,
            constraintKind, superKind,

            -- ** Common Kind type constructors
            liftedTypeKindTyCon, openTypeKindTyCon, unliftedTypeKindTyCon,
            constraintKindTyCon, anyKindTyCon,

            -- * Type free variables
            tyVarsOfType, tyVarsOfTypes, closeOverKinds,
            expandTypeSynonyms,
            typeSize, varSetElemsKvsFirst,

            -- * Type comparison
            eqType, eqTypeX, eqTypes, cmpType, cmpTypes,
            eqPred, eqPredX, cmpPred, eqKind, eqTyVarBndrs,

            -- * Forcing evaluation of types
            seqType, seqTypes,

            -- * Other views onto Types
            coreView, tcView,

            UnaryType, RepType(..), flattenRepType, repType,
            tyConsOfType,

            -- * Type representation for the code generator
            typePrimRep, typeRepArity,

            -- * Main type substitution data types
            TvSubstEnv,     -- Representation widely visible
            TvSubst(..),    -- Representation visible to a few friends

            -- ** Manipulating type substitutions
            emptyTvSubstEnv, emptyTvSubst,

            mkTvSubst, mkOpenTvSubst, zipOpenTvSubst, zipTopTvSubst, mkTopTvSubst, notElemTvSubst,
            getTvSubstEnv, setTvSubstEnv,
            zapTvSubstEnv, getTvInScope,
            extendTvInScope, extendTvInScopeList,
            extendTvSubst, extendTvSubstList,
            isInScope, composeTvSubst, zipTyEnv,
            isEmptyTvSubst, unionTvSubst,

            -- ** Performing substitution on types and kinds
            substTy, substTys, substTyWith, substTysWith, substTheta,
            substTyVar, substTyVars, substTyVarBndr,
            cloneTyVarBndr, deShadowTy, lookupTyVar,
            substKiWith, substKisWith,

            -- * Pretty-printing
            pprType, pprParendType, pprTypeApp, pprTyThingCategory, pprTyThing,
            pprTvBndr, pprTvBndrs, pprForAll, pprUserForAll, pprSigmaType,
            pprTheta, pprThetaArrowTy, pprClassPred,
            pprKind, pprParendKind, pprSourceTyCon,
            TyPrec(..), maybeParen, pprSigmaTypeExtraCts,

            -- * Tidying type related things up for printing
            tidyType,      tidyTypes,
            tidyOpenType,  tidyOpenTypes,
            tidyOpenKind,
            tidyTyVarBndr, tidyTyVarBndrs, tidyFreeTyVars,
            tidyOpenTyVar, tidyOpenTyVars,
            tidyTyVarOcc,
            tidyTopType,
            tidyKind,
        ) where

    #include "HsVersions.h"

    -- We import the representation and primitive functions from TypeRep.
    -- Many things are reexported, but not the representation!

    import Kind
    import TypeRep

    -- friends:
    import Var
    import VarEnv
    import VarSet
    import NameEnv

    import Class
    import TyCon
    import TysPrim
    import {-# SOURCE #-} TysWiredIn ( eqTyCon, coercibleTyCon, typeNatKind, typeSymbolKind )
    import PrelNames ( eqTyConKey, coercibleTyConKey,
                       ipClassNameKey, openTypeKindTyConKey,
                       constraintKindTyConKey, liftedTypeKindTyConKey )
    import CoAxiom

    -- others
    import Unique           ( Unique, hasKey )
    import BasicTypes       ( Arity, RepArity )
    import Util
    import ListSetOps       ( getNth )
    import Outputable
    import FastString

    import Maybes           ( orElse )
    import Data.Maybe       ( isJust )
    import Control.Monad    ( guard )

    infixr 3 `mkFunTy`      -- Associates to the right

    -- $type_classification
    -- #type_classification#
    --
    -- Types are one of:
    --
    -- [Unboxed]            Iff its representation is other than a pointer
    --                      Unboxed types are also unlifted.
    --
    -- [Lifted]             Iff it has bottom as an element.
    --                      Closures always have lifted types: i.e. any
    --                      let-bound identifier in Core must have a lifted
    --                      type. Operationally, a lifted object is one that
    --                      can be entered.
    --                      Only lifted types may be unified with a type variable.
    --
    -- [Algebraic]          Iff it is a type with one or more constructors, whether
    --                      declared with @data@ or @newtype@.
    --                      An algebraic type is one that can be deconstructed
    --                      with a case expression. This is /not/ the same as
    --                      lifted types, because we also include unboxed
    --                      tuples in this classification.
    --
    -- [Data]               Iff it is a type declared with @data@, or a boxed tuple.
    --
    -- [Primitive]          Iff it is a built-in type that can't be expressed in Haskell.
    --
    -- Currently, all primitive types are unlifted, but that's not necessarily
    -- the case: for example, @Int@ could be primitive.
    --
    -- Some primitive types are unboxed, such as @Int#@, whereas some are boxed
    -- but unlifted (such as @ByteArray#@).  The only primitive types that we
    -- classify as algebraic are the unboxed tuples.
    --
    -- Some examples of type classifications that may make this a bit clearer are:
    --
    -- @
    -- Type         primitive       boxed           lifted          algebraic
    -- -----------------------------------------------------------------------------
    -- Int#         Yes             No              No              No
    -- ByteArray#   Yes             Yes             No              No
    -- (\# a, b \#)   Yes             No              No              Yes
    -- (  a, b  )   No              Yes             Yes             Yes
    -- [a]          No              Yes             Yes             Yes
    -- @

    -- $representation_types
    -- A /source type/ is a type that is a separate type as far as the type checker is
    -- concerned, but which has a more low-level representation as far as Core-to-Core
    -- passes and the rest of the back end is concerned.
    --
    -- You don't normally have to worry about this, as the utility functions in
    -- this module will automatically convert a source into a representation type
    -- if they are spotted, to the best of it's abilities. If you don't want this
    -- to happen, use the equivalent functions from the "TcType" module.

    {-
    ************************************************************************
    *                                                                      *
                    Type representation
    *                                                                      *
    ************************************************************************
    -}

    {-# INLINE coreView #-}
    coreView :: Type -> Maybe Type
    -- ^ In Core, we \"look through\" non-recursive newtypes and 'PredTypes': this
    -- function tries to obtain a different view of the supplied type given this
    --
    -- Strips off the /top layer only/ of a type to give
    -- its underlying representation type.
    -- Returns Nothing if there is nothing to look through.
    --
    -- By being non-recursive and inlined, this case analysis gets efficiently
    -- joined onto the case analysis that the caller is already doing
    coreView (TyConApp tc tys) | Just (tenv, rhs, tys') <- coreExpandTyCon_maybe tc tys
                  = Just (mkAppTys (substTy (mkTopTvSubst tenv) rhs) tys')
                   -- Its important to use mkAppTys, rather than (foldl AppTy),
                   -- because the function part might well return a
                   -- partially-applied type constructor; indeed, usually will!
    coreView _                 = Nothing

    -----------------------------------------------
    {-# INLINE tcView #-}
    tcView :: Type -> Maybe Type
    -- ^ Similar to 'coreView', but for the type checker, which just looks through synonyms
    tcView (TyConApp tc tys) | Just (tenv, rhs, tys') <- tcExpandTyCon_maybe tc tys
                             = Just (mkAppTys (substTy (mkTopTvSubst tenv) rhs) tys')
    tcView _                 = Nothing
      -- You might think that tcView belows in TcType rather than Type, but unfortunately
      -- it is needed by Unify, which is turn imported by Coercion (for MatchEnv and matchList).
      -- So we will leave it here to avoid module loops.

    -----------------------------------------------
    expandTypeSynonyms :: Type -> Type
    -- ^ Expand out all type synonyms.  Actually, it'd suffice to expand out
    -- just the ones that discard type variables (e.g.  type Funny a = Int)
    -- But we don't know which those are currently, so we just expand all.
    expandTypeSynonyms ty
      = go ty
      where
        go (TyConApp tc tys)
          | Just (tenv, rhs, tys') <- tcExpandTyCon_maybe tc tys
          = go (mkAppTys (substTy (mkTopTvSubst tenv) rhs) tys')
          | otherwise
          = TyConApp tc (map go tys)
        go (LitTy l)       = LitTy l
        go (TyVarTy tv)    = TyVarTy tv
        go (AppTy t1 t2)   = mkAppTy (go t1) (go t2)
        go (FunTy t1 t2)   = FunTy (go t1) (go t2)
        go (ForAllTy tv t) = ForAllTy tv (go t)

    {-
    ************************************************************************
    *                                                                      *
    \subsection{Constructor-specific functions}
    *                                                                      *
    ************************************************************************

    ---------------------------------------------------------------------
                                    TyVarTy
                                    ~~~~~~~
    -}

    -- | Attempts to obtain the type variable underlying a 'Type', and panics with the
    -- given message if this is not a type variable type. See also 'getTyVar_maybe'
    getTyVar :: String -> Type -> TyVar
    getTyVar msg ty = case getTyVar_maybe ty of
                        Just tv -> tv
                        Nothing -> panic ("getTyVar: " ++ msg)

    isTyVarTy :: Type -> Bool
    isTyVarTy ty = isJust (getTyVar_maybe ty)

    -- | Attempts to obtain the type variable underlying a 'Type'
    getTyVar_maybe :: Type -> Maybe TyVar
    getTyVar_maybe ty | Just ty' <- coreView ty = getTyVar_maybe ty'
    getTyVar_maybe (TyVarTy tv)                 = Just tv
    getTyVar_maybe _                            = Nothing

    allDistinctTyVars :: [KindOrType] -> Bool
    allDistinctTyVars tkvs = go emptyVarSet tkvs
      where
        go _      [] = True
        go so_far (ty : tys)
           = case getTyVar_maybe ty of
                 Nothing -> False
                 Just tv | tv `elemVarSet` so_far -> False
                         | otherwise -> go (so_far `extendVarSet` tv) tys

    {-
    ---------------------------------------------------------------------
                                    AppTy
                                    ~~~~~
    We need to be pretty careful with AppTy to make sure we obey the
    invariant that a TyConApp is always visibly so.  mkAppTy maintains the
    invariant: use it.

    Note [Decomposing fat arrow c=>t]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Can we unify (a b) with (Eq a => ty)?   If we do so, we end up with
    a partial application like ((=>) Eq a) which doesn't make sense in
    source Haskell.  In constrast, we *can* unify (a b) with (t1 -> t2).
    Here's an example (Trac #9858) of how you might do it:
       i :: (Typeable a, Typeable b) => Proxy (a b) -> TypeRep
       i p = typeRep p

       j = i (Proxy :: Proxy (Eq Int => Int))
    The type (Proxy (Eq Int => Int)) is only accepted with -XImpredicativeTypes,
    but suppose we want that.  But then in the call to 'i', we end
    up decomposing (Eq Int => Int), and we definitely don't want that.

    This really only applies to the type checker; in Core, '=>' and '->'
    are the same, as are 'Constraint' and '*'.  But for now I've put
    the test in repSplitAppTy_maybe, which applies throughout, because
    the other calls to splitAppTy are in Unify, which is also used by
    the type checker (e.g. when matching type-function equations).
    -}

    -- | Applies a type to another, as in e.g. @k a@
    mkAppTy :: Type -> Type -> Type
    mkAppTy (TyConApp tc tys) ty2 = mkTyConApp tc (tys ++ [ty2])
    mkAppTy ty1               ty2 = AppTy ty1 ty2
            -- Note that the TyConApp could be an
            -- under-saturated type synonym.  GHC allows that; e.g.
            --      type Foo k = k a -> k a
            --      type Id x = x
            --      foo :: Foo Id -> Foo Id
            --
            -- Here Id is partially applied in the type sig for Foo,
            -- but once the type synonyms are expanded all is well

    mkAppTys :: Type -> [Type] -> Type
    mkAppTys ty1                []   = ty1
    mkAppTys (TyConApp tc tys1) tys2 = mkTyConApp tc (tys1 ++ tys2)
    mkAppTys ty1                tys2 = foldl AppTy ty1 tys2

    -------------
    splitAppTy_maybe :: Type -> Maybe (Type, Type)
    -- ^ Attempt to take a type application apart, whether it is a
    -- function, type constructor, or plain type application. Note
    -- that type family applications are NEVER unsaturated by this!
    splitAppTy_maybe ty | Just ty' <- coreView ty
                        = splitAppTy_maybe ty'
    splitAppTy_maybe ty = repSplitAppTy_maybe ty

    -------------
    repSplitAppTy_maybe :: Type -> Maybe (Type,Type)
    -- ^ Does the AppTy split as in 'splitAppTy_maybe', but assumes that
    -- any Core view stuff is already done
    repSplitAppTy_maybe (FunTy ty1 ty2)
      | isConstraintKind (typeKind ty1)   = Nothing  -- See Note [Decomposing fat arrow c=>t]
      | otherwise                         = Just (TyConApp funTyCon [ty1], ty2)
    repSplitAppTy_maybe (AppTy ty1 ty2)   = Just (ty1, ty2)
    repSplitAppTy_maybe (TyConApp tc tys)
      | isDecomposableTyCon tc || tys `lengthExceeds` tyConArity tc
      , Just (tys', ty') <- snocView tys
      = Just (TyConApp tc tys', ty')    -- Never create unsaturated type family apps!
    repSplitAppTy_maybe _other = Nothing
    -------------
    splitAppTy :: Type -> (Type, Type)
    -- ^ Attempts to take a type application apart, as in 'splitAppTy_maybe',
    -- and panics if this is not possible
    splitAppTy ty = case splitAppTy_maybe ty of
                    Just pr -> pr
                    Nothing -> panic "splitAppTy"

    -------------
    splitAppTys :: Type -> (Type, [Type])
    -- ^ Recursively splits a type as far as is possible, leaving a residual
    -- type being applied to and the type arguments applied to it. Never fails,
    -- even if that means returning an empty list of type applications.
    splitAppTys ty = split ty ty []
      where
        split orig_ty ty args | Just ty' <- coreView ty = split orig_ty ty' args
        split _       (AppTy ty arg)        args = split ty ty (arg:args)
        split _       (TyConApp tc tc_args) args
          = let -- keep type families saturated
                n | isDecomposableTyCon tc = 0
                  | otherwise              = tyConArity tc
                (tc_args1, tc_args2) = splitAt n tc_args
            in
            (TyConApp tc tc_args1, tc_args2 ++ args)
        split _       (FunTy ty1 ty2)       args = ASSERT( null args )
                                                   (TyConApp funTyCon [], [ty1,ty2])
        split orig_ty _                     args = (orig_ty, args)

    {-
                          LitTy
                          ~~~~~
    -}

    mkNumLitTy :: Integer -> Type
    mkNumLitTy n = LitTy (NumTyLit n)

    -- | Is this a numeric literal. We also look through type synonyms.
    isNumLitTy :: Type -> Maybe Integer
    isNumLitTy ty | Just ty1 <- tcView ty = isNumLitTy ty1
    isNumLitTy (LitTy (NumTyLit n)) = Just n
    isNumLitTy _                    = Nothing

    mkStrLitTy :: FastString -> Type
    mkStrLitTy s = LitTy (StrTyLit s)

    -- | Is this a symbol literal. We also look through type synonyms.
    isStrLitTy :: Type -> Maybe FastString
    isStrLitTy ty | Just ty1 <- tcView ty = isStrLitTy ty1
    isStrLitTy (LitTy (StrTyLit s)) = Just s
    isStrLitTy _                    = Nothing

    {-
    ---------------------------------------------------------------------
                                    FunTy
                                    ~~~~~
    -}

    mkFunTy :: Type -> Type -> Type
    -- ^ Creates a function type from the given argument and result type
    mkFunTy arg res = FunTy arg res

    mkFunTys :: [Type] -> Type -> Type
    mkFunTys tys ty = foldr mkFunTy ty tys

    isFunTy :: Type -> Bool
    isFunTy ty = isJust (splitFunTy_maybe ty)

    splitFunTy :: Type -> (Type, Type)
    -- ^ Attempts to extract the argument and result types from a type, and
    -- panics if that is not possible. See also 'splitFunTy_maybe'
    splitFunTy ty | Just ty' <- coreView ty = splitFunTy ty'
    splitFunTy (FunTy arg res) = (arg, res)
    splitFunTy other           = pprPanic "splitFunTy" (ppr other)

    splitFunTy_maybe :: Type -> Maybe (Type, Type)
    -- ^ Attempts to extract the argument and result types from a type
    splitFunTy_maybe ty | Just ty' <- coreView ty = splitFunTy_maybe ty'
    splitFunTy_maybe (FunTy arg res)   = Just (arg, res)
    splitFunTy_maybe _                 = Nothing

    splitFunTys :: Type -> ([Type], Type)
    splitFunTys ty = split [] ty ty
      where
        split args orig_ty ty | Just ty' <- coreView ty = split args orig_ty ty'
        split args _       (FunTy arg res)   = split (arg:args) res res
        split args orig_ty _                 = (reverse args, orig_ty)

    splitFunTysN :: Int -> Type -> ([Type], Type)
    -- ^ Split off exactly the given number argument types, and panics if that is not possible
    splitFunTysN 0 ty = ([], ty)
    splitFunTysN n ty = ASSERT2( isFunTy ty, int n <+> ppr ty )
                        case splitFunTy ty of { (arg, res) ->
                        case splitFunTysN (n-1) res of { (args, res) ->
                        (arg:args, res) }}

    -- | Splits off argument types from the given type and associating
    -- them with the things in the input list from left to right. The
    -- final result type is returned, along with the resulting pairs of
    -- objects and types, albeit with the list of pairs in reverse order.
    -- Panics if there are not enough argument types for the input list.
    zipFunTys :: Outputable a => [a] -> Type -> ([(a, Type)], Type)
    zipFunTys orig_xs orig_ty = split [] orig_xs orig_ty orig_ty
      where
        split acc []     nty _                 = (reverse acc, nty)
        split acc xs     nty ty
              | Just ty' <- coreView ty        = split acc xs nty ty'
        split acc (x:xs) _   (FunTy arg res)   = split ((x,arg):acc) xs res res
        split _   _      _   _                 = pprPanic "zipFunTys" (ppr orig_xs <+> ppr orig_ty)

    funResultTy :: Type -> Type
    -- ^ Extract the function result type and panic if that is not possible
    funResultTy ty | Just ty' <- coreView ty = funResultTy ty'
    funResultTy (FunTy _arg res)  = res
    funResultTy ty                = pprPanic "funResultTy" (ppr ty)

    funArgTy :: Type -> Type
    -- ^ Extract the function argument type and panic if that is not possible
    funArgTy ty | Just ty' <- coreView ty = funArgTy ty'
    funArgTy (FunTy arg _res)  = arg
    funArgTy ty                = pprPanic "funArgTy" (ppr ty)

    {-
    ---------------------------------------------------------------------
                                    TyConApp
                                    ~~~~~~~~
    -}

    -- | A key function: builds a 'TyConApp' or 'FunTy' as appropriate to
    -- its arguments.  Applies its arguments to the constructor from left to right.
    mkTyConApp :: TyCon -> [Type] -> Type
    mkTyConApp tycon tys
      | isFunTyCon tycon, [ty1,ty2] <- tys
      = FunTy ty1 ty2

      | otherwise
      = TyConApp tycon tys

    -- splitTyConApp "looks through" synonyms, because they don't
    -- mean a distinct type, but all other type-constructor applications
    -- including functions are returned as Just ..

    -- | The same as @fst . splitTyConApp@
    tyConAppTyCon_maybe :: Type -> Maybe TyCon
    tyConAppTyCon_maybe ty | Just ty' <- coreView ty = tyConAppTyCon_maybe ty'
    tyConAppTyCon_maybe (TyConApp tc _) = Just tc
    tyConAppTyCon_maybe (FunTy {})      = Just funTyCon
    tyConAppTyCon_maybe _               = Nothing

    tyConAppTyCon :: Type -> TyCon
    tyConAppTyCon ty = tyConAppTyCon_maybe ty `orElse` pprPanic "tyConAppTyCon" (ppr ty)

    -- | The same as @snd . splitTyConApp@
    tyConAppArgs_maybe :: Type -> Maybe [Type]
    tyConAppArgs_maybe ty | Just ty' <- coreView ty = tyConAppArgs_maybe ty'
    tyConAppArgs_maybe (TyConApp _ tys) = Just tys
    tyConAppArgs_maybe (FunTy arg res)  = Just [arg,res]
    tyConAppArgs_maybe _                = Nothing

    tyConAppArgs :: Type -> [Type]
    tyConAppArgs ty = tyConAppArgs_maybe ty `orElse` pprPanic "tyConAppArgs" (ppr ty)

    tyConAppArgN :: Int -> Type -> Type
    -- Executing Nth
    tyConAppArgN n ty
      = case tyConAppArgs_maybe ty of
          Just tys -> ASSERT2( n < length tys, ppr n <+> ppr tys ) tys !! n
          Nothing  -> pprPanic "tyConAppArgN" (ppr n <+> ppr ty)

    -- | Attempts to tease a type apart into a type constructor and the application
    -- of a number of arguments to that constructor. Panics if that is not possible.
    -- See also 'splitTyConApp_maybe'
    splitTyConApp :: Type -> (TyCon, [Type])
    splitTyConApp ty = case splitTyConApp_maybe ty of
                       Just stuff -> stuff
                       Nothing    -> pprPanic "splitTyConApp" (ppr ty)

    -- | Attempts to tease a type apart into a type constructor and the application
    -- of a number of arguments to that constructor
    splitTyConApp_maybe :: Type -> Maybe (TyCon, [Type])
    splitTyConApp_maybe ty | Just ty' <- coreView ty = splitTyConApp_maybe ty'
    splitTyConApp_maybe (TyConApp tc tys) = Just (tc, tys)
    splitTyConApp_maybe (FunTy arg res)   = Just (funTyCon, [arg,res])
    splitTyConApp_maybe _                 = Nothing

    -- | What is the role assigned to the next parameter of this type? Usually,
    -- this will be 'Nominal', but if the type is a 'TyConApp', we may be able to
    -- do better. The type does *not* have to be well-kinded when applied for this
    -- to work!
    nextRole :: Type -> Role
    nextRole ty
      | Just (tc, tys) <- splitTyConApp_maybe ty
      , let num_tys = length tys
      , num_tys < tyConArity tc
      = tyConRoles tc `getNth` num_tys

      | otherwise
      = Nominal

    newTyConInstRhs :: TyCon -> [Type] -> Type
    -- ^ Unwrap one 'layer' of newtype on a type constructor and its
    -- arguments, using an eta-reduced version of the @newtype@ if possible.
    -- This requires tys to have at least @newTyConInstArity tycon@ elements.
    newTyConInstRhs tycon tys
        = ASSERT2( tvs `leLength` tys, ppr tycon $$ ppr tys $$ ppr tvs )
          applyTysX tvs rhs tys
      where
        (tvs, rhs) = newTyConEtadRhs tycon

    {-
    ---------------------------------------------------------------------
                                    SynTy
                                    ~~~~~

    Notes on type synonyms
    ~~~~~~~~~~~~~~~~~~~~~~
    The various "split" functions (splitFunTy, splitRhoTy, splitForAllTy) try
    to return type synonyms wherever possible. Thus

            type Foo a = a -> a

    we want
            splitFunTys (a -> Foo a) = ([a], Foo a)
    not                                ([a], a -> a)

    The reason is that we then get better (shorter) type signatures in
    interfaces.  Notably this plays a role in tcTySigs in TcBinds.lhs.

                    Representation types
                    ~~~~~~~~~~~~~~~~~~~~

    Note [Nullary unboxed tuple]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    We represent the nullary unboxed tuple as the unary (but void) type
    Void#.  The reason for this is that the ReprArity is never
    less than the Arity (as it would otherwise be for a function type like
    (# #) -> Int).

    As a result, ReprArity is always strictly positive if Arity is. This
    is important because it allows us to distinguish at runtime between a
    thunk and a function takes a nullary unboxed tuple as an argument!
    -}

    type UnaryType = Type

    data RepType = UbxTupleRep [UnaryType] -- INVARIANT: never an empty list (see Note [Nullary unboxed tuple])
                 | UnaryRep UnaryType

    flattenRepType :: RepType -> [UnaryType]
    flattenRepType (UbxTupleRep tys) = tys
    flattenRepType (UnaryRep ty)     = [ty]

    -- | Looks through:
    --
    --      1. For-alls
    --      2. Synonyms
    --      3. Predicates
    --      4. All newtypes, including recursive ones, but not newtype families
    --
    -- It's useful in the back end of the compiler.
    repType :: Type -> RepType
    repType ty
      = go initRecTc ty
      where
        go :: RecTcChecker -> Type -> RepType
        go rec_nts ty                       -- Expand predicates and synonyms
          | Just ty' <- coreView ty
          = go rec_nts ty'

        go rec_nts (ForAllTy _ ty)          -- Drop foralls
            = go rec_nts ty

        go rec_nts (TyConApp tc tys)        -- Expand newtypes
          | isNewTyCon tc
          , tys `lengthAtLeast` tyConArity tc
          , Just rec_nts' <- checkRecTc rec_nts tc   -- See Note [Expanding newtypes] in TyCon
          = go rec_nts' (newTyConInstRhs tc tys)

          | isUnboxedTupleTyCon tc
          = if null tys
             then UnaryRep voidPrimTy -- See Note [Nullary unboxed tuple]
             else UbxTupleRep (concatMap (flattenRepType . go rec_nts) tys)

        go _ ty = UnaryRep ty

    -- | All type constructors occurring in the type; looking through type
    --   synonyms, but not newtypes.
    --  When it finds a Class, it returns the class TyCon.
    tyConsOfType :: Type -> NameEnv TyCon
    tyConsOfType ty
      = go ty
      where
         go :: Type -> NameEnv TyCon  -- The NameEnv does duplicate elim
         go ty | Just ty' <- tcView ty = go ty'
         go (TyVarTy {})               = emptyNameEnv
         go (LitTy {})                 = emptyNameEnv
         go (TyConApp tc tys)          = go_tc tc tys
         go (AppTy a b)                = go a `plusNameEnv` go b
         go (FunTy a b)                = go a `plusNameEnv` go b
         go (ForAllTy _ ty)            = go ty

         go_tc tc tys = extendNameEnv (go_s tys) (tyConName tc) tc
         go_s tys = foldr (plusNameEnv . go) emptyNameEnv tys

    -- ToDo: this could be moved to the code generator, using splitTyConApp instead
    -- of inspecting the type directly.

    -- | Discovers the primitive representation of a more abstract 'UnaryType'
    typePrimRep :: UnaryType -> PrimRep
    typePrimRep ty
      = case repType ty of
          UbxTupleRep _ -> pprPanic "typePrimRep: UbxTupleRep" (ppr ty)
          UnaryRep rep -> case rep of
            TyConApp tc _ -> tyConPrimRep tc
            FunTy _ _     -> PtrRep
            AppTy _ _     -> PtrRep      -- See Note [AppTy rep]
            TyVarTy _     -> PtrRep
            _             -> pprPanic "typePrimRep: UnaryRep" (ppr ty)

    typeRepArity :: Arity -> Type -> RepArity
    typeRepArity 0 _ = 0
    typeRepArity n ty = case repType ty of
      UnaryRep (FunTy ty1 ty2) -> length (flattenRepType (repType ty1)) + typeRepArity (n - 1) ty2
      _                        -> pprPanic "typeRepArity: arity greater than type can handle" (ppr (n, ty))

    isVoidTy :: Type -> Bool
    -- True if the type has zero width
    isVoidTy ty = case repType ty of
                    UnaryRep (TyConApp tc _) -> isVoidRep (tyConPrimRep tc)
                    _                        -> False

    {-
    Note [AppTy rep]
    ~~~~~~~~~~~~~~~~
    Types of the form 'f a' must be of kind *, not #, so we are guaranteed
    that they are represented by pointers.  The reason is that f must have
    kind (kk -> kk) and kk cannot be unlifted; see Note [The kind invariant]
    in TypeRep.

    ---------------------------------------------------------------------
                                    ForAllTy
                                    ~~~~~~~~
    -}

    mkForAllTy :: TyVar -> Type -> Type
    mkForAllTy tyvar ty
      = ForAllTy tyvar ty

    -- | Wraps foralls over the type using the provided 'TyVar's from left to right
    mkForAllTys :: [TyVar] -> Type -> Type
    mkForAllTys tyvars ty = foldr ForAllTy ty tyvars

    mkPiKinds :: [TyVar] -> Kind -> Kind
    -- mkPiKinds [k1, k2, (a:k1 -> *)] k2
    -- returns forall k1 k2. (k1 -> *) -> k2
    mkPiKinds [] res = res
    mkPiKinds (tv:tvs) res
      | isKindVar tv = ForAllTy tv          (mkPiKinds tvs res)
      | otherwise    = FunTy (tyVarKind tv) (mkPiKinds tvs res)

    mkPiType  :: Var -> Type -> Type
    -- ^ Makes a @(->)@ type or a forall type, depending
    -- on whether it is given a type variable or a term variable.
    mkPiTypes :: [Var] -> Type -> Type
    -- ^ 'mkPiType' for multiple type or value arguments

    mkPiType v ty
       | isId v    = mkFunTy (varType v) ty
       | otherwise = mkForAllTy v ty

    mkPiTypes vs ty = foldr mkPiType ty vs

    isForAllTy :: Type -> Bool
    isForAllTy (ForAllTy _ _) = True
    isForAllTy _              = False

    -- | Attempts to take a forall type apart, returning the bound type variable
    -- and the remainder of the type
    splitForAllTy_maybe :: Type -> Maybe (TyVar, Type)
    splitForAllTy_maybe ty = splitFAT_m ty
      where
        splitFAT_m ty | Just ty' <- coreView ty = splitFAT_m ty'
        splitFAT_m (ForAllTy tyvar ty)          = Just(tyvar, ty)
        splitFAT_m _                            = Nothing

    -- | Attempts to take a forall type apart, returning all the immediate such bound
    -- type variables and the remainder of the type. Always suceeds, even if that means
    -- returning an empty list of 'TyVar's
    splitForAllTys :: Type -> ([TyVar], Type)
    splitForAllTys ty = split ty ty []
       where
         split orig_ty ty tvs | Just ty' <- coreView ty = split orig_ty ty' tvs
         split _       (ForAllTy tv ty)  tvs = split ty ty (tv:tvs)
         split orig_ty _                 tvs = (reverse tvs, orig_ty)

    -- | Equivalent to @snd . splitForAllTys@
    dropForAlls :: Type -> Type
    dropForAlls ty = snd (splitForAllTys ty)

    {-
    -- (mkPiType now in CoreUtils)

    applyTy, applyTys
    ~~~~~~~~~~~~~~~~~
    -}

    -- | Instantiate a forall type with one or more type arguments.
    -- Used when we have a polymorphic function applied to type args:
    --
    -- > f t1 t2
    --
    -- We use @applyTys type-of-f [t1,t2]@ to compute the type of the expression.
    -- Panics if no application is possible.
    applyTy :: Type -> KindOrType -> Type
    applyTy ty arg | Just ty' <- coreView ty = applyTy ty' arg
    applyTy (ForAllTy tv ty) arg = substTyWith [tv] [arg] ty
    applyTy _                _   = panic "applyTy"

    applyTys :: Type -> [KindOrType] -> Type
    -- ^ This function is interesting because:
    --
    --      1. The function may have more for-alls than there are args
    --
    --      2. Less obviously, it may have fewer for-alls
    --
    -- For case 2. think of:
    --
    -- > applyTys (forall a.a) [forall b.b, Int]
    --
    -- This really can happen, but only (I think) in situations involving
    -- undefined.  For example:
    --       undefined :: forall a. a
    -- Term: undefined @(forall b. b->b) @Int
    -- This term should have type (Int -> Int), but notice that
    -- there are more type args than foralls in 'undefined's type.

    -- If you edit this function, you may need to update the GHC formalism
    -- See Note [GHC Formalism] in coreSyn/CoreLint.lhs
    applyTys ty args = applyTysD empty ty args

    applyTysD :: SDoc -> Type -> [Type] -> Type     -- Debug version
    applyTysD _   orig_fun_ty []      = orig_fun_ty
    applyTysD doc orig_fun_ty arg_tys
      | n_tvs == n_args     -- The vastly common case
      = substTyWith tvs arg_tys rho_ty
      | n_tvs > n_args      -- Too many for-alls
      = substTyWith (take n_args tvs) arg_tys
                    (mkForAllTys (drop n_args tvs) rho_ty)
      | otherwise           -- Too many type args
      = ASSERT2( n_tvs > 0, doc $$ ppr orig_fun_ty $$ ppr arg_tys )        -- Zero case gives infinite loop!
        applyTysD doc (substTyWith tvs (take n_tvs arg_tys) rho_ty)
                      (drop n_tvs arg_tys)
      where
        (tvs, rho_ty) = splitForAllTys orig_fun_ty
        n_tvs  = length tvs
        n_args = length arg_tys

    applyTysX :: [TyVar] -> Type -> [Type] -> Type
    -- applyTyxX beta-reduces (/\tvs. body_ty) arg_tys
    applyTysX tvs body_ty arg_tys
      = ASSERT2( length arg_tys >= n_tvs, ppr tvs $$ ppr body_ty $$ ppr arg_tys )
        mkAppTys (substTyWith tvs (take n_tvs arg_tys) body_ty)
                 (drop n_tvs arg_tys)
      where
        n_tvs = length tvs

    {-
    ************************************************************************
    *                                                                      *
                             Pred
    *                                                                      *
    ************************************************************************

    Predicates on PredType
    -}

    isPredTy :: Type -> Bool
      -- NB: isPredTy is used when printing types, which can happen in debug printing
      --     during type checking of not-fully-zonked types.  So it's not cool to say
      --     isConstraintKind (typeKind ty) because absent zonking the type might
      --     be ill-kinded, and typeKind crashes
      --     Hence the rather tiresome story here
    isPredTy ty = go ty []
      where
        go :: Type -> [KindOrType] -> Bool
        go (AppTy ty1 ty2)   args = go ty1 (ty2 : args)
        go (TyConApp tc tys) args = go_k (tyConKind tc) (tys ++ args)
        go (TyVarTy tv)      args = go_k (tyVarKind tv) args
        go _                 _    = False

        go_k :: Kind -> [KindOrType] -> Bool
        -- True <=> kind is k1 -> .. -> kn -> Constraint
        go_k k                [] = isConstraintKind k
        go_k (FunTy _ k1)     (_ :args) = go_k k1 args
        go_k (ForAllTy kv k1) (k2:args) = go_k (substKiWith [kv] [k2] k1) args
        go_k _ _ = False                  -- Typeable * Int :: Constraint

    isClassPred, isEqPred, isIPPred :: PredType -> Bool
    isClassPred ty = case tyConAppTyCon_maybe ty of
        Just tyCon | isClassTyCon tyCon -> True
        _                               -> False
    isEqPred ty = case tyConAppTyCon_maybe ty of
        Just tyCon -> tyCon `hasKey` eqTyConKey
        _          -> False

    isIPPred ty = case tyConAppTyCon_maybe ty of
        Just tc -> isIPTyCon tc
        _       -> False

    isIPTyCon :: TyCon -> Bool
    isIPTyCon tc = tc `hasKey` ipClassNameKey

    isIPClass :: Class -> Bool
    isIPClass cls = cls `hasKey` ipClassNameKey
      -- Class and it corresponding TyCon have the same Unique

    isIPPred_maybe :: Type -> Maybe (FastString, Type)
    isIPPred_maybe ty =
      do (tc,[t1,t2]) <- splitTyConApp_maybe ty
         guard (isIPTyCon tc)
         x <- isStrLitTy t1
         return (x,t2)

    {-
    Make PredTypes

    --------------------- Equality types ---------------------------------
    -}

    -- | Creates a type equality predicate
    mkEqPred :: Type -> Type -> PredType
    mkEqPred ty1 ty2
      = WARN( not (k `eqKind` typeKind ty2), ppr ty1 $$ ppr ty2 $$ ppr k $$ ppr (typeKind ty2) )
        TyConApp eqTyCon [k, ty1, ty2]
      where
        k = typeKind ty1

    mkCoerciblePred :: Type -> Type -> PredType
    mkCoerciblePred ty1 ty2
      = WARN( not (k `eqKind` typeKind ty2), ppr ty1 $$ ppr ty2 $$ ppr k $$ ppr (typeKind ty2) )
        TyConApp coercibleTyCon [k, ty1, ty2]
      where
        k = typeKind ty1

    mkPrimEqPred :: Type -> Type -> Type
    mkPrimEqPred ty1  ty2
      = WARN( not (k `eqKind` typeKind ty2), ppr ty1 $$ ppr ty2 )
        TyConApp eqPrimTyCon [k, ty1, ty2]
      where
        k = typeKind ty1

    mkReprPrimEqPred :: Type -> Type -> Type
    mkReprPrimEqPred ty1  ty2
      = WARN( not (k `eqKind` typeKind ty2), ppr ty1 $$ ppr ty2 )
        TyConApp eqReprPrimTyCon [k, ty1, ty2]
      where
        k = typeKind ty1

    -- --------------------- Dictionary types ---------------------------------

    mkClassPred :: Class -> [Type] -> PredType
    mkClassPred clas tys = TyConApp (classTyCon clas) tys

    isDictTy :: Type -> Bool
    isDictTy = isClassPred

    isDictLikeTy :: Type -> Bool
    -- Note [Dictionary-like types]
    isDictLikeTy ty | Just ty' <- coreView ty = isDictLikeTy ty'
    isDictLikeTy ty = case splitTyConApp_maybe ty of
            Just (tc, tys) | isClassTyCon tc -> True
                           | isTupleTyCon tc -> all isDictLikeTy tys
            _other                           -> False

    {-
    Note [Dictionary-like types]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Being "dictionary-like" means either a dictionary type or a tuple thereof.
    In GHC 6.10 we build implication constraints which construct such tuples,
    and if we land up with a binding
        t :: (C [a], Eq [a])
        t = blah
    then we want to treat t as cheap under "-fdicts-cheap" for example.
    (Implication constraints are normally inlined, but sadly not if the
    occurrence is itself inside an INLINE function!  Until we revise the
    handling of implication constraints, that is.)  This turned out to
    be important in getting good arities in DPH code.  Example:

        class C a
        class D a where { foo :: a -> a }
        instance C a => D (Maybe a) where { foo x = x }

        bar :: (C a, C b) => a -> b -> (Maybe a, Maybe b)
        {-# INLINE bar #-}
        bar x y = (foo (Just x), foo (Just y))

    Then 'bar' should jolly well have arity 4 (two dicts, two args), but
    we ended up with something like
       bar = __inline_me__ (\d1,d2. let t :: (D (Maybe a), D (Maybe b)) = ...
                                    in \x,y. <blah>)

    This is all a bit ad-hoc; eg it relies on knowing that implication
    constraints build tuples.

    Decomposing PredType
    -}

    -- | A choice of equality relation. This is separate from the type 'Role'
    -- because 'Phantom' does not define a (non-trivial) equality relation.
    data EqRel = NomEq | ReprEq
      deriving (Eq, Ord)

    instance Outputable EqRel where
      ppr NomEq  = text "nominal equality"
      ppr ReprEq = text "representational equality"

    eqRelRole :: EqRel -> Role
    eqRelRole NomEq  = Nominal
    eqRelRole ReprEq = Representational

    data PredTree = ClassPred Class [Type]
                  | EqPred EqRel Type Type
                  | TuplePred [PredType]
                  | IrredPred PredType

    classifyPredType :: PredType -> PredTree
    classifyPredType ev_ty = case splitTyConApp_maybe ev_ty of
        Just (tc, tys) | tc `hasKey` coercibleTyConKey
                       , let [_, ty1, ty2] = tys
                       -> EqPred ReprEq ty1 ty2
        Just (tc, tys) | tc `hasKey` eqTyConKey
                       , let [_, ty1, ty2] = tys
                       -> EqPred NomEq ty1 ty2
         -- NB: Coercible is also a class, so this check must come *after*
         -- the Coercible check
        Just (tc, tys) | Just clas <- tyConClass_maybe tc
                       -> ClassPred clas tys
        Just (tc, tys) | isTupleTyCon tc
                       -> TuplePred tys
        _ -> IrredPred ev_ty

    getClassPredTys :: PredType -> (Class, [Type])
    getClassPredTys ty = case getClassPredTys_maybe ty of
            Just (clas, tys) -> (clas, tys)
            Nothing          -> pprPanic "getClassPredTys" (ppr ty)

    getClassPredTys_maybe :: PredType -> Maybe (Class, [Type])
    getClassPredTys_maybe ty = case splitTyConApp_maybe ty of
            Just (tc, tys) | Just clas <- tyConClass_maybe tc -> Just (clas, tys)
            _ -> Nothing

    getEqPredTys :: PredType -> (Type, Type)
    getEqPredTys ty
      = case splitTyConApp_maybe ty of
          Just (tc, (_ : ty1 : ty2 : tys)) ->
            ASSERT( null tys && (tc `hasKey` eqTyConKey
                                 || tc `hasKey` coercibleTyConKey) )
            (ty1, ty2)
          _ -> pprPanic "getEqPredTys" (ppr ty)

    getEqPredTys_maybe :: PredType -> Maybe (Role, Type, Type)
    getEqPredTys_maybe ty
      = case splitTyConApp_maybe ty of
          Just (tc, [_, ty1, ty2])
            | tc `hasKey` eqTyConKey -> Just (Nominal, ty1, ty2)
            | tc `hasKey` coercibleTyConKey -> Just (Representational, ty1, ty2)
          _ -> Nothing

    getEqPredRole :: PredType -> Role
    getEqPredRole ty
      = case splitTyConApp_maybe ty of
          Just (tc, [_, _, _])
            | tc `hasKey` eqTyConKey -> Nominal
            | tc `hasKey` coercibleTyConKey -> Representational
          _ -> pprPanic "getEqPredRole" (ppr ty)

    -- | Get the equality relation relevant for a pred type.
    predTypeEqRel :: PredType -> EqRel
    predTypeEqRel ty
      | Just (tc, _) <- splitTyConApp_maybe ty
      , tc `hasKey` coercibleTyConKey
      = ReprEq
      | otherwise
      = NomEq

    {-
    %************************************************************************
    %*                                                                      *
                       Size
    *                                                                      *
    ************************************************************************
    -}

    typeSize :: Type -> Int
    typeSize (LitTy {})      = 1
    typeSize (TyVarTy {})    = 1
    typeSize (AppTy t1 t2)   = typeSize t1 + typeSize t2
    typeSize (FunTy t1 t2)   = typeSize t1 + typeSize t2
    typeSize (ForAllTy _ t)  = 1 + typeSize t
    typeSize (TyConApp _ ts) = 1 + sum (map typeSize ts)

    {-
    ************************************************************************
    *                                                                      *
    \subsection{Type families}
    *                                                                      *
    ************************************************************************
    -}

    mkFamilyTyConApp :: TyCon -> [Type] -> Type
    -- ^ Given a family instance TyCon and its arg types, return the
    -- corresponding family type.  E.g:
    --
    -- > data family T a
    -- > data instance T (Maybe b) = MkT b
    --
    -- Where the instance tycon is :RTL, so:
    --
    -- > mkFamilyTyConApp :RTL Int  =  T (Maybe Int)
    mkFamilyTyConApp tc tys
      | Just (fam_tc, fam_tys) <- tyConFamInst_maybe tc
      , let tvs = tyConTyVars tc
            fam_subst = ASSERT2( length tvs == length tys, ppr tc <+> ppr tys )
                        zipTopTvSubst tvs tys
      = mkTyConApp fam_tc (substTys fam_subst fam_tys)
      | otherwise
      = mkTyConApp tc tys

    -- | Get the type on the LHS of a coercion induced by a type/data
    -- family instance.
    coAxNthLHS :: CoAxiom br -> Int -> Type
    coAxNthLHS ax ind =
      mkTyConApp (coAxiomTyCon ax) (coAxBranchLHS (coAxiomNthBranch ax ind))

    -- | Pretty prints a 'TyCon', using the family instance in case of a
    -- representation tycon.  For example:
    --
    -- > data T [a] = ...
    --
    -- In that case we want to print @T [a]@, where @T@ is the family 'TyCon'
    pprSourceTyCon :: TyCon -> SDoc
    pprSourceTyCon tycon
      | Just (fam_tc, tys) <- tyConFamInst_maybe tycon
      = ppr $ fam_tc `TyConApp` tys        -- can't be FunTyCon
      | otherwise
      = ppr tycon

    {-
    ************************************************************************
    *                                                                      *
    \subsection{Liftedness}
    *                                                                      *
    ************************************************************************
    -}

    -- | See "Type#type_classification" for what an unlifted type is
    isUnLiftedType :: Type -> Bool
            -- isUnLiftedType returns True for forall'd unlifted types:
            --      x :: forall a. Int#
            -- I found bindings like these were getting floated to the top level.
            -- They are pretty bogus types, mind you.  It would be better never to
            -- construct them

    isUnLiftedType ty | Just ty' <- coreView ty = isUnLiftedType ty'
    isUnLiftedType (ForAllTy _ ty)      = isUnLiftedType ty
    isUnLiftedType (TyConApp tc _)      = isUnLiftedTyCon tc
    isUnLiftedType _                    = False

    isUnboxedTupleType :: Type -> Bool
    isUnboxedTupleType ty = case tyConAppTyCon_maybe ty of
                               Just tc -> isUnboxedTupleTyCon tc
                               _       -> False

    -- | See "Type#type_classification" for what an algebraic type is.
    -- Should only be applied to /types/, as opposed to e.g. partially
    -- saturated type constructors
    isAlgType :: Type -> Bool
    isAlgType ty
      = case splitTyConApp_maybe ty of
          Just (tc, ty_args) -> ASSERT( ty_args `lengthIs` tyConArity tc )
                                isAlgTyCon tc
          _other             -> False

    -- | See "Type#type_classification" for what an algebraic type is.
    -- Should only be applied to /types/, as opposed to e.g. partially
    -- saturated type constructors. Closed type constructors are those
    -- with a fixed right hand side, as opposed to e.g. associated types
    isClosedAlgType :: Type -> Bool
    isClosedAlgType ty
      = case splitTyConApp_maybe ty of
          Just (tc, ty_args) | isAlgTyCon tc && not (isFamilyTyCon tc)
                 -> ASSERT2( ty_args `lengthIs` tyConArity tc, ppr ty ) True
          _other -> False

    -- | Computes whether an argument (or let right hand side) should
    -- be computed strictly or lazily, based only on its type.
    -- Currently, it's just 'isUnLiftedType'.

    isStrictType :: Type -> Bool
    isStrictType = isUnLiftedType

    isPrimitiveType :: Type -> Bool
    -- ^ Returns true of types that are opaque to Haskell.
    isPrimitiveType ty = case splitTyConApp_maybe ty of
                            Just (tc, ty_args) -> ASSERT( ty_args `lengthIs` tyConArity tc )
                                                  isPrimTyCon tc
                            _                  -> False

    {-
    ************************************************************************
    *                                                                      *
    \subsection{Sequencing on types}
    *                                                                      *
    ************************************************************************
    -}

    seqType :: Type -> ()
    seqType (LitTy n)         = n `seq` ()
    seqType (TyVarTy tv)      = tv `seq` ()
    seqType (AppTy t1 t2)     = seqType t1 `seq` seqType t2
    seqType (FunTy t1 t2)     = seqType t1 `seq` seqType t2
    seqType (TyConApp tc tys) = tc `seq` seqTypes tys
    seqType (ForAllTy tv ty)  = seqType (tyVarKind tv) `seq` seqType ty

    seqTypes :: [Type] -> ()
    seqTypes []       = ()
    seqTypes (ty:tys) = seqType ty `seq` seqTypes tys

    {-
    ************************************************************************
    *                                                                      *
                    Comparison for types
            (We don't use instances so that we know where it happens)
    *                                                                      *
    ************************************************************************
    -}

    eqKind :: Kind -> Kind -> Bool
    -- Watch out for horrible hack: See Note [Comparison with OpenTypeKind]
    eqKind = eqType

    eqType :: Type -> Type -> Bool
    -- ^ Type equality on source types. Does not look through @newtypes@ or
    -- 'PredType's, but it does look through type synonyms.
    -- Watch out for horrible hack: See Note [Comparison with OpenTypeKind]
    eqType t1 t2 = isEqual $ cmpType t1 t2

    instance Eq Type where
      (==) = eqType

    eqTypeX :: RnEnv2 -> Type -> Type -> Bool
    eqTypeX env t1 t2 = isEqual $ cmpTypeX env t1 t2

    eqTypes :: [Type] -> [Type] -> Bool
    eqTypes tys1 tys2 = isEqual $ cmpTypes tys1 tys2

    eqPred :: PredType -> PredType -> Bool
    eqPred = eqType

    eqPredX :: RnEnv2 -> PredType -> PredType -> Bool
    eqPredX env p1 p2 = isEqual $ cmpTypeX env p1 p2

    eqTyVarBndrs :: RnEnv2 -> [TyVar] -> [TyVar] -> Maybe RnEnv2
    -- Check that the tyvar lists are the same length
    -- and have matching kinds; if so, extend the RnEnv2
    -- Returns Nothing if they don't match
    eqTyVarBndrs env [] []
     = Just env
    eqTyVarBndrs env (tv1:tvs1) (tv2:tvs2)
     | eqTypeX env (tyVarKind tv1) (tyVarKind tv2)
     = eqTyVarBndrs (rnBndr2 env tv1 tv2) tvs1 tvs2
    eqTyVarBndrs _ _ _= Nothing

    -- Now here comes the real worker

    cmpType :: Type -> Type -> Ordering
    -- Watch out for horrible hack: See Note [Comparison with OpenTypeKind]
    cmpType t1 t2 = cmpTypeX rn_env t1 t2
      where
        rn_env = mkRnEnv2 (mkInScopeSet (tyVarsOfType t1 `unionVarSet` tyVarsOfType t2))

    cmpTypes :: [Type] -> [Type] -> Ordering
    cmpTypes ts1 ts2 = cmpTypesX rn_env ts1 ts2
      where
        rn_env = mkRnEnv2 (mkInScopeSet (tyVarsOfTypes ts1 `unionVarSet` tyVarsOfTypes ts2))

    cmpPred :: PredType -> PredType -> Ordering
    cmpPred p1 p2 = cmpTypeX rn_env p1 p2
      where
        rn_env = mkRnEnv2 (mkInScopeSet (tyVarsOfType p1 `unionVarSet` tyVarsOfType p2))

    cmpTypeX :: RnEnv2 -> Type -> Type -> Ordering  -- Main workhorse
    cmpTypeX env t1 t2 | Just t1' <- coreView t1 = cmpTypeX env t1' t2
                       | Just t2' <- coreView t2 = cmpTypeX env t1 t2'
    -- We expand predicate types, because in Core-land we have
    -- lots of definitions like
    --      fOrdBool :: Ord Bool
    --      fOrdBool = D:Ord .. .. ..
    -- So the RHS has a data type

    cmpTypeX env (TyVarTy tv1)       (TyVarTy tv2)       = rnOccL env tv1 `compare` rnOccR env tv2
    cmpTypeX env (ForAllTy tv1 t1)   (ForAllTy tv2 t2)   = cmpTypeX env (tyVarKind tv1) (tyVarKind tv2)
                                                           `thenCmp` cmpTypeX (rnBndr2 env tv1 tv2) t1 t2
    cmpTypeX env (AppTy s1 t1)       (AppTy s2 t2)       = cmpTypeX env s1 s2 `thenCmp` cmpTypeX env t1 t2
    cmpTypeX env (FunTy s1 t1)       (FunTy s2 t2)       = cmpTypeX env s1 s2 `thenCmp` cmpTypeX env t1 t2
    cmpTypeX env (TyConApp tc1 tys1) (TyConApp tc2 tys2) = (tc1 `cmpTc` tc2) `thenCmp` cmpTypesX env tys1 tys2
    cmpTypeX _   (LitTy l1)          (LitTy l2)          = compare l1 l2

        -- Deal with the rest: TyVarTy < AppTy < FunTy < LitTy < TyConApp < ForAllTy < PredTy
    cmpTypeX _ (AppTy _ _)    (TyVarTy _)    = GT

    cmpTypeX _ (FunTy _ _)    (TyVarTy _)    = GT
    cmpTypeX _ (FunTy _ _)    (AppTy _ _)    = GT

    cmpTypeX _ (LitTy _)      (TyVarTy _)    = GT
    cmpTypeX _ (LitTy _)      (AppTy _ _)    = GT
    cmpTypeX _ (LitTy _)      (FunTy _ _)    = GT

    cmpTypeX _ (TyConApp _ _) (TyVarTy _)    = GT
    cmpTypeX _ (TyConApp _ _) (AppTy _ _)    = GT
    cmpTypeX _ (TyConApp _ _) (FunTy _ _)    = GT
    cmpTypeX _ (TyConApp _ _) (LitTy _)      = GT

    cmpTypeX _ (ForAllTy _ _) (TyVarTy _)    = GT
    cmpTypeX _ (ForAllTy _ _) (AppTy _ _)    = GT
    cmpTypeX _ (ForAllTy _ _) (FunTy _ _)    = GT
    cmpTypeX _ (ForAllTy _ _) (LitTy _)      = GT
    cmpTypeX _ (ForAllTy _ _) (TyConApp _ _) = GT

    cmpTypeX _ _              _              = LT

    -------------
    cmpTypesX :: RnEnv2 -> [Type] -> [Type] -> Ordering
    cmpTypesX _   []        []        = EQ
    cmpTypesX env (t1:tys1) (t2:tys2) = cmpTypeX env t1 t2 `thenCmp` cmpTypesX env tys1 tys2
    cmpTypesX _   []        _         = LT
    cmpTypesX _   _         []        = GT

    -------------
    cmpTc :: TyCon -> TyCon -> Ordering
    -- Here we treat * and Constraint as equal
    -- See Note [Kind Constraint and kind *] in Kinds.lhs
    --
    -- Also we treat OpenTypeKind as equal to either * or #
    -- See Note [Comparison with OpenTypeKind]
    cmpTc tc1 tc2
      | u1 == openTypeKindTyConKey, isSubOpenTypeKindKey u2 = EQ
      | u2 == openTypeKindTyConKey, isSubOpenTypeKindKey u1 = EQ
      | otherwise = nu1 `compare` nu2
      where
        u1  = tyConUnique tc1
        nu1 = if u1==constraintKindTyConKey then liftedTypeKindTyConKey else u1
        u2  = tyConUnique tc2
        nu2 = if u2==constraintKindTyConKey then liftedTypeKindTyConKey else u2

    {-
    Note [Comparison with OpenTypeKind]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    In PrimOpWrappers we have things like
       PrimOpWrappers.mkWeak# = /\ a b c. Prim.mkWeak# a b c
    where
       Prim.mkWeak# :: forall (a:Open) b c. a -> b -> c
                                         -> State# RealWorld -> (# State# RealWorld, Weak# b #)
    Now, eta reduction will turn the definition into
         PrimOpWrappers.mkWeak# = Prim.mkWeak#
    which is kind-of OK, but now the types aren't really equal.  So HACK HACK
    we pretend (in Core) that Open is equal to * or #.  I hate this.

    Note [cmpTypeX]
    ~~~~~~~~~~~~~~~

    When we compare foralls, we should look at the kinds. But if we do so,
    we get a corelint error like the following (in
    libraries/ghc-prim/GHC/PrimopWrappers.hs):

        Binder's type: forall (o_abY :: *).
                       o_abY
                       -> GHC.Prim.State# GHC.Prim.RealWorld
                       -> GHC.Prim.State# GHC.Prim.RealWorld
        Rhs type: forall (a_12 :: ?).
                  a_12
                  -> GHC.Prim.State# GHC.Prim.RealWorld
                  -> GHC.Prim.State# GHC.Prim.RealWorld

    This is why we don't look at the kind. Maybe we should look if the
    kinds are compatible.

    -- cmpTypeX env (ForAllTy tv1 t1)   (ForAllTy tv2 t2)
    --   = cmpTypeX env (tyVarKind tv1) (tyVarKind tv2) `thenCmp`
    --     cmpTypeX (rnBndr2 env tv1 tv2) t1 t2

    ************************************************************************
    *                                                                      *
                    Type substitutions
    *                                                                      *
    ************************************************************************
    -}

    emptyTvSubstEnv :: TvSubstEnv
    emptyTvSubstEnv = emptyVarEnv

    composeTvSubst :: InScopeSet -> TvSubstEnv -> TvSubstEnv -> TvSubstEnv
    -- ^ @(compose env1 env2)(x)@ is @env1(env2(x))@; i.e. apply @env2@ then @env1@.
    -- It assumes that both are idempotent.
    -- Typically, @env1@ is the refinement to a base substitution @env2@
    composeTvSubst in_scope env1 env2
      = env1 `plusVarEnv` mapVarEnv (substTy subst1) env2
            -- First apply env1 to the range of env2
            -- Then combine the two, making sure that env1 loses if
            -- both bind the same variable; that's why env1 is the
            --  *left* argument to plusVarEnv, because the right arg wins
      where
        subst1 = TvSubst in_scope env1

    emptyTvSubst :: TvSubst
    emptyTvSubst = TvSubst emptyInScopeSet emptyTvSubstEnv

    isEmptyTvSubst :: TvSubst -> Bool
             -- See Note [Extending the TvSubstEnv] in TypeRep
    isEmptyTvSubst (TvSubst _ tenv) = isEmptyVarEnv tenv

    mkTvSubst :: InScopeSet -> TvSubstEnv -> TvSubst
    mkTvSubst = TvSubst

    getTvSubstEnv :: TvSubst -> TvSubstEnv
    getTvSubstEnv (TvSubst _ env) = env

    getTvInScope :: TvSubst -> InScopeSet
    getTvInScope (TvSubst in_scope _) = in_scope

    isInScope :: Var -> TvSubst -> Bool
    isInScope v (TvSubst in_scope _) = v `elemInScopeSet` in_scope

    notElemTvSubst :: CoVar -> TvSubst -> Bool
    notElemTvSubst v (TvSubst _ tenv) = not (v `elemVarEnv` tenv)

    setTvSubstEnv :: TvSubst -> TvSubstEnv -> TvSubst
    setTvSubstEnv (TvSubst in_scope _) tenv = TvSubst in_scope tenv

    zapTvSubstEnv :: TvSubst -> TvSubst
    zapTvSubstEnv (TvSubst in_scope _) = TvSubst in_scope emptyVarEnv

    extendTvInScope :: TvSubst -> Var -> TvSubst
    extendTvInScope (TvSubst in_scope tenv) var = TvSubst (extendInScopeSet in_scope var) tenv

    extendTvInScopeList :: TvSubst -> [Var] -> TvSubst
    extendTvInScopeList (TvSubst in_scope tenv) vars = TvSubst (extendInScopeSetList in_scope vars) tenv

    extendTvSubst :: TvSubst -> TyVar -> Type -> TvSubst
    extendTvSubst (TvSubst in_scope tenv) tv ty = TvSubst in_scope (extendVarEnv tenv tv ty)

    extendTvSubstList :: TvSubst -> [TyVar] -> [Type] -> TvSubst
    extendTvSubstList (TvSubst in_scope tenv) tvs tys
      = TvSubst in_scope (extendVarEnvList tenv (tvs `zip` tys))

    unionTvSubst :: TvSubst -> TvSubst -> TvSubst
    -- Works when the ranges are disjoint
    unionTvSubst (TvSubst in_scope1 tenv1) (TvSubst in_scope2 tenv2)
      = ASSERT( not (tenv1 `intersectsVarEnv` tenv2) )
        TvSubst (in_scope1 `unionInScope` in_scope2)
                (tenv1     `plusVarEnv`   tenv2)

    -- mkOpenTvSubst and zipOpenTvSubst generate the in-scope set from
    -- the types given; but it's just a thunk so with a bit of luck
    -- it'll never be evaluated

    -- Note [Generating the in-scope set for a substitution]
    -- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    -- If we want to substitute [a -> ty1, b -> ty2] I used to
    -- think it was enough to generate an in-scope set that includes
    -- fv(ty1,ty2).  But that's not enough; we really should also take the
    -- free vars of the type we are substituting into!  Example:
    --      (forall b. (a,b,x)) [a -> List b]
    -- Then if we use the in-scope set {b}, there is a danger we will rename
    -- the forall'd variable to 'x' by mistake, getting this:
    --      (forall x. (List b, x, x)
    -- Urk!  This means looking at all the calls to mkOpenTvSubst....

    -- | Generates the in-scope set for the 'TvSubst' from the types in the incoming
    -- environment, hence "open"
    mkOpenTvSubst :: TvSubstEnv -> TvSubst
    mkOpenTvSubst tenv = TvSubst (mkInScopeSet (tyVarsOfTypes (varEnvElts tenv))) tenv

    -- | Generates the in-scope set for the 'TvSubst' from the types in the incoming
    -- environment, hence "open"
    zipOpenTvSubst :: [TyVar] -> [Type] -> TvSubst
    zipOpenTvSubst tyvars tys
      | debugIsOn && (length tyvars /= length tys)
      = pprTrace "zipOpenTvSubst" (ppr tyvars $$ ppr tys) emptyTvSubst
      | otherwise
      = TvSubst (mkInScopeSet (tyVarsOfTypes tys)) (zipTyEnv tyvars tys)

    -- | Called when doing top-level substitutions. Here we expect that the
    -- free vars of the range of the substitution will be empty.
    mkTopTvSubst :: [(TyVar, Type)] -> TvSubst
    mkTopTvSubst prs = TvSubst emptyInScopeSet (mkVarEnv prs)

    zipTopTvSubst :: [TyVar] -> [Type] -> TvSubst
    zipTopTvSubst tyvars tys
      | debugIsOn && (length tyvars /= length tys)
      = pprTrace "zipTopTvSubst" (ppr tyvars $$ ppr tys) emptyTvSubst
      | otherwise
      = TvSubst emptyInScopeSet (zipTyEnv tyvars tys)

    zipTyEnv :: [TyVar] -> [Type] -> TvSubstEnv
    zipTyEnv tyvars tys
      | debugIsOn && (length tyvars /= length tys)
      = pprTrace "zipTyEnv" (ppr tyvars $$ ppr tys) emptyVarEnv
      | otherwise
      = zip_ty_env tyvars tys emptyVarEnv

    -- Later substitutions in the list over-ride earlier ones,
    -- but there should be no loops
    zip_ty_env :: [TyVar] -> [Type] -> TvSubstEnv -> TvSubstEnv
    zip_ty_env []       []       env = env
    zip_ty_env (tv:tvs) (ty:tys) env = zip_ty_env tvs tys (extendVarEnv env tv ty)
            -- There used to be a special case for when
            --      ty == TyVarTy tv
            -- (a not-uncommon case) in which case the substitution was dropped.
            -- But the type-tidier changes the print-name of a type variable without
            -- changing the unique, and that led to a bug.   Why?  Pre-tidying, we had
            -- a type {Foo t}, where Foo is a one-method class.  So Foo is really a newtype.
            -- And it happened that t was the type variable of the class.  Post-tiding,
            -- it got turned into {Foo t2}.  The ext-core printer expanded this using
            -- sourceTypeRep, but that said "Oh, t == t2" because they have the same unique,
            -- and so generated a rep type mentioning t not t2.
            --
            -- Simplest fix is to nuke the "optimisation"
    zip_ty_env tvs      tys      env   = pprTrace "Var/Type length mismatch: " (ppr tvs $$ ppr tys) env
    -- zip_ty_env _ _ env = env

    instance Outputable TvSubst where
      ppr (TvSubst ins tenv)
        = brackets $ sep[ ptext (sLit "TvSubst"),
                          nest 2 (ptext (sLit "In scope:") <+> ppr ins),
                          nest 2 (ptext (sLit "Type env:") <+> ppr tenv) ]

    {-
    ************************************************************************
    *                                                                      *
                    Performing type or kind substitutions
    *                                                                      *
    ************************************************************************
    -}

    -- | Type substitution making use of an 'TvSubst' that
    -- is assumed to be open, see 'zipOpenTvSubst'
    substTyWith :: [TyVar] -> [Type] -> Type -> Type
    substTyWith tvs tys = ASSERT( length tvs == length tys )
                          substTy (zipOpenTvSubst tvs tys)

    substKiWith :: [KindVar] -> [Kind] -> Kind -> Kind
    substKiWith = substTyWith

    -- | Type substitution making use of an 'TvSubst' that
    -- is assumed to be open, see 'zipOpenTvSubst'
    substTysWith :: [TyVar] -> [Type] -> [Type] -> [Type]
    substTysWith tvs tys = ASSERT( length tvs == length tys )
                           substTys (zipOpenTvSubst tvs tys)

    substKisWith :: [KindVar] -> [Kind] -> [Kind] -> [Kind]
    substKisWith = substTysWith

    -- | Substitute within a 'Type'
    substTy :: TvSubst -> Type  -> Type
    substTy subst ty | isEmptyTvSubst subst = ty
                     | otherwise            = subst_ty subst ty

    -- | Substitute within several 'Type's
    substTys :: TvSubst -> [Type] -> [Type]
    substTys subst tys | isEmptyTvSubst subst = tys
                       | otherwise            = map (subst_ty subst) tys

    -- | Substitute within a 'ThetaType'
    substTheta :: TvSubst -> ThetaType -> ThetaType
    substTheta subst theta
      | isEmptyTvSubst subst = theta
      | otherwise            = map (substTy subst) theta

    -- | Remove any nested binders mentioning the 'TyVar's in the 'TyVarSet'
    deShadowTy :: TyVarSet -> Type -> Type
    deShadowTy tvs ty
      = subst_ty (mkTvSubst in_scope emptyTvSubstEnv) ty
      where
        in_scope = mkInScopeSet tvs

    subst_ty :: TvSubst -> Type -> Type
    -- subst_ty is the main workhorse for type substitution
    --
    -- Note that the in_scope set is poked only if we hit a forall
    -- so it may often never be fully computed
    subst_ty subst ty
       = go ty
      where
        go (LitTy n)         = n `seq` LitTy n
        go (TyVarTy tv)      = substTyVar subst tv
        go (TyConApp tc tys) = let args = map go tys
                               in  args `seqList` TyConApp tc args

        go (FunTy arg res)   = (FunTy $! (go arg)) $! (go res)
        go (AppTy fun arg)   = mkAppTy (go fun) $! (go arg)
                    -- The mkAppTy smart constructor is important
                    -- we might be replacing (a Int), represented with App
                    -- by [Int], represented with TyConApp
        go (ForAllTy tv ty)  = case substTyVarBndr subst tv of
                                  (subst', tv') ->
                                     ForAllTy tv' $! (subst_ty subst' ty)

    substTyVar :: TvSubst -> TyVar  -> Type
    substTyVar (TvSubst _ tenv) tv
      | Just ty  <- lookupVarEnv tenv tv      = ty  -- See Note [Apply Once]
      | otherwise = ASSERT( isTyVar tv ) TyVarTy tv --     in TypeRep
      -- We do not require that the tyvar is in scope
      -- Reason: we do quite a bit of (substTyWith [tv] [ty] tau)
      -- and it's a nuisance to bring all the free vars of tau into
      -- scope --- and then force that thunk at every tyvar
      -- Instead we have an ASSERT in substTyVarBndr to check for capture

    substTyVars :: TvSubst -> [TyVar] -> [Type]
    substTyVars subst tvs = map (substTyVar subst) tvs

    lookupTyVar :: TvSubst -> TyVar  -> Maybe Type
            -- See Note [Extending the TvSubst] in TypeRep
    lookupTyVar (TvSubst _ tenv) tv = lookupVarEnv tenv tv

    substTyVarBndr :: TvSubst -> TyVar -> (TvSubst, TyVar)
    substTyVarBndr subst@(TvSubst in_scope tenv) old_var
      = ASSERT2( _no_capture, ppr old_var $$ ppr subst )
        (TvSubst (in_scope `extendInScopeSet` new_var) new_env, new_var)
      where
        new_env | no_change = delVarEnv tenv old_var
                | otherwise = extendVarEnv tenv old_var (TyVarTy new_var)

        _no_capture = not (new_var `elemVarSet` tyVarsOfTypes (varEnvElts tenv))
        -- Assertion check that we are not capturing something in the substitution

        old_ki = tyVarKind old_var
        no_kind_change = isEmptyVarSet (tyVarsOfType old_ki) -- verify that kind is closed
        no_change = no_kind_change && (new_var == old_var)
            -- no_change means that the new_var is identical in
            -- all respects to the old_var (same unique, same kind)
            -- See Note [Extending the TvSubst] in TypeRep
            --
            -- In that case we don't need to extend the substitution
            -- to map old to new.  But instead we must zap any
            -- current substitution for the variable. For example:
            --      (\x.e) with id_subst = [x |-> e']
            -- Here we must simply zap the substitution for x

        new_var | no_kind_change = uniqAway in_scope old_var
                | otherwise = uniqAway in_scope $ updateTyVarKind (substTy subst) old_var
            -- The uniqAway part makes sure the new variable is not already in scope

    cloneTyVarBndr :: TvSubst -> TyVar -> Unique -> (TvSubst, TyVar)
    cloneTyVarBndr (TvSubst in_scope tv_env) tv uniq
      = (TvSubst (extendInScopeSet in_scope tv')
                 (extendVarEnv tv_env tv (mkTyVarTy tv')), tv')
      where
        tv' = setVarUnique tv uniq  -- Simply set the unique; the kind
                                    -- has no type variables to worry about

    {-
    ----------------------------------------------------
    -- Kind Stuff

    Kinds
    ~~~~~

    For the description of subkinding in GHC, see
      http://ghc.haskell.org/trac/ghc/wiki/Commentary/Compiler/TypeType#Kinds
    -}

    type MetaKindVar = TyVar  -- invariant: MetaKindVar will always be a
                              -- TcTyVar with details MetaTv (TauTv ...) ...
    -- meta kind var constructors and functions are in TcType

    type SimpleKind = Kind

    {-
    ************************************************************************
    *                                                                      *
            The kind of a type
    *                                                                      *
    ************************************************************************
    -}

    typeKind :: Type -> Kind
    typeKind orig_ty = go orig_ty
      where

        go ty@(TyConApp tc tys)
          | isPromotedTyCon tc
          = ASSERT( tyConArity tc == length tys ) superKind
          | otherwise
          = kindAppResult (ptext (sLit "typeKind 1") <+> ppr ty $$ ppr orig_ty)
                          (tyConKind tc) tys

        go ty@(AppTy fun arg)   = kindAppResult (ptext (sLit "typeKind 2") <+> ppr ty $$ ppr orig_ty)
                                                (go fun) [arg]
        go (LitTy l)            = typeLiteralKind l
        go (ForAllTy _ ty)      = go ty
        go (TyVarTy tyvar)      = tyVarKind tyvar
        go _ty@(FunTy _arg res)
            -- Hack alert.  The kind of (Int -> Int#) is liftedTypeKind (*),
            --              not unliftedTypeKind (#)
            -- The only things that can be after a function arrow are
            --   (a) types (of kind openTypeKind or its sub-kinds)
            --   (b) kinds (of super-kind TY) (e.g. * -> (* -> *))
            | isSuperKind k         = k
            | otherwise             = ASSERT2( isSubOpenTypeKind k, ppr _ty $$ ppr k ) liftedTypeKind
            where
              k = go res

    typeLiteralKind :: TyLit -> Kind
    typeLiteralKind l =
      case l of
        NumTyLit _ -> typeNatKind
        StrTyLit _ -> typeSymbolKind

    {-
    Kind inference
    ~~~~~~~~~~~~~~
    During kind inference, a kind variable unifies only with
    a "simple kind", sk
            sk ::= * | sk1 -> sk2
    For example
            data T a = MkT a (T Int#)
    fails.  We give T the kind (k -> *), and the kind variable k won't unify
    with # (the kind of Int#).

    Type inference
    ~~~~~~~~~~~~~~
    When creating a fresh internal type variable, we give it a kind to express
    constraints on it.  E.g. in (\x->e) we make up a fresh type variable for x,
    with kind ??.

    During unification we only bind an internal type variable to a type
    whose kind is lower in the sub-kind hierarchy than the kind of the tyvar.

    When unifying two internal type variables, we collect their kind constraints by
    finding the GLB of the two.  Since the partial order is a tree, they only
    have a glb if one is a sub-kind of the other.  In that case, we bind the
    less-informative one to the more informative one.  Neat, eh?
    -}