types/Type.hs

    -- (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, VisibilityFlag(..), KindOrType, PredType, ThetaType,
            Var, TyVar, isTyVar, TyCoVar, TyBinder,

            -- ** Constructing and deconstructing types
            mkTyVarTy, mkTyVarTys, getTyVar, getTyVar_maybe, repGetTyVar_maybe,
            getCastedTyVar_maybe, tyVarKind,

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

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

            mkTyConApp, mkTyConTy,
            tyConAppTyCon_maybe, tyConAppTyConPicky_maybe,
            tyConAppArgs_maybe, tyConAppTyCon, tyConAppArgs,
            splitTyConApp_maybe, splitTyConApp, tyConAppArgN, nextRole,
            splitListTyConApp_maybe,
            repSplitTyConApp_maybe,

            mkForAllTy, mkForAllTys, mkInvForAllTys, mkSpecForAllTys,
            mkVisForAllTys,
            mkNamedForAllTy,
            splitForAllTy_maybe, splitForAllTys, splitForAllTy,
            splitPiTy_maybe, splitPiTys, splitPiTy,
            splitNamedPiTys,
            mkPiType, mkPiTypes, mkPiTypesPreferFunTy,
            piResultTy, piResultTys,
            applyTys, applyTysD, applyTysX, dropForAlls,

            mkNumLitTy, isNumLitTy,
            mkStrLitTy, isStrLitTy,

            mkCastTy, mkCoercionTy, splitCastTy_maybe,

            userTypeError_maybe, pprUserTypeErrorTy,

            coAxNthLHS,
            stripCoercionTy, splitCoercionType_maybe,

            splitPiTysInvisible, filterOutInvisibleTypes,
            filterOutInvisibleTyVars, partitionInvisibles,
            synTyConResKind,
            tyConBinders,

            -- Analyzing types
            TyCoMapper(..), mapType, mapCoercion,

            -- (Newtypes)
            newTyConInstRhs,

            -- Pred types
            mkFamilyTyConApp,
            isDictLikeTy,
            mkPrimEqPred, mkReprPrimEqPred, mkPrimEqPredRole,
            equalityTyCon,
            mkHeteroPrimEqPred, mkHeteroReprPrimEqPred,
            mkClassPred,
            isClassPred, isEqPred, isNomEqPred,
            isIPPred, isIPPred_maybe, isIPTyCon, isIPClass,
            isCTupleClass,

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

            -- ** Binders
            sameVis,
            mkNamedBinder, mkAnonBinder, isNamedBinder, isAnonBinder,
            isIdLikeBinder, binderVisibility, binderVar_maybe,
            binderVar, binderRelevantType_maybe, caseBinder,
            partitionBinders, partitionBindersIntoBinders,
            binderType, isVisibleBinder, isInvisibleBinder,

            -- ** Common type constructors
            funTyCon,

            -- ** Predicates on types
            allDistinctTyVars,
            isTyVarTy, isFunTy, isDictTy, isPredTy, isVoidTy, isCoercionTy,
            isCoercionTy_maybe, isCoercionType, isForAllTy,
            isPiTy,

            -- (Lifting and boxity)
            isUnLiftedType, isUnboxedTupleType, isAlgType, isClosedAlgType,
            isPrimitiveType, isStrictType,
            isLevityTy, isLevityVar, getLevity, getLevityFromKind,

            -- * Main data types representing Kinds
            Kind,

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

            -- ** Common Kind
            liftedTypeKind, unliftedTypeKind,

            -- * Type free variables
            tyCoVarsOfType, tyCoVarsOfTypes, tyCoVarsOfTypeAcc,
            tyCoVarsOfTypeDSet,
            coVarsOfType,
            coVarsOfTypes, closeOverKinds,
            splitDepVarsOfType, splitDepVarsOfTypes,
            splitVisVarsOfType, splitVisVarsOfTypes,
            expandTypeSynonyms,
            typeSize,

            -- * Well-scoped lists of variables
            varSetElemsWellScoped, toposortTyVars, tyCoVarsOfTypeWellScoped,

            -- * Type comparison
            eqType, eqTypeX, eqTypes, cmpType, cmpTypes, cmpTypeX, cmpTypesX, cmpTc,
            eqVarBndrs,

            -- * Forcing evaluation of types
            seqType, seqTypes,

            -- * Other views onto Types
            coreView, coreViewOneStarKind,

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

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

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

            -- ** Manipulating type substitutions
            emptyTvSubstEnv, emptyTCvSubst, mkEmptyTCvSubst,

            mkTCvSubst, zipTvSubst, mkTvSubstPrs,
            notElemTCvSubst,
            getTvSubstEnv, setTvSubstEnv,
            zapTCvSubst, getTCvInScope,
            extendTCvInScope, extendTCvInScopeList, extendTCvInScopeSet,
            extendTCvSubst, extendTCvSubstList, extendTCvSubstAndInScope,
            isInScope, composeTCvSubstEnv, composeTCvSubst, zipTyEnv, zipCoEnv,
            isEmptyTCvSubst, unionTCvSubst,

            -- ** Performing substitution on types and kinds
            substTy, substTys, substTyWith, substTysWith, substTheta,
            substTyAddInScope, substTyUnchecked,
            substTyVarBndr, substTyVar, substTyVars,
            cloneTyVarBndr, cloneTyVarBndrs, lookupTyVar, substTelescope,

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

            -- * Tidying type related things up for printing
            tidyType,      tidyTypes,
            tidyOpenType,  tidyOpenTypes,
            tidyOpenKind,
            tidyTyCoVarBndr, tidyTyCoVarBndrs, tidyFreeTyCoVars,
            tidyOpenTyCoVar, tidyOpenTyCoVars,
            tidyTyVarOcc,
            tidyTopType,
            tidyKind
        ) where

    #include "HsVersions.h"

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

    import Kind
    import TyCoRep

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

    import Class
    import TyCon
    import TysPrim
    import {-# SOURCE #-} TysWiredIn ( listTyCon, typeNatKind
                                     , typeSymbolKind, liftedTypeKind )
    import PrelNames
    import CoAxiom
    import {-# SOURCE #-} Coercion

    -- others
    import BasicTypes       ( Arity, RepArity )
    import Util
    import Outputable
    import FastString
    import Pair
    import ListSetOps
    import Digraph

    import Maybes           ( orElse )
    import Data.Maybe       ( isJust, mapMaybe )
    import Control.Monad    ( guard )
    import Control.Arrow    ( first, second )

    #if __GLASGOW_HASKELL__ < 709
    import Control.Applicative ( Applicative, (<*>), (<$>), pure )
    import Data.Monoid         ( Monoid(..) )
    import Data.Foldable       ( foldMap )
    #endif

    -- $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
    -- ^ This function Strips off the /top layer only/ of a type synonym
    -- application (if any) 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') <- expandSynTyCon_maybe tc tys
      = Just (mkAppTys (substTy (mkTvSubstPrs tenv) rhs) tys')
                   -- The free vars of 'rhs' should all be bound by 'tenv', so it's
                   -- ok to use 'substTy' here.
                   -- See also Note [The substitution invariant] in TyCoRep.
                   -- 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

    -- | Like 'coreView', but it also "expands" @Constraint@ to become
    -- @TYPE Lifted@.
    coreViewOneStarKind :: Type -> Maybe Type
    coreViewOneStarKind = go Nothing
      where
        go _ t | Just t' <- coreView t                    = go (Just t') t'
        go _ (TyConApp tc []) | isStarKindSynonymTyCon tc = go (Just t') t'
          where t' = liftedTypeKind
        go res _ = res

    -----------------------------------------------
    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' only expands out type synonyms mentioned in the type,
    -- not in the kinds of any TyCon or TyVar mentioned in the type.
    expandTypeSynonyms ty
      = go (mkEmptyTCvSubst (mkTyCoInScopeSet [ty] [])) ty
      where
        go subst (TyConApp tc tys)
          | Just (tenv, rhs, tys') <- expandSynTyCon_maybe tc tys
          = let subst' = unionTCvSubst subst (mkTvSubstPrs tenv) in
            go subst' (mkAppTys rhs tys')
          | otherwise
          = TyConApp tc (map (go subst) tys)
        go _     (LitTy l)     = LitTy l
        go subst (TyVarTy tv)  = substTyVar subst tv
        go subst (AppTy t1 t2) = mkAppTy (go subst t1) (go subst t2)
        go subst (ForAllTy (Anon arg) res)
          = mkFunTy (go subst arg) (go subst res)
        go subst (ForAllTy (Named tv vis) t)
          = let (subst', tv') = substTyVarBndrCallback go subst tv in
            ForAllTy (Named tv' vis) (go subst' t)
        go subst (CastTy ty co)  = mkCastTy (go subst ty) (go_co subst co)
        go subst (CoercionTy co) = mkCoercionTy (go_co subst co)

        go_co subst (Refl r ty)
          = mkReflCo r (go subst ty)
           -- NB: coercions are always expanded upon creation
        go_co subst (TyConAppCo r tc args)
          = mkTyConAppCo r tc (map (go_co subst) args)
        go_co subst (AppCo co arg)
          = mkAppCo (go_co subst co) (go_co subst arg)
        go_co subst (ForAllCo tv kind_co co)
          = let (subst', tv', kind_co') = go_cobndr subst tv kind_co in
            mkForAllCo tv' kind_co' (go_co subst' co)
        go_co subst (CoVarCo cv)
          = substCoVar subst cv
        go_co subst (AxiomInstCo ax ind args)
          = mkAxiomInstCo ax ind (map (go_co subst) args)
        go_co subst (UnivCo p r t1 t2)
          = mkUnivCo (go_prov subst p) r (go subst t1) (go subst t2)
        go_co subst (SymCo co)
          = mkSymCo (go_co subst co)
        go_co subst (TransCo co1 co2)
          = mkTransCo (go_co subst co1) (go_co subst co2)
        go_co subst (NthCo n co)
          = mkNthCo n (go_co subst co)
        go_co subst (LRCo lr co)
          = mkLRCo lr (go_co subst co)
        go_co subst (InstCo co arg)
          = mkInstCo (go_co subst co) (go_co subst arg)
        go_co subst (CoherenceCo co1 co2)
          = mkCoherenceCo (go_co subst co1) (go_co subst co2)
        go_co subst (KindCo co)
          = mkKindCo (go_co subst co)
        go_co subst (SubCo co)
          = mkSubCo (go_co subst co)
        go_co subst (AxiomRuleCo ax cs) = AxiomRuleCo ax (map (go_co subst) cs)

        go_prov _     UnsafeCoerceProv    = UnsafeCoerceProv
        go_prov subst (PhantomProv co)    = PhantomProv (go_co subst co)
        go_prov subst (ProofIrrelProv co) = ProofIrrelProv (go_co subst co)
        go_prov _     p@(PluginProv _)    = p
        go_prov _     (HoleProv h)        = pprPanic "expandTypeSynonyms hit a hole" (ppr h)

          -- the "False" and "const" are to accommodate the type of
          -- substForAllCoBndrCallback, which is general enough to
          -- handle coercion optimization (which sometimes swaps the
          -- order of a coercion)
        go_cobndr subst = substForAllCoBndrCallback False (go_co subst) subst

    {-
    ************************************************************************
    *                                                                      *
       Analyzing types
    *                                                                      *
    ************************************************************************

    These functions do a map-like operation over types, performing some operation
    on all variables and binding sites. Primarily used for zonking.

    Note [Efficiency for mapCoercion ForAllCo case]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    As noted in Note [Forall coercions] in TyCoRep, a ForAllCo is a bit redundant.
    It stores a TyVar and a Coercion, where the kind of the TyVar always matches
    the left-hand kind of the coercion. This is convenient lots of the time, but
    not when mapping a function over a coercion.

    The problem is that tcm_tybinder will affect the TyVar's kind and
    mapCoercion will affect the Coercion, and we hope that the results will be
    the same. Even if they are the same (which should generally happen with
    correct algorithms), then there is an efficiency issue. In particular,
    this problem seems to make what should be a linear algorithm into a potentially
    exponential one. But it's only going to be bad in the case where there's
    lots of foralls in the kinds of other foralls. Like this:

      forall a : (forall b : (forall c : ...). ...). ...

    This construction seems unlikely. So we'll do the inefficient, easy way
    for now.

    Note [Specialising mappers]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~
    These INLINABLE pragmas are indispensable. mapType/mapCoercion are used
    to implement zonking, and it's vital that they get specialised to the TcM
    monad. This specialisation happens automatically (that is, without a
    SPECIALISE pragma) as long as the definitions are INLINABLE. For example,
    this one change made a 20% allocation difference in perf/compiler/T5030.

    -}

    -- | This describes how a "map" operation over a type/coercion should behave
    data TyCoMapper env m
      = TyCoMapper
          { tcm_smart :: Bool -- ^ Should the new type be created with smart
                             -- constructors?
          , tcm_tyvar :: env -> TyVar -> m Type
          , tcm_covar :: env -> CoVar -> m Coercion
          , tcm_hole  :: env -> CoercionHole -> Role
                      -> Type -> Type -> m Coercion
              -- ^ What to do with coercion holes. See Note [Coercion holes] in
              -- TyCoRep.

          , tcm_tybinder :: env -> TyVar -> VisibilityFlag -> m (env, TyVar)
              -- ^ The returned env is used in the extended scope
          }

    {-# INLINABLE mapType #-}  -- See Note [Specialising mappers]
    mapType :: (Applicative m, Monad m) => TyCoMapper env m -> env -> Type -> m Type
    mapType mapper@(TyCoMapper { tcm_smart = smart, tcm_tyvar = tyvar
                               , tcm_tybinder = tybinder })
            env ty
      = go ty
      where
        go (TyVarTy tv) = tyvar env tv
        go (AppTy t1 t2) = mkappty <$> go t1 <*> go t2
        go (TyConApp tc tys) = mktyconapp tc <$> mapM go tys
        go (ForAllTy (Anon arg) res) = mkfunty <$> go arg <*> go res
        go (ForAllTy (Named tv vis) inner)
          = do { (env', tv') <- tybinder env tv vis
               ; inner' <- mapType mapper env' inner
               ; return $ ForAllTy (Named tv' vis) inner' }
        go ty@(LitTy {}) = return ty
        go (CastTy ty co) = mkcastty <$> go ty <*> mapCoercion mapper env co
        go (CoercionTy co) = CoercionTy <$> mapCoercion mapper env co

        (mktyconapp, mkappty, mkcastty, mkfunty)
          | smart     = (mkTyConApp, mkAppTy, mkCastTy, mkFunTy)
          | otherwise = (TyConApp,   AppTy,   CastTy,   ForAllTy . Anon)

    {-# INLINABLE mapCoercion #-}  -- See Note [Specialising mappers]
    mapCoercion :: (Applicative m, Monad m)
                => TyCoMapper env m -> env -> Coercion -> m Coercion
    mapCoercion mapper@(TyCoMapper { tcm_smart = smart, tcm_covar = covar
                                   , tcm_hole = cohole, tcm_tybinder = tybinder })
                env co
      = go co
      where
        go (Refl r ty) = Refl r <$> mapType mapper env ty
        go (TyConAppCo r tc args)
          = mktyconappco r tc <$> mapM go args
        go (AppCo c1 c2) = mkappco <$> go c1 <*> go c2
        go (ForAllCo tv kind_co co)
          = do { kind_co' <- go kind_co
               ; (env', tv') <- tybinder env tv Invisible
               ; co' <- mapCoercion mapper env' co
               ; return $ mkforallco tv' kind_co' co' }
            -- See Note [Efficiency for mapCoercion ForAllCo case]
        go (CoVarCo cv) = covar env cv
        go (AxiomInstCo ax i args)
          = mkaxiominstco ax i <$> mapM go args
        go (UnivCo (HoleProv hole) r t1 t2)
          = cohole env hole r t1 t2
        go (UnivCo p r t1 t2)
          = mkunivco <$> go_prov p <*> pure r
                     <*> mapType mapper env t1 <*> mapType mapper env t2
        go (SymCo co) = mksymco <$> go co
        go (TransCo c1 c2) = mktransco <$> go c1 <*> go c2
        go (AxiomRuleCo r cos) = AxiomRuleCo r <$> mapM go cos
        go (NthCo i co)        = mknthco i <$> go co
        go (LRCo lr co)        = mklrco lr <$> go co
        go (InstCo co arg)     = mkinstco <$> go co <*> go arg
        go (CoherenceCo c1 c2) = mkcoherenceco <$> go c1 <*> go c2
        go (KindCo co)         = mkkindco <$> go co
        go (SubCo co)          = mksubco <$> go co

        go_prov UnsafeCoerceProv    = return UnsafeCoerceProv
        go_prov (PhantomProv co)    = PhantomProv <$> go co
        go_prov (ProofIrrelProv co) = ProofIrrelProv <$> go co
        go_prov p@(PluginProv _)    = return p
        go_prov (HoleProv _)        = panic "mapCoercion"

        ( mktyconappco, mkappco, mkaxiominstco, mkunivco
          , mksymco, mktransco, mknthco, mklrco, mkinstco, mkcoherenceco
          , mkkindco, mksubco, mkforallco)
          | smart
          = ( mkTyConAppCo, mkAppCo, mkAxiomInstCo, mkUnivCo
            , mkSymCo, mkTransCo, mkNthCo, mkLRCo, mkInstCo, mkCoherenceCo
            , mkKindCo, mkSubCo, mkForAllCo )
          | otherwise
          = ( TyConAppCo, AppCo, AxiomInstCo, UnivCo
            , SymCo, TransCo, NthCo, LRCo, InstCo, CoherenceCo
            , KindCo, SubCo, ForAllCo )

    {-
    ************************************************************************
    *                                                                      *
    \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'
                      | otherwise               = repGetTyVar_maybe ty

    -- | If the type is a tyvar, possibly under a cast, returns it, along
    -- with the coercion. Thus, the co is :: kind tv ~R kind type
    getCastedTyVar_maybe :: Type -> Maybe (TyVar, Coercion)
    getCastedTyVar_maybe ty | Just ty' <- coreView ty = getCastedTyVar_maybe ty'
    getCastedTyVar_maybe (CastTy (TyVarTy tv) co)     = Just (tv, co)
    getCastedTyVar_maybe (TyVarTy tv)
      = Just (tv, mkReflCo Nominal (tyVarKind tv))
    getCastedTyVar_maybe _                            = Nothing

    -- | Attempts to obtain the type variable underlying a 'Type', without
    -- any expansion
    repGetTyVar_maybe :: Type -> Maybe TyVar
    repGetTyVar_maybe (TyVarTy tv) = Just tv
    repGetTyVar_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 (ForAllTy (Anon ty1) ty2)
                                          = Just (TyConApp funTyCon [ty1], ty2)
    repSplitAppTy_maybe (AppTy ty1 ty2)   = Just (ty1, ty2)
    repSplitAppTy_maybe (TyConApp tc tys)
      | mightBeUnsaturatedTyCon 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

    -- this one doesn't braek apart (c => t).
    -- See Note [Decomposing fat arrow c=>t]
    -- Defined here to avoid module loops between Unify and TcType.
    tcRepSplitAppTy_maybe :: Type -> Maybe (Type,Type)
    -- ^ Does the AppTy split as in 'tcSplitAppTy_maybe', but assumes that
    -- any coreView stuff is already done. Refuses to look through (c => t)
    tcRepSplitAppTy_maybe (ForAllTy (Anon ty1) ty2)
      | isConstraintKind (typeKind ty1)     = Nothing  -- See Note [Decomposing fat arrow c=>t]
      | otherwise                           = Just (TyConApp funTyCon [ty1], ty2)
    tcRepSplitAppTy_maybe (AppTy ty1 ty2)   = Just (ty1, ty2)
    tcRepSplitAppTy_maybe (TyConApp tc tys)
      | mightBeUnsaturatedTyCon tc || tys `lengthExceeds` tyConArity tc
      , Just (tys', ty') <- snocView tys
      = Just (TyConApp tc tys', ty')    -- Never create unsaturated type family apps!
    tcRepSplitAppTy_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 | mightBeUnsaturatedTyCon tc = 0
                  | otherwise                  = tyConArity tc
                (tc_args1, tc_args2) = splitAt n tc_args
            in
            (TyConApp tc tc_args1, tc_args2 ++ args)
        split _   (ForAllTy (Anon 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 <- coreView 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 <- coreView ty = isStrLitTy ty1
    isStrLitTy (LitTy (StrTyLit s)) = Just s
    isStrLitTy _                    = Nothing

    -- | Is this type a custom user error?
    -- If so, give us the kind and the error message.
    userTypeError_maybe :: Type -> Maybe Type
    userTypeError_maybe t
      = do { (tc, _kind : msg : _) <- splitTyConApp_maybe t
              -- There may be more than 2 arguments, if the type error is
              -- used as a type constructor (e.g. at kind `Type -> Type`).

           ; guard (tyConName tc == errorMessageTypeErrorFamName)
           ; return msg }

    -- | Render a type corresponding to a user type error into a SDoc.
    pprUserTypeErrorTy :: Type -> SDoc
    pprUserTypeErrorTy ty =
      case splitTyConApp_maybe ty of

        -- Text "Something"
        Just (tc,[txt])
          | tyConName tc == typeErrorTextDataConName
          , Just str <- isStrLitTy txt -> ftext str

        -- ShowType t
        Just (tc,[_k,t])
          | tyConName tc == typeErrorShowTypeDataConName -> ppr t

        -- t1 :<>: t2
        Just (tc,[t1,t2])
          | tyConName tc == typeErrorAppendDataConName ->
            pprUserTypeErrorTy t1 <> pprUserTypeErrorTy t2

        -- t1 :$$: t2
        Just (tc,[t1,t2])
          | tyConName tc == typeErrorVAppendDataConName ->
            pprUserTypeErrorTy t1 $$ pprUserTypeErrorTy t2

        -- An uneavaluated type function
        _ -> ppr ty

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

    Function types are represented with (ForAllTy (Anon ...) ...)
    -}

    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 (ForAllTy (Anon 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 (ForAllTy (Anon 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 _       (ForAllTy (Anon 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) }}

    funResultTy :: Type -> Type
    -- ^ Extract the function result type and panic if that is not possible
    funResultTy ty = piResultTy ty (pprPanic "funResultTy" (ppr ty))

    -- | Essentially 'funResultTy' on kinds handling pi-types too
    piResultTy :: Type -> Type -> Type
    piResultTy ty arg | Just ty' <- coreView ty = piResultTy ty' arg
    piResultTy (ForAllTy (Anon _) res)     _   = res
    piResultTy (ForAllTy (Named tv _) res) arg = substTyWith [tv] [arg] res
    piResultTy ty arg                          = pprPanic "piResultTy"
                                                     (ppr ty $$ ppr arg)

    -- | Fold 'piResultTy' over many types
    piResultTys :: Type -> [Type] -> Type
    piResultTys = foldl piResultTy

    funArgTy :: Type -> Type
    -- ^ Extract the function argument type and panic if that is not possible
    funArgTy ty | Just ty' <- coreView ty = funArgTy ty'
    funArgTy (ForAllTy (Anon 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
      = ForAllTy (Anon 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 ..

    -- | Retrieve the tycon heading this type, if there is one. Does /not/
    -- look through synonyms.
    tyConAppTyConPicky_maybe :: Type -> Maybe TyCon
    tyConAppTyConPicky_maybe (TyConApp tc _)       = Just tc
    tyConAppTyConPicky_maybe (ForAllTy (Anon _) _) = Just funTyCon
    tyConAppTyConPicky_maybe _                     = Nothing

    -- | 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 (ForAllTy (Anon _) _) = 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 (ForAllTy (Anon 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 `getNth` 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 ty                           = repSplitTyConApp_maybe ty

    -- | Like 'splitTyConApp_maybe', but doesn't look through synonyms. This
    -- assumes the synonyms have already been dealt with.
    repSplitTyConApp_maybe :: Type -> Maybe (TyCon, [Type])
    repSplitTyConApp_maybe (TyConApp tc tys)         = Just (tc, tys)
    repSplitTyConApp_maybe (ForAllTy (Anon arg) res) = Just (funTyCon, [arg,res])
    repSplitTyConApp_maybe _                         = Nothing

    -- | Attempts to tease a list type apart and gives the type of the elements if
    -- successful (looks through type synonyms)
    splitListTyConApp_maybe :: Type -> Maybe Type
    splitListTyConApp_maybe ty = case splitTyConApp_maybe ty of
      Just (tc,[e]) | tc == listTyCon -> Just e
      _other                          -> 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

    {-
    ---------------------------------------------------------------------
                               CastTy
                               ~~~~~~
    A casted type has its *kind* casted into something new.

    Note [Weird typing rule for ForAllTy]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Here is the (truncated) typing rule for the dependent ForAllTy:

    inner : kind
    ------------------------------------
    ForAllTy (Named tv vis) inner : kind

    Note that neither the inner type nor for ForAllTy itself have to have
    kind *! But, it means that we should push any kind casts through the
    ForAllTy. The only trouble is avoiding capture.

    -}

    splitCastTy_maybe :: Type -> Maybe (Type, Coercion)
    splitCastTy_maybe ty | Just ty' <- coreView ty = splitCastTy_maybe ty'
    splitCastTy_maybe (CastTy ty co)               = Just (ty, co)
    splitCastTy_maybe _                            = Nothing

    -- | Make a 'CastTy'. The Coercion must be nominal. This function looks
    -- at the entire structure of the type and coercion in an attempt to
    -- maintain representation invariance (that is, any two types that are `eqType`
    -- look the same). Be very wary of calling this in a loop.
    mkCastTy :: Type -> Coercion -> Type
    -- Running example:
    --   T :: forall k1. k1 -> forall k2. k2 -> Bool -> Maybe k1 -> *
    --   co :: * ~R X    (maybe X is a newtype around *)
    --   ty = T Nat 3 Symbol "foo" True (Just 2)
    --
    -- We wish to "push" the cast down as far as possible. See also
    -- Note [Pushing down casts] in TyCoRep. Here is where we end
    -- up:
    --
    --   (T Nat 3 Symbol |> <Symbol> -> <Bool> -> <Maybe Nat> -> co)
    --      "foo" True (Just 2)
    --
    mkCastTy ty co | isReflexiveCo co = ty
    -- NB: Do the slow check here. This is important to keep the splitXXX
    -- functions working properly. Otherwise, we may end up with something
    -- like (((->) |> something_reflexive_but_not_obviously_so) biz baz)
    -- fails under splitFunTy_maybe. This happened with the cheaper check
    -- in test dependent/should_compile/dynamic-paper.

    mkCastTy (CastTy ty co1) co2 = mkCastTy ty (co1 `mkTransCo` co2)
    -- See Note [Weird typing rule for ForAllTy]
    mkCastTy (ForAllTy (Named tv vis) inner_ty) co
      = -- have to make sure that pushing the co in doesn't capture the bound var
        let fvs = tyCoVarsOfCo co
            empty_subst = mkEmptyTCvSubst (mkInScopeSet fvs)
            (subst, tv') = substTyVarBndr empty_subst tv
        in
        ForAllTy (Named tv' vis) (substTy subst inner_ty `mkCastTy` co)
    mkCastTy ty co = -- NB: don't check if the coercion "from" type matches here;
                     -- there may be unzonked variables about
                     let result = split_apps [] ty co in
                     ASSERT2( CastTy ty co `eqType` result
                            , ppr ty <+> dcolon <+> ppr (typeKind ty) $$
                              ppr co <+> dcolon <+> ppr (coercionKind co) $$
                              ppr result <+> dcolon <+> ppr (typeKind result) )
                     result
      where
        -- split_apps breaks apart any type applications, so we can see how far down
        -- to push the cast
        split_apps args (AppTy t1 t2) co
          = split_apps (t2:args) t1 co
        split_apps args (TyConApp tc tc_args) co
          | mightBeUnsaturatedTyCon tc
          = affix_co (tyConKind tc) (mkTyConTy tc) (tc_args `chkAppend` args) co
          | otherwise -- not decomposable... but it may still be oversaturated
          = let (non_decomp_args, decomp_args) = splitAt (tyConArity tc) tc_args
                saturated_tc = mkTyConApp tc non_decomp_args
            in
            affix_co (typeKind saturated_tc) saturated_tc (decomp_args `chkAppend` args) co

        split_apps args (ForAllTy (Anon arg) res) co
          = affix_co (tyConKind funTyCon) (mkTyConTy funTyCon)
                     (arg : res : args) co
        split_apps args ty co
          = affix_co (typeKind ty) ty args co

        -- having broken everything apart, this figures out the point at which there
        -- are no more dependent quantifications, and puts the cast there
        affix_co _ ty [] co = no_double_casts ty co
        affix_co kind ty args co
          -- if kind contains any dependent quantifications, we can't push.
          -- apply arguments until it doesn't
          = let (bndrs, _inner_ki) = splitPiTys kind
                (no_dep_bndrs, some_dep_bndrs) = spanEnd isAnonBinder bndrs
                (some_dep_args, rest_args) = splitAtList some_dep_bndrs args
                dep_subst = zipTyBinderSubst some_dep_bndrs some_dep_args
                used_no_dep_bndrs = takeList rest_args no_dep_bndrs
                rest_arg_tys = substTys dep_subst (map binderType used_no_dep_bndrs)
                co' = mkFunCos Nominal
                               (map (mkReflCo Nominal) rest_arg_tys)
                               co
            in
            ((ty `mkAppTys` some_dep_args) `no_double_casts` co') `mkAppTys` rest_args

        no_double_casts (CastTy ty co1) co2 = CastTy ty (co1 `mkTransCo` co2)
        no_double_casts ty              co  = CastTy ty co

    {-
    --------------------------------------------------------------------
                                CoercionTy
                                ~~~~~~~~~~
    CoercionTy allows us to inject coercions into types. A CoercionTy
    should appear only in the right-hand side of an application.
    -}

    mkCoercionTy :: Coercion -> Type
    mkCoercionTy = CoercionTy

    isCoercionTy :: Type -> Bool
    isCoercionTy (CoercionTy _) = True
    isCoercionTy _              = False

    isCoercionTy_maybe :: Type -> Maybe Coercion
    isCoercionTy_maybe (CoercionTy co) = Just co
    isCoercionTy_maybe _               = Nothing

    stripCoercionTy :: Type -> Coercion
    stripCoercionTy (CoercionTy co) = co
    stripCoercionTy ty              = pprPanic "stripCoercionTy" (ppr ty)

    {-
    ---------------------------------------------------------------------
                                    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.hs.

                    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

    instance Outputable RepType where
      ppr (UbxTupleRep tys) = text "UbxTupleRep" <+> ppr tys
      ppr (UnaryRep ty)     = text "UnaryRep"    <+> ppr ty

    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
    --      5. Casts
    --
    -- 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 (Named {}) ty2)  -- Drop type foralls
          = go rec_nts ty2

        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) non_levity_tys)
          where
              -- See Note [Unboxed tuple levity vars] in TyCon
            non_levity_tys = drop (length tys `div` 2) tys

        go rec_nts (CastTy ty _)
          = go rec_nts ty

        go _ ty@(CoercionTy _)
          = pprPanic "repType" (ppr ty)

        go _ ty = UnaryRep ty

    -- 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 -> go rep
        where go (TyConApp tc _)   = tyConPrimRep tc
              go (ForAllTy _ _)    = PtrRep
              go (AppTy _ _)       = PtrRep      -- See Note [AppTy rep]
              go (TyVarTy _)       = PtrRep
              go (CastTy ty _)     = go ty
              go _                 = pprPanic "typePrimRep: UnaryRep" (ppr ty)

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

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

    mkForAllTy :: TyBinder -> Type -> Type
    mkForAllTy = ForAllTy

    -- | Make a dependent forall.
    mkNamedForAllTy :: TyVar -> VisibilityFlag -> Type -> Type
    mkNamedForAllTy tv vis = ASSERT( isTyVar tv )
                             ForAllTy (Named tv vis)

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

    -- | Like mkForAllTys, but assumes all variables are dependent and invisible,
    -- a common case
    mkInvForAllTys :: [TyVar] -> Type -> Type
    mkInvForAllTys tvs = ASSERT( all isTyVar tvs )
                         mkForAllTys (map (flip Named Invisible) tvs)

    -- | Like mkForAllTys, but assumes all variables are dependent and specified,
    -- a common case
    mkSpecForAllTys :: [TyVar] -> Type -> Type
    mkSpecForAllTys tvs = ASSERT( all isTyVar tvs )
                          mkForAllTys (map (flip Named Specified) tvs)

    -- | Like mkForAllTys, but assumes all variables are dependent and visible
    mkVisForAllTys :: [TyVar] -> Type -> Type
    mkVisForAllTys tvs = ASSERT( all isTyVar tvs )
                         mkForAllTys (map (flip Named Visible) tvs)

    mkPiType  :: Var -> Type -> Type
    -- ^ Makes a @(->)@ type or an implicit forall type, depending
    -- on whether it is given a type variable or a term variable.
    -- This is used, for example, when producing the type of a lambda.
    -- Always uses Invisible binders.
    mkPiTypes :: [Var] -> Type -> Type
    -- ^ 'mkPiType' for multiple type or value arguments

    mkPiType v ty
       | isTyVar v = mkForAllTy (Named v Invisible) ty
       | otherwise = mkForAllTy (Anon (varType v)) ty

    mkPiTypes vs ty = foldr mkPiType ty vs

    -- | Given a list of type-level vars, makes ForAllTys, preferring
    -- anonymous binders if the variable is, in fact, not dependent.
    -- All binders are /visible/.
    mkPiTypesPreferFunTy :: [TyVar] -> Type -> Type
    mkPiTypesPreferFunTy vars inner_ty = fst $ go vars inner_ty
      where
        go :: [TyVar] -> Type -> (Type, VarSet) -- also returns the free vars
        go [] ty = (ty, tyCoVarsOfType ty)
        go (v:vs) ty
          = if v `elemVarSet` fvs
            then ( mkForAllTy (Named v Visible) qty
                 , fvs `delVarSet` v `unionVarSet` kind_vars )
            else ( mkForAllTy (Anon (tyVarKind v)) qty
                 , fvs `unionVarSet` kind_vars )
          where
            (qty, fvs) = go vs ty
            kind_vars  = tyCoVarsOfType $ tyVarKind v

    -- | Take a ForAllTy apart, returning the list of tyvars and the result type.
    -- This always succeeds, even if it returns only an empty list. Note that the
    -- result type returned may have free variables that were bound by a forall.
    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 (Named tv _) ty) tvs = split ty ty (tv:tvs)
        split orig_ty _                          tvs = (reverse tvs, orig_ty)

    -- | Split off all TyBinders to a type, splitting both proper foralls
    -- and functions
    splitPiTys :: Type -> ([TyBinder], Type)
    splitPiTys ty = split ty ty []
      where
        split orig_ty ty bs | Just ty' <- coreView ty = split orig_ty ty' bs
        split _       (ForAllTy b res) bs  = split res res (b:bs)
        split orig_ty _                bs  = (reverse bs, orig_ty)

    -- | Like 'splitPiTys' but split off only /named/ binders.
    splitNamedPiTys :: Type -> ([TyBinder], Type)
    splitNamedPiTys ty = split ty ty []
      where
        split orig_ty ty bs | Just ty' <- coreView ty = split orig_ty ty' bs
        split _       (ForAllTy b@(Named {}) res) bs  = split res res (b:bs)
        split orig_ty _                           bs  = (reverse bs, orig_ty)

    -- | Checks whether this is a proper forall (with a named binder)
    isForAllTy :: Type -> Bool
    isForAllTy (ForAllTy (Named {}) _) = True
    isForAllTy _                       = False

    -- | Is this a function or forall?
    isPiTy :: Type -> Bool
    isPiTy (ForAllTy {}) = True
    isPiTy _             = False

    -- | Take a forall type apart, or panics if that is not possible.
    splitForAllTy :: Type -> (TyVar, Type)
    splitForAllTy ty
      | Just answer <- splitForAllTy_maybe ty = answer
      | otherwise                             = pprPanic "splitForAllTy" (ppr ty)

    -- | Attempts to take a forall type apart, but only if it's a proper forall,
    -- with a named binder
    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 (Named tv _) ty) = Just (tv, ty)
        splitFAT_m _                          = Nothing

    -- | Attempts to take a forall type apart; works with proper foralls and
    -- functions
    splitPiTy_maybe :: Type -> Maybe (TyBinder, Type)
    splitPiTy_maybe ty = go ty
      where
        go ty | Just ty' <- coreView ty = go ty'
        go (ForAllTy bndr ty) = Just (bndr, ty)
        go _                  = Nothing

    -- | Takes a forall type apart, or panics
    splitPiTy :: Type -> (TyBinder, Type)
    splitPiTy ty
      | Just answer <- splitPiTy_maybe ty = answer
      | otherwise                         = pprPanic "splitPiTy" (ppr ty)

    -- | Drops all non-anonymous ForAllTys
    dropForAlls :: Type -> Type
    dropForAlls ty | Just ty' <- coreView ty = dropForAlls ty'
                   | otherwise = go ty
      where
        go (ForAllTy (Named {}) res) = go res
        go res                       = res

    -- | Given a tycon and its arguments, filters out any invisible arguments
    filterOutInvisibleTypes :: TyCon -> [Type] -> [Type]
    filterOutInvisibleTypes tc tys = snd $ partitionInvisibles tc id tys

    -- | Like 'filterOutInvisibles', but works on 'TyVar's
    filterOutInvisibleTyVars :: TyCon -> [TyVar] -> [TyVar]
    filterOutInvisibleTyVars tc tvs = snd $ partitionInvisibles tc mkTyVarTy tvs

    -- | Given a tycon and a list of things (which correspond to arguments),
    -- partitions the things into the invisible ones and the visible ones.
    -- The callback function is necessary for this scenario:
    --
    -- > T :: forall k. k -> k
    -- > partitionInvisibles T [forall m. m -> m -> m, S, R, Q]
    --
    -- After substituting, we get
    --
    -- > T (forall m. m -> m -> m) :: (forall m. m -> m -> m) -> forall n. n -> n -> n
    --
    -- Thus, the first argument is invisible, @S@ is visible, @R@ is invisible again,
    -- and @Q@ is visible.
    --
    -- If you're absolutely sure that your tycon's kind doesn't end in a variable,
    -- it's OK if the callback function panics, as that's the only time it's
    -- consulted.
    partitionInvisibles :: TyCon -> (a -> Type) -> [a] -> ([a], [a])
    partitionInvisibles tc get_ty = go emptyTCvSubst (tyConKind tc)
      where
        go _ _ [] = ([], [])
        go subst (ForAllTy bndr res_ki) (x:xs)
          | isVisibleBinder bndr = second (x :) (go subst' res_ki xs)
          | otherwise            = first  (x :) (go subst' res_ki xs)
          where
            subst' = extendTCvSubstBinder subst bndr (get_ty x)
        go subst (TyVarTy tv) xs
          | Just ki <- lookupTyVar subst tv = go subst ki xs
        go _ _ xs = ([], xs)  -- something is ill-kinded. But this can happen
                              -- when printing errors. Assume everything is visible.

    -- like splitPiTys, but returns only *invisible* binders, including constraints
    splitPiTysInvisible :: Type -> ([TyBinder], Type)
    splitPiTysInvisible ty = split ty ty []
       where
         split orig_ty ty bndrs
           | Just ty' <- coreView ty = split orig_ty ty' bndrs
         split _       (ForAllTy bndr ty) bndrs
           |  isInvisibleBinder bndr
           = split ty ty (bndr:bndrs)

         split orig_ty _ bndrs
           = (reverse bndrs, orig_ty)

    tyConBinders :: TyCon -> [TyBinder]
    tyConBinders = fst . splitPiTys . tyConKind

    {-
    applyTys
    ~~~~~~~~~~~~~~~~~
    -}

    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.hs
    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_bndrs == n_args     -- The vastly common case
      = substTyWithBinders bndrs arg_tys rho_ty
      | n_bndrs > n_args      -- Too many for-alls
      = substTyWithBinders (take n_args bndrs) arg_tys
                           (mkForAllTys (drop n_args bndrs) rho_ty)
      | otherwise           -- Too many type args
      = ASSERT2( n_bndrs > 0, doc $$ ppr orig_fun_ty $$ ppr arg_tys )       -- Zero case gives infinite loop!
        applyTysD doc (substTyWithBinders bndrs (take n_bndrs arg_tys) rho_ty)
                      (drop n_bndrs arg_tys)
      where
        (bndrs, rho_ty) = splitPiTys orig_fun_ty
        n_bndrs = length bndrs
        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

    {-
    %************************************************************************
    %*                                                                      *
       TyBinders
    %*                                                                      *
    %************************************************************************
    -}

    -- | Make a named binder
    mkNamedBinder :: Var -> VisibilityFlag -> TyBinder
    mkNamedBinder = Named

    -- | Make an anonymous binder
    mkAnonBinder :: Type -> TyBinder
    mkAnonBinder = Anon

    -- | Does this binder bind a variable that is /not/ erased? Returns
    -- 'True' for anonymous binders.
    isIdLikeBinder :: TyBinder -> Bool
    isIdLikeBinder (Named {}) = False
    isIdLikeBinder (Anon {})  = True

    -- | Does this type, when used to the left of an arrow, require
    -- a visible argument? This checks to see if the kind of the type
    -- is constraint.
    isVisibleType :: Type -> Bool
    isVisibleType = not . isPredTy

    binderVisibility :: TyBinder -> VisibilityFlag
    binderVisibility (Named _ vis) = vis
    binderVisibility (Anon ty)
      | isVisibleType ty = Visible
      | otherwise        = Invisible

    -- | Extract a bound variable in a binder, if any
    binderVar_maybe :: TyBinder -> Maybe Var
    binderVar_maybe (Named v _) = Just v
    binderVar_maybe (Anon {})   = Nothing

    -- | Extract a bound variable in a binder, or panics
    binderVar :: String   -- ^ printed if there is a panic
              -> TyBinder -> Var
    binderVar _ (Named v _) = v
    binderVar e (Anon t)    = pprPanic ("binderVar (" ++ e ++ ")") (ppr t)

    -- | Extract a relevant type, if there is one.
    binderRelevantType_maybe :: TyBinder -> Maybe Type
    binderRelevantType_maybe (Named {}) = Nothing
    binderRelevantType_maybe (Anon ty)  = Just ty

    -- | Like 'maybe', but for binders.
    caseBinder :: TyBinder       -- ^ binder to scrutinize
               -> (TyVar -> a) -- ^ named case
               -> (Type -> a)  -- ^ anonymous case
               -> a
    caseBinder (Named v _) f _ = f v
    caseBinder (Anon t) _ d    = d t

    -- | Break apart a list of binders into tyvars and anonymous types.
    partitionBinders :: [TyBinder] -> ([TyVar], [Type])
    partitionBinders = partitionWith named_or_anon
      where
        named_or_anon bndr = caseBinder bndr Left Right

    -- | Break apart a list of binders into a list of named binders and
    -- a list of anonymous types.
    partitionBindersIntoBinders :: [TyBinder] -> ([TyBinder], [Type])
    partitionBindersIntoBinders = partitionWith named_or_anon
      where
        named_or_anon bndr = caseBinder bndr (\_ -> Left bndr) Right

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

    Predicates on PredType
    -}

    -- | Is the type suitable to classify a given/wanted in the typechecker?
    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
      --
      -- NB: This must return "True" to *unlifted* coercions, which are not
      --     of kind Constraint!
    isPredTy ty = go ty []
      where
        go :: Type -> [KindOrType] -> Bool
        go (AppTy ty1 ty2)   args = go ty1 (ty2 : args)
        go (TyConApp tc tys) args
          | tc `hasKey` eqPrimTyConKey || tc `hasKey` eqReprPrimTyConKey
          , [_,_,_,_] <- all_args
          = True

          | otherwise
          = go_k (tyConKind tc) all_args
          where
            all_args = 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 (ForAllTy bndr k1) (arg:args)
          = go_k (substTyWithBinders [bndr] [arg] k1) args
        go_k _ _ = False                  -- Typeable * Int :: Constraint

    isClassPred, isEqPred, isNomEqPred, 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` eqPrimTyConKey
                   || tyCon `hasKey` eqReprPrimTyConKey
        _          -> False

    isNomEqPred ty = case tyConAppTyCon_maybe ty of
        Just tyCon -> tyCon `hasKey` eqPrimTyConKey
        _          -> False

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

    isIPTyCon :: TyCon -> Bool
    isIPTyCon tc = tc `hasKey` ipClassKey
      -- Class and its corresponding TyCon have the same Unique

    isIPClass :: Class -> Bool
    isIPClass cls = cls `hasKey` ipClassKey

    isCTupleClass :: Class -> Bool
    isCTupleClass cls = isTupleTyCon (classTyCon cls)

    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 ---------------------------------
    -}

    -- | Makes a lifted equality predicate at the given role
    mkPrimEqPredRole :: Role -> Type -> Type -> PredType
    mkPrimEqPredRole Nominal          = mkPrimEqPred
    mkPrimEqPredRole Representational = mkReprPrimEqPred
    mkPrimEqPredRole Phantom          = panic "mkPrimEqPredRole phantom"

    -- | Creates a primitive type equality predicate.
    -- Invariant: the types are not Coercions
    mkPrimEqPred :: Type -> Type -> Type
    mkPrimEqPred ty1 ty2
      = TyConApp eqPrimTyCon [k1, k2, ty1, ty2]
      where
        k1 = typeKind ty1
        k2 = typeKind ty2

    -- | Creates a primite type equality predicate with explicit kinds
    mkHeteroPrimEqPred :: Kind -> Kind -> Type -> Type -> Type
    mkHeteroPrimEqPred k1 k2 ty1 ty2 = TyConApp eqPrimTyCon [k1, k2, ty1, ty2]

    -- | Creates a primitive representational type equality predicate
    -- with explicit kinds
    mkHeteroReprPrimEqPred :: Kind -> Kind -> Type -> Type -> Type
    mkHeteroReprPrimEqPred k1 k2 ty1 ty2
      = TyConApp eqReprPrimTyCon [k1, k2, ty1, ty2]

    -- | Try to split up a coercion type into the types that it coerces
    splitCoercionType_maybe :: Type -> Maybe (Type, Type)
    splitCoercionType_maybe ty
      = do { (tc, [_, _, ty1, ty2]) <- splitTyConApp_maybe ty
           ; guard $ tc `hasKey` eqPrimTyConKey || tc `hasKey` eqReprPrimTyConKey
           ; return (ty1, ty2) }

    mkReprPrimEqPred :: Type -> Type -> Type
    mkReprPrimEqPred ty1  ty2
      = TyConApp eqReprPrimTyCon [k1, k2, ty1, ty2]
      where
        k1 = typeKind ty1
        k2 = typeKind ty2

    equalityTyCon :: Role -> TyCon
    equalityTyCon Nominal          = eqPrimTyCon
    equalityTyCon Representational = eqReprPrimTyCon
    equalityTyCon Phantom          = eqPhantPrimTyCon

    -- --------------------- 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
                  | IrredPred PredType

    classifyPredType :: PredType -> PredTree
    classifyPredType ev_ty = case splitTyConApp_maybe ev_ty of
        Just (tc, [_, _, ty1, ty2])
          | tc `hasKey` eqReprPrimTyConKey    -> EqPred ReprEq ty1 ty2
          | tc `hasKey` eqPrimTyConKey        -> EqPred NomEq ty1 ty2
        Just (tc, tys)
          | Just clas <- tyConClass_maybe tc  -> ClassPred clas 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])
            |  tc `hasKey` eqPrimTyConKey
            || tc `hasKey` eqReprPrimTyConKey
            -> (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` eqPrimTyConKey     -> Just (Nominal, ty1, ty2)
            | tc `hasKey` eqReprPrimTyConKey -> Just (Representational, ty1, ty2)
          _ -> Nothing

    getEqPredRole :: PredType -> Role
    getEqPredRole ty = eqRelRole (predTypeEqRel ty)

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

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

    -- NB: This function does not respect `eqType`, in that two types that
    -- are `eqType` may return different sizes. This is OK, because this
    -- function is used only in reporting, not decision-making.
    typeSize :: Type -> Int
    typeSize (LitTy {})       = 1
    typeSize (TyVarTy {})     = 1
    typeSize (AppTy t1 t2)    = typeSize t1 + typeSize t2
    typeSize (ForAllTy b t)   = typeSize (binderType b) + typeSize t
    typeSize (TyConApp _ ts)  = 1 + sum (map typeSize ts)
    typeSize (CastTy ty co)   = typeSize ty + coercionSize co
    typeSize (CoercionTy co)  = coercionSize co

    {-
    %************************************************************************
    %*                                                                      *
             Well-scoped tyvars
    *                                                                      *
    ************************************************************************
    -}

    -- | Do a topological sort on a list of tyvars. This is a deterministic
    -- sorting operation (that is, doesn't depend on Uniques).
    toposortTyVars :: [TyVar] -> [TyVar]
    toposortTyVars tvs = reverse $
                         [ tv | (tv, _, _) <- topologicalSortG $
                                              graphFromEdgedVertices nodes ]
      where
        var_ids :: VarEnv Int
        var_ids = mkVarEnv (zip tvs [1..])

        nodes = [ ( tv
                  , lookupVarEnv_NF var_ids tv
                  , mapMaybe (lookupVarEnv var_ids)
                             (tyCoVarsOfTypeList (tyVarKind tv)) )
                | tv <- tvs ]

    -- | Extract a well-scoped list of variables from a set of variables.
    varSetElemsWellScoped :: VarSet -> [Var]
    varSetElemsWellScoped = toposortTyVars . varSetElems

    -- | Get the free vars of a type in scoped order
    tyCoVarsOfTypeWellScoped :: Type -> [TyVar]
    tyCoVarsOfTypeWellScoped = toposortTyVars . tyCoVarsOfTypeList

    {-
    ************************************************************************
    *                                                                      *
    \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 )
                        zipTvSubst 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 (Named {}) ty) = isUnLiftedType ty
    isUnLiftedType (TyConApp tc _)          = isUnLiftedTyCon tc
    isUnLiftedType _                        = False

    -- | Extract the levity classifier of a type. Panics if this is not possible.
    getLevity :: String   -- ^ Printed in case of an error
              -> Type -> Type
    getLevity err ty = getLevityFromKind err (typeKind ty)

    -- | Extract the levity classifier of a type from its kind.
    -- For example, getLevityFromKind * = Lifted; getLevityFromKind # = Unlifted.
    -- Panics if this is not possible.
    getLevityFromKind :: String  -- ^ Printed in case of an error
                      -> Type -> Type
    getLevityFromKind err = go
      where
        go k | Just k' <- coreViewOneStarKind k = go k'
        go k
          | Just (tc, [arg]) <- splitTyConApp_maybe k
          , tc `hasKey` tYPETyConKey
          = arg
        go k = pprPanic "getLevity" (text err $$
                                     ppr k <+> dcolon <+> ppr (typeKind k))

    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 (TyConApp tc tys)    = tc `seq` seqTypes tys
    seqType (ForAllTy bndr ty)   = seqType (binderType bndr) `seq` seqType ty
    seqType (CastTy ty co)       = seqType ty `seq` seqCo co
    seqType (CoercionTy co)      = seqCo co

    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)
    *                                                                      *
    ************************************************************************

    Note [Equality on AppTys]
    ~~~~~~~~~~~~~~~~~~~~~~~~~
    In our cast-ignoring equality, we want to say that the following two
    are equal:

      (Maybe |> co) (Int |> co')   ~?       Maybe Int

    But the left is an AppTy while the right is a TyConApp. The solution is
    to use repSplitAppTy_maybe to break up the TyConApp into its pieces and
    then continue. Easy to do, but also easy to forget to do.

    -}

    eqType :: Type -> Type -> Bool
    -- ^ Type equality on source types. Does not look through @newtypes@ or
    -- 'PredType's, but it does look through type synonyms.
    -- This first checks that the kinds of the types are equal and then
    -- checks whether the types are equal, ignoring casts and coercions.
    -- (The kind check is a recursive call, but since all kinds have type
    -- @Type@, there is no need to check the types of kinds.)
    -- See also Note [Non-trivial definitional equality] in TyCoRep.
    eqType t1 t2 = isEqual $ cmpType t1 t2

    -- | Compare types with respect to a (presumably) non-empty 'RnEnv2'.
    eqTypeX :: RnEnv2 -> Type -> Type -> Bool
    eqTypeX env t1 t2 = isEqual $ cmpTypeX env t1 t2

    -- | Type equality on lists of types, looking through type synonyms
    -- but not newtypes.
    eqTypes :: [Type] -> [Type] -> Bool
    eqTypes tys1 tys2 = isEqual $ cmpTypes tys1 tys2

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

    -- Now here comes the real worker

    cmpType :: Type -> Type -> Ordering
    cmpType t1 t2
      -- we know k1 and k2 have the same kind, because they both have kind *.
      = cmpTypeX rn_env t1 t2
      where
        rn_env = mkRnEnv2 (mkInScopeSet (tyCoVarsOfTypes [t1, t2]))

    cmpTypes :: [Type] -> [Type] -> Ordering
    cmpTypes ts1 ts2 = cmpTypesX rn_env ts1 ts2
      where
        rn_env = mkRnEnv2 (mkInScopeSet (tyCoVarsOfTypes (ts1 ++ ts2)))

    cmpTypeX :: RnEnv2 -> Type -> Type -> Ordering  -- Main workhorse
        -- See Note [Non-trivial definitional equality] in TyCoRep
    cmpTypeX env orig_t1 orig_t2 = go env k1 k2 `thenCmp` go env orig_t1 orig_t2
      where
        k1 = typeKind orig_t1
        k2 = typeKind orig_t2

        go env t1 t2 | Just t1' <- coreViewOneStarKind t1 = go env t1' t2
        go env t1 t2 | Just t2' <- coreViewOneStarKind t2 = go env t1 t2'

        go env (TyVarTy tv1)       (TyVarTy tv2)
          = rnOccL env tv1 `compare` rnOccR env tv2
        go env (ForAllTy (Named tv1 _) t1) (ForAllTy (Named tv2 _) t2)
          = go env (tyVarKind tv1) (tyVarKind tv2)
            `thenCmp` go (rnBndr2 env tv1 tv2) t1 t2
            -- See Note [Equality on AppTys]
        go env (AppTy s1 t1) ty2
          | Just (s2, t2) <- repSplitAppTy_maybe ty2
          = go env s1 s2 `thenCmp` go env t1 t2
        go env ty1 (AppTy s2 t2)
          | Just (s1, t1) <- repSplitAppTy_maybe ty1
          = go env s1 s2 `thenCmp` go env t1 t2
        go env (ForAllTy (Anon s1) t1) (ForAllTy (Anon s2) t2)
          = go env s1 s2 `thenCmp` go env t1 t2
        go env (TyConApp tc1 tys1) (TyConApp tc2 tys2)
          = (tc1 `cmpTc` tc2) `thenCmp` gos env tys1 tys2
        go _   (LitTy l1)          (LitTy l2)          = compare l1 l2
        go env (CastTy t1 _)       t2                  = go env t1 t2
        go env t1                  (CastTy t2 _)       = go env t1 t2
        go _   (CoercionTy {})     (CoercionTy {})     = EQ

            -- Deal with the rest: TyVarTy < CoercionTy < AppTy < LitTy < TyConApp < ForAllTy
        go _ ty1 ty2
          = (get_rank ty1) `compare` (get_rank ty2)
          where get_rank :: Type -> Int
                get_rank (CastTy {})
                  = pprPanic "cmpTypeX.get_rank" (ppr [ty1,ty2])
                get_rank (TyVarTy {})            = 0
                get_rank (CoercionTy {})         = 1
                get_rank (AppTy {})              = 3
                get_rank (LitTy {})              = 4
                get_rank (TyConApp {})           = 5
                get_rank (ForAllTy (Anon {}) _)  = 6
                get_rank (ForAllTy (Named {}) _) = 7

        gos _   []         []         = EQ
        gos _   []         _          = LT
        gos _   _          []         = GT
        gos env (ty1:tys1) (ty2:tys2) = go env ty1 ty2 `thenCmp` gos env tys1 tys2

    -------------
    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

    -------------
    -- | Compare two 'TyCon's. NB: This should /never/ see the "star synonyms",
    -- as recognized by Kind.isStarKindSynonymTyCon. See Note
    -- [Kind Constraint and kind *] in Kind.
    cmpTc :: TyCon -> TyCon -> Ordering
    cmpTc tc1 tc2
      = ASSERT( not (isStarKindSynonymTyCon tc1) && not (isStarKindSynonymTyCon tc2) )
        u1 `compare` u2
      where
        u1  = tyConUnique tc1
        u2  = tyConUnique tc2

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

    typeKind :: Type -> Kind
    typeKind (TyConApp tc tys)     = piResultTys (tyConKind tc) tys
    typeKind (AppTy fun arg)       = piResultTy (typeKind fun) arg
    typeKind (LitTy l)             = typeLiteralKind l
    typeKind (ForAllTy (Anon _) _) = liftedTypeKind
    typeKind (ForAllTy _ ty)       = typeKind ty
    typeKind (TyVarTy tyvar)       = tyVarKind tyvar
    typeKind (CastTy _ty co)       = pSnd $ coercionKind co
    typeKind (CoercionTy co)       = coercionType co

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

    -- | Print a tyvar with its kind
    pprTyVar :: TyVar -> SDoc
    pprTyVar tv = ppr tv <+> dcolon <+> ppr (tyVarKind tv)

    {-
    %************************************************************************
    %*                                                                      *
            Miscellaneous functions
    %*                                                                      *
    %************************************************************************

    -}
    -- | 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' <- coreView ty = go ty'
         go (TyVarTy {})               = emptyNameEnv
         go (LitTy {})                 = emptyNameEnv
         go (TyConApp tc tys)          = go_tc tc `plusNameEnv` go_s tys
         go (AppTy a b)                = go a `plusNameEnv` go b
         go (ForAllTy (Anon a) b)      = go a `plusNameEnv` go b `plusNameEnv` go_tc funTyCon
         go (ForAllTy (Named tv _) ty) = go ty `plusNameEnv` go (tyVarKind tv)
         go (CastTy ty co)             = go ty `plusNameEnv` go_co co
         go (CoercionTy co)            = go_co co

         go_co (Refl _ ty)             = go ty
         go_co (TyConAppCo _ tc args)  = go_tc tc `plusNameEnv` go_cos args
         go_co (AppCo co arg)          = go_co co `plusNameEnv` go_co arg
         go_co (ForAllCo _ kind_co co) = go_co kind_co `plusNameEnv` go_co co
         go_co (CoVarCo {})            = emptyNameEnv
         go_co (AxiomInstCo ax _ args) = go_ax ax `plusNameEnv` go_cos args
         go_co (UnivCo p _ t1 t2)      = go_prov p `plusNameEnv` go t1 `plusNameEnv` go t2
         go_co (SymCo co)              = go_co co
         go_co (TransCo co1 co2)       = go_co co1 `plusNameEnv` go_co co2
         go_co (NthCo _ co)            = go_co co
         go_co (LRCo _ co)             = go_co co
         go_co (InstCo co arg)         = go_co co `plusNameEnv` go_co arg
         go_co (CoherenceCo co1 co2)   = go_co co1 `plusNameEnv` go_co co2
         go_co (KindCo co)             = go_co co
         go_co (SubCo co)              = go_co co
         go_co (AxiomRuleCo _ cs)      = go_cos cs

         go_prov UnsafeCoerceProv    = emptyNameEnv
         go_prov (PhantomProv co)    = go_co co
         go_prov (ProofIrrelProv co) = go_co co
         go_prov (PluginProv _)      = emptyNameEnv
         go_prov (HoleProv h)        = pprPanic "tyConsOfType hit a hole" (ppr h)

         go_s tys     = foldr (plusNameEnv . go)     emptyNameEnv tys
         go_cos cos   = foldr (plusNameEnv . go_co)  emptyNameEnv cos

         go_tc tc = unitNameEnv (tyConName tc) tc
         go_ax ax = go_tc $ coAxiomTyCon ax

    -- | Find the result 'Kind' of a type synonym,
    -- after applying it to its 'arity' number of type variables
    -- Actually this function works fine on data types too,
    -- but they'd always return '*', so we never need to ask
    synTyConResKind :: TyCon -> Kind
    synTyConResKind tycon = piResultTys (tyConKind tycon) (mkTyVarTys (tyConTyVars tycon))

    -- | Retrieve the free variables in this type, splitting them based
    -- on whether the variable was used in a dependent context. It's possible
    -- for a variable to be reported twice, if it's used both dependently
    -- and non-dependently. (This isn't the most precise analysis, because
    -- it's used in the typechecking knot. It might list some dependent
    -- variables as also non-dependent.)
    splitDepVarsOfType :: Type -> Pair TyCoVarSet
    splitDepVarsOfType = go
      where
        go (TyVarTy tv)              = Pair (tyCoVarsOfType $ tyVarKind tv)
                                            (unitVarSet tv)
        go (AppTy t1 t2)             = go t1 `mappend` go t2
        go (TyConApp _ tys)          = foldMap go tys
        go (ForAllTy (Anon arg) res) = go arg `mappend` go res
        go (ForAllTy (Named tv _) ty)
          = let Pair kvs tvs = go ty in
            Pair (kvs `delVarSet` tv `unionVarSet` tyCoVarsOfType (tyVarKind tv))
                 (tvs `delVarSet` tv)
        go (LitTy {})                = mempty
        go (CastTy ty co)            = go ty `mappend` Pair (tyCoVarsOfCo co)
                                                            emptyVarSet
        go (CoercionTy co)           = go_co co

        go_co co = let Pair ty1 ty2 = coercionKind co in
                   go ty1 `mappend` go ty2  -- NB: the Pairs separate along different
                                            -- dimensions here. Be careful!

    -- | Like 'splitDepVarsOfType', but over a list of types
    splitDepVarsOfTypes :: [Type] -> Pair TyCoVarSet
    splitDepVarsOfTypes = foldMap splitDepVarsOfType

    -- | Retrieve the free variables in this type, splitting them based
    -- on whether they are used visibly or invisibly. Invisible ones come
    -- first.
    splitVisVarsOfType :: Type -> Pair TyCoVarSet
    splitVisVarsOfType orig_ty = Pair invis_vars vis_vars
      where
        Pair invis_vars1 vis_vars = go orig_ty
        invis_vars = invis_vars1 `minusVarSet` vis_vars

        go (TyVarTy tv)  = Pair (tyCoVarsOfType $ tyVarKind tv) (unitVarSet tv)
        go (AppTy t1 t2) = go t1 `mappend` go t2
        go (TyConApp tc tys) = go_tc tc tys
        go (ForAllTy (Anon t1) t2) = go t1 `mappend` go t2
        go (ForAllTy (Named tv _) ty)
          = ((`delVarSet` tv) <$> go ty) `mappend`
            (invisible (tyCoVarsOfType $ tyVarKind tv))
        go (LitTy {}) = mempty
        go (CastTy ty co) = go ty `mappend` invisible (tyCoVarsOfCo co)
        go (CoercionTy co) = invisible $ tyCoVarsOfCo co

        invisible vs = Pair vs emptyVarSet

        go_tc tc tys = let (invis, vis) = partitionInvisibles tc id tys in
                       invisible (tyCoVarsOfTypes invis) `mappend` foldMap go vis

    splitVisVarsOfTypes :: [Type] -> Pair TyCoVarSet
    splitVisVarsOfTypes = foldMap splitVisVarsOfType