content: -- (c) The University of Glasgow 2006
  -- (c) The GRASP/AQUA Project, Glasgow University, 1998
  --
  -- Type - public interface
  
  {-# LANGUAGE FlexibleContexts, PatternSynonyms, ViewPatterns, MultiWayIf, RankNTypes #-}
  {-# OPTIONS_GHC -fno-warn-orphans #-}
  
  -- | Main functions for manipulating types and type-related things
  module GHC.Core.Type (
          -- Note some of this is just re-exports from TyCon..
  
          -- * Main data types representing Types
          -- $type_classification
  
          -- $representation_types
          Type, ForAllTyFlag(..), FunTyFlag(..),
          Specificity(..),
          KindOrType, PredType, ThetaType, FRRType,
          Var, TyVar, isTyVar, TyCoVar, PiTyBinder, ForAllTyBinder, TyVarBinder,
          Mult, Scaled,
          KnotTied, RuntimeRepType,
  
          -- ** Constructing and deconstructing types
          mkTyVarTy, mkTyVarTys, getTyVar, getTyVar_maybe, repGetTyVar_maybe,
          getCastedTyVar_maybe, tyVarKind, varType,
  
          mkAppTy, mkAppTys, splitAppTy, splitAppTys, splitAppTysNoView,
          splitAppTy_maybe, splitAppTyNoView_maybe, tcSplitAppTyNoView_maybe,
  
          mkFunTy, mkVisFunTy,
          mkVisFunTyMany, mkVisFunTysMany,
          mkScaledFunTys,
          mkInvisFunTy, mkInvisFunTys,
          tcMkVisFunTy, tcMkScaledFunTys, tcMkInvisFunTy,
          splitFunTy, splitFunTy_maybe, splitVisibleFunTy_maybe,
          splitFunTys, funResultTy, funArgTy,
          funTyConAppTy_maybe, funTyFlagTyCon,
          tyConAppFunTy_maybe, tyConAppFunCo_maybe,
          mkFunctionType, mkScaledFunctionTys, chooseFunTyFlag,
  
          mkTyConApp, mkTyConTy,
          tyConAppTyCon_maybe, tyConAppTyConPicky_maybe,
          tyConAppArgs_maybe, tyConAppTyCon, tyConAppArgs,
  
          splitTyConApp_maybe, splitTyConAppNoView_maybe, splitTyConApp,
          tcSplitTyConApp, tcSplitTyConApp_maybe,
  
          mkForAllTy, mkForAllTys, mkInvisForAllTys, mkTyCoInvForAllTys,
          mkSpecForAllTy, mkSpecForAllTys,
          mkVisForAllTys, mkTyCoForAllTy, mkTyCoForAllTys, mkTyCoInvForAllTy,
          mkInfForAllTy, mkInfForAllTys,
          splitForAllTyCoVars, splitForAllTyVars,
          splitForAllReqTyBinders, splitForAllInvisTyBinders,
          splitForAllForAllTyBinders, splitForAllForAllTyBinder_maybe,
          splitForAllTyCoVar_maybe, splitForAllTyCoVar,
          splitForAllTyVar_maybe, splitForAllCoVar_maybe,
          splitPiTy_maybe, splitPiTy, splitPiTys, collectPiTyBinders,
          getRuntimeArgTys,
          mkTyConBindersPreferAnon,
          mkPiTy, mkPiTys,
          piResultTy, piResultTys,
          applyTysX, dropForAlls,
          mkFamilyTyConApp,
          buildSynTyCon,
  
          mkNumLitTy, isNumLitTy,
          mkStrLitTy, isStrLitTy,
          mkCharLitTy, isCharLitTy,
          isLitTy,
  
          getRuntimeRep, splitRuntimeRep_maybe, kindRep_maybe, kindRep,
          getLevity, levityType_maybe,
  
          mkCastTy, mkCoercionTy, splitCastTy_maybe,
  
          ErrorMsgType, pprUserTypeErrorTy,
  
          coAxNthLHS,
          stripCoercionTy,
  
          splitInvisPiTys, splitInvisPiTysN, invisibleBndrCount,
          filterOutInvisibleTypes, filterOutInferredTypes,
          partitionInvisibleTypes, partitionInvisibles,
          tyConForAllTyFlags, appTyForAllTyFlags,
  
          -- ** Analyzing types
          TyCoMapper(..), mapTyCo, mapTyCoX,
          TyCoFolder(..), foldTyCo, noView,
  
          -- (Newtypes)
          newTyConInstRhs,
  
          -- ** Binders
          mkForAllTyBinder, mkForAllTyBinders,
          mkTyVarBinder, mkTyVarBinders,
          tyVarSpecToBinders,
          isAnonPiTyBinder,
          binderVar, binderVars, binderType, binderFlag, binderFlags,
          piTyBinderType, namedPiTyBinder_maybe,
          anonPiTyBinderType_maybe,
          isVisibleForAllTyFlag, isInvisibleForAllTyFlag, isVisiblePiTyBinder,
          isInvisiblePiTyBinder, isNamedPiTyBinder,
          tyConBindersPiTyBinders,
  
          -- ** Predicates on types
          isTyVarTy, isFunTy, isCoercionTy,
          isCoercionTy_maybe, isForAllTy,
          isForAllTy_ty, isForAllTy_co,
          isForAllTy_invis_ty,
          isPiTy, isTauTy, isFamFreeTy,
          isAtomicTy,
  
          isValidJoinPointType,
          tyConAppNeedsKindSig,
  
          -- * Space-saving construction
          mkTYPEapp, mkTYPEapp_maybe,
          mkCONSTRAINTapp, mkCONSTRAINTapp_maybe,
          mkBoxedRepApp_maybe, mkTupleRepApp_maybe,
          typeOrConstraintKind, liftedTypeOrConstraintKind,
  
          -- *** Levity and boxity
          sORTKind_maybe, typeTypeOrConstraint,
          typeLevity, typeLevity_maybe, tyConIsTYPEorCONSTRAINT,
          isLiftedTypeKind, isUnliftedTypeKind, pickyIsLiftedTypeKind,
          isLiftedRuntimeRep, isUnliftedRuntimeRep, runtimeRepLevity_maybe,
          isBoxedRuntimeRep,
          isLiftedLevity, isUnliftedLevity,
          isUnliftedType, isBoxedType, isUnboxedTupleType, isUnboxedSumType,
          kindBoxedRepLevity_maybe,
          mightBeLiftedType, mightBeUnliftedType,
          definitelyLiftedType, definitelyUnliftedType,
          isAlgType, isDataFamilyApp, isSatTyFamApp,
          isPrimitiveType, isStrictType, isTerminatingType,
          isLevityTy, isLevityVar,
          isRuntimeRepTy, isRuntimeRepVar, isRuntimeRepKindedTy,
          dropRuntimeRepArgs,
  
          -- * Multiplicity
  
          isMultiplicityTy, isMultiplicityVar,
          unrestricted, linear, tymult,
          mkScaled, irrelevantMult, scaledSet,
          pattern OneTy, pattern ManyTy,
          isOneTy, isManyTy,
          isLinearType,
  
          -- * Main data types representing Kinds
          Kind,
  
          -- ** Finding the kind of a type
          typeKind, typeHasFixedRuntimeRep,
          tcIsLiftedTypeKind,
          isConstraintKind, isConstraintLikeKind, returnsConstraintKind,
          tcIsBoxedTypeKind, isTypeLikeKind,
  
          -- ** Common Kind
          liftedTypeKind, unliftedTypeKind,
  
          -- * Type free variables
          tyCoFVsOfType, tyCoFVsBndr, tyCoFVsVarBndr, tyCoFVsVarBndrs,
          tyCoVarsOfType, tyCoVarsOfTypes,
          tyCoVarsOfTypeDSet,
          coVarsOfType,
          coVarsOfTypes,
  
          anyFreeVarsOfType, anyFreeVarsOfTypes,
          noFreeVarsOfType,
          expandTypeSynonyms, expandSynTyConApp_maybe,
          typeSize, occCheckExpand,
  
          -- ** Closing over kinds
          closeOverKindsDSet, closeOverKindsList,
          closeOverKinds,
  
          -- * Forcing evaluation of types
          seqType, seqTypes,
  
          -- * Other views onto Types
          coreView, coreFullView, rewriterView,
  
          tyConsOfType,
  
          -- * Main type substitution data types
          TvSubstEnv,     -- Representation widely visible
          IdSubstEnv,
          Subst(..),    -- Representation visible to a few friends
  
          -- ** Manipulating type substitutions
          emptyTvSubstEnv, emptySubst, mkEmptySubst,
  
          mkTCvSubst, zipTvSubst, mkTvSubstPrs,
          zipTCvSubst,
          notElemSubst,
          getTvSubstEnv,
          zapSubst, substInScopeSet, setInScope, getSubstRangeTyCoFVs,
          extendSubstInScope, extendSubstInScopeList, extendSubstInScopeSet,
          extendTCvSubst, extendCvSubst,
          extendTvSubst, extendTvSubstList, extendTvSubstAndInScope,
          extendTCvSubstList,
          extendTvSubstWithClone,
          extendTCvSubstWithClone,
          isInScope, composeTCvSubst, zipTyEnv, zipCoEnv,
          isEmptySubst, unionSubst, isEmptyTCvSubst,
  
          -- ** Performing substitution on types and kinds
          substTy, substTys, substScaledTy, substScaledTys, substTyWith, substTysWith, substTheta,
          substTyAddInScope,
          substTyUnchecked, substTysUnchecked, substScaledTyUnchecked, substScaledTysUnchecked,
          substThetaUnchecked, substTyWithUnchecked,
          substCo, substCoWithInScope,
          substTyVarBndr, substTyVarBndrs, substTyVar, substTyVars,
          substVarBndr, substVarBndrs,
          substTyCoBndr, substTyVarToTyVar,
          cloneTyVarBndr, cloneTyVarBndrs, lookupTyVar,
  
          -- * Kinds
          isTYPEorCONSTRAINT,
          isConcreteType,
          isFixedRuntimeRepKind
      ) where
  
  import GHC.Prelude
  
  import GHC.Types.Basic
  
  -- We import the representation and primitive functions from GHC.Core.TyCo.Rep.
  -- Many things are reexported, but not the representation!
  
  import GHC.Core.TyCo.Rep
  import GHC.Core.TyCo.Subst
  import GHC.Core.TyCo.FVs
  
  -- friends:
  import GHC.Types.Var
  import GHC.Types.Var.Env
  import GHC.Types.Var.Set
  
  import GHC.Core.TyCon
  import GHC.Builtin.Types.Prim
  
  import {-# SOURCE #-} GHC.Builtin.Types
     ( charTy, naturalTy
     , typeSymbolKind, liftedTypeKind, unliftedTypeKind
     , constraintKind, zeroBitTypeKind
     , manyDataConTy, oneDataConTy
     , liftedRepTy, unliftedRepTy, zeroBitRepTy )
  
  import GHC.Types.Name( Name )
  import GHC.Builtin.Names
  import GHC.Core.Coercion.Axiom
  
  import {-# SOURCE #-} GHC.Core.Coercion
     ( mkNomReflCo, mkGReflCo, mkReflCo
     , mkTyConAppCo, mkAppCo
     , mkForAllCo, mkFunCo2, mkAxiomCo, mkUnivCo
     , mkSymCo, mkTransCo, mkSelCo, mkLRCo, mkInstCo
     , mkKindCo, mkSubCo, mkFunCo, funRole
     , decomposePiCos, coercionKind
     , coercionRKind, coercionType
     , isReflexiveCo, seqCo
     , topNormaliseNewType_maybe
     )
  import {-# SOURCE #-} GHC.Tc.Utils.TcType ( isConcreteTyVar )
  
  -- others
  import GHC.Utils.Misc
  import GHC.Utils.FV
  import GHC.Utils.Outputable
  import GHC.Utils.Panic
  import GHC.Data.FastString
  
  import GHC.Data.Maybe   ( orElse, isJust )
  import GHC.List (build)
  
  -- $type_classification
  -- #type_classification#
  --
  -- Types are any, but at least one, of:
  --
  -- [Boxed]              Iff its representation is a pointer to an object on the
  --                      GC'd heap. Operationally, heap objects can be entered as
  --                      a means of evaluation.
  --
  -- [Lifted]             Iff it has bottom as an element: An instance of a
  --                      lifted type might diverge when evaluated.
  --                      GHC Haskell's unboxed types are unlifted.
  --                      An unboxed, but lifted type is not very useful.
  --                      (Example: A byte-represented type, where evaluating 0xff
  --                      computes the 12345678th collatz number modulo 0xff.)
  --                      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. There are algebraic types that
  --                      are not lifted types, like unlifted data types or
  --                      unboxed tuples.
  --
  -- [Data]               Iff it is a type declared with @data@, or a boxed tuple.
  --                      There are also /unlifted/ data types.
  --
  -- [Primitive]          Iff it is a built-in type that can't be expressed in Haskell.
  --
  -- [Unlifted]           Anything that isn't lifted is considered unlifted.
  --
  -- 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 \#) 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 its abilities. If you don't want this
  -- to happen, use the equivalent functions from the "TcType" module.
  
  {-
  ************************************************************************
  *                                                                      *
                  Type representation
  *                                                                      *
  ************************************************************************
  -}
  
  rewriterView :: Type -> Maybe Type
  -- Unwrap a type synonym only when either:
  --   The type synonym is forgetful, or
  --   the type synonym mentions a type family in its expansion
  -- See Note [Rewriting synonyms]
  {-# INLINE rewriterView #-}
  rewriterView (TyConApp tc tys)
    | isTypeSynonymTyCon tc
    , isForgetfulSynTyCon tc || not (isFamFreeTyCon tc)
    = expandSynTyConApp_maybe tc tys
  rewriterView _other
    = Nothing
  
  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.
  --
  -- This function does not look through type family applications.
  --
  -- 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) = expandSynTyConApp_maybe tc tys
  coreView _                 = Nothing
  -- See Note [Inlining coreView].
  {-# INLINE coreView #-}
  
  coreFullView, core_full_view :: Type -> Type
  -- ^ Iterates 'coreView' until there is no more to synonym to expand.
  -- NB: coreFullView is non-recursive and can be inlined;
  --     core_full_view is the recursive one
  -- See Note [Inlining coreView].
  coreFullView ty@(TyConApp tc _)
    | isTypeSynonymTyCon tc = core_full_view ty
  coreFullView ty = ty
  {-# INLINE coreFullView #-}
  
  core_full_view ty
    | Just ty' <- coreView ty = core_full_view ty'
    | otherwise               = ty
  
  -----------------------------------------------
  -- | @expandSynTyConApp_maybe tc tys@ expands the RHS of type synonym @tc@
  -- instantiated at arguments @tys@, or returns 'Nothing' if @tc@ is not a
  -- synonym.
  expandSynTyConApp_maybe :: TyCon -> [Type] -> Maybe Type
  {-# INLINE expandSynTyConApp_maybe #-}
  -- This INLINE will inline the call to expandSynTyConApp_maybe in coreView,
  -- which will eliminate the allocation Just/Nothing in the result
  -- Don't be tempted to make `expand_syn` (which is NOINLINE) return the
  -- Just/Nothing, else you'll increase allocation
  expandSynTyConApp_maybe tc arg_tys
    | Just (tvs, rhs) <- synTyConDefn_maybe tc
    , arg_tys `saturates` tyConArity tc
    = Just $! (expand_syn tvs rhs arg_tys)
      -- Why strict application? Because every client of this function will evaluat
      -- that (expand_syn ...) thunk, so it's more efficient not to build a thunk.
      -- Mind you, this function is always INLINEd, so the client context is probably
      -- enough to avoid thunk construction and so the $! is just belt-and-braces.
    | otherwise
    = Nothing
  
  saturates :: [Type] -> Arity -> Bool
  saturates _       0 = True
  saturates []      _ = False
  saturates (_:tys) n = assert( n >= 0 ) $ saturates tys (n-1)
                         -- Arities are always positive; the assertion just checks
                         -- that, to avoid an infinite loop in the bad case
  
  -- | A helper for 'expandSynTyConApp_maybe' to avoid inlining this cold path
  -- into call-sites.
  --
  -- Precondition: the call is saturated or over-saturated;
  --               i.e. length tvs <= length arg_tys
  expand_syn :: [TyVar]  -- ^ the variables bound by the synonym
             -> Type     -- ^ the RHS of the synonym
             -> [Type]   -- ^ the type arguments the synonym is instantiated at.
             -> Type
  {-# NOINLINE expand_syn #-} -- We never want to inline this cold-path.
  
  expand_syn tvs rhs arg_tys
    -- No substitution necessary if either tvs or tys is empty
    -- This is both more efficient, and steers clear of an infinite
    -- loop; see Note [Care using synonyms to compress types]
    | null arg_tys  = assert (null tvs) rhs
    | null tvs      = mkAppTys rhs arg_tys
    | otherwise     = go empty_subst tvs arg_tys
    where
      empty_subst = mkEmptySubst in_scope
      in_scope = mkInScopeSet $ shallowTyCoVarsOfTypes $ arg_tys
        -- The free vars of 'rhs' should all be bound by 'tenv',
        -- so we only need the free vars of tys
        -- See also Note [The substitution invariant] in GHC.Core.TyCo.Subst.
  
      go subst [] tys
        | null tys  = rhs'  -- Exactly Saturated
        | otherwise = mkAppTys rhs' tys
            -- Its important to use mkAppTys, rather than (foldl AppTy),
            -- because the function part might well return a
            -- partially-applied type constructor; indeed, usually will!
        where
          rhs' = substTy subst rhs
  
      go subst (tv:tvs) (ty:tys) = go (extendTvSubst subst tv ty) tvs tys
  
      go _ (_:_) [] = pprPanic "expand_syn" (ppr tvs $$ ppr rhs $$ ppr arg_tys)
                     -- Under-saturated, precondition failed
  
  {- Note [Inlining coreView]
  ~~~~~~~~~~~~~~~~~~~~~~~~~~~
  It is very common to have a function
  
    f :: Type -> ...
    f ty | Just ty' <- coreView ty = f ty'
    f (TyVarTy ...) = ...
    f ...           = ...
  
  If f is not otherwise recursive, the initial call to coreView
  causes f to become recursive, which kills the possibility of
  inlining. Instead, for non-recursive functions, we prefer to
  use coreFullView, which guarantees to unwrap top-level type
  synonyms. It can be inlined and is efficient and non-allocating
  in its fast path. For this to really be fast, all calls made
  on its fast path must also be inlined, linked back to this Note.
  -}
  
  
  {- *********************************************************************
  *                                                                      *
                  expandTypeSynonyms
  *                                                                      *
  ********************************************************************* -}
  
  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.
  --
  -- Keep this synchronized with 'synonymTyConsOfType'
  expandTypeSynonyms ty
    = go (mkEmptySubst in_scope) ty
    where
      in_scope = mkInScopeSet (tyCoVarsOfType ty)
  
      go subst (TyConApp tc tys)
        | ExpandsSyn tenv rhs tys' <- expandSynTyCon_maybe tc expanded_tys
        = let subst' = mkTvSubst in_scope (mkVarEnv tenv)
              -- Make a fresh substitution; rhs has nothing to
              -- do with anything that has happened so far
              -- NB: if you make changes here, be sure to build an
              --     /idempotent/ substitution, even in the nested case
              --        type T a b = a -> b
              --        type S x y = T y x
              -- (#11665)
          in  mkAppTys (go subst' rhs) tys'
        | otherwise
        = TyConApp tc expanded_tys
        where
          expanded_tys = (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 ty@(FunTy _ mult arg res)
        = ty { ft_mult = go subst mult, ft_arg = go subst arg, ft_res = go subst res }
      go subst (ForAllTy (Bndr tv vis) t)
        = let (subst', tv') = substVarBndrUsing go subst tv in
          ForAllTy (Bndr 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_mco _     MRefl    = MRefl
      go_mco subst (MCo co) = MCo (go_co subst co)
  
      go_co subst (Refl ty)
        = mkNomReflCo (go subst ty)
      go_co subst (GRefl r ty mco)
        = mkGReflCo r (go subst ty) (go_mco subst mco)
         -- 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 { fco_tcv = tcv, fco_visL = visL, fco_visR = visR
                            , fco_kind = kind_co, fco_body = co })
        = mkForAllCo tcv' visL visR
                     (go_mco subst kind_co)
                     (go_co subst' co)
        where
          (subst', tcv') = substVarBndr subst tcv
      go_co subst (FunCo r afl afr w co1 co2)
        = mkFunCo2 r afl afr (go_co subst w) (go_co subst co1) (go_co subst co2)
      go_co subst (CoVarCo cv)
        = substCoVar subst cv
      go_co subst (AxiomCo ax cs)
        = mkAxiomCo ax (map (go_co subst) cs)
      go_co subst co@(UnivCo { uco_lty = lty, uco_rty = rty })
        = co { uco_lty = go subst lty, uco_rty = go subst rty }
      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 (SelCo n co)
        = mkSelCo 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 (KindCo co)
        = mkKindCo (go_co subst co)
      go_co subst (SubCo co)
        = mkSubCo (go_co subst co)
      go_co _ (HoleCo h)
        = pprPanic "expandTypeSynonyms hit a hole" (ppr h)
  
  {- 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 GHC.Tc.Gen.Bind.
  -}
  
  {- *********************************************************************
  *                                                                      *
                  Random functions (todo: organise)
  *                                                                      *
  ********************************************************************* -}
  
  -- | An INLINE helper for function such as 'kindRep_maybe' below.
  --
  -- @isTyConKeyApp_maybe key ty@ returns @Just tys@ iff
  -- the type @ty = T tys@, where T's unique = key
  -- key must not be `fUNTyConKey`; to test for functions, use `splitFunTy_maybe`.
  -- Thanks to this fact, we don't have to pattern match on `FunTy` here.
  isTyConKeyApp_maybe :: Unique -> Type -> Maybe [Type]
  isTyConKeyApp_maybe key ty
    | TyConApp tc args <- coreFullView ty
    , tc `hasKey` key
    = Just args
    | otherwise
    = Nothing
  {-# INLINE isTyConKeyApp_maybe #-}
  
  -- | Extract the RuntimeRep classifier of a type from its kind. For example,
  -- @kindRep * = LiftedRep@; Panics if this is not possible.
  -- Treats * and Constraint as the same
  kindRep :: HasDebugCallStack => Kind -> RuntimeRepType
  kindRep k = case kindRep_maybe k of
                Just r  -> r
                Nothing -> pprPanic "kindRep" (ppr k)
  
  -- | Given a kind (TYPE rr) or (CONSTRAINT rr), extract its RuntimeRep classifier rr.
  -- For example, @kindRep_maybe * = Just LiftedRep@
  -- Returns 'Nothing' if the kind is not of form (TYPE rr)
  kindRep_maybe :: HasDebugCallStack => Kind -> Maybe RuntimeRepType
  kindRep_maybe kind
    | Just (_, rep) <- sORTKind_maybe kind = Just rep
    | otherwise                            = Nothing
  
  -- | Returns True if the argument is (lifted) Type or Constraint
  -- See Note [TYPE and CONSTRAINT] in GHC.Builtin.Types.Prim
  isLiftedTypeKind :: Kind -> Bool
  isLiftedTypeKind kind
    = case kindRep_maybe kind of
        Just rep -> isLiftedRuntimeRep rep
        Nothing  -> False
  
  -- | Returns True if the kind classifies unlifted types (like 'Int#') and False
  -- otherwise. Note that this returns False for representation-polymorphic
  -- kinds, which may be specialized to a kind that classifies unlifted types.
  isUnliftedTypeKind :: Kind -> Bool
  isUnliftedTypeKind kind
    = case kindRep_maybe kind of
        Just rep -> isUnliftedRuntimeRep rep
        Nothing  -> False
  
  pickyIsLiftedTypeKind :: Kind -> Bool
  -- Checks whether the kind is literally
  --      TYPE LiftedRep
  -- or   TYPE ('BoxedRep 'Lifted)
  -- or   Type
  -- without expanding type synonyms or anything
  -- Used only when deciding whether to suppress the ":: *" in
  -- (a :: *) when printing kinded type variables
  -- See Note [Suppressing * kinds] in GHC.Core.TyCo.Ppr
  pickyIsLiftedTypeKind kind
    | TyConApp tc [arg] <- kind
    , tc `hasKey` tYPETyConKey
    , TyConApp rr_tc rr_args <- arg = case rr_args of
        [] -> rr_tc `hasKey` liftedRepTyConKey
        [rr_arg]
          | rr_tc `hasKey` boxedRepDataConKey
          , TyConApp lev [] <- rr_arg
          , lev `hasKey` liftedDataConKey -> True
        _ -> False
    | TyConApp tc [] <- kind
    , tc `hasKey` liftedTypeKindTyConKey = True
    | otherwise                          = False
  
  -- | Check whether a kind is of the form `TYPE (BoxedRep Lifted)`
  -- or `TYPE (BoxedRep Unlifted)`.
  --
  -- Returns:
  --
  --  - `Just Lifted` for `TYPE (BoxedRep Lifted)` and `Type`,
  --  - `Just Unlifted` for `TYPE (BoxedRep Unlifted)` and `UnliftedType`,
  --  - `Nothing` for anything else, e.g. `TYPE IntRep`, `TYPE (BoxedRep l)`, etc.
  kindBoxedRepLevity_maybe :: Type -> Maybe Levity
  kindBoxedRepLevity_maybe ty
    | Just rep <- kindRep_maybe ty
    , isBoxedRuntimeRep rep
    = runtimeRepLevity_maybe rep
    | otherwise
    = Nothing
  
  -- | Check whether a type of kind 'RuntimeRep' is lifted.
  --
  -- 'isLiftedRuntimeRep' is:
  --
  --  * True of @LiftedRep :: RuntimeRep@
  --  * False of type variables, type family applications,
  --    and of other reps such as @IntRep :: RuntimeRep@.
  isLiftedRuntimeRep :: RuntimeRepType -> Bool
  isLiftedRuntimeRep rep
    = runtimeRepLevity_maybe rep == Just Lifted
  
  -- | Check whether a type of kind 'RuntimeRep' is unlifted.
  --
  --  * True of definitely unlifted 'RuntimeRep's such as
  --    'UnliftedRep', 'IntRep', 'FloatRep', ...
  --  * False of 'LiftedRep',
  --  * False for type variables and type family applications.
  isUnliftedRuntimeRep :: RuntimeRepType -> Bool
  isUnliftedRuntimeRep rep =
    runtimeRepLevity_maybe rep == Just Unlifted
  
  -- | An INLINE helper for functions such as 'isLiftedLevity' and 'isUnliftedLevity'.
  --
  -- Checks whether the type is a nullary 'TyCon' application,
  -- for a 'TyCon' with the given 'Unique'.
  isNullaryTyConKeyApp :: Unique -> Type -> Bool
  isNullaryTyConKeyApp key ty
    | Just args <- isTyConKeyApp_maybe key ty
    = assert (null args) True
    | otherwise
    = False
  {-# INLINE isNullaryTyConKeyApp #-}
  
  isLiftedLevity :: Type -> Bool
  isLiftedLevity = isNullaryTyConKeyApp liftedDataConKey
  
  isUnliftedLevity :: Type -> Bool
  isUnliftedLevity = isNullaryTyConKeyApp unliftedDataConKey
  
  -- | Is this the type 'Levity'?
  isLevityTy :: Type -> Bool
  isLevityTy = isNullaryTyConKeyApp levityTyConKey
  
  -- | Is this the type 'RuntimeRep'?
  isRuntimeRepTy :: Type -> Bool
  isRuntimeRepTy = isNullaryTyConKeyApp runtimeRepTyConKey
  
  -- | Is a tyvar of type 'RuntimeRep'?
  isRuntimeRepVar :: TyVar -> Bool
  isRuntimeRepVar = isRuntimeRepTy . tyVarKind
  
  -- | Is a tyvar of type 'Levity'?
  isLevityVar :: TyVar -> Bool
  isLevityVar = isLevityTy . tyVarKind
  
  -- | Is this the type 'Multiplicity'?
  isMultiplicityTy :: Type -> Bool
  isMultiplicityTy  = isNullaryTyConKeyApp multiplicityTyConKey
  
  -- | Is a tyvar of type 'Multiplicity'?
  isMultiplicityVar :: TyVar -> Bool
  isMultiplicityVar = isMultiplicityTy . tyVarKind
  
  --------------------------------------------
  --  Splitting RuntimeRep
  --------------------------------------------
  
  -- | (splitRuntimeRep_maybe rr) takes a Type rr :: RuntimeRep, and
  --   returns the (TyCon,[Type]) for the RuntimeRep, if possible, where
  --   the TyCon is one of the promoted DataCons of RuntimeRep.
  -- Remember: the unique on TyCon that is a a promoted DataCon is the
  --           same as the unique on the DataCon
  --           See Note [Promoted data constructors] in GHC.Core.TyCon
  -- May not be possible if `rr` is a type variable or type
  --   family application
  splitRuntimeRep_maybe :: RuntimeRepType -> Maybe (TyCon, [Type])
  splitRuntimeRep_maybe rep
    | TyConApp rr_tc args <- coreFullView rep
    , isPromotedDataCon rr_tc
      -- isPromotedDataCon: be careful of type families (F tys) :: RuntimeRep,
    = Just (rr_tc, args)
    | otherwise
    = Nothing
  
  -- | See 'isBoxedRuntimeRep_maybe'.
  isBoxedRuntimeRep :: RuntimeRepType -> Bool
  isBoxedRuntimeRep rep = isJust (isBoxedRuntimeRep_maybe rep)
  
  -- | `isBoxedRuntimeRep_maybe (rep :: RuntimeRep)` returns `Just lev` if `rep`
  -- expands to `Boxed lev` and returns `Nothing` otherwise.
  --
  -- Types with this runtime rep are represented by pointers on the GC'd heap.
  isBoxedRuntimeRep_maybe :: RuntimeRepType -> Maybe LevityType
  isBoxedRuntimeRep_maybe rep
    | Just (rr_tc, args) <- splitRuntimeRep_maybe rep
    , rr_tc `hasKey` boxedRepDataConKey
    , [lev] <- args
    = Just lev
    | otherwise
    = Nothing
  
  -- | Check whether a type (usually of kind 'RuntimeRep') is lifted, unlifted,
  --   or unknown.  Returns Nothing if the type isn't of kind 'RuntimeRep'.
  --
  -- `runtimeRepLevity_maybe rr` returns:
  --
  --   * `Just Lifted` if `rr` is `LiftedRep :: RuntimeRep`
  --   * `Just Unlifted` if `rr` is definitely unlifted, e.g. `IntRep`
  --   * `Nothing` if not known (e.g. it's a type variable or a type family application).
  runtimeRepLevity_maybe :: RuntimeRepType -> Maybe Levity
  runtimeRepLevity_maybe rep
    | Just (rr_tc, args) <- splitRuntimeRep_maybe rep
    =       -- NB: args might be non-empty e.g. TupleRep [r1, .., rn]
      if (rr_tc `hasKey` boxedRepDataConKey)
      then case args of
              [lev] -> levityType_maybe lev
              _     -> Nothing  -- Type isn't of kind RuntimeRep
                       -- The latter case happens via the call to isLiftedRuntimeRep
                       -- in GHC.Tc.Errors.Ppr.pprMismatchMsg (#22742)
      else Just Unlifted
          -- Avoid searching all the unlifted RuntimeRep type cons
          -- In the RuntimeRep data type, only LiftedRep is lifted
    | otherwise
    = Nothing
  
  --------------------------------------------
  --  Splitting Levity
  --------------------------------------------
  
  -- | `levityType_maybe` takes a Type of kind Levity, and returns its levity
  -- May not be possible for a type variable or type family application
  levityType_maybe :: LevityType -> Maybe Levity
  levityType_maybe lev
    | TyConApp lev_tc args <- coreFullView lev
    = if | lev_tc `hasKey` liftedDataConKey   -> assert( null args) $ Just Lifted
         | lev_tc `hasKey` unliftedDataConKey -> assert( null args) $ Just Unlifted
         | otherwise                          -> Nothing
    | otherwise
    = Nothing
  
  
  {- *********************************************************************
  *                                                                      *
                 mapType
  *                                                                      *
  ************************************************************************
  
  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 ForAllCo case of mapTyCoX]
  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  As noted in Note [ForAllCo] in GHC.Core.TyCo.Rep, a ForAllCo is a bit redundant.
  It stores a TyCoVar and a Coercion, where the kind of the TyCoVar 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 TyCoVar'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 INLINE pragmas are indispensable. mapTyCo and mapTyCoX are used
  to implement zonking, and it's vital that they get specialised to the TcM
  monad and the particular mapper in use.
  
  Even specialising to the monad alone made a 20% allocation difference
  in perf/compiler/T5030.
  
  See Note [Specialising foldType] in "GHC.Core.TyCo.Rep" for more details of this
  idiom.
  -}
  
  -- | This describes how a "map" operation over a type/coercion should behave
  data TyCoMapper env m
    = TyCoMapper
        { tcm_tyvar :: env -> TyVar -> m Type
        , tcm_covar :: env -> CoVar -> m Coercion
        , tcm_hole  :: env -> CoercionHole -> m Coercion
            -- ^ What to do with coercion holes.
            -- See Note [Coercion holes] in "GHC.Core.TyCo.Rep".
  
        , tcm_tycobinder :: forall r. env -> TyCoVar -> ForAllTyFlag
                         -> (env -> TyCoVar -> m r) -> m r
            -- ^ The returned env is used in the extended scope
  
        , tcm_tycon :: TyCon -> m TyCon
            -- ^ This is used only for TcTyCons
            -- a) To zonk TcTyCons
            -- b) To turn TcTyCons into TyCons.
            --    See Note [Type checking recursive type and class declarations]
            --    in "GHC.Tc.TyCl"
        }
  
  {-# INLINE mapTyCo #-}  -- See Note [Specialising mappers]
  mapTyCo :: Monad m => TyCoMapper () m
          -> ( Type       -> m  Type
             , [Type]     -> m  [Type]
             , Coercion   -> m  Coercion
             , [Coercion] -> m [Coercion] )
  mapTyCo mapper
    = case mapTyCoX mapper of
       (go_ty, go_tys, go_co, go_cos)
          -> (go_ty (), go_tys (), go_co (), go_cos ())
  
  {-# INLINE mapTyCoX #-}  -- See Note [Specialising mappers]
  mapTyCoX :: forall m env. Monad m
           => TyCoMapper env m
           -> ( env -> Type       -> m Type
              , env -> [Type]     -> m [Type]
              , env -> Coercion   -> m Coercion
              , env -> [Coercion] -> m [Coercion] )
  mapTyCoX (TyCoMapper { tcm_tyvar = tyvar
                       , tcm_tycobinder = tycobinder
                       , tcm_tycon = tycon
                       , tcm_covar = covar
                       , tcm_hole = cohole })
    = (go_ty, go_tys, go_co, go_cos)
    where
      -- See Note [Use explicit recursion in mapTyCo]
      go_tys !_   []       = return []
      go_tys !env (ty:tys) = (:) <$> go_ty env ty <*> go_tys env tys
  
      go_ty !env (TyVarTy tv)    = tyvar env tv
      go_ty !env (AppTy t1 t2)   = mkAppTy <$> go_ty env t1 <*> go_ty env t2
      go_ty !_   ty@(LitTy {})   = return ty
      go_ty !env (CastTy ty co)  = mkCastTy <$> go_ty env ty <*> go_co env co
      go_ty !env (CoercionTy co) = CoercionTy <$> go_co env co
  
      go_ty !env ty@(FunTy _ w arg res)
        = do { w' <- go_ty env w; arg' <- go_ty env arg; res' <- go_ty env res
             ; return (ty { ft_mult = w', ft_arg = arg', ft_res = res' }) }
  
      go_ty !env ty@(TyConApp tc tys)
        | isTcTyCon tc
        = do { tc' <- tycon tc
             ; mkTyConApp tc' <$> go_tys env tys }
  
        -- Not a TcTyCon
        | null tys    -- Avoid allocation in this very
        = return ty   -- common case (E.g. Int, LiftedRep etc)
  
        | otherwise
        = mkTyConApp tc <$> go_tys env tys
  
      go_ty !env (ForAllTy (Bndr tv vis) inner)
        = do { tycobinder env tv vis $ \env' tv' -> do
             ; inner' <- go_ty env' inner
             ; return $ ForAllTy (Bndr tv' vis) inner' }
  
      -- See Note [Use explicit recursion in mapTyCo]
      go_cos !_   []       = return []
      go_cos !env (co:cos) = (:) <$> go_co env co <*> go_cos env cos
  
      go_mco !_   MRefl    = return MRefl
      go_mco !env (MCo co) = MCo <$> (go_co env co)
  
      go_co :: env -> Coercion -> m Coercion
      go_co !env (Refl ty)                  = Refl <$> go_ty env ty
      go_co !env (GRefl r ty mco)           = mkGReflCo r <$> go_ty env ty <*> go_mco env mco
      go_co !env (AppCo c1 c2)              = mkAppCo <$> go_co env c1 <*> go_co env c2
      go_co !env (FunCo r afl afr cw c1 c2) = mkFunCo2 r afl afr <$> go_co env cw
                                             <*> go_co env c1 <*> go_co env c2
      go_co !env (CoVarCo cv)               = covar env cv
      go_co !env (HoleCo hole)              = cohole env hole
      go_co !env (UnivCo { uco_prov = p, uco_role = r
                         , uco_lty = t1, uco_rty = t2, uco_deps = deps })
                                            = mkUnivCo <$> pure p
                                                       <*> go_cos env deps
                                                       <*> pure r
                                                       <*> go_ty env t1 <*> go_ty env t2
      go_co !env (SymCo co)                 = mkSymCo <$> go_co env co
      go_co !env (TransCo c1 c2)            = mkTransCo <$> go_co env c1 <*> go_co env c2
      go_co !env (AxiomCo r cos)            = mkAxiomCo r <$> go_cos env cos
      go_co !env (SelCo i co)               = mkSelCo i <$> go_co env co
      go_co !env (LRCo lr co)               = mkLRCo lr <$> go_co env co
      go_co !env (InstCo co arg)            = mkInstCo <$> go_co env co <*> go_co env arg
      go_co !env (KindCo co)                = mkKindCo <$> go_co env co
      go_co !env (SubCo co)                 = mkSubCo <$> go_co env co
      go_co !env co@(TyConAppCo r tc cos)
        | isTcTyCon tc
        = do { tc' <- tycon tc
             ; mkTyConAppCo r tc' <$> go_cos env cos }
  
        -- Not a TcTyCon
        | null cos    -- Avoid allocation in this very
        = return co   -- common case (E.g. Int, LiftedRep etc)
  
        | otherwise
        = mkTyConAppCo r tc <$> go_cos env cos
      go_co !env (ForAllCo { fco_tcv = tv, fco_visL = visL, fco_visR = visR
                           , fco_kind = kind_co, fco_body = co })
        = do { kind_co' <- go_mco env kind_co
             ; tycobinder env tv visL $ \env' tv' ->  do
             ; co' <- go_co env' co
             ; return $ mkForAllCo tv' visL visR kind_co' co' }
          -- See Note [Efficiency for ForAllCo case of mapTyCoX]
  
  
  {- Note [Use explicit recursion in mapTyCo]
  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  We use explicit recursion in `mapTyCo`, rather than calling, say, `strictFoldDVarSet`,
  for exactly the same reason as in Note [Use explicit recursion in foldTyCo] in
  GHC.Core.TyCo.Rep. We are in a monadic context, and using too-clever higher order
  functions makes the strictness analyser produce worse results.
  
  We could probably use `foldr`, since it is inlined bodily, fairly early; but
  I'm doing the simple thing and inlining it by hand.
  
  See !12037 for performance glitches caused by using `strictFoldDVarSet` (which is
  definitely not inlined bodily).
  -}
  
  {- *********************************************************************
  *                                                                      *
                        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 :: HasDebugCallStack => Type -> TyVar
  getTyVar ty = case getTyVar_maybe ty of
                      Just tv -> tv
                      Nothing -> pprPanic "getTyVar" (ppr ty)
  
  -- | Attempts to obtain the type variable underlying a 'Type'
  getTyVar_maybe :: Type -> Maybe TyVar
  getTyVar_maybe = repGetTyVar_maybe . coreFullView
  
  -- | 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
  
  isTyVarTy :: Type -> Bool
  isTyVarTy ty = isJust (getTyVar_maybe ty)
  
  -- | If the type is a tyvar, possibly under a cast, returns it, along
  -- with the coercion. Thus, the co is :: kind tv ~N kind ty
  getCastedTyVar_maybe :: Type -> Maybe (TyVar, CoercionN)
  getCastedTyVar_maybe ty = case coreFullView ty of
    CastTy (TyVarTy tv) co -> Just (tv, co)
    TyVarTy tv             -> Just (tv, mkReflCo Nominal (tyVarKind tv))
    _                      -> Nothing
  
  
  {- *********************************************************************
  *                                                                      *
                        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 contrast, we *can* unify (a b) with (t1 -> t2).
  Here's an example (#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.
  
  We are willing to split (t1 -=> t2) because the argument is still of
  kind Type, not Constraint.  So the criterion is isVisibleFunArg.
  
  In Core there is no real reason to avoid such decomposition.  But for now I've
  put the test in splitAppTyNoView_maybe, which applies throughout, because the
  other calls to splitAppTy are in GHC.Core.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
    -- See Note [Respecting definitional equality], invariant (EQ1).
  mkAppTy (CastTy fun_ty co) arg_ty
    | ([arg_co], res_co) <- decomposePiCos co (coercionKind co) [arg_ty]
    = (fun_ty `mkAppTy` (arg_ty `mkCastTy` arg_co)) `mkCastTy` res_co
  
  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
          --
          -- Moreover in GHC.Tc.Types.tcInferTyApps we build up a type
          --   (T t1 t2 t3) one argument at a type, thus forming
          --   (T t1), (T t1 t2), etc
  
  mkAppTys :: Type -> [Type] -> Type
  mkAppTys ty1                []   = ty1
  mkAppTys (CastTy fun_ty co) arg_tys  -- much more efficient then nested mkAppTy
                                       -- Why do this? See (EQ1) of
                                       -- Note [Respecting definitional equality]
                                       -- in GHC.Core.TyCo.Rep
    = foldl' AppTy ((mkAppTys fun_ty casted_arg_tys) `mkCastTy` res_co) leftovers
    where
      (arg_cos, res_co) = decomposePiCos co (coercionKind co) arg_tys
      (args_to_cast, leftovers) = splitAtList arg_cos arg_tys
      casted_arg_tys = zipWith mkCastTy args_to_cast arg_cos
  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 = splitAppTyNoView_maybe . coreFullView
  
  splitAppTy :: Type -> (Type, Type)
  -- ^ Attempts to take a type application apart, as in 'splitAppTy_maybe',
  -- and panics if this is not possible
  splitAppTy ty = splitAppTy_maybe ty `orElse` pprPanic "splitAppTy" (ppr ty)
  
  -------------
  splitAppTyNoView_maybe :: HasDebugCallStack => Type -> Maybe (Type,Type)
  -- ^ Does the AppTy split as in 'splitAppTy_maybe', but assumes that
  -- any coreView stuff is already done
  splitAppTyNoView_maybe (AppTy ty1 ty2)
    = Just (ty1, ty2)
  
  splitAppTyNoView_maybe (FunTy af w ty1 ty2)
    | Just (tc, tys)   <- funTyConAppTy_maybe af w ty1 ty2
    , Just (tys', ty') <- snocView tys
    = Just (TyConApp tc tys', ty')
  
  splitAppTyNoView_maybe (TyConApp tc tys)
    | not (tyConMustBeSaturated tc) || tys `lengthExceeds` tyConArity tc
    , Just (tys', ty') <- snocView tys
    = Just (TyConApp tc tys', ty')    -- Never create unsaturated type family apps!
  
  splitAppTyNoView_maybe _other = Nothing
  
  tcSplitAppTyNoView_maybe :: Type -> Maybe (Type,Type)
  -- ^ Just like splitAppTyNoView_maybe, but does not split (c => t)
  -- See Note [Decomposing fat arrow c=>t]
  tcSplitAppTyNoView_maybe ty
    | FunTy { ft_af = af } <- ty
    , not (isVisibleFunArg af)  -- See Note [Decomposing fat arrow c=>t]
    = Nothing
    | otherwise
    = splitAppTyNoView_maybe ty
  
  -------------
  splitAppTys :: HasDebugCallStack => 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 | tyConMustBeSaturated tc = tyConArity tc
                | otherwise               = 0
              (tc_args1, tc_args2) = splitAt n tc_args
          in
          (TyConApp tc tc_args1, tc_args2 ++ args)
      split _   (FunTy af w ty1 ty2) args
        | Just (tc,tys) <- funTyConAppTy_maybe af w ty1 ty2
        = assert (null args )
          (TyConApp tc [], tys)
  
      split orig_ty _ args  = (orig_ty, args)
  
  -- | Like 'splitAppTys', but doesn't look through type synonyms
  splitAppTysNoView :: HasDebugCallStack => Type -> (Type, [Type])
  splitAppTysNoView ty = split ty []
    where
      split (AppTy ty arg) args = split ty (arg:args)
      split (TyConApp tc tc_args) args
        = let n | tyConMustBeSaturated tc = tyConArity tc
                | otherwise               = 0
              (tc_args1, tc_args2) = splitAt n tc_args
          in
          (TyConApp tc tc_args1, tc_args2 ++ args)
      split (FunTy af w ty1 ty2) args
        | Just (tc, tys) <- funTyConAppTy_maybe af w ty1 ty2
        = assert (null args )
          (TyConApp tc [], tys)
  
      split ty args = (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
    | LitTy (NumTyLit n) <- coreFullView ty = Just n
    | otherwise                             = 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
    | LitTy (StrTyLit s) <- coreFullView ty = Just s
    | otherwise                             = Nothing
  
  mkCharLitTy :: Char -> Type
  mkCharLitTy c = LitTy (CharTyLit c)
  
  -- | Is this a char literal? We also look through type synonyms.
  isCharLitTy :: Type -> Maybe Char
  isCharLitTy ty
    | LitTy (CharTyLit s) <- coreFullView ty = Just s
    | otherwise                              = Nothing
  
  
  -- | Is this a type literal (symbol, numeric, or char)?
  isLitTy :: Type -> Maybe TyLit
  isLitTy ty
    | LitTy l <- coreFullView ty = Just l
    | otherwise                  = Nothing
  
  -- | A type of kind 'ErrorMessage' (from the 'GHC.TypeError' module).
  type ErrorMsgType = Type
  
  -- | Render a type corresponding to a user type error into a SDoc.
  pprUserTypeErrorTy :: ErrorMsgType -> 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 unevaluated type function
      _ -> ppr ty
  
  {- *********************************************************************
  *                                                                      *
                        FunTy
  *                                                                      *
  ********************************************************************* -}
  
  {- Note [Representation of function types]
  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  Functions (e.g. Int -> Char) can be thought of as being applications
  of funTyCon (known in Haskell surface syntax as (->)), (note that
  `RuntimeRep' quantifiers are left inferred)
  
      (->) :: forall {r1 :: RuntimeRep} {r2 :: RuntimeRep}
                     (a :: TYPE r1) (b :: TYPE r2).
              a -> b -> Type
  
  However, for efficiency's sake we represent saturated applications of (->)
  with FunTy. For instance, the type,
  
      (->) r1 r2 a b
  
  is equivalent to,
  
      FunTy (Anon a) b
  
  Note how the RuntimeReps are implied in the FunTy representation. For this
  reason we must be careful when reconstructing the TyConApp representation (see,
  for instance, splitTyConApp_maybe).
  
  In the compiler we maintain the invariant that all saturated applications of
  (->) are represented with FunTy.
  
  See #11714.
  -}
  
  -----------------------------------------------
  funTyConAppTy_maybe :: FunTyFlag -> Type -> Type -> Type
                      -> Maybe (TyCon, [Type])
  -- ^ Given the components of a FunTy
  -- figure out the corresponding TyConApp.
  funTyConAppTy_maybe af mult arg res
    | Just arg_rep <- getRuntimeRep_maybe arg
    , Just res_rep <- getRuntimeRep_maybe res
    -- If you're changing the lines below, you'll probably want to adapt the
    -- `fUNTyCon` case of GHC.Core.Unify.unify_ty correspondingly.
    , let args | isFUNArg af = [mult, arg_rep, res_rep, arg, res]
               | otherwise   = [      arg_rep, res_rep, arg, res]
    = Just $ (funTyFlagTyCon af, args)
    | otherwise
    = Nothing
  
  tyConAppFunTy_maybe :: HasDebugCallStack => TyCon -> [Type] -> Maybe Type
  -- ^ Return Just if this TyConApp should be represented as a FunTy
  tyConAppFunTy_maybe tc tys
    | Just (af, mult, arg, res) <- ty_con_app_fun_maybe manyDataConTy tc tys
              = Just (FunTy { ft_af = af, ft_mult = mult, ft_arg = arg, ft_res = res })
    | otherwise = Nothing
  
  tyConAppFunCo_maybe :: HasDebugCallStack => Role -> TyCon -> [Coercion]
                      -> Maybe Coercion
  -- ^ Return Just if this TyConAppCo should be represented as a FunCo
  tyConAppFunCo_maybe r tc cos
    | Just (af, mult, arg, res) <- ty_con_app_fun_maybe mult_refl tc cos
    = Just (mkFunCo r af mult arg res)
    | otherwise
    = Nothing
    where
      mult_refl = mkReflCo (funRole r SelMult) manyDataConTy
  
  ty_con_app_fun_maybe :: (HasDebugCallStack, Outputable a) => a -> TyCon -> [a]
                       -> Maybe (FunTyFlag, a, a, a)
  {-# INLINE ty_con_app_fun_maybe #-}
  -- Specialise this function for its two call sites
  ty_con_app_fun_maybe many_ty_co tc args
    | tc_uniq == fUNTyConKey     = fUN_case
    | tc_uniq == tcArrowTyConKey = non_FUN_case FTF_T_C
    | tc_uniq == ctArrowTyConKey = non_FUN_case FTF_C_T
    | tc_uniq == ccArrowTyConKey = non_FUN_case FTF_C_C
    | otherwise                  = Nothing
    where
      tc_uniq = tyConUnique tc
  
      fUN_case
        | (w:_r1:_r2:a1:a2:rest) <- args
        = assertPpr (null rest) (ppr tc <+> ppr args) $
          Just (FTF_T_T, w, a1, a2)
        | otherwise = Nothing
  
      non_FUN_case ftf
        | (_r1:_r2:a1:a2:rest) <- args
        = assertPpr (null rest) (ppr tc <+> ppr args) $
          Just (ftf, many_ty_co, a1, a2)
        | otherwise
        = Nothing
  
  mkFunctionType :: HasDebugCallStack => Mult -> Type -> Type -> Type
  -- ^ This one works out the FunTyFlag from the argument type
  -- See GHC.Types.Var Note [FunTyFlag]
  mkFunctionType mult arg_ty res_ty
   = FunTy { ft_af = af, ft_arg = arg_ty, ft_res = res_ty
           , ft_mult = assertPpr mult_ok (ppr [mult, arg_ty, res_ty]) $
                       mult }
    where
      af = chooseFunTyFlag arg_ty res_ty
      mult_ok = isVisibleFunArg af || isManyTy mult
  
  mkScaledFunctionTys :: [Scaled Type] -> Type -> Type
  -- ^ Like mkFunctionType, compute the FunTyFlag from the arguments
  mkScaledFunctionTys arg_tys res_ty
    = foldr mk res_ty arg_tys
    where
      mk (Scaled mult arg_ty) res_ty
        = mkFunTy (chooseFunTyFlag arg_ty res_ty)
                  mult arg_ty res_ty
  
  chooseFunTyFlag :: HasDebugCallStack => Type -> Type -> FunTyFlag
  -- ^ See GHC.Types.Var Note [FunTyFlag]
  chooseFunTyFlag arg_ty res_ty
    = mkFunTyFlag (typeTypeOrConstraint arg_ty) (typeTypeOrConstraint res_ty)
  
  splitFunTy :: Type -> (Mult, Type, Type)
  -- ^ Attempts to extract the multiplicity, argument and result types from a type,
  -- and panics if that is not possible. See also 'splitFunTy_maybe'
  splitFunTy ty = case splitFunTy_maybe ty of
                     Just (_af, mult, arg, res) -> (mult,arg,res)
                     Nothing                    -> pprPanic "splitFunTy" (ppr ty)
  
  {-# INLINE splitFunTy_maybe #-}
  splitFunTy_maybe :: Type -> Maybe (FunTyFlag, Mult, Type, Type)
  -- ^ Attempts to extract the multiplicity, argument and result types from a type
  splitFunTy_maybe ty
    | FunTy af w arg res <- coreFullView ty = Just (af, w, arg, res)
    | otherwise                             = Nothing
  
  {-# INLINE splitVisibleFunTy_maybe #-}
  splitVisibleFunTy_maybe :: Type -> Maybe (Type, Type)
  -- ^ Works on visible function types only (t1 -> t2), and
  --   returns t1 and t2, but not the multiplicity
  splitVisibleFunTy_maybe ty
    | FunTy af _ arg res <- coreFullView ty
    , isVisibleFunArg af = Just (arg, res)
    | otherwise          = Nothing
  
  splitFunTys :: Type -> ([Scaled Type], Type)
  splitFunTys ty = split [] ty ty
    where
        -- common case first
      split args _       (FunTy _ w arg res) = split (Scaled w arg : args) res res
      split args orig_ty ty | Just ty' <- coreView ty = split args orig_ty ty'
      split args orig_ty _                   = (reverse args, orig_ty)
  
  funResultTy :: HasDebugCallStack => Type -> Type
  -- ^ Extract the function result type and panic if that is not possible
  funResultTy ty
    | FunTy { ft_res = res } <- coreFullView ty = res
    | otherwise                                 = pprPanic "funResultTy" (ppr ty)
  
  funArgTy :: HasDebugCallStack => Type -> Type
  -- ^ Extract the function argument type and panic if that is not possible
  funArgTy ty
    | FunTy { ft_arg = arg } <- coreFullView ty = arg
    | otherwise                                 = pprPanic "funArgTy" (ppr ty)
  
  -- ^ Just like 'piResultTys' but for a single argument
  -- Try not to iterate 'piResultTy', because it's inefficient to substitute
  -- one variable at a time; instead use 'piResultTys"
  piResultTy :: HasDebugCallStack => Type -> Type ->  Type
  piResultTy ty arg = case piResultTy_maybe ty arg of
                        Just res -> res
                        Nothing  -> pprPanic "piResultTy" (ppr ty $$ ppr arg)
  
  piResultTy_maybe :: Type -> Type -> Maybe Type
  -- We don't need a 'tc' version, because
  -- this function behaves the same for Type and Constraint
  piResultTy_maybe ty arg = case coreFullView ty of
    FunTy { ft_res = res } -> Just res
  
    ForAllTy (Bndr tv _) res
      -> let empty_subst = mkEmptySubst $ mkInScopeSet $
                           tyCoVarsOfTypes [arg,res]
         in Just (substTy (extendTCvSubst empty_subst tv arg) res)
  
    _ -> Nothing
  
  -- | (piResultTys f_ty [ty1, .., tyn]) gives the type of (f ty1 .. tyn)
  --   where f :: f_ty
  -- 'piResultTys' is interesting because:
  --      1. 'f_ty' may have more for-alls than there are args
  --      2. Less obviously, it may have fewer for-alls
  -- For case 2. think of:
  --   piResultTys (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 GHC.Core.Lint
  
  -- This is a heavily used function (e.g. from typeKind),
  -- so we pay attention to efficiency, especially in the special case
  -- where there are no for-alls so we are just dropping arrows from
  -- a function type/kind.
  piResultTys :: HasDebugCallStack => Type -> [Type] -> Type
  piResultTys ty [] = ty
  piResultTys ty orig_args@(arg:args)
    | FunTy { ft_res = res } <- ty
    = piResultTys res args
  
    | ForAllTy (Bndr tcv _) res <- ty
    = -- Both type and coercion variables
      go (extendTCvSubst init_subst tcv arg) res args
  
    | Just ty' <- coreView ty
    = piResultTys ty' orig_args
  
    | otherwise
    = pprPanic "piResultTys1" (ppr ty $$ ppr orig_args)
    where
      init_subst = mkEmptySubst $ mkInScopeSet (tyCoVarsOfTypes (ty:orig_args))
  
      go :: Subst -> Type -> [Type] -> Type
      go subst ty [] = substTyUnchecked subst ty
  
      go subst ty all_args@(arg:args)
        | FunTy { ft_res = res } <- ty
        = go subst res args
  
        | ForAllTy (Bndr tv _) res <- ty
        = go (extendTCvSubst subst tv arg) res args
  
        | Just ty' <- coreView ty
        = go subst ty' all_args
  
        | not (isEmptyTCvSubst subst)  -- See Note [Care with kind instantiation]
        = go init_subst
            (substTy subst ty)
            all_args
  
        | otherwise
        = -- We have not run out of arguments, but the function doesn't
          -- have the right kind to apply to them; so panic.
          -- Without the explicit isEmptyVarEnv test, an ill-kinded type
          -- would give an infinite loop, which is very unhelpful
          -- c.f. #15473
          pprPanic "piResultTys2" (ppr ty $$ ppr orig_args $$ ppr all_args)
  
  applyTysX :: HasDebugCallStack => [TyVar] -> Type -> [Type] -> Type
  -- applyTysX beta-reduces (/\tvs. body_ty) arg_tys
  -- Assumes that (/\tvs. body_ty) is closed
  applyTysX tvs body_ty arg_tys
    = assertPpr (tvs `leLength` arg_tys) pp_stuff $
      assertPpr (tyCoVarsOfType body_ty `subVarSet` mkVarSet tvs) pp_stuff $
      mkAppTys (substTyWith tvs arg_tys_prefix body_ty)
               arg_tys_rest
    where
      pp_stuff = vcat [ppr tvs, ppr body_ty, ppr arg_tys]
      (arg_tys_prefix, arg_tys_rest) = splitAtList tvs arg_tys
  
  
  {- Note [Care with kind instantiation]
  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  Suppose we have
    T :: forall k. k
  and we are finding the kind of
    T (forall b. b -> b) * Int
  Then
    T (forall b. b->b) :: k[ k :-> forall b. b->b]
                       :: forall b. b -> b
  So
    T (forall b. b->b) * :: (b -> b)[ b :-> *]
                         :: * -> *
  
  In other words we must instantiate the forall!
  
  Similarly (#15428)
     S :: forall k f. k -> f k
  and we are finding the kind of
     S * (* ->) Int Bool
  We have
     S * (* ->) :: (k -> f k)[ k :-> *, f :-> (* ->)]
                :: * -> * -> *
  So again we must instantiate.
  
  The same thing happens in GHC.CoreToIface.toIfaceAppArgsX.
  -}
  
  
  {- *********************************************************************
  *                                                                      *
                        TyConApp
  *                                                                      *
  ********************************************************************* -}
  
  -- 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 (FunTy { ft_af = af }) = Just (funTyFlagTyCon af)
  tyConAppTyConPicky_maybe _                      = Nothing
  
  
  -- | The same as @fst . splitTyConApp@
  -- We can short-cut the FunTy case
  {-# INLINE tyConAppTyCon_maybe #-}
  tyConAppTyCon_maybe :: Type -> Maybe TyCon
  tyConAppTyCon_maybe ty = case coreFullView ty of
    TyConApp tc _        -> Just tc
    FunTy { ft_af = af } -> Just (funTyFlagTyCon af)
    _                    -> Nothing
  
  tyConAppTyCon :: HasDebugCallStack => 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 = case splitTyConApp_maybe ty of
                            Just (_, tys) -> Just tys
                            Nothing       -> Nothing
  
  tyConAppArgs :: HasDebugCallStack => Type -> [Type]
  tyConAppArgs ty = tyConAppArgs_maybe ty `orElse` pprPanic "tyConAppArgs" (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 = splitTyConApp_maybe ty `orElse` 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 :: HasDebugCallStack => Type -> Maybe (TyCon, [Type])
  splitTyConApp_maybe ty = splitTyConAppNoView_maybe (coreFullView ty)
  
  splitTyConAppNoView_maybe :: HasDebugCallStack => Type -> Maybe (TyCon, [Type])
  -- Same as splitTyConApp_maybe but without looking through synonyms
  splitTyConAppNoView_maybe ty
    = case ty of
        FunTy { ft_af = af, ft_mult = w, ft_arg = arg, ft_res = res}
                        -> funTyConAppTy_maybe af w arg res
        TyConApp tc tys -> Just (tc, tys)
        _               -> Nothing
  
  -- | tcSplitTyConApp_maybe splits a type constructor application into
  -- its type constructor and applied types.
  --
  -- Differs from splitTyConApp_maybe in that it does *not* split types
  -- headed with (=>), as that's not a TyCon in the type-checker.
  --
  -- Note that this may fail (in funTyConAppTy_maybe) in the case
  -- of a 'FunTy' with an argument of unknown kind 'FunTy'
  -- (e.g. `FunTy (a :: k) Int`, since the kind of @a@ isn't of
  -- the form `TYPE rep`.  This isn't usually a problem but may
  -- be temporarily the case during canonicalization:
  --     see Note [Decomposing FunTy] in GHC.Tc.Solver.Equality
  --     and Note [The Purely Kinded Type Invariant (PKTI)] in GHC.Tc.Gen.HsType,
  --         Wrinkle around FunTy
  --
  -- Consequently, you may need to zonk your type before
  -- using this function.
  tcSplitTyConApp_maybe :: HasDebugCallStack => Type -> Maybe (TyCon, [Type])
  -- Defined here to avoid module loops between Unify and TcType.
  tcSplitTyConApp_maybe ty
    = case coreFullView ty of
        FunTy { ft_af = af, ft_mult = w, ft_arg = arg, ft_res = res}
                        | isVisibleFunArg af    -- Visible args only
                          -- See Note [Decomposing fat arrow c=>t]
                        -> funTyConAppTy_maybe af w arg res
        TyConApp tc tys -> Just (tc, tys)
        _               -> Nothing
  
  tcSplitTyConApp :: Type -> (TyCon, [Type])
  tcSplitTyConApp ty
    = tcSplitTyConApp_maybe ty `orElse` pprPanic "tcSplitTyConApp" (ppr ty)
  
  ---------------------------
  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
      = assertPpr (tvs `leLength` tys) (ppr tycon $$ ppr tys $$ ppr tvs) $
        applyTysX tvs rhs tys
    where
      (tvs, rhs) = newTyConEtadRhs tycon
  
  
  {- *********************************************************************
  *                                                                      *
                        CastTy
  *                                                                      *
  ********************************************************************* -}
  
  splitCastTy_maybe :: Type -> Maybe (Type, Coercion)
  splitCastTy_maybe ty
    | CastTy ty' co <- coreFullView ty = Just (ty', co)
    | otherwise                        = Nothing
  
  -- | Make a 'CastTy'. The Coercion must be nominal. Checks the
  -- Coercion for reflexivity, dropping it if it's reflexive.
  -- See @Note [Respecting definitional equality]@ in "GHC.Core.TyCo.Rep"
  mkCastTy :: Type -> Coercion -> Type
  mkCastTy orig_ty co | isReflexiveCo co = orig_ty  -- (EQ2) from the Note
  -- 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 orig_ty co = mk_cast_ty orig_ty co
  
  -- | Like 'mkCastTy', but avoids checking the coercion for reflexivity,
  -- as that can be expensive.
  mk_cast_ty :: Type -> Coercion -> Type
  mk_cast_ty orig_ty co = go orig_ty
    where
      go :: Type -> Type
      -- See Note [Using coreView in mk_cast_ty]
      go ty | Just ty' <- coreView ty = go ty'
  
      go (CastTy ty co1)
        -- (EQ3) from the Note
        = mkCastTy ty (co1 `mkTransCo` co)
            -- call mkCastTy again for the reflexivity check
  
      go (ForAllTy (Bndr tv vis) inner_ty)
        -- (EQ4) from the Note
        -- See Note [Weird typing rule for ForAllTy] in GHC.Core.TyCo.Rep.
        | isTyVar tv
        , let fvs = tyCoVarsOfCo co
        = -- have to make sure that pushing the co in doesn't capture the bound var!
          if tv `elemVarSet` fvs
          then let empty_subst = mkEmptySubst (mkInScopeSet fvs)
                   (subst, tv') = substVarBndr empty_subst tv
               in ForAllTy (Bndr tv' vis) (substTy subst inner_ty `mk_cast_ty` co)
          else ForAllTy (Bndr tv vis) (inner_ty `mk_cast_ty` co)
  
      go _ = CastTy orig_ty co -- NB: orig_ty: preserve synonyms if possible
  
  {-
  Note [Using coreView in mk_cast_ty]
  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  Invariants (EQ3) and (EQ4) of Note [Respecting definitional equality] in
  GHC.Core.TyCo.Rep must apply regardless of type synonyms. For instance,
  consider this example (#19742):
  
     type EqSameNat = () |> co
     useNatEq :: EqSameNat |> sym co
  
  (Those casts aren't visible in the user-source code, of course; see #19742 for
  what the user might write.)
  
  The type `EqSameNat |> sym co` looks as if it satisfies (EQ3), as it has no
  nested casts, but if we expand EqSameNat, we see that it doesn't.
  And then Bad Things happen.
  
  The solution is easy: just use `coreView` when establishing (EQ3) and (EQ4) in
  `mk_cast_ty`.
  -}
  
  {- *********************************************************************
  *                                                                      *
                       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)
  
  
  {- *********************************************************************
  *                                                                      *
                        ForAllTy
  *                                                                      *
  ********************************************************************* -}
  
  tyConBindersPiTyBinders :: [TyConBinder] -> [PiTyBinder]
  -- Return the tyConBinders in PiTyBinder form
  tyConBindersPiTyBinders = map to_tyb
    where
      to_tyb (Bndr tv (NamedTCB vis)) = Named (Bndr tv vis)
      to_tyb (Bndr tv AnonTCB)        = Anon (tymult (varType tv)) FTF_T_T
  
  -- | Make a dependent forall over a TyCoVar
  mkTyCoForAllTy :: TyCoVar -> ForAllTyFlag -> Type -> Type
  mkTyCoForAllTy tv vis ty
    | isCoVar tv
    , not (tv `elemVarSet` tyCoVarsOfType ty)
     -- Maintain ForAllTy's invariants
      -- See Note [Unused coercion variable in ForAllTy] in GHC.Core.TyCo.Rep
    = mkVisFunTyMany (varType tv) ty
    | otherwise
    = ForAllTy (mkForAllTyBinder vis tv) ty
  
  -- | Make a dependent forall over a TyCoVar
  mkTyCoForAllTys :: [ForAllTyBinder] -> Type -> Type
  mkTyCoForAllTys bndrs ty
    = foldr (\(Bndr var vis) -> mkTyCoForAllTy var vis) ty bndrs
  
  -- | Make a dependent forall over an 'Inferred' variable
  mkTyCoInvForAllTy :: TyCoVar -> Type -> Type
  mkTyCoInvForAllTy tv ty = mkTyCoForAllTy tv Inferred ty
  
  -- | Like 'mkTyCoInvForAllTy', but tv should be a tyvar
  mkInfForAllTy :: TyVar -> Type -> Type
  mkInfForAllTy tv ty = assert (isTyVar tv )
                        ForAllTy (Bndr tv Inferred) ty
  
  -- | Like 'mkForAllTys', but assumes all variables are dependent and
  -- 'Inferred', a common case
  mkTyCoInvForAllTys :: [TyCoVar] -> Type -> Type
  mkTyCoInvForAllTys tvs ty = foldr mkTyCoInvForAllTy ty tvs
  
  -- | Like 'mkTyCoInvForAllTys', but tvs should be a list of tyvar
  mkInfForAllTys :: [TyVar] -> Type -> Type
  mkInfForAllTys tvs ty = foldr mkInfForAllTy ty tvs
  
  -- | Like 'mkForAllTy', but assumes the variable is dependent and 'Specified',
  -- a common case
  mkSpecForAllTy :: TyVar -> Type -> Type
  mkSpecForAllTy tv ty = assert (isTyVar tv )
                         -- covar is always Inferred, so input should be tyvar
                         ForAllTy (Bndr tv Specified) ty
  
  -- | Like 'mkForAllTys', but assumes all variables are dependent and
  -- 'Specified', a common case
  mkSpecForAllTys :: [TyVar] -> Type -> Type
  mkSpecForAllTys tvs ty = foldr mkSpecForAllTy ty tvs
  
  -- | Like mkForAllTys, but assumes all variables are dependent and visible
  mkVisForAllTys :: [TyVar] -> Type -> Type
  mkVisForAllTys tvs = assert (all isTyVar tvs )
                       -- covar is always Inferred, so all inputs should be tyvar
                       mkForAllTys [ Bndr tv Required | tv <- tvs ]
  
  -- | Given a list of type-level vars and the free vars of a result kind,
  -- makes PiTyBinders, preferring anonymous binders
  -- if the variable is, in fact, not dependent.
  -- e.g.    mkTyConBindersPreferAnon [(k:*),(b:k),(c:k)] (k->k)
  -- We want (k:*) Named, (b:k) Anon, (c:k) Anon
  --
  -- All non-coercion binders are /visible/.
  mkTyConBindersPreferAnon :: [TyVar]      -- ^ binders
                           -> TyCoVarSet   -- ^ free variables of result
                           -> [TyConBinder]
  mkTyConBindersPreferAnon vars inner_tkvs = assert (all isTyVar vars)
                                             fst (go vars)
    where
      go :: [TyVar] -> ([TyConBinder], VarSet) -- also returns the free vars
      go [] = ([], inner_tkvs)
      go (v:vs) | v `elemVarSet` fvs
                = ( Bndr v (NamedTCB Required) : binders
                  , fvs `delVarSet` v `unionVarSet` kind_vars )
                | otherwise
                = ( Bndr v AnonTCB : binders
                  , fvs `unionVarSet` kind_vars )
        where
          (binders, fvs) = go vs
          kind_vars      = tyCoVarsOfType $ tyVarKind v
  
  -- | Take a ForAllTy apart, returning the binders and result type
  splitForAllForAllTyBinders :: Type -> ([ForAllTyBinder], Type)
  splitForAllForAllTyBinders ty = split ty ty []
    where
      split _ (ForAllTy b res) bs                   = split res res (b:bs)
      split orig_ty ty bs | Just ty' <- coreView ty = split orig_ty ty' bs
      split orig_ty _                bs             = (reverse bs, orig_ty)
  {-# INLINE splitForAllForAllTyBinders #-}
  
  -- | Take a ForAllTy apart, returning the list of tycovars 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.
  splitForAllTyCoVars :: Type -> ([TyCoVar], Type)
  splitForAllTyCoVars ty = split ty ty []
    where
      split _       (ForAllTy (Bndr tv _) ty)    tvs = split ty ty (tv:tvs)
      split orig_ty ty tvs | Just ty' <- coreView ty = split orig_ty ty' tvs
      split orig_ty _                            tvs = (reverse tvs, orig_ty)
  
  -- | Like 'splitForAllTyCoVars', but split only for tyvars.
  -- 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.
  splitForAllTyVars :: Type -> ([TyVar], Type)
  splitForAllTyVars ty = split ty ty []
    where
      split _ (ForAllTy (Bndr tv _) ty) tvs | isTyVar tv = split ty ty (tv:tvs)
      split orig_ty ty tvs | Just ty' <- coreView ty     = split orig_ty ty' tvs
      split orig_ty _                   tvs              = (reverse tvs, orig_ty)
  
  -- | Like 'splitForAllTyCoVars', but only splits 'ForAllTy's with 'Required' type
  -- variable binders. Furthermore, each returned tyvar is annotated with '()'.
  splitForAllReqTyBinders :: Type -> ([ReqTyBinder], Type)
  splitForAllReqTyBinders ty = split ty ty []
    where
      split _ (ForAllTy (Bndr tv Required) ty) tvs   = split ty ty (Bndr tv ():tvs)
      split orig_ty ty tvs | Just ty' <- coreView ty = split orig_ty ty' tvs
      split orig_ty _                   tvs          = (reverse tvs, orig_ty)
  
  -- | Like 'splitForAllTyCoVars', but only splits 'ForAllTy's with 'Invisible' type
  -- variable binders. Furthermore, each returned tyvar is annotated with its
  -- 'Specificity'.
  splitForAllInvisTyBinders :: Type -> ([InvisTyBinder], Type)
  splitForAllInvisTyBinders ty = split ty ty []
    where
      split _ (ForAllTy (Bndr tv (Invisible spec)) ty) tvs = split ty ty (Bndr tv spec:tvs)
      split orig_ty ty tvs | Just ty' <- coreView ty       = split orig_ty ty' tvs
      split orig_ty _                   tvs                = (reverse tvs, orig_ty)
  
  -- | Checks whether this is a proper forall (with a named binder)
  isForAllTy :: Type -> Bool
  isForAllTy ty
    | ForAllTy {} <- coreFullView ty = True
    | otherwise                      = False
  
  -- | Like `isForAllTy`, but returns True only if it is a tyvar binder
  isForAllTy_ty :: Type -> Bool
  isForAllTy_ty ty
    | ForAllTy (Bndr tv _) _ <- coreFullView ty
    , isTyVar tv
    = True
  
    | otherwise = False
  
  -- | Like `isForAllTy`, but returns True only if it is an inferred tyvar binder
  isForAllTy_invis_ty :: Type -> Bool
  isForAllTy_invis_ty  ty
    | ForAllTy (Bndr tv (Invisible InferredSpec)) _ <- coreFullView ty
    , isTyVar tv
    = True
  
    | otherwise = False
  
  -- | Like `isForAllTy`, but returns True only if it is a covar binder
  isForAllTy_co :: Type -> Bool
  isForAllTy_co ty
    | ForAllTy (Bndr tv _) _ <- coreFullView ty
    , isCoVar tv
    = True
  
    | otherwise = False
  
  -- | Is this a function or forall?
  isPiTy :: Type -> Bool
  isPiTy ty = case coreFullView ty of
    ForAllTy {} -> True
    FunTy {}    -> True
    _           -> False
  
  -- | Is this a function?
  -- Note: `forall {b}. Show b => b -> IO b` will not be considered a function by this function.
  --       It would merely be a forall wrapping a function type.
  isFunTy :: Type -> Bool
  isFunTy ty
    | FunTy {} <- coreFullView ty = True
    | otherwise                   = False
  
  -- | Take a forall type apart, or panics if that is not possible.
  splitForAllTyCoVar :: Type -> (TyCoVar, Type)
  splitForAllTyCoVar ty
    | Just answer <- splitForAllTyCoVar_maybe ty = answer
    | otherwise                                  = pprPanic "splitForAllTyCoVar" (ppr ty)
  
  -- | Drops all ForAllTys
  dropForAlls :: Type -> Type
  dropForAlls ty = go ty
    where
      go (ForAllTy _ res)            = go res
      go ty | Just ty' <- coreView ty = go ty'
      go res                         = res
  
  -- | Attempts to take a ForAllTy apart, returning the full ForAllTyBinder
  splitForAllForAllTyBinder_maybe :: Type -> Maybe (ForAllTyBinder, Type)
  splitForAllForAllTyBinder_maybe ty
    | ForAllTy bndr inner_ty <- coreFullView ty = Just (bndr, inner_ty)
    | otherwise                                 = Nothing
  
  
  -- | Attempts to take a ForAllTy apart, returning the Var
  splitForAllTyCoVar_maybe :: Type -> Maybe (TyCoVar, Type)
  splitForAllTyCoVar_maybe ty
    | ForAllTy (Bndr tv _) inner_ty <- coreFullView ty = Just (tv, inner_ty)
    | otherwise                                        = Nothing
  
  -- | Attempts to take a ForAllTy apart, but only if the binder is a TyVar
  splitForAllTyVar_maybe :: Type -> Maybe (TyVar, Type)
  splitForAllTyVar_maybe ty
    | ForAllTy (Bndr tv _) inner_ty <- coreFullView ty
    , isTyVar tv
    = Just (tv, inner_ty)
  
    | otherwise = Nothing
  
  -- | Like 'splitForAllTyCoVar_maybe', but only returns Just if it is a covar binder.
  splitForAllCoVar_maybe :: Type -> Maybe (CoVar, Type)
  splitForAllCoVar_maybe ty
    | ForAllTy (Bndr tv _) inner_ty <- coreFullView ty
    , isCoVar tv
    = Just (tv, inner_ty)
  
    | otherwise = Nothing
  
  -- | Attempts to take a forall type apart; works with proper foralls and
  -- functions
  {-# INLINE splitPiTy_maybe #-}  -- callers will immediately deconstruct
  splitPiTy_maybe :: Type -> Maybe (PiTyBinder, Type)
  splitPiTy_maybe ty = case coreFullView ty of
    ForAllTy bndr ty -> Just (Named bndr, ty)
    FunTy { ft_af = af, ft_mult = w, ft_arg = arg, ft_res = res}
                     -> Just (Anon (mkScaled w arg) af, res)
    _                -> Nothing
  
  -- | Takes a forall type apart, or panics
  splitPiTy :: Type -> (PiTyBinder, Type)
  splitPiTy ty
    | Just answer <- splitPiTy_maybe ty = answer
    | otherwise                         = pprPanic "splitPiTy" (ppr ty)
  
  -- | Split off all PiTyBinders to a type, splitting both proper foralls
  -- and functions
  splitPiTys :: Type -> ([PiTyBinder], Type)
  splitPiTys ty = split ty ty []
    where
      split _       (ForAllTy b res) bs = split res res (Named b  : bs)
      split _       (FunTy { ft_af = af, ft_mult = w, ft_arg = arg, ft_res = res }) bs
                                        = split res res (Anon (Scaled w arg) af : bs)
      split orig_ty ty bs | Just ty' <- coreView ty = split orig_ty ty' bs
      split orig_ty _                bs = (reverse bs, orig_ty)
  
  collectPiTyBinders :: Type -> [PiTyBinder]
  collectPiTyBinders ty = build $ \c n ->
    let
      split (ForAllTy b res) = Named b `c` split res
      split (FunTy { ft_af = af, ft_mult = w, ft_arg = arg, ft_res = res })
                             = Anon (Scaled w arg) af `c` split res
      split ty | Just ty' <- coreView ty = split ty'
      split _                = n
    in
      split ty
  {-# INLINE collectPiTyBinders #-}
  
  -- | Extracts a list of run-time arguments from a function type,
  -- looking through newtypes to the right of arrows.
  --
  -- Examples:
  --
  -- @
  --    newtype Identity a = I a
  --
  --    getRuntimeArgTys (Int -> Bool -> Double) == [(Int, FTF_T_T), (Bool, FTF_T_T)]
  --    getRuntimeArgTys (Identity Int -> Bool -> Double) == [(Identity Int, FTF_T_T), (Bool, FTF_T_T)]
  --    getRuntimeArgTys (Int -> Identity (Bool -> Identity Double)) == [(Int, FTF_T_T), (Bool, FTF_T_T)]
  --    getRuntimeArgTys (forall a. Show a => Identity a -> a -> Int -> Bool)
  --             == [(Show a, FTF_C_T), (Identity a, FTF_T_T),(a, FTF_T_T),(Int, FTF_T_T)]
  -- @
  --
  -- Note that, in the last case, the returned types might mention an out-of-scope
  -- type variable. This function is used only when we really care about the /kinds/
  -- of the returned types, so this is OK.
  --
  -- **Warning**: this function can return an infinite list. For example:
  --
  -- @
  --   newtype N a = MkN (a -> N a)
  --   getRuntimeArgTys (N a) == repeat (a, FTF_T_T)
  -- @
  getRuntimeArgTys :: Type -> [(Scaled Type, FunTyFlag)]
  getRuntimeArgTys = go
    where
      go :: Type -> [(Scaled Type, FunTyFlag)]
      go (ForAllTy _ res)
        = go res
      go (FunTy { ft_mult = w, ft_arg = arg, ft_res = res, ft_af = af })
        = (Scaled w arg, af) : go res
      go ty
        | Just ty' <- coreView ty
        = go ty'
        | Just (_,ty') <- topNormaliseNewType_maybe ty
        = go ty'
        | otherwise
        = []
  
  invisibleBndrCount :: Type -> Int
  -- Returns the number of leading invisible forall'd binders in the type
  -- Includes invisible predicate arguments; e.g. for
  --    e.g.  forall {k}. (k ~ *) => k -> k
  -- returns 2 not 1
  invisibleBndrCount ty = length (fst (splitInvisPiTys ty))
  
  -- | Like 'splitPiTys', but returns only *invisible* binders, including constraints.
  -- Stops at the first visible binder.
  splitInvisPiTys :: Type -> ([PiTyBinder], Type)
  splitInvisPiTys ty = split ty ty []
     where
      split _ (ForAllTy b res) bs
        | Bndr _ vis <- b
        , isInvisibleForAllTyFlag vis   = split res res (Named b  : bs)
      split _ (FunTy { ft_af = af, ft_mult = mult, ft_arg = arg, ft_res = res })  bs
        | isInvisibleFunArg af     = split res res (Anon (mkScaled mult arg) af : bs)
      split orig_ty ty bs
        | Just ty' <- coreView ty  = split orig_ty ty' bs
      split orig_ty _          bs  = (reverse bs, orig_ty)
  
  splitInvisPiTysN :: Int -> Type -> ([PiTyBinder], Type)
  -- ^ Same as 'splitInvisPiTys', but stop when
  --   - you have found @n@ 'PiTyBinder's,
  --   - or you run out of invisible binders
  splitInvisPiTysN n ty = split n ty ty []
     where
      split n orig_ty ty bs
        | n == 0                  = (reverse bs, orig_ty)
        | Just ty' <- coreView ty = split n orig_ty ty' bs
        | ForAllTy b res <- ty
        , Bndr _ vis <- b
        , isInvisibleForAllTyFlag vis  = split (n-1) res res (Named b  : bs)
        | FunTy { ft_af = af, ft_mult = mult, ft_arg = arg, ft_res = res } <- ty
        , isInvisibleFunArg af   = split (n-1) res res (Anon (Scaled mult arg) af : bs)
        | otherwise              = (reverse bs, orig_ty)
  
  -- | Given a 'TyCon' and a list of argument types, filter out any invisible
  -- (i.e., 'Inferred' or 'Specified') arguments.
  filterOutInvisibleTypes :: TyCon -> [Type] -> [Type]
  filterOutInvisibleTypes tc tys = snd $ partitionInvisibleTypes tc tys
  
  -- | Given a 'TyCon' and a list of argument types, filter out any 'Inferred'
  -- arguments.
  filterOutInferredTypes :: TyCon -> [Type] -> [Type]
  filterOutInferredTypes tc tys =
    filterByList (map (/= Inferred) $ tyConForAllTyFlags tc tys) tys
  
  -- | Given a 'TyCon' and a list of argument types, partition the arguments
  -- into:
  --
  -- 1. 'Inferred' or 'Specified' (i.e., invisible) arguments and
  --
  -- 2. 'Required' (i.e., visible) arguments
  partitionInvisibleTypes :: TyCon -> [Type] -> ([Type], [Type])
  partitionInvisibleTypes tc tys =
    partitionByList (map isInvisibleForAllTyFlag $ tyConForAllTyFlags tc tys) tys
  
  -- | Given a list of things paired with their visibilities, partition the
  -- things into (invisible things, visible things).
  partitionInvisibles :: [(a, ForAllTyFlag)] -> ([a], [a])
  partitionInvisibles = partitionWith pick_invis
    where
      pick_invis :: (a, ForAllTyFlag) -> Either a a
      pick_invis (thing, vis) | isInvisibleForAllTyFlag vis = Left thing
                              | otherwise              = Right thing
  
  -- | Given a 'TyCon' and a list of argument types to which the 'TyCon' is
  -- applied, determine each argument's visibility
  -- ('Inferred', 'Specified', or 'Required').
  --
  -- Wrinkle: consider the following scenario:
  --
  -- > T :: forall k. k -> k
  -- > tyConForAllTyFlags 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.
  tyConForAllTyFlags :: TyCon -> [Type] -> [ForAllTyFlag]
  tyConForAllTyFlags tc = fun_kind_arg_flags (tyConKind tc)
  
  -- | Given a 'Type' and a list of argument types to which the 'Type' is
  -- applied, determine each argument's visibility
  -- ('Inferred', 'Specified', or 'Required').
  --
  -- Most of the time, the arguments will be 'Required', but not always. Consider
  -- @f :: forall a. a -> Type@. In @f Type Bool@, the first argument (@Type@) is
  -- 'Specified' and the second argument (@Bool@) is 'Required'. It is precisely
  -- this sort of higher-rank situation in which 'appTyForAllTyFlags' comes in handy,
  -- since @f Type Bool@ would be represented in Core using 'AppTy's.
  -- (See also #15792).
  appTyForAllTyFlags :: Type -> [Type] -> [ForAllTyFlag]
  appTyForAllTyFlags ty = fun_kind_arg_flags (typeKind ty)
  
  -- | Given a function kind and a list of argument types (where each argument's
  -- kind aligns with the corresponding position in the argument kind), determine
  -- each argument's visibility ('Inferred', 'Specified', or 'Required').
  fun_kind_arg_flags :: Kind -> [Type] -> [ForAllTyFlag]
  fun_kind_arg_flags = go emptySubst
    where
      go subst ki arg_tys
        | Just ki' <- coreView ki = go subst ki' arg_tys
      go _ _ [] = []
      go subst (ForAllTy (Bndr tv argf) res_ki) (arg_ty:arg_tys)
        = argf : go subst' res_ki arg_tys
        where
          subst' = extendTvSubst subst tv arg_ty
      go subst (TyVarTy tv) arg_tys
        | Just ki <- lookupTyVar subst tv = go subst ki arg_tys
      -- This FunTy case is important to handle kinds with nested foralls, such
      -- as this kind (inspired by #16518):
      --
      --   forall {k1} k2. k1 -> k2 -> forall k3. k3 -> Type
      --
      -- Here, we want to get the following ForAllTyFlags:
      --
      -- [Inferred,   Specified, Required, Required, Specified, Required]
      -- forall {k1}. forall k2. k1 ->     k2 ->     forall k3. k3 ->     Type
      go subst (FunTy{ft_af = af, ft_res = res_ki}) (_:arg_tys)
        = argf : go subst res_ki arg_tys
        where
          argf | isVisibleFunArg af = Required
               | otherwise          = Inferred
      go _ _ arg_tys = map (const Required) arg_tys
                          -- something is ill-kinded. But this can happen
                          -- when printing errors. Assume everything is Required.
  
  -- @isTauTy@ tests if a type has no foralls or (=>)
  isTauTy :: Type -> Bool
  isTauTy ty | Just ty' <- coreView ty = isTauTy ty'
  isTauTy (TyVarTy _)       = True
  isTauTy (LitTy {})        = True
  isTauTy (TyConApp tc tys) = all isTauTy tys && isTauTyCon tc
  isTauTy (AppTy a b)       = isTauTy a && isTauTy b
  isTauTy (FunTy { ft_af = af, ft_mult = w, ft_arg = a, ft_res = b })
   | isInvisibleFunArg af   = False                               -- e.g., Eq a => b
   | otherwise              = isTauTy w && isTauTy a && isTauTy b -- e.g., a -> b
  isTauTy (ForAllTy {})     = False
  isTauTy (CastTy ty _)     = isTauTy ty
  isTauTy (CoercionTy _)    = False  -- Not sure about this
  
  isAtomicTy :: Type -> Bool
  -- True if the type is just a single token, and can be printed compactly
  -- Used when deciding how to lay out type error messages; see the
  -- call in GHC.Tc.Errors
  isAtomicTy (TyVarTy {})    = True
  isAtomicTy (LitTy {})      = True
  isAtomicTy (TyConApp _ []) = True
  
  isAtomicTy ty | isLiftedTypeKind ty = True
     -- 'Type' prints compactly as *
     -- See GHC.Iface.Type.ppr_kind_type
  
  isAtomicTy _ = False
  
  {-
  ************************************************************************
  *                                                                      *
  \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 = assertPpr (tvs `equalLength` 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))
  
  isFamFreeTy :: Type -> Bool
  isFamFreeTy ty | Just ty' <- coreView ty = isFamFreeTy ty'
  isFamFreeTy (TyVarTy _)       = True
  isFamFreeTy (LitTy {})        = True
  isFamFreeTy (TyConApp tc tys) = all isFamFreeTy tys && isFamFreeTyCon tc
  isFamFreeTy (AppTy a b)       = isFamFreeTy a && isFamFreeTy b
  isFamFreeTy (FunTy _ w a b)   = isFamFreeTy w && isFamFreeTy a && isFamFreeTy b
  isFamFreeTy (ForAllTy _ ty)   = isFamFreeTy ty
  isFamFreeTy (CastTy ty _)     = isFamFreeTy ty
  isFamFreeTy (CoercionTy _)    = False  -- Not sure about this
  
  -- | Check whether a type is a data family type
  isDataFamilyApp :: Type -> Bool
  isDataFamilyApp ty = case tyConAppTyCon_maybe ty of
                             Just tc -> isDataFamilyTyCon tc
                             _       -> False
  
  isSatTyFamApp :: Type -> Maybe (TyCon, [Type])
  -- Return the argument if we have a saturated type family application
  -- Why saturated?  See (ATF4) in Note [Apartness and type families]
  isSatTyFamApp (TyConApp tc tys)
    |  isTypeFamilyTyCon tc
    && not (tys `lengthExceeds` tyConArity tc)  -- Not over-saturated
    = Just (tc, tys)
  isSatTyFamApp _ = Nothing
  
  buildSynTyCon :: Name -> [KnotTied TyConBinder] -> Kind   -- ^ /result/ kind
                -> [Role] -> KnotTied Type -> TyCon
  -- This function is here because here is where we have
  --   isFamFree and isTauTy
  buildSynTyCon name binders res_kind roles rhs
    = mkSynonymTyCon name binders res_kind roles rhs
                     is_tau is_fam_free is_forgetful is_concrete
    where
      qtvs         = mkVarSet (map binderVar binders)
      is_tau       = isTauTy rhs
      is_fam_free  = isFamFreeTy rhs
      is_concrete  = isConcreteTypeWith qtvs rhs
      is_forgetful = not (qtvs `subVarSet` expanded_rhs_tyvars)
  
      expanded_rhs_tyvars = tyCoVarsOfType (expandTypeSynonyms rhs)
         -- See Note [Forgetful type synonyms] in GHC.Core.TyCon
         -- To find out if this TyCon is forgetful, expand the synonyms in its RHS
         -- and check that all of the binders are free in the expanded type.
         -- We really only need to expand the /forgetful/ synonyms on the RHS,
         -- but we don't currently have a function to do that.
         -- Failing to expand the RHS led to #25094, e.g.
         --    type Bucket a b c = Key (a,b,c)
         --    type Key x = Any
         -- Here Bucket is definitely forgetful!
  
  {-
  ************************************************************************
  *                                                                      *
  \subsection{Liftedness}
  *                                                                      *
  ************************************************************************
  -}
  
  -- | Tries to compute the 'Levity' of the given type. Returns either
  -- a definite 'Levity', or 'Nothing' if we aren't sure (e.g. the
  -- type is representation-polymorphic).
  --
  -- Panics if the kind does not have the shape @TYPE r@.
  typeLevity_maybe :: HasDebugCallStack => Type -> Maybe Levity
  typeLevity_maybe ty = runtimeRepLevity_maybe (getRuntimeRep ty)
  
  typeLevity :: HasDebugCallStack => Type -> Levity
  typeLevity ty = case typeLevity_maybe ty of
                     Just lev -> lev
                     Nothing  -> pprPanic "typeLevity" (ppr ty)
  
  -- | Is the given type definitely unlifted?
  -- See "Type#type_classification" for what an unlifted type is.
  --
  -- Panics on representation-polymorphic types; See 'mightBeUnliftedType' for
  -- a more approximate predicate that behaves better in the presence of
  -- representation polymorphism.
  isUnliftedType :: HasDebugCallStack => 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.
  isUnliftedType ty =
    case typeLevity_maybe ty of
      Just Lifted   -> False
      Just Unlifted -> True
      Nothing       -> pprPanic "isUnliftedType" (ppr ty <+> dcolon <+> ppr (typeKind ty))
  
  -- | Returns:
  --
  -- * 'False' if the type is /guaranteed/ unlifted or
  -- * 'True' if it lifted, OR we aren't sure
  --    (e.g. in a representation-polymorphic case)
  mightBeLiftedType :: Type -> Bool
  mightBeLiftedType = mightBeLifted . typeLevity_maybe
  
  definitelyLiftedType :: Type -> Bool
  definitelyLiftedType = not . mightBeUnliftedType
  
  -- | Returns:
  --
  -- * 'False' if the type is /guaranteed/ lifted or
  -- * 'True' if it is unlifted, OR we aren't sure
  --    (e.g. in a representation-polymorphic case)
  mightBeUnliftedType :: Type -> Bool
  mightBeUnliftedType = mightBeUnlifted . typeLevity_maybe
  
  definitelyUnliftedType :: Type -> Bool
  definitelyUnliftedType = not . mightBeLiftedType
  
  -- | See "Type#type_classification" for what a boxed type is.
  -- Panics on representation-polymorphic types; See 'mightBeUnliftedType' for
  -- a more approximate predicate that behaves better in the presence of
  -- representation polymorphism.
  isBoxedType :: Type -> Bool
  isBoxedType ty = isBoxedRuntimeRep (getRuntimeRep ty)
  
  -- | Is this a type of kind RuntimeRep? (e.g. LiftedRep)
  isRuntimeRepKindedTy :: Type -> Bool
  isRuntimeRepKindedTy = isRuntimeRepTy . typeKind
  
  -- | Drops prefix of RuntimeRep constructors in 'TyConApp's. Useful for e.g.
  -- dropping 'LiftedRep arguments of unboxed tuple TyCon applications:
  --
  --   dropRuntimeRepArgs [ 'LiftedRep, 'IntRep
  --                      , String, Int# ] == [String, Int#]
  --
  dropRuntimeRepArgs :: [Type] -> [Type]
  dropRuntimeRepArgs = dropWhile isRuntimeRepKindedTy
  
  -- | Extract the RuntimeRep classifier of a type. For instance,
  -- @getRuntimeRep_maybe Int = Just LiftedRep@. Returns 'Nothing' if this is not
  -- possible.
  getRuntimeRep_maybe :: HasDebugCallStack
                      => Type -> Maybe RuntimeRepType
  getRuntimeRep_maybe = kindRep_maybe . typeKind
  
  -- | Extract the RuntimeRep classifier of a type. For instance,
  -- @getRuntimeRep_maybe Int = LiftedRep@. Panics if this is not possible.
  getRuntimeRep :: HasDebugCallStack => Type -> RuntimeRepType
  getRuntimeRep ty
    = case getRuntimeRep_maybe ty of
        Just r  -> r
        Nothing -> pprPanic "getRuntimeRep" (ppr ty <+> dcolon <+> ppr (typeKind ty))
  
  -- | Extract the 'Levity' of a type. For example, @getLevity_maybe Int = Just Lifted@,
  -- @getLevity (Array# Int) = Just Unlifted@, @getLevity Float# = Nothing@.
  --
  -- Returns 'Nothing' if this is not possible. Does not look through type family applications.
  getLevity_maybe :: HasDebugCallStack => Type -> Maybe Type
  getLevity_maybe ty
    | Just rep <- getRuntimeRep_maybe ty
    -- Directly matching on TyConApp after expanding type synonyms
    -- saves allocations compared to `splitTyConApp_maybe`. See #22254.
    -- Given that this is a pretty hot function we make use of the fact
    -- and use isTyConKeyApp_maybe instead.
    , Just [lev] <- isTyConKeyApp_maybe boxedRepDataConKey rep
    = Just lev
    | otherwise
    = Nothing
  
  -- | Extract the 'Levity' of a type. For example, @getLevity Int = Lifted@,
  -- or @getLevity (Array# Int) = Unlifted@.
  --
  -- Panics if this is not possible. Does not look through type family applications.
  getLevity :: HasDebugCallStack => Type -> Type
  getLevity ty
    | Just lev <- getLevity_maybe ty
    = lev
    | otherwise
    = pprPanic "getLevity" (ppr ty <+> dcolon <+> ppr (typeKind ty))
  
  isUnboxedTupleType :: Type -> Bool
  isUnboxedTupleType ty
    = tyConAppTyCon (getRuntimeRep ty) `hasKey` tupleRepDataConKey
    -- NB: Do not use typePrimRep, as that can't tell the difference between
    -- unboxed tuples and unboxed sums
  
  
  isUnboxedSumType :: Type -> Bool
  isUnboxedSumType ty
    = tyConAppTyCon (getRuntimeRep ty) `hasKey` sumRepDataConKey
  
  -- | 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
  
  -- | 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'.
  -- Panics on representation-polymorphic types.
  isStrictType :: HasDebugCallStack => Type -> Bool
  isStrictType = isUnliftedType
  
  isTerminatingType :: HasDebugCallStack => Type -> Bool
  -- ^ True <=> a term of this type cannot be bottom
  -- This identifies the types described by
  --    Note [NON-BOTTOM-DICTS invariant] in GHC.Core
  -- NB: unlifted types are not terminating types!
  --     e.g. you can write a term (loop 1)::Int# that diverges.
  isTerminatingType ty = case tyConAppTyCon_maybe ty of
      Just tc -> isClassTyCon tc && not (isUnaryClassTyCon tc)
                 -- A non-unary class TyCon is terminating
                 -- See (UCM3) in Note [Unary class magic] in GHC.Core.TyCon
      _       -> False
  
  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{Join points}
  *                                                                      *
  ************************************************************************
  -}
  
  -- | Determine whether a type could be the type of a join point of given total
  -- arity, according to the polymorphism rule. A join point cannot be polymorphic
  -- in its return type, since given
  --   join j @a @b x y z = e1 in e2,
  -- the types of e1 and e2 must be the same, and a and b are not in scope for e2.
  -- (See Note [The polymorphism rule of join points] in "GHC.Core".) Returns False
  -- also if the type simply doesn't have enough arguments.
  --
  -- Note that we need to know how many arguments (type *and* value) the putative
  -- join point takes; for instance, if
  --   j :: forall a. a -> Int
  -- then j could be a binary join point returning an Int, but it could *not* be a
  -- unary join point returning a -> Int.
  --
  -- TODO: See Note [Excess polymorphism and join points]
  isValidJoinPointType :: JoinArity -> Type -> Bool
  isValidJoinPointType arity ty
    = valid_under emptyVarSet arity ty
    where
      valid_under tvs arity ty
        | arity == 0
        = tvs `disjointVarSet` tyCoVarsOfType ty
        | Just (t, ty') <- splitForAllTyCoVar_maybe ty
        = valid_under (tvs `extendVarSet` t) (arity-1) ty'
        | Just (_, _, _, res_ty) <- splitFunTy_maybe ty
        = valid_under tvs (arity-1) res_ty
        | otherwise
        = False
  
  {- Note [Excess polymorphism and join points]
  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  In principle, if a function would be a join point except that it fails
  the polymorphism rule (see Note [The polymorphism rule of join points] in
  GHC.Core), it can still be made a join point with some effort. This is because
  all tail calls must return the same type (they return to the same context!), and
  thus if the return type depends on an argument, that argument must always be the
  same.
  
  For instance, consider:
  
    let f :: forall a. a -> Char -> [a]
        f @a x c = ... f @a y 'a' ...
    in ... f @Int 1 'b' ... f @Int 2 'c' ...
  
  (where the calls are tail calls). `f` fails the polymorphism rule because its
  return type is [a], where [a] is bound. But since the type argument is always
  'Int', we can rewrite it as:
  
    let f' :: Int -> Char -> [Int]
        f' x c = ... f' y 'a' ...
    in ... f' 1 'b' ... f 2 'c' ...
  
  and now we can make f' a join point:
  
    join f' :: Int -> Char -> [Int]
         f' x c = ... jump f' y 'a' ...
    in ... jump f' 1 'b' ... jump f' 2 'c' ...
  
  It's not clear that this comes up often, however. TODO: Measure how often and
  add this analysis if necessary.  See #14620.
  
  
  ************************************************************************
  *                                                                      *
  \subsection{Sequencing on types}
  *                                                                      *
  ************************************************************************
  -}
  
  seqType :: Type -> ()
  seqType (LitTy n)                   = n `seq` ()
  seqType (TyVarTy tv)                = tv `seq` ()
  seqType (AppTy t1 t2)               = seqType t1 `seq` seqType t2
  seqType (FunTy _ w t1 t2)           = seqType w `seq` seqType t1 `seq` seqType t2
  seqType (TyConApp tc tys)           = tc `seq` seqTypes tys
  seqType (ForAllTy (Bndr tv _) ty)   = seqType (varType tv) `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
  
  {-
  ************************************************************************
  *                                                                      *
          The kind of a type
  *                                                                      *
  ************************************************************************
  
  Note [Kinding rules for types]
  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  Here are the key kinding rules for types
  
            torc1 is TYPE or CONSTRAINT
            torc2 is TYPE or CONSTRAINT
            t1 : torc1 rep1
            t2 : torc2 rep2
     (FUN)  ----------------
            t1 -> t2 : torc2 LiftedRep
            -- In fact the arrow varies with torc1/torc2
            -- See Note [Function type constructors and FunTy]
            -- in GHC.Builtin.Types.Prim
  
            torc is TYPE or CONSTRAINT
            ty : body_torc rep
            ki : Type
            `a` is a type variable
            `a` is not free in rep
  (FORALL1) -----------------------
            forall (a::ki). ty : body_torc rep
  
            torc is TYPE or CONSTRAINT
            ty : body_torc rep
            `c` is a coercion variable
            `c` is not free in rep
            `c` is free in ty       -- Surprise 1!
  (FORALL2) -------------------------
            forall (cv::k1 ~#{N,R} k2). ty : body_torc LiftedRep
                                             -- Surprise 2!
  
  Note that:
  * (FORALL1) rejects (forall (a::Maybe). blah)
  
  * (FORALL2) Surprise 1:
    See GHC.Core.TyCo.Rep Note [Unused coercion variable in ForAllTy]
  
  * (FORALL2) Surprise 2: coercion abstractions are not erased, so
    this must be LiftedRep, just like (FUN).  (FORALL2) is just a
    dependent form of (FUN).
  
  
  Note [Phantom type variables in kinds]
  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  Consider
  
    type K (r :: RuntimeRep) = Type   -- Note 'r' is unused
    data T r :: K r                   -- T :: forall r -> K r
    foo :: forall r. T r
  
  The body of the forall in foo's type has kind (K r), and
  normally it would make no sense to have
     forall r. (ty :: K r)
  because the kind of the forall would escape the binding
  of 'r'.  But in this case it's fine because (K r) expands
  to Type, so we explicitly /permit/ the type
     forall r. T r
  
  To accommodate such a type, in typeKind (forall a.ty) we use
  occCheckExpand to expand any type synonyms in the kind of 'ty'
  to eliminate 'a'.  See kinding rule (FORALL) in
  Note [Kinding rules for types]
  
  
  See also
   * GHC.Core.Type.occCheckExpand
   * GHC.Core.Utils.coreAltsType
   * GHC.Tc.Validity.checkEscapingKind
  all of which grapple with the same problem.
  
  See #14939.
  -}
  
  -----------------------------
  typeKind :: HasDebugCallStack => Type -> Kind
  -- No need to expand synonyms
  typeKind (TyConApp tc tys)      = piResultTys (tyConKind tc) tys
  typeKind (LitTy l)              = typeLiteralKind l
  typeKind (FunTy { ft_af = af }) = liftedTypeOrConstraintKind (funTyFlagResultTypeOrConstraint af)
  typeKind (TyVarTy tyvar)        = tyVarKind tyvar
  typeKind (CastTy _ty co)        = coercionRKind co
  typeKind (CoercionTy co)        = coercionType co
  
  typeKind (AppTy fun arg)
    = go fun [arg]
    where
      -- Accumulate the type arguments, so we can call piResultTys,
      -- rather than a succession of calls to piResultTy (which is
      -- asymptotically costly as the number of arguments increases)
      go (AppTy fun arg) args = go fun (arg:args)
      go fun             args = piResultTys (typeKind fun) args
  
  typeKind ty@(ForAllTy {})
    = assertPpr (not (null tcvs)) (ppr ty) $
         -- If tcvs is empty somehow we'll get an infinite loop!
      case occCheckExpand tcvs body_kind of
        -- We must make sure tvs do not occur in kind,
        -- as they would be out of scope!
        -- See Note [Phantom type variables in kinds]
        Nothing -> pprPanic "typeKind"
                    (ppr ty $$ ppr tcvs $$ ppr body <+> dcolon <+> ppr body_kind)
  
        Just k' | all isTyVar tcvs -> k'                     -- Rule (FORALL1)
                | otherwise        -> lifted_kind_from_body  -- Rule (FORALL2)
    where
      (tcvs, body) = splitForAllTyCoVars ty  -- Important: splits both TyVar and CoVar binders
      body_kind    = typeKind body
  
      lifted_kind_from_body  -- Implements (FORALL2)
        = case sORTKind_maybe body_kind of
            Just (torc, _) -> liftedTypeOrConstraintKind torc
            Nothing        -> pprPanic "typeKind" (ppr body_kind)
  
  
  ---------------------------------------------
  
  sORTKind_maybe :: Kind -> Maybe (TypeOrConstraint, Type)
  -- Sees if the argument is of form (TYPE rep) or (CONSTRAINT rep)
  -- and if so returns which, and the runtime rep
  --
  -- This is a "hot" function.  Do not call splitTyConApp_maybe here,
  -- to avoid the faff with FunTy
  sORTKind_maybe (TyConApp tc tys)
    -- First, short-cuts for Type and Constraint that do no allocation
    | tc_uniq == liftedTypeKindTyConKey = assert( null tys ) $ Just (TypeLike,       liftedRepTy)
    | tc_uniq == constraintKindTyConKey = assert( null tys ) $ Just (ConstraintLike, liftedRepTy)
    | tc_uniq == tYPETyConKey           = get_rep TypeLike
    | tc_uniq == cONSTRAINTTyConKey     = get_rep ConstraintLike
    | Just ty' <- expandSynTyConApp_maybe tc tys = sORTKind_maybe ty'
    where
      !tc_uniq = tyConUnique tc
       -- This bang on tc_uniq is important.  It means that sORTKind_maybe starts
       -- by evaluating tc_uniq, and then ends up with a single case with a 4-way branch
  
      get_rep torc = case tys of
                       (rep:_reps) -> assert (null _reps) $ Just (torc, rep)
                       []          -> Nothing
  
  sORTKind_maybe _ = Nothing
  
  typeTypeOrConstraint :: HasDebugCallStack => Type -> TypeOrConstraint
  -- Precondition: expects a type that classifies values.
  -- Returns whether it is TypeLike or ConstraintLike.
  -- Equivalent to calling sORTKind_maybe, but faster in the FunTy case
  typeTypeOrConstraint ty
     = case coreFullView ty of
         FunTy { ft_af = af } -> funTyFlagResultTypeOrConstraint af
         ty' | Just (torc, _) <- sORTKind_maybe (typeKind ty')
            -> torc
            | otherwise
            -> pprPanic "typeOrConstraint" (ppr ty <+> dcolon <+> ppr (typeKind ty))
  
  -- | Does this classify a type allowed to have values? Responds True to things
  -- like *, TYPE Lifted, TYPE IntRep, TYPE v, Constraint.
  isTYPEorCONSTRAINT :: Kind -> Bool
  -- ^ True of a kind `TYPE _` or `CONSTRAINT _`
  isTYPEorCONSTRAINT k = isJust (sORTKind_maybe k)
  
  tyConIsTYPEorCONSTRAINT :: TyCon -> Bool
  tyConIsTYPEorCONSTRAINT tc
    = tc_uniq == tYPETyConKey || tc_uniq == cONSTRAINTTyConKey
    where
      !tc_uniq = tyConUnique tc
  
  isConstraintLikeKind :: Kind -> Bool
  -- True of (CONSTRAINT _)
  isConstraintLikeKind kind
    = case sORTKind_maybe kind of
        Just (ConstraintLike, _) -> True
        _                        -> False
  
  isConstraintKind :: Kind -> Bool
  -- True of (CONSTRAINT LiftedRep)
  isConstraintKind kind
    = case sORTKind_maybe kind of
        Just (ConstraintLike, rep) -> isLiftedRuntimeRep rep
        _                          -> False
  
  tcIsLiftedTypeKind :: Kind -> Bool
  -- ^ Is this kind equivalent to 'Type' i.e. TYPE LiftedRep?
  tcIsLiftedTypeKind kind
    | Just (TypeLike, rep) <- sORTKind_maybe kind
    = isLiftedRuntimeRep rep
    | otherwise
    = False
  
  tcIsBoxedTypeKind :: Kind -> Bool
  -- ^ Is this kind equivalent to @TYPE (BoxedRep l)@ for some @l :: Levity@?
  tcIsBoxedTypeKind kind
    | Just (TypeLike, rep) <- sORTKind_maybe kind
    = isBoxedRuntimeRep rep
    | otherwise
    = False
  
  -- | Is this kind equivalent to @TYPE r@ (for some unknown r)?
  --
  -- This considers 'Constraint' to be distinct from @*@.
  isTypeLikeKind :: Kind -> Bool
  isTypeLikeKind kind
    = case sORTKind_maybe kind of
        Just (TypeLike, _) -> True
        _                  -> False
  
  returnsConstraintKind :: Kind -> Bool
  -- True <=> the Kind ultimately returns a Constraint
  --   E.g.  * -> Constraint
  --         forall k. k -> Constraint
  returnsConstraintKind kind
    | Just kind' <- coreView kind = returnsConstraintKind kind'
  returnsConstraintKind (ForAllTy _ ty)         = returnsConstraintKind ty
  returnsConstraintKind (FunTy { ft_res = ty }) = returnsConstraintKind ty
  returnsConstraintKind kind                    = isConstraintLikeKind kind
  
  --------------------------
  typeLiteralKind :: TyLit -> Kind
  typeLiteralKind (NumTyLit {}) = naturalTy
  typeLiteralKind (StrTyLit {}) = typeSymbolKind
  typeLiteralKind (CharTyLit {}) = charTy
  
  -- | Returns True if a type has a syntactically fixed runtime rep,
  -- as per Note [Fixed RuntimeRep] in GHC.Tc.Utils.Concrete.
  --
  -- This function is equivalent to `isFixedRuntimeRepKind . typeKind`
  -- but much faster.
  --
  -- __Precondition:__ The type has kind @('TYPE' blah)@
  typeHasFixedRuntimeRep :: HasDebugCallStack => Type -> Bool
  typeHasFixedRuntimeRep = go
    where
      go (TyConApp tc _)
        | tcHasFixedRuntimeRep tc = True
      go (FunTy {})               = True
      go (LitTy {})               = True
      go (ForAllTy _ ty)          = go ty
      go ty                       = isFixedRuntimeRepKind (typeKind ty)
  
  -- | Checks that a kind of the form 'Type', 'Constraint'
  -- or @'TYPE r@ is concrete. See 'isConcreteType'.
  --
  -- __Precondition:__ The type has kind `TYPE blah` or `CONSTRAINT blah`
  isFixedRuntimeRepKind :: HasDebugCallStack => Kind -> Bool
  isFixedRuntimeRepKind k
    = assertPpr (isTYPEorCONSTRAINT k) (ppr k) $
      -- the isLiftedTypeKind check is necessary b/c of Constraint
      isConcreteType k
  
  -- | Tests whether the given type is concrete, i.e. it
  -- whether it consists only of concrete type constructors,
  -- concrete type variables, and applications.
  --
  -- See Note [Concrete types] in GHC.Tc.Utils.Concrete.
  isConcreteType :: Type -> Bool
  isConcreteType = isConcreteTypeWith emptyVarSet
  
  -- | Like 'isConcreteType', but allows passing in a set of 'TyVar's that
  -- should be considered concrete.
  --
  -- See Note [Concrete types] in GHC.Tc.Utils.Concrete.
  isConcreteTypeWith :: TyVarSet -> Type -> Bool
  -- This version, with a 'TyVarSet' argument, supports 'mkSynonymTyCon',
  -- which needs to test the RHS for concreteness, under the assumption that
  -- the binders are instantiated to concrete types
  isConcreteTypeWith conc_tvs = go
    where
      go (TyVarTy tv)        = isConcreteTyVar tv || tv `elemVarSet` conc_tvs
      go (AppTy ty1 ty2)     = go ty1 && go ty2
      go (TyConApp tc tys)   = go_tc tc tys
      go ForAllTy{}          = False
      go (FunTy _ w t1 t2)   =  go w
                             && go (typeKind t1) && go t1
                             && go (typeKind t2) && go t2
      go LitTy{}             = True
      go CastTy{}            = False
      go CoercionTy{}        = False
  
      go_tc :: TyCon -> [Type] -> Bool
      go_tc tc tys
        | isForgetfulSynTyCon tc  -- E.g. type S a = Int
                                  -- Then (S x) is concrete even if x isn't
        , Just ty' <- expandSynTyConApp_maybe tc tys
        = go ty'
  
        -- Apart from forgetful synonyms, isConcreteTyCon
        -- is enough; no need to expand.  This is good for e.g
        --      type LiftedRep = BoxedRep Lifted
        | isConcreteTyCon tc
        = all go tys
  
        | otherwise  -- E.g. type families
        = False
  
  {-
  %************************************************************************
  %*                                                                      *
           Pretty-printing
  %*                                                                      *
  %************************************************************************
  
  Most pretty-printing is either in GHC.Core.TyCo.Rep or GHC.Iface.Type.
  
  -}
  
  -- | Does a 'TyCon' (that is applied to some number of arguments) need to be
  -- ascribed with an explicit kind signature to resolve ambiguity if rendered as
  -- a source-syntax type?
  -- (See @Note [When does a tycon application need an explicit kind signature?]@
  -- for a full explanation of what this function checks for.)
  tyConAppNeedsKindSig
    :: Bool  -- ^ Should specified binders count towards injective positions in
             --   the kind of the TyCon? (If you're using visible kind
             --   applications, then you want True here.
    -> TyCon
    -> Int   -- ^ The number of args the 'TyCon' is applied to.
    -> Bool  -- ^ Does @T t_1 ... t_n@ need a kind signature? (Where @n@ is the
             --   number of arguments)
  tyConAppNeedsKindSig spec_inj_pos tc n_args
    | LT <- listLengthCmp tc_binders n_args
    = False
    | otherwise
    = let (dropped_binders, remaining_binders)
            = splitAt n_args tc_binders
          result_kind  = mkTyConKind remaining_binders tc_res_kind
          result_vars  = tyCoVarsOfType result_kind
          dropped_vars = fvVarSet $
                         mapUnionFV injective_vars_of_binder dropped_binders
  
      in not (subVarSet result_vars dropped_vars)
    where
      tc_binders  = tyConBinders tc
      tc_res_kind = tyConResKind tc
  
      -- Returns the variables that would be fixed by knowing a TyConBinder. See
      -- Note [When does a tycon application need an explicit kind signature?]
      -- for a more detailed explanation of what this function does.
      injective_vars_of_binder :: TyConBinder -> FV
      injective_vars_of_binder (Bndr tv vis) =
        case vis of
          AnonTCB        -> injectiveVarsOfType False -- conservative choice
                                                (varType tv)
          NamedTCB argf  | source_of_injectivity argf
                         -> unitFV tv `unionFV`
                            injectiveVarsOfType False (varType tv)
          _              -> emptyFV
  
      source_of_injectivity Required  = True
      source_of_injectivity Specified = spec_inj_pos
      source_of_injectivity Inferred  = False
  
  {-
  Note [When does a tycon application need an explicit kind signature?]
  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  There are a couple of places in GHC where we convert Core Types into forms that
  more closely resemble user-written syntax. These include:
  
  1. Template Haskell Type reification (see, for instance, GHC.Tc.Gen.Splice.reify_tc_app)
  2. Converting Types to LHsTypes (such as in Haddock.Convert in haddock)
  
  This conversion presents a challenge: how do we ensure that the resulting type
  has enough kind information so as not to be ambiguous? To better motivate this
  question, consider the following Core type:
  
    -- Foo :: Type -> Type
    type Foo = Proxy Type
  
  There is nothing ambiguous about the RHS of Foo in Core. But if we were to,
  say, reify it into a TH Type, then it's tempting to just drop the invisible
  Type argument and simply return `Proxy`. But now we've lost crucial kind
  information: we don't know if we're dealing with `Proxy Type` or `Proxy Bool`
  or `Proxy Int` or something else! We've inadvertently introduced ambiguity.
  
  Unlike in other situations in GHC, we can't just turn on
  -fprint-explicit-kinds, as we need to produce something which has the same
  structure as a source-syntax type. Moreover, we can't rely on visible kind
  application, since the first kind argument to Proxy is inferred, not specified.
  Our solution is to annotate certain tycons with their kinds whenever they
  appear in applied form in order to resolve the ambiguity. For instance, we
  would reify the RHS of Foo like so:
  
    type Foo = (Proxy :: Type -> Type)
  
  We need to devise an algorithm that determines precisely which tycons need
  these explicit kind signatures. We certainly don't want to annotate _every_
  tycon with a kind signature, or else we might end up with horribly bloated
  types like the following:
  
    (Either :: Type -> Type -> Type) (Int :: Type) (Char :: Type)
  
  We only want to annotate tycons that absolutely require kind signatures in
  order to resolve some sort of ambiguity, and nothing more.
  
  Suppose we have a tycon application (T ty_1 ... ty_n). Why might this type
  require a kind signature? It might require it when we need to fill in any of
  T's omitted arguments. By "omitted argument", we mean one that is dropped when
  reifying ty_1 ... ty_n. Sometimes, the omitted arguments are inferred and
  specified arguments (e.g., TH reification in GHC.Tc.Gen.Splice), and sometimes the
  omitted arguments are only the inferred ones (e.g., in situations where
  specified arguments are reified through visible kind application).
  Regardless, the key idea is that _some_ arguments are going to be omitted after
  reification, and the only mechanism we have at our disposal for filling them in
  is through explicit kind signatures.
  
  What do we mean by "fill in"? Let's consider this small example:
  
    T :: forall {k}. Type -> (k -> Type) -> k
  
  Moreover, we have this application of T:
  
    T @{j} Int aty
  
  When we reify this type, we omit the inferred argument @{j}. Is it fixed by the
  other (non-inferred) arguments? Yes! If we know the kind of (aty :: blah), then
  we'll generate an equality constraint (kappa -> Type) and, assuming we can
  solve it, that will fix `kappa`. (Here, `kappa` is the unification variable
  that we instantiate `k` with.)
  
  Therefore, for any application of a tycon T to some arguments, the Question We
  Must Answer is:
  
  * Given the first n arguments of T, do the kinds of the non-omitted arguments
    fill in the omitted arguments?
  
  (This is still a bit hand-wavy, but we'll refine this question incrementally
  as we explain more of the machinery underlying this process.)
  
  Answering this question is precisely the role that the `injectiveVarsOfType`
  and `injective_vars_of_binder` functions exist to serve. If an omitted argument
  `a` appears in the set returned by `injectiveVarsOfType ty`, then knowing
  `ty` determines (i.e., fills in) `a`. (More on `injective_vars_of_binder` in a
  bit.)
  
  More formally, if
  `a` is in `injectiveVarsOfType ty`
  and  S1(ty) ~ S2(ty),
  then S1(a)  ~ S2(a),
  where S1 and S2 are arbitrary substitutions.
  
  For example, is `F` is a non-injective type family, then
  
    injectiveVarsOfType(Either c (Maybe (a, F b c))) = {a, c}
  
  Now that we know what this function does, here is a second attempt at the
  Question We Must Answer:
  
  * Given the first n arguments of T (ty_1 ... ty_n), consider the binders
    of T that are instantiated by non-omitted arguments. Do the injective
    variables of these binders fill in the remainder of T's kind?
  
  Alright, we're getting closer. Next, we need to clarify what the injective
  variables of a tycon binder are. This the role that the
  `injective_vars_of_binder` function serves. Here is what this function does for
  each form of tycon binder:
  
  * Anonymous binders are injective positions. For example, in the promoted data
    constructor '(:):
  
      '(:) :: forall a. a -> [a] -> [a]
  
    The second and third tyvar binders (of kinds `a` and `[a]`) are both
    anonymous, so if we had '(:) 'True '[], then the kinds of 'True and
    '[] would contribute to the kind of '(:) 'True '[]. Therefore,
    injective_vars_of_binder(_ :: a) = injectiveVarsOfType(a) = {a}.
    (Similarly, injective_vars_of_binder(_ :: [a]) = {a}.)
  * Named binders:
    - Inferred binders are never injective positions. For example, in this data
      type:
  
        data Proxy a
        Proxy :: forall {k}. k -> Type
  
      If we had Proxy 'True, then the kind of 'True would not contribute to the
      kind of Proxy 'True. Therefore,
      injective_vars_of_binder(forall {k}. ...) = {}.
    - Required binders are injective positions. For example, in this data type:
  
        data Wurble k (a :: k) :: k
        Wurble :: forall k -> k -> k
  
    The first tyvar binder (of kind `forall k`) has required visibility, so if
    we had Wurble (Maybe a) Nothing, then the kind of Maybe a would
    contribute to the kind of Wurble (Maybe a) Nothing. Hence,
    injective_vars_of_binder(forall a -> ...) = {a}.
    - Specified binders /might/ be injective positions, depending on how you
      approach things. Continuing the '(:) example:
  
        '(:) :: forall a. a -> [a] -> [a]
  
      Normally, the (forall a. ...) tyvar binder wouldn't contribute to the kind
      of '(:) 'True '[], since it's not explicitly instantiated by the user. But
      if visible kind application is enabled, then this is possible, since the
      user can write '(:) @Bool 'True '[]. (In that case,
      injective_vars_of_binder(forall a. ...) = {a}.)
  
      There are some situations where using visible kind application is appropriate
      and others where it is not (e.g., TH
      reification), so the `injective_vars_of_binder` function is parameterized by
      a Bool which decides if specified binders should be counted towards
      injective positions or not.
  
  Now that we've defined injective_vars_of_binder, we can refine the Question We
  Must Answer once more:
  
  * Given the first n arguments of T (ty_1 ... ty_n), consider the binders
    of T that are instantiated by non-omitted arguments. For each such binder
    b_i, take the union of all injective_vars_of_binder(b_i). Is this set a
    superset of the free variables of the remainder of T's kind?
  
  If the answer to this question is "no", then (T ty_1 ... ty_n) needs an
  explicit kind signature, since T's kind has kind variables leftover that
  aren't fixed by the non-omitted arguments.
  
  One last sticking point: what does "the remainder of T's kind" mean? You might
  be tempted to think that it corresponds to all of the arguments in the kind of
  T that would normally be instantiated by omitted arguments. But this isn't
  quite right, strictly speaking. Consider the following (silly) example:
  
    S :: forall {k}. Type -> Type
  
  And suppose we have this application of S:
  
    S Int Bool
  
  The Int argument would be omitted, and
  injective_vars_of_binder(_ :: Type) = {}. This is not a superset of {k}, which
  might suggest that (S Bool) needs an explicit kind signature. But
  (S Bool :: Type) doesn't actually fix `k`! This is because the kind signature
  only affects the /result/ of the application, not all of the individual
  arguments. So adding a kind signature here won't make a difference. Therefore,
  the fourth (and final) iteration of the Question We Must Answer is:
  
  * Given the first n arguments of T (ty_1 ... ty_n), consider the binders
    of T that are instantiated by non-omitted arguments. For each such binder
    b_i, take the union of all injective_vars_of_binder(b_i). Is this set a
    superset of the free variables of the kind of (T ty_1 ... ty_n)?
  
  Phew, that was a lot of work!
  
  How can be sure that this is correct? That is, how can we be sure that in the
  event that we leave off a kind annotation, that one could infer the kind of the
  tycon application from its arguments? It's essentially a proof by induction: if
  we can infer the kinds of every subtree of a type, then the whole tycon
  application will have an inferrable kind--unless, of course, the remainder of
  the tycon application's kind has uninstantiated kind variables.
  
  What happens if T is oversaturated? That is, if T's kind has fewer than n
  arguments, in the case that the concrete application instantiates a result
  kind variable with an arrow kind? If we run out of arguments, we do not attach
  a kind annotation. This should be a rare case, indeed. Here is an example:
  
     data T1 :: k1 -> k2 -> *
     data T2 :: k1 -> k2 -> *
  
     type family G (a :: k) :: k
     type instance G T1 = T2
  
     type instance F Char = (G T1 Bool :: (* -> *) -> *)   -- F from above
  
  Here G's kind is (forall k. k -> k), and the desugared RHS of that last
  instance of F is (G (* -> (* -> *) -> *) (T1 * (* -> *)) Bool). According to
  the algorithm above, there are 3 arguments to G so we should peel off 3
  arguments in G's kind. But G's kind has only two arguments. This is the
  rare special case, and we choose not to annotate the application of G with
  a kind signature. After all, we needn't do this, since that instance would
  be reified as:
  
     type instance F Char = G (T1 :: * -> (* -> *) -> *) Bool
  
  So the kind of G isn't ambiguous anymore due to the explicit kind annotation
  on its argument. See #8953 and test th/T8953.
  -}
  
  {-
  ************************************************************************
  *                                                                      *
          Multiplicities
  *                                                                      *
  ************************************************************************
  
  These functions would prefer to be in GHC.Core.Multiplicity, but
  they some are used elsewhere in this module, and wanted to bring
  their friends here with them.
  -}
  
  unrestricted, linear, tymult :: a -> Scaled a
  
  -- | Scale a payload by Many
  unrestricted = Scaled ManyTy
  
  -- | Scale a payload by One
  linear = Scaled OneTy
  
  -- | Scale a payload by Many; used for type arguments in core
  tymult = Scaled ManyTy
  
  irrelevantMult :: Scaled a -> a
  irrelevantMult = scaledThing
  
  mkScaled :: Mult -> a -> Scaled a
  mkScaled = Scaled
  
  scaledSet :: Scaled a -> b -> Scaled b
  scaledSet (Scaled m _) b = Scaled m b
  
  pattern OneTy :: Mult
  pattern OneTy <- (isOneTy -> True)
    where OneTy = oneDataConTy
  
  pattern ManyTy :: Mult
  pattern ManyTy <- (isManyTy -> True)
    where ManyTy = manyDataConTy
  
  isManyTy :: Mult -> Bool
  isManyTy ty
    | Just tc <- tyConAppTyCon_maybe ty
    = tc `hasKey` manyDataConKey
  isManyTy _ = False
  
  isOneTy :: Mult -> Bool
  isOneTy ty
    | Just tc <- tyConAppTyCon_maybe ty
    = tc `hasKey` oneDataConKey
  isOneTy _ = False
  
  isLinearType :: Type -> Bool
  -- ^ @isLinear t@ returns @True@ of a if @t@ is a type of (curried) function
  -- where at least one argument is linear (or otherwise non-unrestricted). We use
  -- this function to check whether it is safe to eta reduce an Id in CorePrep. It
  -- is always safe to return 'True', because 'True' deactivates the optimisation.
  isLinearType ty = case ty of
                        FunTy _ ManyTy _ res -> isLinearType res
                        FunTy _ _ _ _        -> True
                        ForAllTy _ res       -> isLinearType res
                        _ -> False
  
  {- *********************************************************************
  *                                                                      *
                      Space-saving construction
  *                                                                      *
  ********************************************************************* -}
  
  {- Note [Using synonyms to compress types]
  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  Was: [Prefer Type over TYPE (BoxedRep Lifted)]
  
  The Core of nearly any program will have numerous occurrences of the Types
  
     TyConApp BoxedRep [TyConApp Lifted []]    -- Synonym LiftedRep
     TyConApp BoxedRep [TyConApp Unlifted []]  -- Synonym UnliftedREp
     TyConApp TYPE [TyConApp LiftedRep []]     -- Synonym Type
     TyConApp TYPE [TyConApp UnliftedRep []]   -- Synonym UnliftedType
  
  While investigating #17292 we found that these constituted a majority
  of all TyConApp constructors on the heap:
  
      (From a sample of 100000 TyConApp closures)
      0x45f3523    - 28732 - `Type`
      0x420b840702 - 9629  - generic type constructors
      0x42055b7e46 - 9596
      0x420559b582 - 9511
      0x420bb15a1e - 9509
      0x420b86c6ba - 9501
      0x42055bac1e - 9496
      0x45e68fd    - 538   - `TYPE ...`
  
  Consequently, we try hard to ensure that operations on such types are
  efficient. Specifically, we strive to
  
   a. Avoid heap allocation of such types; use a single static TyConApp
   b. Use a small (shallow in the tree-depth sense) representation
      for such types
  
  Goal (b) is particularly useful as it makes traversals (e.g. free variable
  traversal, substitution, and comparison) more efficient.
  Comparison in particular takes special advantage of nullary type synonym
  applications (e.g. things like @TyConApp typeTyCon []@). See
  * Note [Comparing type synonyms] in "GHC.Core.TyCo.Compare"
  * Note [Unifying type synonyms] in "GHC.Core.Unify"
  
  To accomplish these we use a number of tricks, implemented by mkTyConApp.
  
   1. Instead of (TyConApp BoxedRep [TyConApp Lifted []]),
      we prefer a statically-allocated (TyConApp LiftedRep [])
      where `LiftedRep` is a type synonym:
         type LiftedRep = BoxedRep Lifted
      Similarly for UnliftedRep
  
   2. Instead of (TyConApp TYPE [TyConApp LiftedRep []])
      we prefer the statically-allocated (TyConApp Type [])
      where `Type` is a type synonym
         type Type = TYPE LiftedRep
      Similarly for UnliftedType
  
  These serve goal (b) since there are no applied type arguments to traverse,
  e.g., during comparison.
  
   3. We have a single, statically allocated top-level binding to
      represent `TyConApp GHC.Types.Type []` (namely
      'GHC.Builtin.Types.Prim.liftedTypeKind'), ensuring that we don't
      need to allocate such types (goal (a)).  See functions
      mkTYPEapp and mkBoxedRepApp
  
   4. We use the sharing mechanism described in Note [Sharing nullary TyConApps]
      in GHC.Core.TyCon to ensure that we never need to allocate such
      nullary applications (goal (a)).
  
  See #17958, #20541
  -}
  
  -- | 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 []
    = -- See Note [Sharing nullary TyConApps] in GHC.Core.TyCon
      mkTyConTy tycon
  
  mkTyConApp tycon tys@(ty1:rest)
    | Just fun_ty <- tyConAppFunTy_maybe tycon tys
    = fun_ty
  
    -- See Note [Using synonyms to compress types]
    | key == tYPETyConKey
    , Just ty <- mkTYPEapp_maybe ty1
    = assert (null rest) ty
  
    | key == cONSTRAINTTyConKey
    , Just ty <- mkCONSTRAINTapp_maybe ty1
    = assert (null rest) ty
  
    -- See Note [Using synonyms to compress types]
    | key == boxedRepDataConTyConKey
    , Just ty <- mkBoxedRepApp_maybe ty1
    = assert (null rest) ty
  
    | key == tupleRepDataConTyConKey
    , Just ty <- mkTupleRepApp_maybe ty1
    = assert (null rest) ty
  
    -- The catch-all case
    | otherwise
    = TyConApp tycon tys
    where
      key = tyConUnique tycon
  
  
  {- Note [Care using synonyms to compress types]
  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  Using a synonym to compress a types has a tricky wrinkle. Consider
  coreView applied to (TyConApp LiftedRep [])
  
  * coreView expands the LiftedRep synonym:
       type LiftedRep = BoxedRep Lifted
  
  * Danger: we might apply the empty substitution to the RHS of the
    synonym.  And substTy calls mkTyConApp BoxedRep [Lifted]. And
    mkTyConApp compresses that back to LiftedRep.  Loop!
  
  * Solution: in expandSynTyConApp_maybe, don't call substTy for nullary
    type synonyms.  That's more efficient anyway.
  -}
  
  
  mkTYPEapp :: RuntimeRepType -> Type
  mkTYPEapp rr
    = case mkTYPEapp_maybe rr of
         Just ty -> ty
         Nothing -> TyConApp tYPETyCon [rr]
  
  mkTYPEapp_maybe :: RuntimeRepType -> Maybe Type
  -- ^ Given a @RuntimeRep@, applies @TYPE@ to it.
  -- On the fly it rewrites
  --      TYPE LiftedRep      -->   liftedTypeKind    (a synonym)
  --      TYPE UnliftedRep    -->   unliftedTypeKind  (ditto)
  --      TYPE ZeroBitRep     -->   zeroBitTypeKind   (ditto)
  -- NB: no need to check for TYPE (BoxedRep Lifted), TYPE (BoxedRep Unlifted)
  --     because those inner types should already have been rewritten
  --     to LiftedRep and UnliftedRep respectively, by mkTyConApp
  --
  -- see Note [TYPE and CONSTRAINT] in GHC.Builtin.Types.Prim.
  -- See Note [Using synonyms to compress types] in GHC.Core.Type
  {-# NOINLINE mkTYPEapp_maybe #-}
  mkTYPEapp_maybe (TyConApp tc args)
    | key == liftedRepTyConKey    = assert (null args) $ Just liftedTypeKind   -- TYPE LiftedRep
    | key == unliftedRepTyConKey  = assert (null args) $ Just unliftedTypeKind -- TYPE UnliftedRep
    | key == zeroBitRepTyConKey   = assert (null args) $ Just zeroBitTypeKind  -- TYPE ZeroBitRep
    where
      key = tyConUnique tc
  mkTYPEapp_maybe _ = Nothing
  
  ------------------
  mkCONSTRAINTapp :: RuntimeRepType -> Type
  -- ^ Just like mkTYPEapp
  mkCONSTRAINTapp rr
    = case mkCONSTRAINTapp_maybe rr of
         Just ty -> ty
         Nothing -> TyConApp cONSTRAINTTyCon [rr]
  
  mkCONSTRAINTapp_maybe :: RuntimeRepType -> Maybe Type
  -- ^ Just like mkTYPEapp_maybe
  {-# NOINLINE mkCONSTRAINTapp_maybe #-}
  mkCONSTRAINTapp_maybe (TyConApp tc args)
    | tc `hasKey` liftedRepTyConKey = assert (null args) $
                                      Just constraintKind   -- CONSTRAINT LiftedRep
  mkCONSTRAINTapp_maybe _ = Nothing
  
  ------------------
  mkBoxedRepApp_maybe :: LevityType -> Maybe Type
  -- ^ Given a `Levity`, apply `BoxedRep` to it
  -- On the fly, rewrite
  --      BoxedRep Lifted     -->   liftedRepTy    (a synonym)
  --      BoxedRep Unlifted   -->   unliftedRepTy  (ditto)
  -- See Note [TYPE and CONSTRAINT] in GHC.Builtin.Types.Prim.
  -- See Note [Using synonyms to compress types] in GHC.Core.Type
  {-# NOINLINE mkBoxedRepApp_maybe #-}
  mkBoxedRepApp_maybe (TyConApp tc args)
    | key == liftedDataConKey   = assert (null args) $ Just liftedRepTy    -- BoxedRep Lifted
    | key == unliftedDataConKey = assert (null args) $ Just unliftedRepTy  -- BoxedRep Unlifted
    where
      key = tyConUnique tc
  mkBoxedRepApp_maybe _ = Nothing
  
  mkTupleRepApp_maybe :: Type -> Maybe Type
  -- ^ Given a `[RuntimeRep]`, apply `TupleRep` to it
  -- On the fly, rewrite
  --      TupleRep [] -> zeroBitRepTy   (a synonym)
  -- See Note [TYPE and CONSTRAINT] in GHC.Builtin.Types.Prim.
  -- See Note [Using synonyms to compress types] in GHC.Core.Type
  {-# NOINLINE mkTupleRepApp_maybe #-}
  mkTupleRepApp_maybe (TyConApp tc args)
    | key == nilDataConKey = assert (isSingleton args) $ Just zeroBitRepTy  -- ZeroBitRep
    where
      key = tyConUnique tc
  mkTupleRepApp_maybe _ = Nothing
  
  typeOrConstraintKind :: TypeOrConstraint -> RuntimeRepType -> Kind
  typeOrConstraintKind TypeLike       rep = mkTYPEapp       rep
  typeOrConstraintKind ConstraintLike rep = mkCONSTRAINTapp rep
  
  liftedTypeOrConstraintKind :: TypeOrConstraint -> Kind
  liftedTypeOrConstraintKind TypeLike       = liftedTypeKind
  liftedTypeOrConstraintKind ConstraintLike = constraintKind
  