% % (c) The University of Glasgow 2006 % (c) The GRASP/AQUA Project, Glasgow University, 1992-1998 % Arity and ete expansion \begin{code}

    -- | Arit and eta expansion
    module CoreArity (
    	manifestArity, exprArity, exprBotStrictness_maybe,
    	exprEtaExpandArity, CheapFun, etaExpand
        ) where

    #include "HsVersions.h"

    import CoreSyn
    import CoreFVs
    import CoreUtils
    import CoreSubst
    import Demand
    import Var
    import VarEnv
    import Id
    import Type
    import TyCon	( isRecursiveTyCon, isClassTyCon )
    import Coercion
    import BasicTypes
    import Unique
    import Outputable
    import FastString

\end{code} %************************************************************************ %* * manifestArity and exprArity %* * %************************************************************************ exprArity is a cheap-and-cheerful version of exprEtaExpandArity. It tells how many things the expression can be applied to before doing any work. It doesn't look inside cases, lets, etc. The idea is that exprEtaExpandArity will do the hard work, leaving something that's easy for exprArity to grapple with. In particular, Simplify uses exprArity to compute the ArityInfo for the Id. Originally I thought that it was enough just to look for top-level lambdas, but it isn't. I've seen this foo = PrelBase.timesInt We want foo to get arity 2 even though the eta-expander will leave it unchanged, in the expectation that it'll be inlined. But occasionally it isn't, because foo is blacklisted (used in a rule). Similarly, see the ok_note check in exprEtaExpandArity. So f = __inline_me (\x -> e) won't be eta-expanded. And in any case it seems more robust to have exprArity be a bit more intelligent. But note that (\x y z -> f x y z) should have arity 3, regardless of f's arity. \begin{code}

    manifestArity :: CoreExpr -> Arity
    -- ^ manifestArity sees how many leading value lambdas there are
    manifestArity (Lam v e) | isId v    	= 1 + manifestArity e
    			| otherwise 	= manifestArity e
    manifestArity (Note n e) | notSccNote n	= manifestArity e
    manifestArity (Cast e _)            	= manifestArity e
    manifestArity _                     	= 0

    ---------------
    exprArity :: CoreExpr -> Arity
    -- ^ An approximate, fast, version of 'exprEtaExpandArity'
    exprArity e = go e
      where
        go (Var v) 	       	           = idArity v
        go (Lam x e) | isId x    	   = go e + 1
        		 | otherwise 	   = go e
        go (Note n e) | notSccNote n   = go e
        go (Cast e co)                 = go e `min` length (typeArity (snd (coercionKind co)))
           	       			     	-- Note [exprArity invariant]
        go (App e (Type _))            = go e
        go (App f a) | exprIsTrivial a = (go f - 1) `max` 0
            -- See Note [exprArity for applications]
        go _		       	   = 0

    ---------------
    typeArity :: Type -> [OneShot]
    -- How many value arrows are visible in the type?
    -- We look through foralls, and newtypes
    -- See Note [exprArity invariant]
    typeArity ty
      | Just (_, ty')  <- splitForAllTy_maybe ty
      = typeArity ty'

      | Just (arg,res) <- splitFunTy_maybe ty
      = isStateHackType arg : typeArity res

      | Just (tc,tys) <- splitTyConApp_maybe ty
      , Just (ty', _) <- instNewTyCon_maybe tc tys
      , not (isRecursiveTyCon tc)
      , not (isClassTyCon tc)	-- Do not eta-expand through newtype classes
        		      		-- See Note [Newtype classes and eta expansion]
      = typeArity ty'
      	-- Important to look through non-recursive newtypes, so that, eg
      	--	(f x)   where f has arity 2, f :: Int -> IO ()
      	-- Here we want to get arity 1 for the result!

      | otherwise
      = []

    ---------------
    exprBotStrictness_maybe :: CoreExpr -> Maybe (Arity, StrictSig)
    -- A cheap and cheerful function that identifies bottoming functions
    -- and gives them a suitable strictness signatures.  It's used during
    -- float-out
    exprBotStrictness_maybe e
      = case getBotArity (arityType is_cheap e) of
    	Nothing -> Nothing
    	Just ar -> Just (ar, mkStrictSig (mkTopDmdType (replicate ar topDmd) BotRes))
      where
        is_cheap _ _ = False  -- Irrelevant for this purpose

\end{code} Note [exprArity invariant] ~~~~~~~~~~~~~~~~~~~~~~~~~~ exprArity has the following invariant: * If typeArity (exprType e) = n, then manifestArity (etaExpand e n) = n That is, etaExpand can always expand as much as typeArity says So the case analysis in etaExpand and in typeArity must match * exprArity e <= typeArity (exprType e) * Hence if (exprArity e) = n, then manifestArity (etaExpand e n) = n That is, if exprArity says "the arity is n" then etaExpand really can get "n" manifest lambdas to the top. Why is this important? Because - In TidyPgm we use exprArity to fix the *final arity* of each top-level Id, and in - In CorePrep we use etaExpand on each rhs, so that the visible lambdas actually match that arity, which in turn means that the StgRhs has the right number of lambdas An alternative would be to do the eta-expansion in TidyPgm, at least for top-level bindings, in which case we would not need the trim_arity in exprArity. That is a less local change, so I'm going to leave it for today! Note [Newtype classes and eta expansion] ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ We have to be careful when eta-expanding through newtypes. In general it's a good idea, but annoyingly it interacts badly with the class-op rule mechanism. Consider class C a where { op :: a -> a } instance C b => C [b] where op x = ... These translate to co :: forall a. (a->a) ~ C a $copList :: C b -> [b] -> [b] $copList d x = ... $dfList :: C b -> C [b] {-# DFunUnfolding = [$copList] #-} $dfList d = $copList d |> co@[b] Now suppose we have: dCInt :: C Int blah :: [Int] -> [Int] blah = op ($dfList dCInt) Now we want the built-in op/$dfList rule will fire to give blah = $copList dCInt But with eta-expansion 'blah' might (and in Trac #3772, which is slightly more complicated, does) turn into blah = op (\eta. ($dfList dCInt |> sym co) eta) and now it is *much* harder for the op/$dfList rule to fire, becuase exprIsConApp_maybe won't hold of the argument to op. I considered trying to *make* it hold, but it's tricky and I gave up. The test simplCore/should_compile/T3722 is an excellent example. Note [exprArity for applications] ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ When we come to an application we check that the arg is trivial. eg f (fac x) does not have arity 2, even if f has arity 3! * We require that is trivial rather merely cheap. Suppose f has arity 2. Then f (Just y) has arity 0, because if we gave it arity 1 and then inlined f we'd get let v = Just y in \w. which has arity 0. And we try to maintain the invariant that we don't have arity decreases. * The `max 0` is important! (\x y -> f x) has arity 2, even if f is unknown, hence arity 0 %************************************************************************ %* * Computing the "arity" of an expression %* * %************************************************************************ Note [Definition of arity] ~~~~~~~~~~~~~~~~~~~~~~~~~~ The "arity" of an expression 'e' is n if applying 'e' to *fewer* than n *value* arguments converges rapidly Or, to put it another way there is no work lost in duplicating the partial application (e x1 .. x(n-1)) In the divegent case, no work is lost by duplicating because if the thing is evaluated once, that's the end of the program. Or, to put it another way, in any context C C[ (\x1 .. xn. e x1 .. xn) ] is as efficient as C[ e ] It's all a bit more subtle than it looks: Note [Arity of case expressions] ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ We treat the arity of case x of p -> \s -> ... as 1 (or more) because for I/O ish things we really want to get that \s to the top. We are prepared to evaluate x each time round the loop in order to get that. This isn't really right in the presence of seq. Consider f = \x -> case x of True -> \y -> x+y False -> \y -> x-y Can we eta-expand here? At first the answer looks like "yes of course", but consider (f bot) `seq` 1 This should diverge! But if we eta-expand, it won't. Again, we ignore this "problem", because being scrupulous would lose an important transformation for many programs. 1. Note [One-shot lambdas] ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Consider one-shot lambdas let x = expensive in \y z -> E We want this to have arity 1 if the \y-abstraction is a 1-shot lambda. 3. Note [Dealing with bottom] ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Consider f = \x -> error "foo" Here, arity 1 is fine. But if it is f = \x -> case x of True -> error "foo" False -> \y -> x+y then we want to get arity 2. Technically, this isn't quite right, because (f True) `seq` 1 should diverge, but it'll converge if we eta-expand f. Nevertheless, we do so; it improves some programs significantly, and increasing convergence isn't a bad thing. Hence the ABot/ATop in ArityType. 4. Note [Newtype arity] ~~~~~~~~~~~~~~~~~~~~~~~~ Non-recursive newtypes are transparent, and should not get in the way. We do (currently) eta-expand recursive newtypes too. So if we have, say newtype T = MkT ([T] -> Int) Suppose we have e = coerce T f where f has arity 1. Then: etaExpandArity e = 1; that is, etaExpandArity looks through the coerce. When we eta-expand e to arity 1: eta_expand 1 e T we want to get: coerce T (\x::[T] -> (coerce ([T]->Int) e) x) HOWEVER, note that if you use coerce bogusly you can ge coerce Int negate And since negate has arity 2, you might try to eta expand. But you can't decopose Int to a function type. Hence the final case in eta_expand. Note [The state-transformer hack] ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Suppose we have f = e where e has arity n. Then, if we know from the context that f has a usage type like t1 -> ... -> tn -1-> t(n+1) -1-> ... -1-> tm -> ... then we can expand the arity to m. This usage type says that any application (x e1 .. en) will be applied to uniquely to (m-n) more args Consider f = \x. let y = <expensive> in case x of True -> foo False -> \(s:RealWorld) -> e where foo has arity 1. Then we want the state hack to apply to foo too, so we can eta expand the case. Then we expect that if f is applied to one arg, it'll be applied to two (that's the hack -- we don't really know, and sometimes it's false) See also Id.isOneShotBndr. Note [State hack and bottoming functions] ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ It's a terrible idea to use the state hack on a bottoming function. Here's what happens (Trac #2861): f :: String -> IO T f = \p. error "..." Eta-expand, using the state hack: f = \p. (\s. ((error "...") |> g1) s) |> g2 g1 :: IO T ~ (S -> (S,T)) g2 :: (S -> (S,T)) ~ IO T Extrude the g2 f' = \p. \s. ((error "...") |> g1) s f = f' |> (String -> g2) Discard args for bottomming function f' = \p. \s. ((error "...") |> g1 |> g3 g3 :: (S -> (S,T)) ~ (S,T) Extrude g1.g3 f'' = \p. \s. (error "...") f' = f'' |> (String -> S -> g1.g3) And now we can repeat the whole loop. Aargh! The bug is in applying the state hack to a function which then swallows the argument. This arose in another guise in Trac #3959. Here we had catch# (throw exn >> return ()) Note that (throw :: forall a e. Exn e => e -> a) is called with [a = IO ()]. After inlining (>>) we get catch# (\_. throw {IO ()} exn) We must *not* eta-expand to catch# (\_ _. throw {...} exn) because 'catch#' expects to get a (# _,_ #) after applying its argument to a State#, not another function! In short, we use the state hack to allow us to push let inside a lambda, but not to introduce a new lambda. Note [ArityType] ~~~~~~~~~~~~~~~~ ArityType is the result of a compositional analysis on expressions, from which we can decide the real arity of the expression (extracted with function exprEtaExpandArity). Here is what the fields mean. If an arbitrary expression 'f' has ArityType 'at', then * If at = ABot n, then (f x1..xn) definitely diverges. Partial applications to fewer than n args may *or may not* diverge. We allow ourselves to eta-expand bottoming functions, even if doing so may lose some `seq` sharing, let x = <expensive> in \y. error (g x y) ==> \y. let x = <expensive> in error (g x y) * If at = ATop as, and n=length as, then expanding 'f' to (\x1..xn. f x1 .. xn) loses no sharing, assuming the calls of f respect the one-shot-ness of of its definition. NB 'f' is an arbitary expression, eg (f = g e1 e2). This 'f' can have ArityType as ATop, with length as > 0, only if e1 e2 are themselves. * In both cases, f, (f x1), ... (f x1 ... f(n-1)) are definitely really functions, or bottom, but *not* casts from a data type, in at least one case branch. (If it's a function in one case branch but an unsafe cast from a data type in another, the program is bogus.) So eta expansion is dynamically ok; see Note [State hack and bottoming functions], the part about catch# Example: f = \x\y. let v = <expensive> in \s(one-shot) \t(one-shot). blah 'f' has ArityType [ManyShot,ManyShot,OneShot,OneShot] The one-shot-ness means we can, in effect, push that 'let' inside the \st. Suppose f = \xy. x+y Then f :: AT [False,False] ATop f v :: AT [False] ATop f <expensive> :: AT [] ATop -------------------- Main arity code ---------------------------- \begin{code}

    -- See Note [ArityType]
    data ArityType = ATop [OneShot] | ABot Arity
         -- There is always an explicit lambda
         -- to justify the [OneShot], or the Arity

    type OneShot = Bool    -- False <=> Know nothing
                           -- True  <=> Can definitely float inside this lambda
    	               -- The 'True' case can arise either because a binder
    		       -- is marked one-shot, or because it's a state lambda
    		       -- and we have the state hack on

    vanillaArityType :: ArityType
    vanillaArityType = ATop []	-- Totally uninformative

    -- ^ The Arity returned is the number of value args the [_$_]
    -- expression can be applied to without doing much work
    exprEtaExpandArity :: CheapFun -> CoreExpr -> Arity
    -- exprEtaExpandArity is used when eta expanding
    -- 	e  ==>  \xy -> e x y
    exprEtaExpandArity cheap_fun e
      = case (arityType cheap_fun e) of
          ATop (os:oss)
            | os || has_lam e -> 1 + length oss	-- Note [Eta expanding thunks]
            | otherwise       -> 0
          ATop []             -> 0
          ABot n              -> n
      where
        has_lam (Note _ e) = has_lam e
        has_lam (Lam b e)  = isId b || has_lam e
        has_lam _          = False

    getBotArity :: ArityType -> Maybe Arity
    -- Arity of a divergent function
    getBotArity (ABot n) = Just n
    getBotArity _        = Nothing

\end{code} Note [Eta expanding thunks] ~~~~~~~~~~~~~~~~~~~~~~~~~~~ When we see f = case y of p -> \x -> blah should we eta-expand it? Well, if 'x' is a one-shot state token then 'yes' because 'f' will only be applied once. But otherwise we (conservatively) say no. My main reason is to avoid expanding PAPSs f = g d ==> f = \x. g d x because that might in turn make g inline (if it has an inline pragma), which we might not want. After all, INLINE pragmas say "inline only when saturate" so we don't want to be too gung-ho about saturating! \begin{code}

    arityLam :: Id -> ArityType -> ArityType
    arityLam id (ATop as) = ATop (isOneShotBndr id : as)
    arityLam _  (ABot n)  = ABot (n+1)

    floatIn :: Bool -> ArityType -> ArityType
    -- We have something like (let x = E in b),
    -- where b has the given arity type.
    floatIn _     (ABot n)  = ABot n
    floatIn True  (ATop as) = ATop as
    floatIn False (ATop as) = ATop (takeWhile id as)
       -- If E is not cheap, keep arity only for one-shots

    arityApp :: ArityType -> Bool -> ArityType
    -- Processing (fun arg) where at is the ArityType of fun,
    -- Knock off an argument and behave like 'let'
    arityApp (ABot 0)      _     = ABot 0
    arityApp (ABot n)      _     = ABot (n-1)
    arityApp (ATop [])     _     = ATop []
    arityApp (ATop (_:as)) cheap = floatIn cheap (ATop as)

    andArityType :: ArityType -> ArityType -> ArityType   -- Used for branches of a 'case'
    andArityType (ABot n1) (ABot n2)
      = ABot (n1 `min` n2)
    andArityType (ATop as)  (ABot _)  = ATop as
    andArityType (ABot _)   (ATop bs) = ATop bs
    andArityType (ATop as)  (ATop bs) = ATop (as `combine` bs)
      where	     -- See Note [Combining case branches]
        combine (a:as) (b:bs) = (a && b) : combine as bs
        combine []     bs     = take_one_shots bs
        combine as     []     = take_one_shots as

        take_one_shots [] = []
        take_one_shots (one_shot : as)
          | one_shot  = True : take_one_shots as
          | otherwise = []

\end{code} Note [Combining case branches] ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Consider go = \x. let z = go e0 go2 = \x. case x of True -> z False -> \s(one-shot). e1 in go2 x We *really* want to eta-expand go and go2. When combining the barnches of the case we have ATop [] `andAT` ATop [True] and we want to get ATop [True]. But if the inner lambda wasn't one-shot we don't want to do this. (We need a proper arity analysis to justify that.) \begin{code}

    ---------------------------
    type CheapFun = CoreExpr -> Maybe Type -> Bool
         	-- How to decide if an expression is cheap
    	-- If the Maybe is Just, the type is the type
    	-- of the expression; Nothing means "don't know"

    arityType :: CheapFun -> CoreExpr -> ArityType
    arityType _ (Var v)
      | Just strict_sig <- idStrictness_maybe v
      , (ds, res) <- splitStrictSig strict_sig
      , let arity = length ds
      = if isBotRes res then ABot arity
                        else ATop (take arity one_shots)
      | otherwise
      = ATop (take (idArity v) one_shots)
      where
        one_shots :: [Bool]	    -- One-shot-ness derived from the type
        one_shots = typeArity (idType v)

    	-- Lambdas; increase arity
    arityType cheap_fn (Lam x e)
      | isId x    = arityLam x (arityType cheap_fn e)
      | otherwise = arityType cheap_fn e

    	-- Applications; decrease arity
    arityType cheap_fn (App fun (Type _))
       = arityType cheap_fn fun
    arityType cheap_fn (App fun arg )
       = arityApp (arityType cheap_fn fun) (cheap_fn arg Nothing)

    	-- Case/Let; keep arity if either the expression is cheap
    	-- or it's a 1-shot lambda
    	-- The former is not really right for Haskell
    	--	f x = case x of { (a,b) -> \y. e }
    	--  ===>
    	--	f x y = case x of { (a,b) -> e }
    	-- The difference is observable using 'seq'
    arityType cheap_fn (Case scrut bndr _ alts)
      = floatIn (cheap_fn scrut (Just (idType bndr)))
    	    (foldr1 andArityType [arityType cheap_fn rhs | (_,_,rhs) <- alts])

    arityType cheap_fn (Let b e)
      = floatIn (cheap_bind b) (arityType cheap_fn e)
      where
        cheap_bind (NonRec b e) = is_cheap (b,e)
        cheap_bind (Rec prs)    = all is_cheap prs
        is_cheap (b,e) = cheap_fn e (Just (idType b))

    arityType cheap_fn (Note n e)
      | notSccNote n              = arityType cheap_fn e
    arityType cheap_fn (Cast e _) = arityType cheap_fn e
    arityType _           _       = vanillaArityType

\end{code} %************************************************************************ %* * The main eta-expander %* * %************************************************************************ We go for: f = \x1..xn -> N ==> f = \x1..xn y1..ym -> N y1..ym (n >= 0) where (in both cases) * The xi can include type variables * The yi are all value variables * N is a NORMAL FORM (i.e. no redexes anywhere) wanting a suitable number of extra args. The biggest reason for doing this is for cases like f = \x -> case x of True -> \y -> e1 False -> \y -> e2 Here we want to get the lambdas together. A good exmaple is the nofib program fibheaps, which gets 25% more allocation if you don't do this eta-expansion. We may have to sandwich some coerces between the lambdas to make the types work. exprEtaExpandArity looks through coerces when computing arity; and etaExpand adds the coerces as necessary when actually computing the expansion. Note [No crap in eta-expanded code] ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The eta expander is careful not to introduce "crap". In particular, given a CoreExpr satisfying the 'CpeRhs' invariant (in CorePrep), it returns a CoreExpr satisfying the same invariant. See Note [Eta expansion and the CorePrep invariants] in CorePrep. This means the eta-expander has to do a bit of on-the-fly simplification but it's not too hard. The alernative, of relying on a subsequent clean-up phase of the Simplifier to de-crapify the result, means you can't really use it in CorePrep, which is painful. Note [Eta expansion and SCCs] ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Note that SCCs are not treated specially by etaExpand. If we have etaExpand 2 (\x -> scc "foo" e) = (\xy -> (scc "foo" e) y) So the costs of evaluating 'e' (not 'e y') are attributed to "foo" \begin{code}

    -- | @etaExpand n us e ty@ returns an expression with
    -- the same meaning as @e@, but with arity @n@.
    --
    -- Given:
    --
    -- > e' = etaExpand n us e ty
    --
    -- We should have that:
    --
    -- > ty = exprType e = exprType e'
    etaExpand :: Arity	  	-- ^ Result should have this number of value args
    	  -> CoreExpr	        -- ^ Expression to expand
    	  -> CoreExpr
    -- etaExpand deals with for-alls. For example:
    --		etaExpand 1 E
    -- where  E :: forall a. a -> a
    -- would return
    --	(/\b. \y::a -> E b y)
    --
    -- It deals with coerces too, though they are now rare
    -- so perhaps the extra code isn't worth it

    etaExpand n orig_expr
      = go n orig_expr
      where
          -- Strip off existing lambdas and casts
          -- Note [Eta expansion and SCCs]
        go 0 expr = expr
        go n (Lam v body) | isTyCoVar v = Lam v (go n     body)
           	              | otherwise   = Lam v (go (n-1) body)
        go n (Cast expr co) = Cast (go n expr) co
        go n expr           = -- pprTrace "ee" (vcat [ppr orig_expr, ppr expr, ppr etas]) $
           	 		  etaInfoAbs etas (etaInfoApp subst' expr etas)
        	  		where
    			    in_scope = mkInScopeSet (exprFreeVars expr)
    			    (in_scope', etas) = mkEtaWW n in_scope (exprType expr)
    			    subst' = mkEmptySubst in_scope'

          	      	       	        -- Wrapper    Unwrapper
    --------------
    data EtaInfo = EtaVar Var	-- /\a. [],   [] a
         	                 	-- \x.  [],   [] x
    	     | EtaCo Coercion   -- [] |> co,  [] |> (sym co)

    instance Outputable EtaInfo where
       ppr (EtaVar v) = ptext (sLit "EtaVar") <+> ppr v
       ppr (EtaCo co) = ptext (sLit "EtaCo")  <+> ppr co

    pushCoercion :: Coercion -> [EtaInfo] -> [EtaInfo]
    pushCoercion co1 (EtaCo co2 : eis)
      | isIdentityCoercion co = eis
      | otherwise	       	  = EtaCo co : eis
      where
        co = co1 `mkTransCoercion` co2

    pushCoercion co eis = EtaCo co : eis

    --------------
    etaInfoAbs :: [EtaInfo] -> CoreExpr -> CoreExpr
    etaInfoAbs []               expr = expr
    etaInfoAbs (EtaVar v : eis) expr = Lam v (etaInfoAbs eis expr)
    etaInfoAbs (EtaCo co : eis) expr = Cast (etaInfoAbs eis expr) (mkSymCoercion co)

    --------------
    etaInfoApp :: Subst -> CoreExpr -> [EtaInfo] -> CoreExpr
    -- (etaInfoApp s e eis) returns something equivalent to
    -- 	       ((substExpr s e) `appliedto` eis)

    etaInfoApp subst (Lam v1 e) (EtaVar v2 : eis)
      = etaInfoApp subst' e eis
      where
        subst' | isTyCoVar v1 = CoreSubst.extendTvSubst subst v1 (mkTyVarTy v2)
        	   | otherwise  = CoreSubst.extendIdSubst subst v1 (Var v2)

    etaInfoApp subst (Cast e co1) eis
      = etaInfoApp subst e (pushCoercion co' eis)
      where
        co' = CoreSubst.substTy subst co1

    etaInfoApp subst (Case e b _ alts) eis
      = Case (subst_expr subst e) b1 (coreAltsType alts') alts'
      where
        (subst1, b1) = substBndr subst b
        alts' = map subst_alt alts
        subst_alt (con, bs, rhs) = (con, bs', etaInfoApp subst2 rhs eis)
        	      where
    	      	 (subst2,bs') = substBndrs subst1 bs

    etaInfoApp subst (Let b e) eis
      = Let b' (etaInfoApp subst' e eis)
      where
        (subst', b') = subst_bind subst b

    etaInfoApp subst (Note note e) eis
      = Note note (etaInfoApp subst e eis)

    etaInfoApp subst e eis
      = go (subst_expr subst e) eis
      where
        go e []                  = e
        go e (EtaVar v    : eis) = go (App e (varToCoreExpr v)) eis
        go e (EtaCo co    : eis) = go (Cast e co) eis

    --------------
    mkEtaWW :: Arity -> InScopeSet -> Type
    	-> (InScopeSet, [EtaInfo])
    	-- EtaInfo contains fresh variables,
    	--   not free in the incoming CoreExpr
    	-- Outgoing InScopeSet includes the EtaInfo vars
    	--   and the original free vars

    mkEtaWW orig_n in_scope orig_ty
      = go orig_n empty_subst orig_ty []
      where
        empty_subst = mkTvSubst in_scope emptyTvSubstEnv

        go n subst ty eis	    -- See Note [exprArity invariant]
           | n == 0
           = (getTvInScope subst, reverse eis)

           | Just (tv,ty') <- splitForAllTy_maybe ty
           , let (subst', tv') = substTyVarBndr subst tv
               -- Avoid free vars of the original expression
           = go n subst' ty' (EtaVar tv' : eis)

           | Just (arg_ty, res_ty) <- splitFunTy_maybe ty
           , let (subst', eta_id') = freshEtaId n subst arg_ty
               -- Avoid free vars of the original expression
           = go (n-1) subst' res_ty (EtaVar eta_id' : eis)

           | Just(ty',co) <- splitNewTypeRepCo_maybe ty
           = 	-- Given this:
           		-- 	newtype T = MkT ([T] -> Int)
           		-- Consider eta-expanding this
           		--  	eta_expand 1 e T
           		-- We want to get
           		--	coerce T (\x::[T] -> (coerce ([T]->Int) e) x)
             go n subst ty' (EtaCo (Type.substTy subst co) : eis)

           | otherwise	 -- We have an expression of arity > 0,
           	 		 -- but its type isn't a function.
           = WARN( True, ppr orig_n <+> ppr orig_ty )
             (getTvInScope subst, reverse eis)
        	-- This *can* legitmately happen:
        	-- e.g.  coerce Int (\x. x) Essentially the programmer is
    	-- playing fast and loose with types (Happy does this a lot).
    	-- So we simply decline to eta-expand.  Otherwise we'd end up
    	-- with an explicit lambda having a non-function type

    --------------
    -- Avoiding unnecessary substitution; use short-cutting versions

    subst_expr :: Subst -> CoreExpr -> CoreExpr
    subst_expr = substExprSC (text "CoreArity:substExpr")

    subst_bind :: Subst -> CoreBind -> (Subst, CoreBind)
    subst_bind = substBindSC

    --------------
    freshEtaId :: Int -> TvSubst -> Type -> (TvSubst, Id)
    -- Make a fresh Id, with specified type (after applying substitution)
    -- It should be "fresh" in the sense that it's not in the in-scope set
    -- of the TvSubstEnv; and it should itself then be added to the in-scope
    -- set of the TvSubstEnv
    --
    -- The Int is just a reasonable starting point for generating a unique;
    -- it does not necessarily have to be unique itself.
    freshEtaId n subst ty
          = (subst', eta_id')
          where
            ty'     = Type.substTy subst ty
    	eta_id' = uniqAway (getTvInScope subst) $
    		  mkSysLocal (fsLit "eta") (mkBuiltinUnique n) ty'
    	subst'  = extendTvInScope subst eta_id'

\end{code}