{-
(c) The University of Glasgow 2006
(c) The GRASP/AQUA Project, Glasgow University, 1992-1998


        Arity and eta expansion
-}

{-# LANGUAGE CPP #-}

-- | Arity and eta expansion
module GHC.Core.Opt.Arity
   ( -- Finding arity
     manifestArity, joinRhsArity, exprArity
   , findRhsArity, cheapArityType
   , ArityOpts(..)

   -- ** Eta expansion
   , exprEtaExpandArity, etaExpand, etaExpandAT

   -- ** Eta reduction
   , tryEtaReduce

   -- ** ArityType
   , ArityType, mkBotArityType
   , arityTypeArity, idArityType

   -- ** Bottoming things
   , exprIsDeadEnd, exprBotStrictness_maybe, arityTypeBotSigs_maybe

   -- ** typeArity and the state hack
   , typeArity, typeOneShots, typeOneShot
   , isOneShotBndr
   , isStateHackType

   -- * Lambdas
   , zapLamBndrs


   -- ** Join points
   , etaExpandToJoinPoint, etaExpandToJoinPointRule

   -- ** Coercions and casts
   , pushCoArg, pushCoArgs, pushCoValArg, pushCoTyArg
   , pushCoercionIntoLambda, pushCoDataCon, collectBindersPushingCo
   )
where

import GHC.Prelude

import GHC.Core
import GHC.Core.FVs
import GHC.Core.Utils
import GHC.Core.DataCon
import GHC.Core.TyCon     ( tyConArity )
import GHC.Core.TyCon.RecWalk     ( initRecTc, checkRecTc )
import GHC.Core.Predicate ( isDictTy, isEvVar, isCallStackPredTy )
import GHC.Core.Multiplicity

-- We have two sorts of substitution:
--   GHC.Core.Subst.Subst, and GHC.Core.TyCo.Subst
-- Both have substTy, substCo  Hence need for qualification
import GHC.Core.Subst    as Core
import GHC.Core.Type     as Type
import GHC.Core.Coercion as Type
import GHC.Core.TyCo.Compare( eqType )

import GHC.Types.Demand
import GHC.Types.Cpr( CprSig, mkCprSig, botCpr )
import GHC.Types.Id
import GHC.Types.Var.Env
import GHC.Types.Var.Set
import GHC.Types.Basic
import GHC.Types.Tickish

import GHC.Builtin.Types.Prim
import GHC.Builtin.Uniques

import GHC.Data.FastString
import GHC.Data.Graph.UnVar
import GHC.Data.Pair

import GHC.Utils.GlobalVars( unsafeHasNoStateHack )
import GHC.Utils.Constants (debugIsOn)
import GHC.Utils.Outputable
import GHC.Utils.Panic
import GHC.Utils.Panic.Plain
import GHC.Utils.Misc

import Data.Maybe( isJust )

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

manifestArity :: CoreExpr -> Arity
-- ^ manifestArity sees how many leading value lambdas there are,
--   after looking through casts
manifestArity (Lam v e) | isId v        = 1 + manifestArity e
                        | otherwise     = manifestArity e
manifestArity (Tick t e) | not (tickishIsCode t) =  manifestArity e
manifestArity (Cast e _)                = manifestArity e
manifestArity _                         = 0

joinRhsArity :: CoreExpr -> JoinArity
-- Join points are supposed to have manifestly-visible
-- lambdas at the top: no ticks, no casts, nothing
-- Moreover, type lambdas count in JoinArity
joinRhsArity (Lam _ e) = 1 + joinRhsArity e
joinRhsArity _         = 0


---------------
exprBotStrictness_maybe :: CoreExpr -> Maybe (Arity, DmdSig, CprSig)
-- A cheap and cheerful function that identifies bottoming functions
-- and gives them a suitable strictness and CPR signatures.
-- It's used during float-out
exprBotStrictness_maybe e = arityTypeBotSigs_maybe (cheapArityType e)

arityTypeBotSigs_maybe :: ArityType ->  Maybe (Arity, DmdSig, CprSig)
-- Arity of a divergent function
arityTypeBotSigs_maybe (AT lams div)
  | isDeadEndDiv div = Just ( arity
                            , mkVanillaDmdSig arity botDiv
                            , mkCprSig arity botCpr)
  | otherwise        = Nothing
  where
    arity = length lams


{- 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. <f-body>
  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


************************************************************************
*                                                                      *
              typeArity and the "state hack"
*                                                                      *
********************************************************************* -}


typeArity :: Type -> Arity
-- ^ (typeArity ty) says how many arrows GHC can expose in 'ty', after
-- looking through newtypes.  More generally, (typeOneShots ty) returns
-- ty's [OneShotInfo], based only on the type itself, using typeOneShot
-- on the argument type to access the "state hack".
typeArity = length . typeOneShots

typeOneShots :: Type -> [OneShotInfo]
-- How many value arrows are visible in the type?
-- We look through foralls, and newtypes
-- See Note [Arity invariants for bindings]
typeOneShots ty
  = go initRecTc ty
  where
    go rec_nts ty
      | Just (_, ty')  <- splitForAllTyCoVar_maybe ty
      = go rec_nts ty'

      | Just (_,_,arg,res) <- splitFunTy_maybe ty
      = typeOneShot arg : go rec_nts res

      | Just (tc,tys) <- splitTyConApp_maybe ty
      , Just (ty', _) <- instNewTyCon_maybe tc tys
      , Just rec_nts' <- checkRecTc rec_nts tc  -- See Note [Expanding newtypes and products]
                                                -- in GHC.Core.TyCon
--   , not (isClassTyCon tc)    -- Do not eta-expand through newtype classes
--                              -- See Note [Newtype classes and eta expansion]
--                              (no longer required)
      = go rec_nts' 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!
        --
        -- AND through a layer of recursive newtypes
        -- e.g. newtype Stream m a b = Stream (m (Either b (a, Stream m a b)))

      | otherwise
      = []

typeOneShot :: Type -> OneShotInfo
typeOneShot ty
   | isStateHackType ty = OneShotLam
   | otherwise          = NoOneShotInfo

-- | Like 'idOneShotInfo', but taking the Horrible State Hack in to account
-- See Note [The state-transformer hack] in "GHC.Core.Opt.Arity"
idStateHackOneShotInfo :: Id -> OneShotInfo
idStateHackOneShotInfo id
    | isStateHackType (idType id) = OneShotLam
    | otherwise                   = idOneShotInfo id

-- | Returns whether the lambda associated with the 'Id' is
--   certainly applied at most once
-- This one is the "business end", called externally.
-- It works on type variables as well as Ids, returning True
-- Its main purpose is to encapsulate the Horrible State Hack
-- See Note [The state-transformer hack] in "GHC.Core.Opt.Arity"
isOneShotBndr :: Var -> Bool
isOneShotBndr var
  | isTyVar var                              = True
  | OneShotLam <- idStateHackOneShotInfo var = True
  | otherwise                                = False

isStateHackType :: Type -> Bool
isStateHackType ty
  | unsafeHasNoStateHack   -- Switch off with -fno-state-hack
  = False
  | otherwise
  = case tyConAppTyCon_maybe ty of
        Just tycon -> tycon == statePrimTyCon
        _          -> False
        -- This is a gross hack.  It claims that
        -- every function over realWorldStatePrimTy is a one-shot
        -- function.  This is pretty true in practice, and makes a big
        -- difference.  For example, consider
        --      a `thenST` \ r -> ...E...
        -- The early full laziness pass, if it doesn't know that r is one-shot
        -- will pull out E (let's say it doesn't mention r) to give
        --      let lvl = E in a `thenST` \ r -> ...lvl...
        -- When `thenST` gets inlined, we end up with
        --      let lvl = E in \s -> case a s of (r, s') -> ...lvl...
        -- and we don't re-inline E.
        --
        -- It would be better to spot that r was one-shot to start with, but
        -- I don't want to rely on that.
        --
        -- Another good example is in fill_in in PrelPack.hs.  We should be able to
        -- spot that fill_in has arity 2 (and when Keith is done, we will) but we can't yet.


{- Note [Arity invariants for bindings]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
We have the following invariants for let-bindings

  (1) In any binding f = e,
         idArity f <= typeArity (idType f)
      We enforce this with trimArityType, called in findRhsArity;
      see Note [Arity trimming].

      Note that we enforce this only for /bindings/.  We do /not/ insist that
         arityTypeArity (arityType e) <= typeArity (exprType e)
      because that is quite a bit more expensive to guaranteed; it would
      mean checking at every Cast in the recursive arityType, for example.

  (2) If typeArity (exprType e) = n,
      then manifestArity (etaExpand e n) = n

      That is, etaExpand can always expand as much as typeArity says
      (or less, of course). So the case analysis in etaExpand and in
      typeArity must match.

      Consequence: because of (1), if we eta-expand to (idArity f), we will
      end up with n manifest lambdas.

   (3) In any binding f = e,
         idArity f <= arityTypeArity (safeArityType (arityType e))
       That is, we call safeArityType before attributing e's arityType to f.
       See Note [SafeArityType].

       So we call safeArityType in findRhsArity.

Suppose we have
   f :: Int -> Int -> Int
   f x y = x+y    -- Arity 2

   g :: F Int
   g = case <cond> of { True  -> f |> co1
                      ; False -> g |> co2 }

where F is a type family.  Now, we can't eta-expand g to have arity 2,
because etaExpand, which works off the /type/ of the expression
(albeit looking through newtypes), doesn't know how to make an
eta-expanded binding
   g = (\a b. case x of ...) |> co
because it can't make up `co` or the types of `a` and `b`.

So invariant (1) ensures that every binding has an arity that is no greater
than the typeArity of the RHS; and invariant (2) ensures that etaExpand
and handle what typeArity says.

Why is this important?  Because

  - In GHC.Iface.Tidy we use exprArity/manifestArity 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 a number of lambdas that precisely matches the arity.

Note [Arity trimming]
~~~~~~~~~~~~~~~~~~~~~
Invariant (1) of Note [Arity invariants for bindings] is upheld by findRhsArity,
which calls trimArityType to trim the ArityType to match the Arity of the
binding.  Failing to do so, and hence breaking invariant (1) led to #5441.

How to trim?  If we end in topDiv, it's easy.  But we must take great care with
dead ends (i.e. botDiv). Suppose the expression was (\x y. error "urk"),
we'll get \??.⊥.  We absolutely must not trim that to \?.⊥, because that
claims that ((\x y. error "urk") |> co) diverges when given one argument,
which it absolutely does not. And Bad Things happen if we think something
returns bottom when it doesn't (#16066).

So, if we need to trim a dead-ending arity type, switch (conservatively) to
topDiv.

Historical note: long ago, we unconditionally switched to topDiv when we
encountered a cast, but that is far too conservative: see #5475

Note [Newtype classes and eta expansion]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    NB: this nasty special case is no longer required, because
    for newtype classes we don't use the class-op rule mechanism
    at all.  See Note [Single-method classes] in GHC.Tc.TyCl.Instance. SLPJ May 2013

-------- Old out of date comments, just for interest -----------
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 #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, because
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.
-------- End of old out of date comments, just for interest -----------
-}

{- ********************************************************************
*                                                                      *
                  Zapping lambda binders
*                                                                      *
********************************************************************* -}

zapLamBndrs :: FullArgCount -> [Var] -> [Var]
-- If (\xyz. t) appears under-applied to only two arguments,
-- we must zap the occ-info on x,y, because they appear (in 't') under the \z.
-- See Note [Occurrence analysis for lambda binders] in GHc.Core.Opt.OccurAnal
--
-- NB: both `arg_count` and `bndrs` include both type and value args/bndrs
zapLamBndrs arg_count bndrs
  | no_need_to_zap = bndrs
  | otherwise      = zap_em arg_count bndrs
  where
    no_need_to_zap = all isOneShotBndr (drop arg_count bndrs)

    zap_em :: FullArgCount -> [Var] -> [Var]
    zap_em 0 bs = bs
    zap_em _ [] = []
    zap_em n (b:bs) | isTyVar b = b              : zap_em (n-1) bs
                    | otherwise = zapLamIdInfo b : zap_em (n-1) bs


{- *********************************************************************
*                                                                      *
           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 divergent 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 [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.

Note [Dealing with bottom]
~~~~~~~~~~~~~~~~~~~~~~~~~~
GHC does some transformations that are technically unsound wrt
bottom, because doing so improves arities... a lot!  We describe
them in this Note.

The flag -fpedantic-bottoms (off by default) restore technically
correct behaviour at the cots of efficiency.

It's mostly to do with eta-expansion.  Consider

   f = \x -> case x of
               True  -> \s -> e1
               False -> \s -> e2

This happens all the time when f :: Bool -> IO ()
In this case we do eta-expand, in order to get that \s to the
top, and give f arity 2.

This isn't really right in the presence of seq.  Consider
        (f bot) `seq` 1

This should diverge!  But if we eta-expand, it won't.  We ignore this
"problem" (unless -fpedantic-bottoms is on), because being scrupulous
would lose an important transformation for many programs. (See
#5587 for an example.)

Consider also
        f = \x -> error "foo"
Here, arity 1 is fine.  But if it looks like this (see #22068)
        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.

So these two transformations aren't always the Right Thing, and we
have several tickets reporting unexpected behaviour resulting from
this transformation.  So we try to limit it as much as possible:

 (1) Do NOT move a lambda outside a known-bottom case expression
       case undefined of { (a,b) -> \y -> e }
     This showed up in #5557

 (2) Do NOT move a lambda outside a case unless
     (a) The scrutinee is ok-for-speculation, or
     (b) more liberally: the scrutinee is cheap (e.g. a variable), and
         -fpedantic-bottoms is not enforced (see #2915 for an example)

Of course both (1) and (2) are readily defeated by disguising the bottoms.

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
  decompose 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 (#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 bottoming 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 #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 can be thought of as an abstraction of an expression.
The ArityType
   AT [ (IsCheap,     NoOneShotInfo)
      , (IsExpensive, OneShotLam)
      , (IsCheap,     OneShotLam) ] Dunno)

abstracts an expression like
   \x. let <expensive> in
       \y{os}.
       \z{os}. blah

In general we have (AT lams div).  Then
* In lams :: [(Cost,OneShotInfo)]
  * The Cost flag describes the part of the expression down
    to the first (value) lambda.
  * The OneShotInfo flag gives the one-shot info on that lambda.

* If 'div' is dead-ending ('isDeadEndDiv'), then application to
  'length lams' arguments will surely diverge, similar to the situation
  with 'DmdType'.

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).

We use the following notation:
  at  ::= \p1..pn.div
  div ::= T | x | ⊥
  p   ::= (c o)
  c   ::= X | C    -- Expensive or Cheap
  o   ::= ? | 1    -- NotOneShot or OneShotLam
We may omit the \. if n = 0.
And ⊥ stands for `AT [] botDiv`

Here is an example demonstrating the notation:
  \(C?)(X1)(C1).T
stands for
   AT [ (IsCheap,NoOneShotInfo)
      , (IsExpensive,OneShotLam)
      , (IsCheap,OneShotLam) ]
      topDiv

See the 'Outputable' instance for more information. It's pretty simple.

How can we use ArityType?  Example:
      f = \x\y. let v = <expensive> in
          \s(one-shot) \t(one-shot). blah
      'f' has arity type \(C?)(C?)(X1)(C1).T
      The one-shot-ness means we can, in effect, push that
      'let' inside the \st, and expand to arity 4

Suppose f = \xy. x+y
Then  f             :: \(C?)(C?).T
      f v           :: \(C?).T
      f <expensive> :: \(X?).T

Here is what the fields mean. If an arbitrary expression 'f' has
ArityType 'at', then

 * If @at = AT [o1,..,on] botDiv@ (notation: \o1..on.⊥), then @f x1..xn@
   definitely diverges. Partial applications to fewer than n args may *or
   may not* diverge.  Ditto exnDiv.

 * If `f` has ArityType `at` we can eta-expand `f` to have (aritTypeOneShots at)
   arguments without losing sharing. This function checks that the either
   there are no expensive expressions, or the lambdas are one-shots.

   NB 'f' is an arbitrary expression, eg @f = g e1 e2@.  This 'f' can have
   arity type @AT oss _@, with @length oss > 0@, only if e1 e2 are themselves
   cheap.

 * In both cases, @f@, @f x1@, ... @f x1 ... x(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#

Wrinkles

* Wrinkle [Bottoming functions]: see function 'arityLam'.
  We treat bottoming functions as one-shot, because there is no point
  in floating work outside the lambda, and it's fine to float it inside.

  For example, this is fine (see test stranal/sigs/BottomFromInnerLambda)
       let x = <expensive> in \y. error (g x y)
       ==> \y. let x = <expensive> in error (g x y)

  Idea: perhaps we could enforce this invariant with
     data Arity Type = TopAT [(Cost, OneShotInfo)] | DivAT [Cost]


Note [SafeArityType]
~~~~~~~~~~~~~~~~~~~~
The function safeArityType trims an ArityType to return a "safe" ArityType,
for which we use a type synonym SafeArityType.  It is "safe" in the sense
that (arityTypeArity at) really reflects the arity of the expression, whereas
a regular ArityType might have more lambdas in its [ATLamInfo] that the
(cost-free) arity of the expression.

For example
   \x.\y.let v = expensive in \z. blah
has
   arityType = AT [C?, C?, X?, C?] Top
But the expression actually has arity 2, not 4, because of the X.
So safeArityType will trim it to (AT [C?, C?] Top), whose [ATLamInfo]
now reflects the (cost-free) arity of the expression

Why do we ever need an "unsafe" ArityType, such as the example above?
Because its (cost-free) arity may increased by combineWithDemandOneShots
in findRhsArity. See Note [Combining arity type with demand info].

Thus the function `arityType` returns a regular "unsafe" ArityType, that
goes deeply into the lambdas (including under IsExpensive). But that is
very local; most ArityTypes are indeed "safe".  We use the type synonym
SafeArityType to indicate where we believe the ArityType is safe.
-}

-- | The analysis lattice of arity analysis. It is isomorphic to
--
-- @
--    data ArityType'
--      = AEnd Divergence
--      | ALam OneShotInfo ArityType'
-- @
--
-- Which is easier to display the Hasse diagram for:
--
-- @
--  ALam OneShotLam at
--          |
--      AEnd topDiv
--          |
--  ALam NoOneShotInfo at
--          |
--      AEnd exnDiv
--          |
--      AEnd botDiv
-- @
--
-- where the @at@ fields of @ALam@ are inductively subject to the same order.
-- That is, @ALam os at1 < ALam os at2@ iff @at1 < at2@.
--
-- Why the strange Top element?
--   See Note [Combining case branches: optimistic one-shot-ness]
--
-- We rely on this lattice structure for fixed-point iteration in
-- 'findRhsArity'. For the semantics of 'ArityType', see Note [ArityType].
data ArityType  -- See Note [ArityType]
  = AT ![ATLamInfo] !Divergence
    -- ^ `AT oss div` is an abstraction of the expression, which describes
    -- its lambdas, and how much work appears where.
    -- See Note [ArityType] for more information
    --
    -- If `div` is dead-ending ('isDeadEndDiv'), then application to
    -- `length os` arguments will surely diverge, similar to the situation
    -- with 'DmdType'.
  deriving Eq

type ATLamInfo = (Cost,OneShotInfo)
     -- ^ Info about one lambda in an ArityType
     -- See Note [ArityType]

type SafeArityType = ArityType -- See Note [SafeArityType]

data Cost = IsCheap | IsExpensive
          deriving( Eq )

allCosts :: (a -> Cost) -> [a] -> Cost
allCosts f xs = foldr (addCost . f) IsCheap xs

addCost :: Cost -> Cost -> Cost
addCost IsCheap IsCheap = IsCheap
addCost _       _       = IsExpensive

-- | This is the BNF of the generated output:
--
-- @
-- @
--
-- We format
-- @AT [o1,..,on] topDiv@ as @\o1..on.T@ and
-- @AT [o1,..,on] botDiv@ as @\o1..on.⊥@, respectively.
-- More concretely, @AT [NOI,OS,OS] topDiv@ is formatted as @\?11.T@.
-- If the one-shot info is empty, we omit the leading @\.@.
instance Outputable ArityType where
  ppr (AT oss div)
    | null oss  = pp_div div
    | otherwise = char '\\' <> hcat (map pp_os oss) <> dot <> pp_div div
    where
      pp_div Diverges = char '⊥'
      pp_div ExnOrDiv = char 'x'
      pp_div Dunno    = char 'T'
      pp_os (IsCheap,     OneShotLam)    = text "(C1)"
      pp_os (IsExpensive, OneShotLam)    = text "(X1)"
      pp_os (IsCheap,     NoOneShotInfo) = text "(C?)"
      pp_os (IsExpensive, NoOneShotInfo) = text "(X?)"

mkBotArityType :: [OneShotInfo] -> ArityType
mkBotArityType oss = AT [(IsCheap,os) | os <- oss] botDiv

botArityType :: ArityType
botArityType = mkBotArityType []

topArityType :: ArityType
topArityType = AT [] topDiv

-- | The number of value args for the arity type
arityTypeArity :: SafeArityType -> Arity
arityTypeArity (AT lams _) = length lams

arityTypeOneShots :: SafeArityType -> [OneShotInfo]
-- Returns a list only as long as the arity should be
arityTypeOneShots (AT lams _) = map snd lams

safeArityType :: ArityType -> SafeArityType
-- ^ Assuming this ArityType is all we know, find the arity of
-- the function, and trim the argument info (and Divergence)
-- to match that arity. See Note [SafeArityType]
safeArityType at@(AT lams _)
  = case go 0 IsCheap lams of
      Nothing -> at  -- No trimming needed
      Just ar -> AT (take ar lams) topDiv
 where
   go :: Arity -> Cost -> [(Cost,OneShotInfo)] -> Maybe Arity
   go _ _ [] = Nothing
   go ar ch1 ((ch2,os):lams)
      = case (ch1 `addCost` ch2, os) of
          (IsExpensive, NoOneShotInfo) -> Just ar
          (ch,          _)             -> go (ar+1) ch lams

infixl 2 `trimArityType`

trimArityType :: Arity -> ArityType -> ArityType
-- ^ Trim an arity type so that it has at most the given arity.
-- Any excess 'OneShotInfo's are truncated to 'topDiv', even if
-- they end in 'ABot'.  See Note [Arity trimming]
trimArityType max_arity at@(AT lams _)
  | lams `lengthAtMost` max_arity = at
  | otherwise                     = AT (take max_arity lams) topDiv

data ArityOpts = ArityOpts
  { ao_ped_bot :: !Bool -- See Note [Dealing with bottom]
  , ao_dicts_cheap :: !Bool -- See Note [Eta expanding through dictionaries]
  }

-- | The Arity returned is the number of value args the
-- expression can be applied to without doing much work
exprEtaExpandArity :: ArityOpts -> CoreExpr -> Maybe SafeArityType
-- exprEtaExpandArity is used when eta expanding
--      e  ==>  \xy -> e x y
-- Nothing if the expression has arity 0
exprEtaExpandArity opts e
  | AT [] _ <- arity_type
  = Nothing
  | otherwise
  = Just arity_type
  where
    arity_type = safeArityType (arityType (findRhsArityEnv opts False) e)


{- *********************************************************************
*                                                                      *
                   findRhsArity
*                                                                      *
********************************************************************* -}

findRhsArity :: ArityOpts -> RecFlag -> Id -> CoreExpr
             -> (Bool, SafeArityType)
-- This implements the fixpoint loop for arity analysis
-- See Note [Arity analysis]
--
-- The Bool is True if the returned arity is greater than (exprArity rhs)
--     so the caller should do eta-expansion
-- That Bool is never True for join points, which are never eta-expanded
--
-- Returns an SafeArityType that is guaranteed trimmed to typeArity of 'bndr'
--         See Note [Arity trimming]

findRhsArity opts is_rec bndr rhs
  | isJoinId bndr
  = (False, join_arity_type)
    -- False: see Note [Do not eta-expand join points]
    -- But do return the correct arity and bottom-ness, because
    -- these are used to set the bndr's IdInfo (#15517)
    -- Note [Invariants on join points] invariant 2b, in GHC.Core

  | otherwise
  = (arity_increased, non_join_arity_type)
    -- arity_increased: eta-expand if we'll get more lambdas
    -- to the top of the RHS
  where
    old_arity = exprArity rhs

    init_env :: ArityEnv
    init_env = findRhsArityEnv opts (isJoinId bndr)

    -- Non-join-points only
    non_join_arity_type = case is_rec of
                             Recursive    -> go 0 botArityType
                             NonRecursive -> step init_env
    arity_increased = arityTypeArity non_join_arity_type > old_arity

    -- Join-points only
    -- See Note [Arity for non-recursive join bindings]
    -- and Note [Arity for recursive join bindings]
    join_arity_type = case is_rec of
                         Recursive    -> go 0 botArityType
                         NonRecursive -> trimArityType ty_arity (cheapArityType rhs)

    ty_arity     = typeArity (idType bndr)
    id_one_shots = idDemandOneShots bndr

    step :: ArityEnv -> SafeArityType
    step env = trimArityType ty_arity $
               safeArityType $ -- See Note [Arity invariants for bindings], item (3)
               arityType env rhs `combineWithDemandOneShots` id_one_shots
       -- trimArityType: see Note [Trim arity inside the loop]
       -- combineWithDemandOneShots: take account of the demand on the
       -- binder.  Perhaps it is always called with 2 args
       --   let f = \x. blah in (f 3 4, f 1 9)
       -- f's demand-info says how many args it is called with

    -- The fixpoint iteration (go), done for recursive bindings. We
    -- always do one step, but usually that produces a result equal
    -- to old_arity, and then we stop right away, because old_arity
    -- is assumed to be sound. In other words, arities should never
    -- decrease.  Result: the common case is that there is just one
    -- iteration
    go :: Int -> SafeArityType -> SafeArityType
    go !n cur_at@(AT lams div)
      | not (isDeadEndDiv div)           -- the "stop right away" case
      , length lams <= old_arity = cur_at -- from above
      | next_at == cur_at        = cur_at
      | otherwise
         -- Warn if more than 2 iterations. Why 2? See Note [Exciting arity]
      = warnPprTrace (debugIsOn && n > 2)
            "Exciting arity"
            (nest 2 (ppr bndr <+> ppr cur_at <+> ppr next_at $$ ppr rhs)) $
        go (n+1) next_at
      where
        next_at = step (extendSigEnv init_env bndr cur_at)

infixl 2 `combineWithDemandOneShots`

combineWithDemandOneShots :: ArityType -> [OneShotInfo] -> ArityType
-- See Note [Combining arity type with demand info]
combineWithDemandOneShots at@(AT lams div) oss
  | null lams = at
  | otherwise = AT (zip_lams lams oss) div
  where
    zip_lams :: [ATLamInfo] -> [OneShotInfo] -> [ATLamInfo]
    zip_lams lams []  = lams
    zip_lams []   oss | isDeadEndDiv div = []
                      | otherwise        = [ (IsExpensive,OneShotLam)
                                           | _ <- takeWhile isOneShotInfo oss]
    zip_lams ((ch,os1):lams) (os2:oss)
      = (ch, os1 `bestOneShot` os2) : zip_lams lams oss

idDemandOneShots :: Id -> [OneShotInfo]
idDemandOneShots bndr
  = call_arity_one_shots `zip_lams` dmd_one_shots
  where
    call_arity_one_shots :: [OneShotInfo]
    call_arity_one_shots
      | call_arity == 0 = []
      | otherwise       = NoOneShotInfo : replicate (call_arity-1) OneShotLam
    -- Call Arity analysis says the function is always called
    -- applied to this many arguments.  The first NoOneShotInfo is because
    -- if Call Arity says "always applied to 3 args" then the one-shot info
    -- we get is [NoOneShotInfo, OneShotLam, OneShotLam]
    call_arity = idCallArity bndr

    dmd_one_shots :: [OneShotInfo]
    -- If the demand info is C(x,C(1,C(1,.))) then we know that an
    -- application to one arg is also an application to three
    dmd_one_shots = argOneShots (idDemandInfo bndr)

    -- Take the *longer* list
    zip_lams (lam1:lams1) (lam2:lams2) = (lam1 `bestOneShot` lam2) : zip_lams lams1 lams2
    zip_lams []           lams2        = lams2
    zip_lams lams1        []           = lams1

{- Note [Arity analysis]
~~~~~~~~~~~~~~~~~~~~~~~~
The motivating example for arity analysis is this:

  f = \x. let g = f (x+1)
          in \y. ...g...

What arity does f have?  Really it should have arity 2, but a naive
look at the RHS won't see that.  You need a fixpoint analysis which
says it has arity "infinity" the first time round.

This example happens a lot; it first showed up in Andy Gill's thesis,
fifteen years ago!  It also shows up in the code for 'rnf' on lists
in #4138.

We do the necessary, quite simple fixed-point iteration in 'findRhsArity',
which assumes for a single binding 'ABot' on the first run and iterates
until it finds a stable arity type. Two wrinkles

* We often have to ask (see the Case or Let case of 'arityType') whether some
  expression is cheap. In the case of an application, that depends on the arity
  of the application head! That's why we have our own version of 'exprIsCheap',
  'myExprIsCheap', that will integrate the optimistic arity types we have on
  f and g into the cheapness check.

* Consider this (#18793)

    go = \ds. case ds of
           []     -> id
           (x:ys) -> let acc = go ys in
                     case blah of
                       True  -> acc
                       False -> \ x1 -> acc (negate x1)

  We must propagate go's optimistically large arity to @acc@, so that the
  tail call to @acc@ in the True branch has sufficient arity.  This is done
  by the 'am_sigs' field in 'FindRhsArity', and 'lookupSigEnv' in the Var case
  of 'arityType'.

Note [Exciting arity]
~~~~~~~~~~~~~~~~~~~~~
The fixed-point iteration in 'findRhsArity' stabilises very quickly in almost
all cases. To get notified of cases where we need an usual number of iterations,
we emit a warning in debug mode, so that we can investigate and make sure that
we really can't do better. It's a gross hack, but catches real bugs (#18870).

Now, which number is "unusual"? We pick n > 2. Here's a pretty common and
expected example that takes two iterations and would ruin the specificity
of the warning (from T18937):

  f :: [Int] -> Int -> Int
  f []     = id
  f (x:xs) = let y = sum [0..x]
             in \z -> f xs (y + z)

Fixed-point iteration starts with arity type ⊥ for f. After the first
iteration, we get arity type \??.T, e.g. arity 2, because we unconditionally
'floatIn' the let-binding (see its bottom case).  After the second iteration,
we get arity type \?.T, e.g. arity 1, because now we are no longer allowed
to floatIn the non-cheap let-binding.  Which is all perfectly benign, but
means we do two iterations (well, actually 3 'step's to detect we are stable)
and don't want to emit the warning.

Note [Trim arity inside the loop]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Here's an example (from gadt/nbe.hs) which caused trouble.
  data Exp g t where
     Lam :: Ty a -> Exp (g,a) b -> Exp g (a->b)

  eval :: Exp g t -> g -> t
  eval (Lam _ e) g = \a -> eval e (g,a)

The danger is that we get arity 3 from analysing this; and the
next time arity 4, and so on for ever.  Solution: use trimArityType
on each iteration.

Note [Combining arity type with demand info]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Consider
   let f = \x. let y = <expensive> in \p \q{os}. blah
   in ...(f a b)...(f c d)...

* From the RHS we get an ArityType like
    AT [ (IsCheap,?), (IsExpensive,?), (IsCheap,OneShotLam) ] Dunno
  where "?" means NoOneShotInfo

* From the body, the demand analyser (or Call Arity) will tell us
  that the function is always applied to at least two arguments.

Combining these two pieces of info, we can get the final ArityType
    AT [ (IsCheap,?), (IsExpensive,OneShotLam), (IsCheap,OneShotLam) ] Dunno
result: arity=3, which is better than we could do from either
source alone.

The "combining" part is done by combineWithDemandOneShots.  It
uses info from both Call Arity and demand analysis.

We may have /more/ call demands from the calls than we have lambdas
in the binding.  E.g.
    let f1 = \x. g x x in ...(f1 p q r)...
    -- Demand on f1 is C(x,C(1,C(1,L)))

    let f2 = \y. error y in ...(f2 p q r)...
    -- Demand on f2 is C(x,C(1,C(1,L)))

In both these cases we can eta expand f1 and f2 to arity 3.
But /only/ for called-once demands.  Suppose we had
    let f1 = \y. g x x in ...let h = f1 p q in ...(h r1)...(h r2)...

Now we don't want to eta-expand f1 to have 3 args; only two.
Nor, in the case of f2, do we want to push that error call under
a lambda.  Hence the takeWhile in combineWithDemandDoneShots.

Note [Do not eta-expand join points]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Similarly to CPR (see Note [Don't w/w join points for CPR] in
GHC.Core.Opt.WorkWrap), a join point stands well to gain from its outer binding's
eta-expansion, and eta-expanding a join point is fraught with issues like how to
deal with a cast:

    let join $j1 :: IO ()
             $j1 = ...
             $j2 :: Int -> IO ()
             $j2 n = if n > 0 then $j1
                              else ...

    =>

    let join $j1 :: IO ()
             $j1 = (\eta -> ...)
                     `cast` N:IO :: State# RealWorld -> (# State# RealWorld, ())
                                 ~  IO ()
             $j2 :: Int -> IO ()
             $j2 n = (\eta -> if n > 0 then $j1
                                       else ...)
                     `cast` N:IO :: State# RealWorld -> (# State# RealWorld, ())
                                 ~  IO ()

The cast here can't be pushed inside the lambda (since it's not casting to a
function type), so the lambda has to stay, but it can't because it contains a
reference to a join point. In fact, $j2 can't be eta-expanded at all. Rather
than try and detect this situation (and whatever other situations crop up!), we
don't bother; again, any surrounding eta-expansion will improve these join
points anyway, since an outer cast can *always* be pushed inside. By the time
CorePrep comes around, the code is very likely to look more like this:

    let join $j1 :: State# RealWorld -> (# State# RealWorld, ())
             $j1 = (...) eta
             $j2 :: Int -> State# RealWorld -> (# State# RealWorld, ())
             $j2 = if n > 0 then $j1
                            else (...) eta

Note [Arity for recursive join bindings]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Consider
  f x = joinrec j 0 = \ a b c -> (a,x,b)
                j n = j (n-1)
        in j 20

Obviously `f` should get arity 4.  But it's a bit tricky:

1. Remember, we don't eta-expand join points; see
   Note [Do not eta-expand join points].

2. But even though we aren't going to eta-expand it, we still want `j` to get
   idArity=4, via the findRhsArity fixpoint.  Then when we are doing findRhsArity
   for `f`, we'll call arityType on f's RHS:
    - At the letrec-binding for `j` we'll whiz up an arity-4 ArityType
      for `j` (See Note [arityType for non-recursive let-bindings]
      in GHC.Core.Opt.Arity)b
    - At the occurrence (j 20) that arity-4 ArityType will leave an arity-3
      result.

3. All this, even though j's /join-arity/ (stored in the JoinId) is 1.
   This is is the Main Reason that we want the idArity to sometimes be
   larger than the join-arity c.f. Note [Invariants on join points] item 2b
   in GHC.Core.

4. Be very careful of things like this (#21755):
     g x = let j 0 = \y -> (x,y)
               j n = expensive n `seq` j (n-1)
           in j x
   Here we do /not/ want eta-expand `g`, lest we duplicate all those
   (expensive n) calls.

   But it's fine: the findRhsArity fixpoint calculation will compute arity-1
   for `j` (not arity 2); and that's just what we want. But we do need that
   fixpoint.

   Historical note: an earlier version of GHC did a hack in which we gave
   join points an ArityType of ABot, but that did not work with this #21755
   case.

5. arityType does not usually expect to encounter free join points;
   see GHC.Core.Opt.Arity Note [No free join points in arityType].
   But consider
          f x = join    j1 y = .... in
                joinrec j2 z = ...j1 y... in
                j2 v

   When doing findRhsArity on `j2` we'll encounter the free `j1`.
   But that is fine, because we aren't going to eta-expand `j2`;
   we just want to know its arity.  So we have a flag am_no_eta,
   switched on when doing findRhsArity on a join point RHS. If
   the flag is on, we allow free join points, but not otherwise.


Note [Arity for non-recursive join bindings]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Note [Arity for recursive join bindings] deals with recursive join
bindings. But what about /non-recursive/ones?  If we just call
findRhsArity, it will call arityType.  And that can be expensive when
we have deeply nested join points:
  join j1 x1 = join j2 x2 = join j3 x3 = blah3
                            in blah2
               in blah1
(e.g. test T18698b).

So we call cheapArityType instead.  It's good enough for practical
purposes.

(Side note: maybe we should use cheapArity for the RHS of let bindings
in the main arityType function.)
-}


{- *********************************************************************
*                                                                      *
                   arityType
*                                                                      *
********************************************************************* -}

arityLam :: Id -> ArityType -> ArityType
arityLam id (AT oss div)
  = AT ((IsCheap, one_shot) : oss) div
  where
    one_shot | isDeadEndDiv div = OneShotLam
             | otherwise        = idStateHackOneShotInfo id
    -- If the body diverges, treat it as one-shot: no point
    -- in floating out, and no penalty for floating in
    -- See Wrinkle [Bottoming functions] in Note [ArityType]

floatIn :: Cost -> ArityType -> ArityType
-- We have something like (let x = E in b),
-- where b has the given arity type.
floatIn IsCheap     at = at
floatIn IsExpensive at = addWork at

addWork :: ArityType -> ArityType
-- Add work to the outermost level of the arity type
addWork at@(AT lams div)
  = case lams of
      []      -> at
      lam:lams' -> AT (add_work lam : lams') div

add_work :: ATLamInfo -> ATLamInfo
add_work (_,os) = (IsExpensive,os)

arityApp :: ArityType -> Cost -> ArityType
-- Processing (fun arg) where at is the ArityType of fun,
-- Knock off an argument and behave like 'let'
arityApp (AT ((ch1,_):oss) div) ch2 = floatIn (ch1 `addCost` ch2) (AT oss div)
arityApp at                     _   = at

-- | Least upper bound in the 'ArityType' lattice.
-- See the haddocks on 'ArityType' for the lattice.
--
-- Used for branches of a @case@.
andArityType :: ArityEnv -> ArityType -> ArityType -> ArityType
andArityType env (AT (lam1:lams1) div1) (AT (lam2:lams2) div2)
  | AT lams' div' <- andArityType env (AT lams1 div1) (AT lams2 div2)
  = AT ((lam1 `and_lam` lam2) : lams') div'
  where
    (ch1,os1) `and_lam` (ch2,os2)
      = ( ch1 `addCost` ch2, os1 `bestOneShot` os2)
        -- bestOneShot: see Note [Combining case branches: optimistic one-shot-ness]

andArityType env (AT [] div1) at2 = andWithTail env div1 at2
andArityType env at1 (AT [] div2) = andWithTail env div2 at1

andWithTail :: ArityEnv -> Divergence -> ArityType -> ArityType
andWithTail env div1 at2@(AT lams2 _)
  | isDeadEndDiv div1    -- case x of { T -> error; F -> \y.e }
  = at2                  -- See Note
  | pedanticBottoms env  --    [Combining case branches: andWithTail]
  = AT [] topDiv

  | otherwise  -- case x of { T -> plusInt <expensive>; F -> \y.e }
  = AT (map add_work lams2) topDiv    -- We know div1 = topDiv
    -- See Note [Combining case branches: andWithTail]

{- Note [Combining case branches: optimistic one-shot-ness]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
When combining the ArityTypes for two case branches (with
andArityType) and both ArityTypes have ATLamInfo, then we just combine
their expensive-ness and one-shot info.  The tricky point is when we
have

     case x of True -> \x{one-shot). blah1
               Fale -> \y.           blah2

Since one-shot-ness is about the /consumer/ not the /producer/, we
optimistically assume that if either branch is one-shot, we combine
the best of the two branches, on the (slightly dodgy) basis that if we
know one branch is one-shot, then they all must be.  Surprisingly,
this means that the one-shot arity type is effectively the top element
of the lattice.

Hence the call to `bestOneShot` in `andArityType`.

Here's an example:
  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 respect the one-shot annotation provided by the
user and eta-expand go and go2.  In the first fixpoint iteration of
'go' we'll bind 'go' to botArityType (written \.⊥, see Note
[ArityType]).  So 'z' will get arityType \.⊥; so we end up combining
the True and False branches:

      \.⊥ `andArityType` \1.T

That gives \1.T (see Note [Combining case branches: andWithTail],
first bullet).  So 'go2' gets an arityType of \(C?)(C1).T, which is
what we want.

Note [Combining case branches: andWithTail]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
When combining the ArityTypes for two case branches (with andArityType)
and one side or the other has run out of ATLamInfo; then we get
into `andWithTail`.

* If one branch is guaranteed bottom (isDeadEndDiv), we just take
  the other. Consider   case x of
             True  -> \x.  error "urk"
             False -> \xy. error "urk2"

  Remember: \o1..on.⊥ means "if you apply to n args, it'll definitely
  diverge".  So we need \??.⊥ for the whole thing, the /max/ of both
  arities.

* Otherwise, if pedantic-bottoms is on, we just have to return
  AT [] topDiv.  E.g. if we have
    f x z = case x of True  -> \y. blah
                      False -> z
  then we can't eta-expand, because that would change the behaviour
  of (f False bottom().

* But if pedantic-bottoms is not on, we allow ourselves to push
  `z` under a lambda (much as we allow ourselves to put the `case x`
  under a lambda).  However we know nothing about the expensiveness
  or one-shot-ness of `z`, so we'd better assume it looks like
  (Expensive, NoOneShotInfo) all the way. Remembering
  Note [Combining case branches: optimistic one-shot-ness],
  we just add work to ever ATLamInfo, keeping the one-shot-ness.

Note [Eta expanding through CallStacks]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Just as it's good to eta-expand through dictionaries, so it is good to
do so through CallStacks.  #20103 is a case in point, where we got
  foo :: HasCallStack => Int -> Int
  foo = \(d::CallStack). let d2 = pushCallStack blah d in
        \(x:Int). blah

We really want to eta-expand this!  #20103 is quite convincing!
We do this regardless of -fdicts-cheap; it's not really a dictionary.

Note [Eta expanding through dictionaries]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
If the experimental -fdicts-cheap flag is on, we eta-expand through
dictionary bindings.  This improves arities. Thereby, it also
means that full laziness is less prone to floating out the
application of a function to its dictionary arguments, which
can thereby lose opportunities for fusion.  Example:
        foo :: Ord a => a -> ...
     foo = /\a \(d:Ord a). let d' = ...d... in \(x:a). ....
        -- So foo has arity 1

     f = \x. foo dInt $ bar x

The (foo DInt) is floated out, and makes ineffective a RULE
     foo (bar x) = ...

One could go further and make exprIsCheap reply True to any
dictionary-typed expression, but that's more work.
-}

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

data ArityEnv
  = AE { am_opts :: !ArityOpts

       , am_sigs :: !(IdEnv SafeArityType)
         -- NB `SafeArityType` so we can use this in myIsCheapApp
         -- See Note [Arity analysis] for details about fixed-point iteration.

       , am_free_joins :: !Bool  -- True <=> free join points allowed
         -- Used /only/ to support assertion checks
       }

instance Outputable ArityEnv where
  ppr (AE { am_sigs = sigs, am_free_joins = free_joins })
    = text "AE" <+> braces (sep [ text "free joins:" <+> ppr free_joins
                                , text "sigs:" <+> ppr sigs ])

-- | The @ArityEnv@ used by 'findRhsArity'.
findRhsArityEnv :: ArityOpts -> Bool -> ArityEnv
findRhsArityEnv opts free_joins
  = AE { am_opts       = opts
       , am_free_joins = free_joins
       , am_sigs       = emptyVarEnv }

freeJoinsOK :: ArityEnv -> Bool
freeJoinsOK (AE { am_free_joins = free_joins }) = free_joins

-- First some internal functions in snake_case for deleting in certain VarEnvs
-- of the ArityType. Don't call these; call delInScope* instead!

modifySigEnv :: (IdEnv ArityType -> IdEnv ArityType) -> ArityEnv -> ArityEnv
modifySigEnv f env@(AE { am_sigs = sigs }) = env { am_sigs = f sigs }
{-# INLINE modifySigEnv #-}

del_sig_env :: Id -> ArityEnv -> ArityEnv -- internal!
del_sig_env id = modifySigEnv (\sigs -> delVarEnv sigs id)
{-# INLINE del_sig_env #-}

del_sig_env_list :: [Id] -> ArityEnv -> ArityEnv -- internal!
del_sig_env_list ids = modifySigEnv (\sigs -> delVarEnvList sigs ids)
{-# INLINE del_sig_env_list #-}

-- end of internal deletion functions

extendSigEnv :: ArityEnv -> Id -> SafeArityType -> ArityEnv
extendSigEnv env id ar_ty
  = modifySigEnv (\sigs -> extendVarEnv sigs id ar_ty) $
    env

delInScope :: ArityEnv -> Id -> ArityEnv
delInScope env id = del_sig_env id env

delInScopeList :: ArityEnv -> [Id] -> ArityEnv
delInScopeList env ids = del_sig_env_list ids env

lookupSigEnv :: ArityEnv -> Id -> Maybe SafeArityType
lookupSigEnv (AE { am_sigs = sigs }) id = lookupVarEnv sigs id

-- | Whether the analysis should be pedantic about bottoms.
-- 'exprBotStrictness_maybe' always is.
pedanticBottoms :: ArityEnv -> Bool
pedanticBottoms (AE { am_opts = ArityOpts{ ao_ped_bot = ped_bot }}) = ped_bot

exprCost :: ArityEnv -> CoreExpr -> Maybe Type -> Cost
exprCost env e mb_ty
  | myExprIsCheap env e mb_ty = IsCheap
  | otherwise                 = IsExpensive

-- | A version of 'exprIsCheap' that considers results from arity analysis
-- and optionally the expression's type.
-- Under 'exprBotStrictness_maybe', no expressions are cheap.
myExprIsCheap :: ArityEnv -> CoreExpr -> Maybe Type -> Bool
myExprIsCheap (AE { am_opts = opts, am_sigs = sigs }) e mb_ty
  = cheap_dict || cheap_fun e
  where
    cheap_dict = case mb_ty of
                     Nothing -> False
                     Just ty -> (ao_dicts_cheap opts && isDictTy ty)
                                || isCallStackPredTy ty
        -- See Note [Eta expanding through dictionaries]
        -- See Note [Eta expanding through CallStacks]

    cheap_fun e = exprIsCheapX (myIsCheapApp sigs) e

-- | A version of 'isCheapApp' that considers results from arity analysis.
-- See Note [Arity analysis] for what's in the signature environment and why
-- it's important.
myIsCheapApp :: IdEnv SafeArityType -> CheapAppFun
myIsCheapApp sigs fn n_val_args = case lookupVarEnv sigs fn of

  -- Nothing means not a local function, fall back to regular
  -- 'GHC.Core.Utils.isCheapApp'
  Nothing -> isCheapApp fn n_val_args

  -- `Just at` means local function with `at` as current SafeArityType.
  -- NB the SafeArityType bit: that means we can ignore the cost flags
  --    in 'lams', and just consider the length
  -- Roughly approximate what 'isCheapApp' is doing.
  Just (AT lams div)
    | isDeadEndDiv div -> True -- See Note [isCheapApp: bottoming functions] in GHC.Core.Utils
    | n_val_args == 0          -> True -- Essentially
    | n_val_args < length lams -> True -- isWorkFreeApp
    | otherwise                -> False

----------------
arityType :: HasDebugCallStack => ArityEnv -> CoreExpr -> ArityType
-- Precondition: all the free join points of the expression
--               are bound by the ArityEnv
-- See Note [No free join points in arityType]
--
-- Returns ArityType, not SafeArityType.  The caller must do
-- trimArityType if necessary.
arityType env (Var v)
  | Just at <- lookupSigEnv env v -- Local binding
  = at
  | otherwise
  = assertPpr (freeJoinsOK env || not (isJoinId v)) (ppr v) $
    -- All join-point should be in the ae_sigs
    -- See Note [No free join points in arityType]
    idArityType v

arityType env (Cast e _)
  = arityType env e

        -- Lambdas; increase arity
arityType env (Lam x e)
  | isId x    = arityLam x (arityType env' e)
  | otherwise = arityType env' e
  where
    env' = delInScope env x

        -- Applications; decrease arity, except for types
arityType env (App fun (Type _))
   = arityType env fun
arityType env (App fun arg )
   = arityApp fun_at arg_cost
   where
     fun_at   = arityType env fun
     arg_cost = exprCost env 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 env (Case scrut bndr _ alts)
  | exprIsDeadEnd scrut || null alts
  = botArityType    -- Do not eta expand. See (1) in Note [Dealing with bottom]

  | not (pedanticBottoms env)  -- See (2) in Note [Dealing with bottom]
  , myExprIsCheap env scrut (Just (idType bndr))
  = alts_type

  | exprOkForSpeculation scrut
  = alts_type

  | otherwise            -- In the remaining cases we may not push
  = addWork alts_type -- evaluation of the scrutinee in
  where
    env' = delInScope env bndr
    arity_type_alt (Alt _con bndrs rhs) = arityType (delInScopeList env' bndrs) rhs
    alts_type = foldr1 (andArityType env) (map arity_type_alt alts)

arityType env (Let (NonRec b rhs) e)
  = -- See Note [arityType for non-recursive let-bindings]
    floatIn rhs_cost (arityType env' e)
  where
    rhs_cost = exprCost env rhs (Just (idType b))
    env'     = extendSigEnv env b (safeArityType (arityType env rhs))

arityType env (Let (Rec prs) e)
  = -- See Note [arityType for recursive let-bindings]
    floatIn (allCosts bind_cost prs) (arityType env' e)
  where
    bind_cost (b,e) = exprCost env' e (Just (idType b))
    env'            = foldl extend_rec env prs
    extend_rec :: ArityEnv -> (Id,CoreExpr) -> ArityEnv
    extend_rec env (b,_) = extendSigEnv env b  $
                           idArityType b
      -- See Note [arityType for recursive let-bindings]

arityType env (Tick t e)
  | not (tickishIsCode t)     = arityType env e

arityType _ _ = topArityType

--------------------
idArityType :: Id -> ArityType
idArityType v
  | strict_sig <- idDmdSig v
  , (ds, div) <- splitDmdSig strict_sig
  , isDeadEndDiv div
  = AT (takeList ds one_shots) div

  | isEmptyTy id_ty
  = botArityType

  | otherwise
  = AT (take (idArity v) one_shots) topDiv
  where
    id_ty = idType v

    one_shots :: [(Cost,OneShotInfo)]  -- One-shot-ness derived from the type
    one_shots = repeat IsCheap `zip` typeOneShots id_ty

--------------------
cheapArityType :: HasDebugCallStack => CoreExpr -> ArityType
-- A fast and cheap version of arityType.
-- Returns an ArityType with IsCheap everywhere
-- c.f. GHC.Core.Utils.exprIsDeadEnd
--
-- /Can/ encounter a free join-point Id; e.g. via the call
--   in exprBotStrictness_maybe, which is called in lots
--   of places
--
-- Returns ArityType, not SafeArityType.  The caller must do
-- trimArityType if necessary.
cheapArityType e = go e
  where
    go (Var v)                  = idArityType v
    go (Cast e _)               = go e
    go (Lam x e)  | isId x      = arityLam x (go e)
                  | otherwise   = go e
    go (App e a)  | isTypeArg a = go e
                  | otherwise   = arity_app a (go e)

    go (Tick t e) | not (tickishIsCode t) = go e

    -- Null alts: see Note [Empty case alternatives] in GHC.Core
    go (Case _ _ _ alts) | null alts = botArityType

    -- Give up on let, case.  In particular, unlike arityType,
    -- we make no attempt to look inside let's.
    go _ = topArityType

    -- Specialised version of arityApp; all costs in ArityType are IsCheap
    -- See Note [exprArity for applications]
    -- NB: (1) coercions count as a value argument
    --     (2) we use the super-cheap exprIsTrivial rather than the
    --         more complicated and expensive exprIsCheap
    arity_app _ at@(AT [] _) = at
    arity_app arg at@(AT ((cost,_):lams) div)
       | assertPpr (cost == IsCheap) (ppr at $$ ppr arg) $
         isDeadEndDiv div  = AT lams div
       | exprIsTrivial arg = AT lams topDiv
       | otherwise         = topArityType

---------------
exprArity :: CoreExpr -> Arity
-- ^ An approximate, even faster, version of 'cheapArityType'
-- Roughly   exprArity e = arityTypeArity (cheapArityType e)
-- But it's a bit less clever about bottoms
--
-- We do /not/ guarantee that exprArity e <= typeArity e
-- You may need to do arity trimming after calling exprArity
-- See Note [Arity trimming]
-- Reason: if we do arity trimming here we have take exprType
--         and that can be expensive if there is a large cast
exprArity e = go e
  where
    go (Var v)                     = idArity v
    go (Lam x e) | isId x          = go e + 1
                 | otherwise       = go e
    go (Tick t e) | not (tickishIsCode t) = go e
    go (Cast e _)                  = go e
    go (App e (Type _))            = go e
    go (App f a) | exprIsTrivial a = (go f - 1) `max` 0
        -- See Note [exprArity for applications]
        -- NB: coercions count as a value argument

    go _                           = 0

---------------
exprIsDeadEnd :: CoreExpr -> Bool
-- See Note [Bottoming expressions]
-- This function is, in effect, just a specialised (and hence cheap)
--    version of cheapArityType:
--    exprIsDeadEnd e = case cheapArityType e of
--                         AT lams div -> null lams && isDeadEndDiv div
-- See also exprBotStrictness_maybe, which uses cheapArityType
exprIsDeadEnd e
  = go 0 e
  where
    go :: Arity -> CoreExpr -> Bool
    -- (go n e) = True <=> expr applied to n value args is bottom
    go _ (Lit {})                = False
    go _ (Type {})               = False
    go _ (Coercion {})           = False
    go n (App e a) | isTypeArg a = go n e
                   | otherwise   = go (n+1) e
    go n (Tick _ e)              = go n e
    go n (Cast e _)              = go n e
    go n (Let _ e)               = go n e
    go n (Lam v e) | isTyVar v   = go n e
                   | otherwise   = False

    go _ (Case _ _ _ alts)       = null alts
       -- See Note [Empty case alternatives] in GHC.Core

    go n (Var v) | isDeadEndAppSig (idDmdSig v) n = True
                 | isEmptyTy (idType v)           = True
                 | otherwise                      = False

{- Note [Bottoming expressions]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
A bottoming expression is guaranteed to diverge, or raise an
exception.  We can test for it in two different ways, and exprIsDeadEnd
checks for both of these situations:

* Visibly-bottom computations.  For example
      (error Int "Hello")
  is visibly bottom.  The strictness analyser also finds out if
  a function diverges or raises an exception, and puts that info
  in its strictness signature.

* Empty types.  If a type is empty, its only inhabitant is bottom.
  For example:
      data T
      f :: T -> Bool
      f = \(x:t). case x of Bool {}
  Since T has no data constructors, the case alternatives are of course
  empty.  However note that 'x' is not bound to a visibly-bottom value;
  it's the *type* that tells us it's going to diverge.

A GADT may also be empty even though it has constructors:
        data T a where
          T1 :: a -> T Bool
          T2 :: T Int
        ...(case (x::T Char) of {})...
Here (T Char) is uninhabited.  A more realistic case is (Int ~ Bool),
which is likewise uninhabited.

Note [No free join points in arityType]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Suppose we call arityType on this expression (EX1)
   \x . case x of True  -> \y. e
                  False -> $j 3
where $j is a join point.  It really makes no sense to talk of the arity
of this expression, because it has a free join point.  In particular, we
can't eta-expand the expression because we'd have do the same thing to the
binding of $j, and we can't see that binding.

If we had (EX2)
   \x. join $j y = blah
       case x of True  -> \y. e
                 False -> $j 3
then it would make perfect sense: we can determine $j's ArityType, and
propagate it to the usage site as usual.

But how can we get (EX1)?  It doesn't make much sense, because $j can't
be a join point under the \x anyway.  So we make it a precondition of
arityType that the argument has no free join-point Ids.  (This is checked
with an assert in the Var case of arityType.)

Wrinkles

* We /do/ allow free join point when doing findRhsArity for join-point
  right-hand sides. See Note [Arity for recursive join bindings]
  point (5) in GHC.Core.Opt.Simplify.Utils.

* The invariant (no free join point in arityType) risks being
  invalidated by one very narrow special case: runRW#

   join $j y = blah
   runRW# (\s. case x of True  -> \y. e
                         False -> $j x)

  We have special magic in OccurAnal, and Simplify to allow continuations to
  move into the body of a runRW# call.

  So we are careful never to attempt to eta-expand the (\s.blah) in the
  argument to runRW#, at least not when there is a literal lambda there,
  so that OccurAnal has seen it and allowed join points bound outside.
  See Note [No eta-expansion in runRW#] in GHC.Core.Opt.Simplify.Iteration.

Note [arityType for non-recursive let-bindings]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
For non-recursive let-bindings, we just get the arityType of the RHS,
and extend the environment.  That works nicely for things like this
(#18793):
  go = \ ds. case ds_a2CF of {
               []     -> id
               : y ys -> case y of { GHC.Types.I# x ->
                         let acc = go ys in
                         case x ># 42# of {
                            __DEFAULT -> acc
                            1# -> \x1. acc (negate x2)

Here we want to get a good arity for `acc`, based on the ArityType
of `go`.

All this is particularly important for join points. Consider this (#18328)

  f x = join j y = case y of
                      True -> \a. blah
                      False -> \b. blah
        in case x of
              A -> j True
              B -> \c. blah
              C -> j False

and suppose the join point is too big to inline.  Now, what is the
arity of f?  If we inlined the join point, we'd definitely say "arity
2" because we are prepared to push case-scrutinisation inside a
lambda. It's important that we extend the envt with j's ArityType, so
that we can use that information in the A/C branch of the case.

Note [arityType for recursive let-bindings]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
For /recursive/ bindings it's more difficult, to call arityType
(as we do in Note [arityType for non-recursive let-bindings])
because we don't have an ArityType to put in the envt for the
recursively bound Ids.  So for we satisfy ourselves with whizzing up
up an ArityType from the idArity of the function, via idArityType.

That is nearly equivalent to deleting the binder from the envt, at
which point we'll call idArityType at the occurrences.  But doing it
here means

  (a) we only call idArityType once, no matter how many
      occurrences, and

  (b) we can check (in the arityType (Var v) case) that
      we don't mention free join-point Ids. See
      Note [No free join points in arityType].

But see Note [Arity for recursive join bindings] in
GHC.Core.Opt.Simplify.Utils for dark corners.
-}

{-
%************************************************************************
%*                                                                      *
              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 example 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 alternative, 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 for join points]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The no-crap rule is very tiresome to guarantee when
we have join points. Consider eta-expanding
   let j :: Int -> Int -> Bool
       j x = e
   in b

The simple way is
  \(y::Int). (let j x = e in b) y

The no-crap way is
  \(y::Int). let j' :: Int -> Bool
                 j' x = e y
             in b[j'/j] y
where I have written to stress that j's type has
changed.  Note that (of course!) we have to push the application
inside the RHS of the join as well as into the body.  AND if j
has an unfolding we have to push it into there too.  AND j might
be recursive...

So for now I'm abandoning the no-crap rule in this case. I think
that for the use in CorePrep it really doesn't matter; and if
it does, then CoreToStg.myCollectArgs will fall over.

(Moreover, I think that casts can make the no-crap rule fail too.)

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"

Note [Eta expansion and source notes]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
CorePrep puts floatable ticks outside of value applications, but not
type applications. As a result we might be trying to eta-expand an
expression like

  (src<...> v) @a

which we want to lead to code like

  \x -> src<...> v @a x

This means that we need to look through type applications and be ready
to re-add floats on the top.

Note [Eta expansion with ArityType]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The etaExpandAT function takes an ArityType (not just an Arity) to
guide eta-expansion.  Why? Because we want to preserve one-shot info.
Consider
  foo = \x. case x of
              True  -> (\s{os}. blah) |> co
              False -> wubble
We'll get an ArityType for foo of \?1.T.

Then we want to eta-expand to
  foo = (\x. \eta{os}. (case x of ...as before...) eta) |> some_co

That 'eta' binder is fresh, and we really want it to have the
one-shot flag from the inner \s{os}.  By expanding with the
ArityType gotten from analysing the RHS, we achieve this neatly.

This makes a big difference to the one-shot monad trick;
see Note [The one-shot state monad trick] in GHC.Utils.Monad.
-}

-- | @etaExpand n e@ returns an expression with
-- the same meaning as @e@, but with arity @n@.
--
-- Given:
--
-- > e' = etaExpand n e
--
-- We should have that:
--
-- > ty = exprType e = exprType e'

etaExpand :: Arity -> CoreExpr -> CoreExpr
etaExpand n orig_expr
  = eta_expand in_scope (replicate n NoOneShotInfo) orig_expr
  where
    in_scope = {-#SCC "eta_expand:in-scopeX" #-}
               mkInScopeSet (exprFreeVars orig_expr)

etaExpandAT :: InScopeSet -> SafeArityType -> CoreExpr -> CoreExpr
-- See Note [Eta expansion with ArityType]
--
-- We pass in the InScopeSet from the simplifier to avoid recomputing
-- it here, which can be jolly expensive if the casts are big
-- In #18223 it took 10% of compile time just to do the exprFreeVars!
etaExpandAT in_scope at orig_expr
  = eta_expand in_scope (arityTypeOneShots at) orig_expr

-- etaExpand arity e = res
-- Then 'res' has at least 'arity' lambdas at the top
--    possibly with a cast wrapped around the outside
-- See Note [Eta expansion with ArityType]
--
-- etaExpand deals with for-alls. For example:
--              etaExpand 1 E
-- where  E :: forall a. a -> a
-- would return
--      (/\b. \y::a -> E b y)

eta_expand :: InScopeSet -> [OneShotInfo] -> CoreExpr -> CoreExpr
eta_expand in_scope one_shots (Cast expr co)
  = mkCast (eta_expand in_scope one_shots expr) co
    -- This mkCast is important, because eta_expand might return an
    -- expression with a cast at the outside; and tryCastWorkerWrapper
    -- asssumes that we don't have nested casts. Makes a difference
    -- in compile-time for T18223

eta_expand in_scope one_shots orig_expr
  = go in_scope one_shots [] orig_expr
  where
      -- Strip off existing lambdas and casts before handing off to mkEtaWW
      -- This is mainly to avoid spending time cloning binders and substituting
      -- when there is actually nothing to do.  It's slightly awkward to deal
      -- with casts here, apart from the topmost one, and they are rare, so
      -- if we find one we just hand off to mkEtaWW anyway
      -- Note [Eta expansion and SCCs]
    go _ [] _ _ = orig_expr  -- Already has the specified arity; no-op

    go in_scope oss@(_:oss1) vs (Lam v body)
      | isTyVar v = go (in_scope `extendInScopeSet` v) oss  (v:vs) body
      | otherwise = go (in_scope `extendInScopeSet` v) oss1 (v:vs) body

    go in_scope oss rev_vs expr
      = -- pprTrace "ee" (vcat [ppr in_scope', ppr top_bndrs, ppr eis]) $
        retick $
        etaInfoAbs top_eis $
        etaInfoApp in_scope' sexpr eis
      where
          (in_scope', eis@(EI eta_bndrs mco))
                    = mkEtaWW oss (ppr orig_expr) in_scope (exprType expr)
          top_bndrs = reverse rev_vs
          top_eis   = EI (top_bndrs ++ eta_bndrs) (mkPiMCos top_bndrs mco)

          -- Find ticks behind type apps.
          -- See Note [Eta expansion and source notes]
          -- I don't really understand this code SLPJ May 21
          (expr', args) = collectArgs expr
          (ticks, expr'') = stripTicksTop tickishFloatable expr'
          sexpr = mkApps expr'' args
          retick expr = foldr mkTick expr ticks

{- *********************************************************************
*                                                                      *
              The EtaInfo mechanism
          mkEtaWW, etaInfoAbs, etaInfoApp
*                                                                      *
********************************************************************* -}

{- Note [The EtaInfo mechanism]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Suppose we have (e :: ty) and we want to eta-expand it to arity N.
This what eta_expand does.  We do it in two steps:

1.  mkEtaWW: from 'ty' and 'N' build a EtaInfo which describes
    the shape of the expansion necessary to expand to arity N.

2.  Build the term
       \ v1..vn.  e v1 .. vn
    where those abstractions and applications are described by
    the same EtaInfo.  Specifically we build the term

       etaInfoAbs etas (etaInfoApp in_scope e etas)

   where etas :: EtaInfo
         etaInfoAbs builds the lambdas
         etaInfoApp builds the applications

   Note that the /same/ EtaInfo drives both etaInfoAbs and etaInfoApp

To a first approximation EtaInfo is just [Var].  But
casts complicate the question.  If we have
   newtype N a = MkN (S -> a)
     axN :: N a  ~  S -> a
and
   e :: N (N Int)
then the eta-expansion should look like
   (\(x::S) (y::S) -> (e |> co) x y) |> sym co
where
  co :: N (N Int) ~ S -> S -> Int
  co = axN @(N Int) ; (S -> axN @Int)

We want to get one cast, at the top, to account for all those
nested newtypes. This is expressed by the EtaInfo type:

   data EtaInfo = EI [Var] MCoercionR

Note [Check for reflexive casts in eta expansion]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
It turns out that the casts created by the above mechanism are often Refl.
When casts are very deeply nested (as happens in #18223), the repetition
of types can make the overall term very large.  So there is a big
payoff in cancelling out casts aggressively wherever possible.
(See also Note [No crap in eta-expanded code].)

This matters particularly in etaInfoApp, where we
* Do beta-reduction on the fly
* Use getArg_maybe to get a cast out of the way,
  so that we can do beta reduction
Together this makes a big difference.  Consider when e is
   case x of
      True  -> (\x -> e1) |> c1
      False -> (\p -> e2) |> c2

When we eta-expand this to arity 1, say, etaInfoAbs will wrap
a (\eta) around the outside and use etaInfoApp to apply each
alternative to 'eta'.  We want to beta-reduce all that junk
away.

#18223 was a dramatic example in which the intermediate term was
grotesquely huge, even though the next Simplifier iteration squashed
it.  Better to kill it at birth.

The crucial spots in etaInfoApp are:
* `checkReflexiveMCo` in the (Cast e co) case of `go`
* `checkReflexiveMCo` in `pushCoArg`
* Less important: checkReflexiveMCo in the final case of `go`
Collectively these make a factor-of-5 difference to the total
allocation of T18223, so take care if you change this stuff!

Example:
   newtype N = MkN (Y->Z)
   f :: X -> N
   f = \(x::X). ((\(y::Y). blah) |> fco)

where fco :: (Y->Z) ~ N

mkEtaWW makes an EtaInfo of (EI [(eta1:X), (eta2:Y)] eta_co
  where
    eta_co :: (X->N) ~ (X->Y->Z)
    eta_co =  (<X> -> nco)
    nco :: N ~ (Y->Z)  -- Comes from topNormaliseNewType_maybe

Now, when we push that eta_co inward in etaInfoApp:
* In the (Cast e co) case, the 'fco' and 'nco' will meet, and
  should cancel.
* When we meet the (\y.e) we want no cast on the y.

-}

--------------
data EtaInfo = EI [Var] MCoercionR

-- (EI bs co) describes a particular eta-expansion, as follows:
--  Abstraction:  (\b1 b2 .. bn. []) |> sym co
--  Application:  ([] |> co) b1 b2 .. bn
--
--    e :: T    co :: T ~ (t1 -> t2 -> .. -> tn -> tr)
--    e = (\b1 b2 ... bn. (e |> co) b1 b2 .. bn) |> sym co

instance Outputable EtaInfo where
  ppr (EI vs mco) = text "EI" <+> ppr vs <+> parens (ppr mco)


etaInfoApp :: InScopeSet -> CoreExpr -> EtaInfo -> CoreExpr
-- (etaInfoApp s e (EI bs mco) returns something equivalent to
--             ((substExpr s e) |> mco b1 .. bn)
-- See Note [The EtaInfo mechanism]
--
-- NB: With very deeply nested casts, this function can be expensive
--     In T18223, this function alone costs 15% of allocation, all
--     spent in the calls to substExprSC and substBindSC

etaInfoApp in_scope expr eis
  = go (mkEmptySubst in_scope) expr eis
  where
    go :: Subst -> CoreExpr -> EtaInfo -> CoreExpr
    -- 'go' pushed down the eta-infos into the branch of a case
    -- and the body of a let; and does beta-reduction if possible
    --   go subst fun co [b1,..,bn]  returns  (subst(fun) |> co) b1 .. bn
    go subst (Tick t e) eis
      = Tick (substTickish subst t) (go subst e eis)

    go subst (Cast e co) (EI bs mco)
      = go subst e (EI bs mco')
      where
        mco' = checkReflexiveMCo (Core.substCo subst co `mkTransMCoR` mco)
               -- See Note [Check for reflexive casts in eta expansion]

    go subst (Case e b ty alts) eis
      = Case (Core.substExprSC subst e) b1 ty' alts'
      where
        (subst1, b1) = Core.substBndr subst b
        alts' = map subst_alt alts
        ty'   = etaInfoAppTy (substTyUnchecked subst ty) eis
        subst_alt (Alt con bs rhs) = Alt con bs' (go subst2 rhs eis)
                 where
                  (subst2,bs') = Core.substBndrs subst1 bs

    go subst (Let b e) eis
      | not (isJoinBind b) -- See Note [Eta expansion for join points]
      = Let b' (go subst' e eis)
      where
        (subst', b') = Core.substBindSC subst b

    -- Beta-reduction if possible, pushing any intervening casts past
    -- the argument. See Note [The EtaInfo mechanism]
    go subst (Lam v e) (EI (b:bs) mco)
      | Just (arg,mco') <- pushMCoArg mco (varToCoreExpr b)
      = go (Core.extendSubst subst v arg) e (EI bs mco')

    -- Stop pushing down; just wrap the expression up
    -- See Note [Check for reflexive casts in eta expansion]
    go subst e (EI bs mco) = Core.substExprSC subst e
                             `mkCastMCo` checkReflexiveMCo mco
                             `mkVarApps` bs

--------------
etaInfoAppTy :: Type -> EtaInfo -> Type
-- If                    e :: ty
-- then   etaInfoApp e eis :: etaInfoApp ty eis
etaInfoAppTy ty (EI bs mco)
  = applyTypeToArgs (text "etaInfoAppTy") ty1 (map varToCoreExpr bs)
  where
    ty1 = case mco of
             MRefl  -> ty
             MCo co -> coercionRKind co

--------------
etaInfoAbs :: EtaInfo -> CoreExpr -> CoreExpr
-- See Note [The EtaInfo mechanism]
etaInfoAbs (EI bs mco) expr = (mkLams bs expr) `mkCastMCo` mkSymMCo mco

--------------
-- | @mkEtaWW n _ fvs ty@ will compute the 'EtaInfo' necessary for eta-expanding
-- an expression @e :: ty@ to take @n@ value arguments, where @fvs@ are the
-- free variables of @e@.
--
-- Note that this function is entirely unconcerned about cost centres and other
-- semantically-irrelevant source annotations, so call sites must take care to
-- preserve that info. See Note [Eta expansion and SCCs].
mkEtaWW
  :: [OneShotInfo]
  -- ^ How many value arguments to eta-expand
  -> SDoc
  -- ^ The pretty-printed original expression, for warnings.
  -> InScopeSet
  -- ^ A super-set of the free vars of the expression to eta-expand.
  -> Type
  -> (InScopeSet, EtaInfo)
  -- ^ The variables in 'EtaInfo' are fresh wrt. to the incoming 'InScopeSet'.
  -- The outgoing 'InScopeSet' extends the incoming 'InScopeSet' with the
  -- fresh variables in 'EtaInfo'.

mkEtaWW orig_oss ppr_orig_expr in_scope orig_ty
  = go 0 orig_oss empty_subst orig_ty
  where
    empty_subst = mkEmptySubst in_scope

    go :: Int                -- For fresh names
       -> [OneShotInfo]      -- Number of value args to expand to
       -> Subst -> Type   -- We are really looking at subst(ty)
       -> (InScopeSet, EtaInfo)
    -- (go [o1,..,on] subst ty) = (in_scope, EI [b1,..,bn] co)
    --    co :: subst(ty) ~ b1_ty -> ... -> bn_ty -> tr

    go _ [] subst _
       ----------- Done!  No more expansion needed
       = (getSubstInScope subst, EI [] MRefl)

    go n oss@(one_shot:oss1) subst ty
       ----------- Forall types  (forall a. ty)
       | Just (tcv,ty') <- splitForAllTyCoVar_maybe ty
       , (subst', tcv') <- Type.substVarBndr subst tcv
       , let oss' | isTyVar tcv = oss
                  | otherwise   = oss1
         -- A forall can bind a CoVar, in which case
         -- we consume one of the [OneShotInfo]
       , (in_scope, EI bs mco) <- go n oss' subst' ty'
       = (in_scope, EI (tcv' : bs) (mkHomoForAllMCo tcv' mco))

       ----------- Function types  (t1 -> t2)
       | Just (_af, mult, arg_ty, res_ty) <- splitFunTy_maybe ty
       , typeHasFixedRuntimeRep arg_ty
          -- See Note [Representation polymorphism invariants] in GHC.Core
          -- See also test case typecheck/should_run/EtaExpandLevPoly

       , (subst', eta_id) <- freshEtaId n subst (Scaled mult arg_ty)
          -- Avoid free vars of the original expression

       , let eta_id' = eta_id `setIdOneShotInfo` one_shot
       , (in_scope, EI bs mco) <- go (n+1) oss1 subst' res_ty
       = (in_scope, EI (eta_id' : bs) (mkFunResMCo eta_id' mco))

       ----------- Newtypes
       -- 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)
       | Just (co, ty') <- topNormaliseNewType_maybe ty
       , -- co :: ty ~ ty'
         let co' = Type.substCo subst co
             -- Remember to apply the substitution to co (#16979)
             -- (or we could have applied to ty, but then
             --  we'd have had to zap it for the recursive call)
       , (in_scope, EI bs mco) <- go n oss subst ty'
         -- mco :: subst(ty') ~ b1_ty -> ... -> bn_ty -> tr
       = (in_scope, EI bs (mkTransMCoR co' mco))

       | otherwise       -- We have an expression of arity > 0,
                         -- but its type isn't a function, or a binder
                         -- does not have a fixed runtime representation
       = warnPprTrace True "mkEtaWW" ((ppr orig_oss <+> ppr orig_ty) $$ ppr_orig_expr)
         (getSubstInScope subst, EI [] MRefl)
        -- This *can* legitimately 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


{-
************************************************************************
*                                                                      *
                Eta reduction
*                                                                      *
************************************************************************

Note [Eta reduction makes sense]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
GHC's eta reduction transforms
   \x y. <fun> x y  --->  <fun>
We discuss when this is /sound/ in Note [Eta reduction soundness].
But even assuming it is sound, when is it /desirable/.  That
is what we discuss here.

This test is made by `ok_fun` in tryEtaReduce.

1. We want to eta-reduce only if we get all the way to a trivial
   expression; we don't want to remove extra lambdas unless we are
   going to avoid allocating this thing altogether.

   Trivial means *including* casts and type lambdas:
     * `\x. f x |> co  -->  f |> (ty(x) -> co)` (provided `co` doesn't mention `x`)
     * `/\a. \x. f @(Maybe a) x -->  /\a. f @(Maybe a)`
   See Note [Do not eta reduce PAPs] for why we insist on a trivial head.

Of course, eta reduction is not always sound. See Note [Eta reduction soundness]
for when it is.

When there are multiple arguments, we might get multiple eta-redexes. Example:
   \x y. e x y
   ==> { reduce \y. (e x) y in context \x._ }
   \x. e x
   ==> { reduce \x. e x in context _ }
   e
And (1) implies that we never want to stop with `\x. e x`, because that is not a
trivial expression. So in practice, the implementation works by considering a
whole group of leading lambdas to reduce.

These delicacies are why we don't simply use 'exprIsTrivial' and 'exprIsHNF'
in 'tryEtaReduce'. Alas.

Note [Eta reduction soundness]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
GHC's eta reduction transforms
   \x y. <fun> x y  --->  <fun>
For soundness, we obviously require that `x` and `y`
to not occur free. But what /other/ restrictions are there for
eta reduction to be sound?

We discuss separately what it means for eta reduction to be
/desirable/, in Note [Eta reduction makes sense].

Eta reduction is *not* a sound transformation in general, because it
may change termination behavior if *value* lambdas are involved:
  `bot`  /=  `\x. bot x`   (as can be observed by a simple `seq`)
The past has shown that oversight of this fact can not only lead to endless
loops or exceptions, but also straight out *segfaults*.

Nevertheless, we can give the following criteria for when it is sound to
perform eta reduction on an expression with n leading lambdas `\xs. e xs`
(checked in 'is_eta_reduction_sound' in 'tryEtaReduce', which focuses on the
case where `e` is trivial):

(A) It is sound to eta-reduce n arguments as long as n does not exceed the
    `exprArity` of `e`. (Needs Arity analysis.)
    This criterion exploits information about how `e` is *defined*.

    Example: If `e = \x. bot` then we know it won't diverge until it is called
    with one argument. Hence it is safe to eta-reduce `\x. e x` to `e`.
    By contrast, it would be *unsound* to eta-reduce 2 args, `\x y. e x y` to `e`:
    `e 42` diverges when `(\x y. e x y) 42` does not.

(S) It is sound to eta-reduce n arguments in an evaluation context in which all
    calls happen with at least n arguments. (Needs Strictness analysis.)
    NB: This treats evaluations like a call with 0 args.
    NB: This criterion exploits information about how `e` is *used*.

    Example: Given a function `g` like
      `g c = Just (c 1 2 + c 2 3)`
    it is safe to eta-reduce the arg in `g (\x y. e x y)` to `g e` without
    knowing *anything* about `e` (perhaps it's a parameter occ itself), simply
    because `g` always calls its parameter with 2 arguments.
    It is also safe to eta-reduce just one arg, e.g., `g (\x. e x)` to `g e`.
    By contrast, it would *unsound* to eta-reduce 3 args in a call site
    like `g (\x y z. e x y z)` to `g e`, because that diverges when
    `e = \x y. bot`.

    Could we relax to "*At least one call in the same trace* is with n args"?
    No. Consider what happens for
      ``g2 c = c True `seq` c False 42``
    Here, `g2` will call `c` with 2 arguments (if there is a call at all).
    But it is unsound to eta-reduce the arg in `g2 (\x y. e x y)` to `g2 e`
    when `e = \x. if x then bot else id`, because the latter will diverge when
    the former would not. Fortunately, the strictness analyser will report
    "Not always called with two arguments" for `g2` and we won't eta-expand.

    See Note [Eta reduction based on evaluation context] for the implementation
    details. This criterion is tested extensively in T21261.

(R) Note [Eta reduction in recursive RHSs] tells us that we should not
    eta-reduce `f` in its own RHS and describes our fix.
    There we have `f = \x. f x` and we should not eta-reduce to `f=f`. Which
    might change a terminating program (think @f `seq` e@) to a non-terminating
    one.

(E) (See fun_arity in tryEtaReduce.) As a perhaps special case on the
    boundary of (A) and (S), when we know that a fun binder `f` is in
    WHNF, we simply assume it has arity 1 and apply (A).  Example:
       g f = f `seq` \x. f x
    Here it's sound eta-reduce `\x. f x` to `f`, because `f` can't be bottom
    after the `seq`. This turned up in #7542.

 T. If the binders are all type arguments, it's always safe to eta-reduce,
    regardless of the arity of f.
       /\a b. f @a @b  --> f

2. Type and dictionary abstraction. Regardless of whether 'f' is a value, it
   is always sound to reduce /type lambdas/, thus:
        (/\a -> f a)  -->   f
   Moreover, we always want to, because it makes RULEs apply more often:
      This RULE:    `forall g. foldr (build (/\a -> g a))`
      should match  `foldr (build (/\b -> ...something complex...))`
   and the simplest way to do so is eta-reduce `/\a -> g a` in the RULE to `g`.

   More debatably, we extend this to dictionary arguments too, because the type
   checker can insert these eta-expanded versions, with both type and dictionary
   lambdas; hence the slightly ad-hoc (all ok_lam bndrs).  That is, we eta-reduce
        \(d::Num a). f d   -->   f
   regardless of f's arity. Its not clear whether or not this is important, and
   it is not in general sound.  But that's the way it is right now.

And here are a few more technical criteria for when it is *not* sound to
eta-reduce that are specific to Core and GHC:

(L) With linear types, eta-reduction can break type-checking:
      f :: A ⊸ B
      g :: A -> B
      g = \x. f x
    The above is correct, but eta-reducing g would yield g=f, the linter will
    complain that g and f don't have the same type. NB: Not unsound in the
    dynamic semantics, but unsound according to the static semantics of Core.

(J) We may not undersaturate join points.
    See Note [Invariants on join points] in GHC.Core, and #20599.

(B) We may not undersaturate functions with no binding.
    See Note [Eta expanding primops].

(W) We may not undersaturate StrictWorkerIds.
    See Note [CBV Function Ids] in GHC.Types.Id.Info.

Here is a list of historic accidents surrounding unsound eta-reduction:

* Consider
        f = \x.f x
        h y = case (case y of { True -> f `seq` True; False -> False }) of
                True -> ...; False -> ...
  If we (unsoundly) eta-reduce f to get f=f, the strictness analyser
  says f=bottom, and replaces the (f `seq` True) with just
  (f `cast` unsafe-co).
  [SG in 2022: I don't think worker/wrapper would do this today.]
  BUT, as things stand, 'f' got arity 1, and it *keeps* arity 1 (perhaps also
  wrongly). So CorePrep eta-expands the definition again, so that it does not
  terminate after all.
  Result: seg-fault because the boolean case actually gets a function value.
  See #1947.

* Never *reduce* arity. For example
      f = \xy. g x y
  Then if h has arity 1 we don't want to eta-reduce because then
  f's arity would decrease, and that is bad
  [SG in 2022: I don't understand this point. There is no `h`, perhaps that
   should have been `g`. Even then, this proposed eta-reduction is invalid by
   criterion (A), which might actually be the point this anecdote is trying to
   make. Perhaps the "no arity decrease" idea is also related to
   Note [Arity robustness]?]

Note [Do not eta reduce PAPs]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
I considered eta-reducing if the result is a PAP:
   \x. f e1 e2 x  ==>   f e1 e2

This reduces clutter, sometimes a lot. See Note [Do not eta-expand PAPs]
in GHC.Core.Opt.Simplify.Utils, where we are careful not to eta-expand
a PAP.  If eta-expanding is bad, then eta-reducing is good!

Also the code generator likes eta-reduced PAPs; see GHC.CoreToStg.Prep
Note [No eta reduction needed in rhsToBody].

But note that we don't want to eta-reduce
     \x y.  f <expensive> x y
to
     f <expensive>
The former has arity 2, and repeats <expensive> for every call of the
function; the latter has arity 0, and shares <expensive>.  We don't want
to change behaviour.  Hence the call to exprIsCheap in ok_fun.

I noticed this when examining #18993 and, although it is delicate,
eta-reducing to a PAP happens to fix the regression in #18993.

HOWEVER, if we transform
   \x. f y x   ==>   f y
that might mean that f isn't saturated any more, and does not inline.
This led to some other regressions.

TL;DR currently we do /not/ eta reduce if the result is a PAP.

Note [Eta reduction with casted arguments]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Consider
    (\(x:t3). f (x |> g)) :: t3 -> t2
  where
    f :: t1 -> t2
    g :: t3 ~ t1
This should be eta-reduced to

    f |> (sym g -> t2)

So we need to accumulate a coercion, pushing it inward (past
variable arguments only) thus:
   f (x |> co_arg) |> co  -->  (f |> (sym co_arg -> co)) x
   f (x:t)         |> co  -->  (f |> (t -> co)) x
   f @ a           |> co  -->  (f |> (forall a.co)) @ a
   f @ (g:t1~t2)   |> co  -->  (f |> (t1~t2 => co)) @ (g:t1~t2)
These are the equations for ok_arg.

Note [Eta reduction with casted function]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Since we are pushing a coercion inwards, it is easy to accommodate
    (\xy. (f x |> g) y)
    (\xy. (f x y) |> g)

See the `(Cast e co)` equation for `go` in `tryEtaReduce`.  The
eta-expander pushes those casts outwards, so you might think we won't
ever see a cast here, but if we have
  \xy. (f x y |> g)
we will call tryEtaReduce [x,y] (f x y |> g), and we'd like that to
work.  This happens in GHC.Core.Opt.Simplify.Utils.mkLam, where
eta-expansion may be turned off (by sm_eta_expand).

Note [Eta reduction based on evaluation context]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Note [Eta reduction soundness], criterion (S) allows us to eta-reduce
`g (\x y. e x y)` to `g e` when we know that `g` always calls its parameter with
at least 2 arguments. So how do we read that off `g`'s demand signature?

Let's take the simple example of #21261, where `g` (actually, `f`) is defined as
  g c = c 1 2 + c 3 4
Then this is how the pieces are put together:

  * Demand analysis infers `<SC(S,C(1,L))>` for `g`'s demand signature

  * When the Simplifier next simplifies the argument in `g (\x y. e x y)`, it
    looks up the *evaluation context* of the argument in the form of the
    sub-demand `C(S,C(1,L))` and stores it in the 'SimplCont'.
    (Why does it drop the outer evaluation cardinality of the demand, `S`?
    Because it's irrelevant! When we simplify an expression, we do so under the
    assumption that it is currently under evaluation.)
    This sub-demand literally says "Whenever this expression is evaluated, it
    is called with at least two arguments, potentially multiple times".

  * Then the simplifier takes apart the lambda and simplifies the lambda group
    and then calls 'tryEtaReduce' when rebuilding the lambda, passing the
    evaluation context `C(S,C(1,L))` along. Then we simply peel off 2 call
    sub-demands `Cn` and see whether all of the n's (here: `S=C_1N` and
    `1=C_11`) were strict. And strict they are! Thus, it will eta-reduce
    `\x y. e x y` to `e`.

Note [Eta reduction in recursive RHSs]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Consider the following recursive function:
  f = \x. ....g (\y. f y)....
The recursive call of f in its own RHS seems like a fine opportunity for
eta-reduction because f has arity 1. And often it is!

Alas, that is unsound in general if the eta-reduction happens in a tail context.
Making the arity visible in the RHS allows us to eta-reduce
  f = \x -> f x
to
  f = f
which means we optimise terminating programs like (f `seq` ()) into
non-terminating ones. Nor is this problem just for tail calls.  Consider
  f = id (\x -> f x)
where we have (for some reason) not yet inlined `id`.  We must not eta-reduce to
  f = id f
because that will then simplify to `f = f` as before.

An immediate idea might be to look at whether the called function is a local
loopbreaker and refrain from eta-expanding. But that doesn't work for mutually
recursive function like in #21652:
  f = g
  g* x = f x
Here, g* is the loopbreaker but f isn't.

What can we do?

Fix 1: Zap `idArity` when analysing recursive RHSs and re-attach the info when
    entering the let body.
    Has the disadvantage that other transformations which make use of arity
    (such as dropping of `seq`s when arity > 0) will no longer work in the RHS.
    Plus it requires non-trivial refactorings to both the simple optimiser (in
    the way `subst_opt_bndr` is used) as well as the Simplifier (in the way
    `simplRecBndrs` and `simplRecJoinBndrs` is used), modifying the SimplEnv's
    substitution twice in the process. A very complicated stop-gap.

Fix 2: Pass the set of enclosing recursive binders to `tryEtaReduce`; these are
    the ones we should not eta-reduce. All call-site must maintain this set.
    Example:
      rec { f1 = ....rec { g = ... (\x. g x)...(\y. f2 y)... }...
          ; f2 = ...f1... }
    when eta-reducing those inner lambdas, we need to know that we are in the
    rec group for {f1, f2, g}.
    This is very much like the solution in Note [Speculative evaluation] in
    GHC.CoreToStg.Prep.
    It is a bit tiresome to maintain this info, because it means another field
    in SimplEnv and SimpleOptEnv.

We implement Fix (2) because of it isn't as complicated to maintain as (1).
Plus, it is the correct fix to begin with. After all, the arity is correct,
but doing the transformation isn't. The moving parts are:
  * A field `scRecIds` in `SimplEnv` tracks the enclosing recursive binders
  * We extend the `scRecIds` set in `GHC.Core.Opt.Simplify.simplRecBind`
  * We consult the set in `is_eta_reduction_sound` in `tryEtaReduce`
The situation is very similar to Note [Speculative evaluation] which has the
same fix.
-}

-- | `tryEtaReduce [x,y,z] e sd` returns `Just e'` if `\x y z -> e` is evaluated
-- according to `sd` and can soundly and gainfully be eta-reduced to `e'`.
-- See Note [Eta reduction soundness]
-- and Note [Eta reduction makes sense] when that is the case.
tryEtaReduce :: UnVarSet -> [Var] -> CoreExpr -> SubDemand -> Maybe CoreExpr
-- Return an expression equal to (\bndrs. body)
tryEtaReduce rec_ids bndrs body eval_sd
  = go (reverse bndrs) body (mkRepReflCo (exprType body))
  where
    incoming_arity = count isId bndrs -- See Note [Eta reduction makes sense], point (2)

    go :: [Var]            -- Binders, innermost first, types [a3,a2,a1]
       -> CoreExpr         -- Of type tr
       -> Coercion         -- Of type tr ~ ts
       -> Maybe CoreExpr   -- Of type a1 -> a2 -> a3 -> ts
    -- See Note [Eta reduction with casted arguments]
    -- for why we have an accumulating coercion
    --
    -- Invariant: (go bs body co) returns an expression
    --            equivalent to (\(reverse bs). (body |> co))

    -- See Note [Eta reduction with casted function]
    go bs (Cast e co1) co2
      = go bs e (co1 `mkTransCo` co2)

    go bs (Tick t e) co
      | tickishFloatable t
      = fmap (Tick t) $ go bs e co
      -- Float app ticks: \x -> Tick t (e x) ==> Tick t e

    go (b : bs) (App fun arg) co
      | Just (co', ticks) <- ok_arg b arg co (exprType fun)
      = fmap (flip (foldr mkTick) ticks) $ go bs fun co'
            -- Float arg ticks: \x -> e (Tick t x) ==> Tick t e

    go remaining_bndrs fun co
      | all isTyVar remaining_bndrs
            -- If all the remaining_bnrs are tyvars, then the etad_exp
            --    will be trivial, which is what we want.
            -- e.g. We might have  /\a \b. f [a] b, and we want to
            --      eta-reduce to  /\a. f [a]
            -- We don't want to give up on this one: see #20040
            -- See Note [Eta reduction makes sense], point (1)
      , remaining_bndrs `ltLength` bndrs
            -- Only reply Just if /something/ has happened
      , ok_fun fun
      , let used_vars     = exprFreeVars fun `unionVarSet` tyCoVarsOfCo co
            reduced_bndrs = mkVarSet (dropList remaining_bndrs bndrs)
            -- reduced_bndrs are the ones we are eta-reducing away
      , used_vars `disjointVarSet` reduced_bndrs
          -- Check for any of the reduced_bndrs (about to be dropped)
          -- free in the result, including the accumulated coercion.
          -- See Note [Eta reduction makes sense], intro and point (1)
          -- NB: don't compute used_vars from exprFreeVars (mkCast fun co)
          --     because the latter may be ill formed if the guard fails (#21801)
      = Just (mkLams (reverse remaining_bndrs) (mkCast fun co))

    go _remaining_bndrs _fun  _  = -- pprTrace "tER fail" (ppr _fun $$ ppr _remaining_bndrs) $
                                   Nothing

    ---------------
    -- See Note [Eta reduction makes sense], point (1)
    ok_fun (App fun (Type {})) = ok_fun fun
    ok_fun (Cast fun _)        = ok_fun fun
    ok_fun (Tick _ expr)       = ok_fun expr
    ok_fun (Var fun_id)        = is_eta_reduction_sound fun_id
    ok_fun _fun                = False

    ---------------
    -- See Note [Eta reduction soundness], this is THE place to check soundness!
    is_eta_reduction_sound fun
      | fun `elemUnVarSet` rec_ids          -- Criterion (R)
      = False -- Don't eta-reduce in fun in its own recursive RHSs

      | cantEtaReduceFun fun                -- Criteria (L), (J), (W), (B)
      = False -- Function can't be eta reduced to arity 0
              -- without violating invariants of Core and GHC

      | otherwise
      = -- Check that eta-reduction won't make the program stricter...
        fun_arity fun >= incoming_arity          -- Criterion (A) and (E)
        || all_calls_with_arity incoming_arity   -- Criterion (S)
        || all ok_lam bndrs                      -- Criterion (T)

    all_calls_with_arity n = isStrict (fst $ peelManyCalls n eval_sd)
       -- See Note [Eta reduction based on evaluation context]

    ---------------
    fun_arity fun
       | arity > 0                           = arity
       | isEvaldUnfolding (idUnfolding fun)  = 1
           -- See Note [Eta reduction soundness], criterion (E)
       | otherwise                           = 0
       where
         arity = idArity fun

    ---------------
    ok_lam v = isTyVar v || isEvVar v
    -- See Note [Eta reduction makes sense], point (2)

    ---------------
    ok_arg :: Var              -- Of type bndr_t
           -> CoreExpr         -- Of type arg_t
           -> Coercion         -- Of kind (t1~t2)
           -> Type             -- Type (arg_t -> t1) of the function
                               --      to which the argument is supplied
           -> Maybe (Coercion  -- Of type (arg_t -> t1 ~  bndr_t -> t2)
                               --   (and similarly for tyvars, coercion args)
                    , [CoreTickish])
    -- See Note [Eta reduction with casted arguments]
    ok_arg bndr (Type ty) co _
       | Just tv <- getTyVar_maybe ty
       , bndr == tv  = Just (mkHomoForAllCos [tv] co, [])
    ok_arg bndr (Var v) co fun_ty
       | bndr == v
       , let mult = idMult bndr
       , Just (_af, fun_mult, _, _) <- splitFunTy_maybe fun_ty
       , mult `eqType` fun_mult -- There is no change in multiplicity, otherwise we must abort
       = Just (mkFunResCo Representational bndr co, [])
    ok_arg bndr (Cast e co_arg) co fun_ty
       | (ticks, Var v) <- stripTicksTop tickishFloatable e
       , Just (_, fun_mult, _, _) <- splitFunTy_maybe fun_ty
       , bndr == v
       , fun_mult `eqType` idMult bndr
       = Just (mkFunCoNoFTF Representational (multToCo fun_mult) (mkSymCo co_arg) co, ticks)
       -- The simplifier combines multiple casts into one,
       -- so we can have a simple-minded pattern match here
    ok_arg bndr (Tick t arg) co fun_ty
       | tickishFloatable t, Just (co', ticks) <- ok_arg bndr arg co fun_ty
       = Just (co', t:ticks)

    ok_arg _ _ _ _ = Nothing

-- | Can we eta-reduce the given function
-- See Note [Eta reduction soundness], criteria (B), (J), (W) and (L).
cantEtaReduceFun :: Id -> Bool
cantEtaReduceFun fun
  =    hasNoBinding fun -- (B)
       -- Don't undersaturate functions with no binding.

    ||  isJoinId fun    -- (J)
       -- Don't undersaturate join points.
       -- See Note [Invariants on join points] in GHC.Core, and #20599

    || (isJust (idCbvMarks_maybe fun)) -- (W)
       -- Don't undersaturate StrictWorkerIds.
       -- See Note [CBV Function Ids] in GHC.Types.Id.Info.

    ||  isLinearType (idType fun) -- (L)
       -- Don't perform eta reduction on linear types.
       -- If `f :: A %1-> B` and `g :: A -> B`,
       -- then `g x = f x` is OK but `g = f` is not.


{- *********************************************************************
*                                                                      *
              The "push rules"
*                                                                      *
************************************************************************

Here we implement the "push rules" from FC papers:

* The push-argument rules, where we can move a coercion past an argument.
  We have
      (fun |> co) arg
  and we want to transform it to
    (fun arg') |> co'
  for some suitable co' and transformed arg'.

* The PushK rule for data constructors.  We have
       (K e1 .. en) |> co
  and we want to transform to
       (K e1' .. en')
  by pushing the coercion into the arguments
-}

pushCoArgs :: CoercionR -> [CoreArg] -> Maybe ([CoreArg], MCoercion)
pushCoArgs co []         = return ([], MCo co)
pushCoArgs co (arg:args) = do { (arg',  m_co1) <- pushCoArg  co  arg
                              ; case m_co1 of
                                  MCo co1 -> do { (args', m_co2) <- pushCoArgs co1 args
                                                 ; return (arg':args', m_co2) }
                                  MRefl  -> return (arg':args, MRefl) }

pushMCoArg :: MCoercionR -> CoreArg -> Maybe (CoreArg, MCoercion)
pushMCoArg MRefl    arg = Just (arg, MRefl)
pushMCoArg (MCo co) arg = pushCoArg co arg

pushCoArg :: CoercionR -> CoreArg -> Maybe (CoreArg, MCoercion)
-- We have (fun |> co) arg, and we want to transform it to
--         (fun arg) |> co
-- This may fail, e.g. if (fun :: N) where N is a newtype
-- C.f. simplCast in GHC.Core.Opt.Simplify
-- 'co' is always Representational
pushCoArg co arg
  | Type ty <- arg
  = do { (ty', m_co') <- pushCoTyArg co ty
       ; return (Type ty', m_co') }
  | otherwise
  = do { (arg_mco, m_co') <- pushCoValArg co
       ; let arg_mco' = checkReflexiveMCo arg_mco
             -- checkReflexiveMCo: see Note [Check for reflexive casts in eta expansion]
             -- The coercion is very often (arg_co -> res_co), but without
             -- the argument coercion actually being ReflCo
       ; return (arg `mkCastMCo` arg_mco', m_co') }

pushCoTyArg :: CoercionR -> Type -> Maybe (Type, MCoercionR)
-- We have (fun |> co) @ty
-- Push the coercion through to return
--         (fun @ty') |> co'
-- 'co' is always Representational
-- If the returned coercion is Nothing, then it would have been reflexive;
-- it's faster not to compute it, though.
pushCoTyArg co ty
  -- The following is inefficient - don't do `eqType` here, the coercion
  -- optimizer will take care of it. See #14737.
  -- -- | tyL `eqType` tyR
  -- -- = Just (ty, Nothing)

  | isReflCo co
  = Just (ty, MRefl)

  | isForAllTy_ty tyL
  = assertPpr (isForAllTy_ty tyR) (ppr co $$ ppr ty) $
    Just (ty `mkCastTy` co1, MCo co2)

  | otherwise
  = Nothing
  where
    Pair tyL tyR = coercionKind co
       -- co :: tyL ~R tyR
       -- tyL = forall (a1 :: k1). ty1
       -- tyR = forall (a2 :: k2). ty2

    co1 = mkSymCo (mkSelCo SelForAll co)
       -- co1 :: k2 ~N k1
       -- Note that SelCo extracts a Nominal equality between the
       -- kinds of the types related by a coercion between forall-types.
       -- See the SelCo case in GHC.Core.Lint.

    co2 = mkInstCo co (mkGReflLeftCo Nominal ty co1)
        -- co2 :: ty1[ (ty|>co1)/a1 ] ~R ty2[ ty/a2 ]
        -- Arg of mkInstCo is always nominal, hence Nominal

-- | If @pushCoValArg co = Just (co_arg, co_res)@, then
--
-- > (\x.body) |> co  =  (\y. let { x = y |> co_arg } in body) |> co_res)
--
-- or, equivalently
--
-- > (fun |> co) arg  =  (fun (arg |> co_arg)) |> co_res
--
-- If the LHS is well-typed, then so is the RHS. In particular, the argument
-- @arg |> co_arg@ is guaranteed to have a fixed 'RuntimeRep', in the sense of
-- Note [Fixed RuntimeRep] in GHC.Tc.Utils.Concrete.
pushCoValArg :: CoercionR -> Maybe (MCoercionR, MCoercionR)
pushCoValArg co
  -- The following is inefficient - don't do `eqType` here, the coercion
  -- optimizer will take care of it. See #14737.
  -- -- | tyL `eqType` tyR
  -- -- = Just (mkRepReflCo arg, Nothing)

  | isReflCo co
  = Just (MRefl, MRefl)

  | isFunTy tyL
  , (co_mult, co1, co2) <- decomposeFunCo co
      -- If   co  :: (tyL1 -> tyL2) ~ (tyR1 -> tyR2)
      -- then co1 :: tyL1 ~ tyR1
      --      co2 :: tyL2 ~ tyR2

  , isReflexiveCo co_mult
    -- We can't push the coercion in the case where co_mult isn't reflexivity:
    -- it could be an unsafe axiom, and losing this information could yield
    -- ill-typed terms. For instance (fun x ::(1) Int -> (fun _ -> () |> co) x)
    -- with co :: (Int -> ()) ~ (Int %1 -> ()), would reduce to (fun x ::(1) Int
    -- -> (fun _ ::(Many) Int -> ()) x) which is ill-typed.

  , typeHasFixedRuntimeRep new_arg_ty
    -- We can't push the coercion inside if it would give rise to
    -- a representation-polymorphic argument.

  = assertPpr (isFunTy tyL && isFunTy tyR)
     (vcat [ text "co:" <+> ppr co
           , text "old_arg_ty:" <+> ppr old_arg_ty
           , text "new_arg_ty:" <+> ppr new_arg_ty ]) $
    Just (coToMCo (mkSymCo co1), coToMCo co2)
    -- Critically, coToMCo to checks for ReflCo; the whole coercion may not
    -- be reflexive, but either of its components might be
    -- We could use isReflexiveCo, but it's not clear if the benefit
    -- is worth the cost, and it makes no difference in #18223

  | otherwise
  = Nothing
  where
    old_arg_ty = funArgTy tyR
    new_arg_ty = funArgTy tyL
    Pair tyL tyR = coercionKind co

pushCoercionIntoLambda
    :: HasDebugCallStack => InScopeSet -> Var -> CoreExpr -> CoercionR -> Maybe (Var, CoreExpr)
-- This implements the Push rule from the paper on coercions
--    (\x. e) |> co
-- ===>
--    (\x'. e |> co')
pushCoercionIntoLambda in_scope x e co
    | assert (not (isTyVar x) && not (isCoVar x)) True
    , Pair s1s2 t1t2 <- coercionKind co
    , Just {}              <- splitFunTy_maybe s1s2
    , Just (_, w1, t1,_t2) <- splitFunTy_maybe t1t2
    , (co_mult, co1, co2)  <- decomposeFunCo co
    , isReflexiveCo co_mult
      -- We can't push the coercion in the case where co_mult isn't
      -- reflexivity. See pushCoValArg for more details.
    , typeHasFixedRuntimeRep t1
      -- We can't push the coercion into the lambda if it would create
      -- a representation-polymorphic binder.
    = let
          -- Should we optimize the coercions here?
          -- Otherwise they might not match too well
          x' = x `setIdType` t1 `setIdMult` w1
          in_scope' = in_scope `extendInScopeSet` x'
          subst = extendIdSubst (mkEmptySubst in_scope')
                                x
                                (mkCast (Var x') (mkSymCo co1))
            -- We substitute x' for x, except we need to preserve types.
            -- The types are as follows:
            --   x :: s1,  x' :: t1,  co1 :: s1 ~# t1,
            -- so we extend the substitution with x |-> (x' |> sym co1).
      in Just (x', substExpr subst e `mkCast` co2)
    | otherwise
    = Nothing

pushCoDataCon :: DataCon -> [CoreExpr] -> Coercion
              -> Maybe (DataCon
                       , [Type]      -- Universal type args
                       , [CoreExpr]) -- All other args incl existentials
-- Implement the KPush reduction rule as described in "Down with kinds"
-- The transformation applies iff we have
--      (C e1 ... en) `cast` co
-- where co :: (T t1 .. tn) ~ to_ty
-- The left-hand one must be a T, because exprIsConApp returned True
-- but the right-hand one might not be.  (Though it usually will.)
pushCoDataCon dc dc_args co
  | isReflCo co || from_ty `eqType` to_ty  -- try cheap test first
  , let (univ_ty_args, rest_args) = splitAtList (dataConUnivTyVars dc) dc_args
  = Just (dc, map exprToType univ_ty_args, rest_args)

  | Just (to_tc, to_tc_arg_tys) <- splitTyConApp_maybe to_ty
  , to_tc == dataConTyCon dc
        -- These two tests can fail; we might see
        --      (C x y) `cast` (g :: T a ~ S [a]),
        -- where S is a type function.  In fact, exprIsConApp
        -- will probably not be called in such circumstances,
        -- but there's nothing wrong with it

  = let
        tc_arity       = tyConArity to_tc
        dc_univ_tyvars = dataConUnivTyVars dc
        dc_ex_tcvars   = dataConExTyCoVars dc
        arg_tys        = dataConRepArgTys dc

        non_univ_args  = dropList dc_univ_tyvars dc_args
        (ex_args, val_args) = splitAtList dc_ex_tcvars non_univ_args

        -- Make the "Psi" from the paper
        omegas = decomposeCo tc_arity co (tyConRolesRepresentational to_tc)
        (psi_subst, to_ex_arg_tys)
          = liftCoSubstWithEx Representational
                              dc_univ_tyvars
                              omegas
                              dc_ex_tcvars
                              (map exprToType ex_args)

          -- Cast the value arguments (which include dictionaries)
        new_val_args = zipWith cast_arg (map scaledThing arg_tys) val_args
        cast_arg arg_ty arg = mkCast arg (psi_subst arg_ty)

        to_ex_args = map Type to_ex_arg_tys

        dump_doc = vcat [ppr dc,      ppr dc_univ_tyvars, ppr dc_ex_tcvars,
                         ppr arg_tys, ppr dc_args,
                         ppr ex_args, ppr val_args, ppr co, ppr from_ty, ppr to_ty, ppr to_tc
                         , ppr $ mkTyConApp to_tc (map exprToType $ takeList dc_univ_tyvars dc_args) ]
    in
    assertPpr (eqType from_ty (mkTyConApp to_tc (map exprToType $ takeList dc_univ_tyvars dc_args))) dump_doc $
    assertPpr (equalLength val_args arg_tys) dump_doc $
    Just (dc, to_tc_arg_tys, to_ex_args ++ new_val_args)

  | otherwise
  = Nothing

  where
    Pair from_ty to_ty = coercionKind co

collectBindersPushingCo :: CoreExpr -> ([Var], CoreExpr)
-- Collect lambda binders, pushing coercions inside if possible
-- E.g.   (\x.e) |> g         g :: <Int> -> blah
--        = (\x. e |> SelCo (SelFun SelRes) g)
--
-- That is,
--
-- collectBindersPushingCo ((\x.e) |> g) === ([x], e |> SelCo (SelFun SelRes) g)
collectBindersPushingCo e
  = go [] e
  where
    -- Peel off lambdas until we hit a cast.
    go :: [Var] -> CoreExpr -> ([Var], CoreExpr)
    -- The accumulator is in reverse order
    go bs (Lam b e)   = go (b:bs) e
    go bs (Cast e co) = go_c bs e co
    go bs e           = (reverse bs, e)

    -- We are in a cast; peel off casts until we hit a lambda.
    go_c :: [Var] -> CoreExpr -> CoercionR -> ([Var], CoreExpr)
    -- (go_c bs e c) is same as (go bs e (e |> c))
    go_c bs (Cast e co1) co2 = go_c bs e (co1 `mkTransCo` co2)
    go_c bs (Lam b e)    co  = go_lam bs b e co
    go_c bs e            co  = (reverse bs, mkCast e co)

    -- We are in a lambda under a cast; peel off lambdas and build a
    -- new coercion for the body.
    go_lam :: [Var] -> Var -> CoreExpr -> CoercionR -> ([Var], CoreExpr)
    -- (go_lam bs b e c) is same as (go_c bs (\b.e) c)
    go_lam bs b e co
      | isTyVar b
      , let Pair tyL tyR = coercionKind co
      , assert (isForAllTy_ty tyL) $
        isForAllTy_ty tyR
      , isReflCo (mkSelCo SelForAll co)  -- See Note [collectBindersPushingCo]
      = go_c (b:bs) e (mkInstCo co (mkNomReflCo (mkTyVarTy b)))

      | isCoVar b
      , let Pair tyL tyR = coercionKind co
      , assert (isForAllTy_co tyL) $
        isForAllTy_co tyR
      , isReflCo (mkSelCo SelForAll co)  -- See Note [collectBindersPushingCo]
      , let cov = mkCoVarCo b
      = go_c (b:bs) e (mkInstCo co (mkNomReflCo (mkCoercionTy cov)))

      | isId b
      , let Pair tyL tyR = coercionKind co
      , assert (isFunTy tyL) $ isFunTy tyR
      , (co_mult, co_arg, co_res) <- decomposeFunCo co
      , isReflCo co_mult -- See Note [collectBindersPushingCo]
      , isReflCo co_arg  -- See Note [collectBindersPushingCo]
      = go_c (b:bs) e co_res

      | otherwise = (reverse bs, mkCast (Lam b e) co)

{-

Note [collectBindersPushingCo]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
We just look for coercions of form
   <type> % w -> blah
(and similarly for foralls) to keep this function simple.  We could do
more elaborate stuff, but it'd involve substitution etc.

-}

{- *********************************************************************
*                                                                      *
                Join points
*                                                                      *
********************************************************************* -}

-------------------
-- | Split an expression into the given number of binders and a body,
-- eta-expanding if necessary. Counts value *and* type binders.
etaExpandToJoinPoint :: JoinArity -> CoreExpr -> ([CoreBndr], CoreExpr)
etaExpandToJoinPoint join_arity expr
  = go join_arity [] expr
  where
    go 0 rev_bs e         = (reverse rev_bs, e)
    go n rev_bs (Lam b e) = go (n-1) (b : rev_bs) e
    go n rev_bs e         = case etaBodyForJoinPoint n e of
                              (bs, e') -> (reverse rev_bs ++ bs, e')

etaExpandToJoinPointRule :: JoinArity -> CoreRule -> CoreRule
etaExpandToJoinPointRule _ rule@(BuiltinRule {})
  = warnPprTrace True "Can't eta-expand built-in rule:" (ppr rule)
      -- How did a local binding get a built-in rule anyway? Probably a plugin.
    rule
etaExpandToJoinPointRule join_arity rule@(Rule { ru_bndrs = bndrs, ru_rhs = rhs
                                               , ru_args  = args })
  | need_args == 0
  = rule
  | need_args < 0
  = pprPanic "etaExpandToJoinPointRule" (ppr join_arity $$ ppr rule)
  | otherwise
  = rule { ru_bndrs = bndrs ++ new_bndrs
         , ru_args  = args ++ new_args
         , ru_rhs   = new_rhs }
  -- new_rhs really ought to be occ-analysed (see GHC.Core Note
  -- [OccInfo in unfoldings and rules]), but it makes a module loop to
  -- do so; it doesn't happen often; and it doesn't really matter if
  -- the outer binders have bogus occurrence info; and new_rhs won't
  -- have dead code if rhs didn't.

  where
    need_args = join_arity - length args
    (new_bndrs, new_rhs) = etaBodyForJoinPoint need_args rhs
    new_args = varsToCoreExprs new_bndrs

-- Adds as many binders as asked for; assumes expr is not a lambda
etaBodyForJoinPoint :: Int -> CoreExpr -> ([CoreBndr], CoreExpr)
etaBodyForJoinPoint need_args body
  = go need_args (exprType body) (init_subst body) [] body
  where
    go 0 _  _     rev_bs e
      = (reverse rev_bs, e)
    go n ty subst rev_bs e
      | Just (tv, res_ty) <- splitForAllTyCoVar_maybe ty
      , let (subst', tv') = substVarBndr subst tv
      = go (n-1) res_ty subst' (tv' : rev_bs) (e `App` varToCoreExpr tv')
        -- The varToCoreExpr is important: `tv` might be a coercion variable

      | Just (_, mult, arg_ty, res_ty) <- splitFunTy_maybe ty
      , let (subst', b) = freshEtaId n subst (Scaled mult arg_ty)
      = go (n-1) res_ty subst' (b : rev_bs) (e `App` varToCoreExpr b)
        -- The varToCoreExpr is important: `b` might be a coercion variable

      | otherwise
      = pprPanic "etaBodyForJoinPoint" $ int need_args $$
                                         ppr body $$ ppr (exprType body)

    init_subst e = mkEmptySubst (mkInScopeSet (exprFreeVars e))



--------------
freshEtaId :: Int -> Subst -> Scaled Type -> (Subst, 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
        Scaled mult' ty' = Type.substScaledTyUnchecked subst ty
        eta_id' = uniqAway (getSubstInScope subst) $
                  mkSysLocalOrCoVar (fsLit "eta") (mkBuiltinUnique n) mult' ty'
                  -- "OrCoVar" since this can be used to eta-expand
                  -- coercion abstractions
        subst'  = extendSubstInScope subst eta_id'