GHC Handbook

Note [No free join points in arityType]

GHC/Core/Opt/Arity.hs:1791 compiler 

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.

References 2

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