/ghc-7.0.4/compiler/stranal/DmdAnal.lhs

%
% (c) The GRASP/AQUA Project, Glasgow University, 1993-1998
%

			-----------------
			A demand analysis
			-----------------

\begin{code}
module DmdAnal ( dmdAnalPgm, dmdAnalTopRhs, 
		 both {- needed by WwLib -}
   ) where

#include "HsVersions.h"

import DynFlags		( DynFlags )
import StaticFlags	( opt_MaxWorkerArgs )
import Demand	-- All of it
import CoreSyn
import PprCore	
import CoreUtils	( exprIsHNF, exprIsTrivial )
import CoreArity	( exprArity )
import DataCon		( dataConTyCon, dataConRepStrictness )
import TyCon		( isProductTyCon, isRecursiveTyCon )
import Id		( Id, idType, idInlineActivation,
			  isDataConWorkId, isGlobalId, idArity,
			  idStrictness, 
			  setIdStrictness, idDemandInfo, idUnfolding,
			  idDemandInfo_maybe, setIdDemandInfo
			)
import Var		( Var )
import VarEnv
import TysWiredIn	( unboxedPairDataCon )
import TysPrim		( realWorldStatePrimTy )
import UniqFM		( addToUFM_Directly, lookupUFM_Directly,
			  minusUFM, filterUFM )
import Type		( isUnLiftedType, coreEqType, splitTyConApp_maybe )
import Coercion         ( coercionKind )
import Util		( mapAndUnzip, lengthIs, zipEqual )
import BasicTypes	( Arity, TopLevelFlag(..), isTopLevel, isNeverActive,
			  RecFlag(..), isRec, isMarkedStrict )
import Maybes		( orElse, expectJust )
import Outputable
import Data.List
import FastString
\end{code}

To think about

* set a noinline pragma on bottoming Ids

* Consider f x = x+1 `fatbar` error (show x)
  We'd like to unbox x, even if that means reboxing it in the error case.


%************************************************************************
%*									*
\subsection{Top level stuff}
%*									*
%************************************************************************

\begin{code}
dmdAnalPgm :: DynFlags -> [CoreBind] -> IO [CoreBind]
dmdAnalPgm _ binds
  = do {
	let { binds_plus_dmds = do_prog binds } ;
	return binds_plus_dmds
    }
  where
    do_prog :: [CoreBind] -> [CoreBind]
    do_prog binds = snd $ mapAccumL dmdAnalTopBind emptySigEnv binds

dmdAnalTopBind :: SigEnv
	       -> CoreBind 
	       -> (SigEnv, CoreBind)
dmdAnalTopBind sigs (NonRec id rhs)
  = (sigs2, NonRec id2 rhs2)
  where
    (    _, _, (_,   rhs1)) = dmdAnalRhs TopLevel NonRecursive (virgin sigs)    (id, rhs)
    (sigs2, _, (id2, rhs2)) = dmdAnalRhs TopLevel NonRecursive (nonVirgin sigs) (id, rhs1)
    	-- Do two passes to improve CPR information
    	-- See comments with ignore_cpr_info in mk_sig_ty
    	-- and with extendSigsWithLam

dmdAnalTopBind sigs (Rec pairs)
  = (sigs', Rec pairs')
  where
    (sigs', _, pairs')  = dmdFix TopLevel (virgin sigs) pairs
		-- We get two iterations automatically
		-- c.f. the NonRec case above
\end{code}

\begin{code}
dmdAnalTopRhs :: CoreExpr -> (StrictSig, CoreExpr)
-- Analyse the RHS and return
--	a) appropriate strictness info
--	b) the unfolding (decorated with strictness info)
dmdAnalTopRhs rhs
  = (sig, rhs2)
  where
    call_dmd	   = vanillaCall (exprArity rhs)
    (_,      rhs1) = dmdAnal (virgin emptySigEnv)    call_dmd rhs
    (rhs_ty, rhs2) = dmdAnal (nonVirgin emptySigEnv) call_dmd rhs1
    sig		   = mkTopSigTy rhs rhs_ty
	-- Do two passes; see notes with extendSigsWithLam
	-- Otherwise we get bogus CPR info for constructors like
	-- 	newtype T a = MkT a
	-- The constructor looks like (\x::T a -> x), modulo the coerce
	-- extendSigsWithLam will optimistically give x a CPR tag the 
	-- first time, which is wrong in the end.
\end{code}

%************************************************************************
%*									*
\subsection{The analyser itself}	
%*									*
%************************************************************************

\begin{code}
dmdAnal :: AnalEnv -> Demand -> CoreExpr -> (DmdType, CoreExpr)

dmdAnal _ Abs  e = (topDmdType, e)

dmdAnal env dmd e
  | not (isStrictDmd dmd)
  = let 
	(res_ty, e') = dmdAnal env evalDmd e
    in
    (deferType res_ty, e')
	-- It's important not to analyse e with a lazy demand because
	-- a) When we encounter   case s of (a,b) -> 
	--	we demand s with U(d1d2)... but if the overall demand is lazy
	--	that is wrong, and we'd need to reduce the demand on s,
	--	which is inconvenient
	-- b) More important, consider
	--	f (let x = R in x+x), where f is lazy
	--    We still want to mark x as demanded, because it will be when we
	--    enter the let.  If we analyse f's arg with a Lazy demand, we'll
	--    just mark x as Lazy
	-- c) The application rule wouldn't be right either
	--    Evaluating (f x) in a L demand does *not* cause
	--    evaluation of f in a C(L) demand!


dmdAnal _ _ (Lit lit) = (topDmdType, Lit lit)
dmdAnal _ _ (Type ty) = (topDmdType, Type ty)	-- Doesn't happen, in fact

dmdAnal env dmd (Var var)
  = (dmdTransform env var dmd, Var var)

dmdAnal env dmd (Cast e co)
  = (dmd_ty, Cast e' co)
  where
    (dmd_ty, e') = dmdAnal env dmd' e
    to_co        = snd (coercionKind co)
    dmd'
      | Just (tc, _) <- splitTyConApp_maybe to_co
      , isRecursiveTyCon tc = evalDmd
      | otherwise           = dmd
	-- This coerce usually arises from a recursive
        -- newtype, and we don't want to look inside them
	-- for exactly the same reason that we don't look
	-- inside recursive products -- we might not reach
	-- a fixpoint.  So revert to a vanilla Eval demand

dmdAnal env dmd (Note n e)
  = (dmd_ty, Note n e')
  where
    (dmd_ty, e') = dmdAnal env dmd e

dmdAnal env dmd (App fun (Type ty))
  = (fun_ty, App fun' (Type ty))
  where
    (fun_ty, fun') = dmdAnal env dmd fun

-- Lots of the other code is there to make this
-- beautiful, compositional, application rule :-)
dmdAnal env dmd (App fun arg)	-- Non-type arguments
  = let				-- [Type arg handled above]
	(fun_ty, fun') 	  = dmdAnal env (Call dmd) fun
	(arg_ty, arg') 	  = dmdAnal env arg_dmd arg
	(arg_dmd, res_ty) = splitDmdTy fun_ty
    in
    (res_ty `bothType` arg_ty, App fun' arg')

dmdAnal env dmd (Lam var body)
  | isTyCoVar var
  = let   
	(body_ty, body') = dmdAnal env dmd body
    in
    (body_ty, Lam var body')

  | Call body_dmd <- dmd	-- A call demand: good!
  = let	
	env'		 = extendSigsWithLam env var
	(body_ty, body') = dmdAnal env' body_dmd body
	(lam_ty, var')   = annotateLamIdBndr env body_ty var
    in
    (lam_ty, Lam var' body')

  | otherwise	-- Not enough demand on the lambda; but do the body
  = let		-- anyway to annotate it and gather free var info
	(body_ty, body') = dmdAnal env evalDmd body
	(lam_ty, var')   = annotateLamIdBndr env body_ty var
    in
    (deferType lam_ty, Lam var' body')

dmdAnal env dmd (Case scrut case_bndr ty [alt@(DataAlt dc, _, _)])
  | let tycon = dataConTyCon dc
  , isProductTyCon tycon
  , not (isRecursiveTyCon tycon)
  = let
	env_alt	      = extendAnalEnv NotTopLevel env case_bndr case_bndr_sig
	(alt_ty, alt')	      = dmdAnalAlt env_alt dmd alt
	(alt_ty1, case_bndr') = annotateBndr alt_ty case_bndr
	(_, bndrs', _)	      = alt'
	case_bndr_sig	      = cprSig
		-- Inside the alternative, the case binder has the CPR property.
		-- Meaning that a case on it will successfully cancel.
		-- Example:
		--	f True  x = case x of y { I# x' -> if x' ==# 3 then y else I# 8 }
		--	f False x = I# 3
		--	
		-- We want f to have the CPR property:
		--	f b x = case fw b x of { r -> I# r }
		--	fw True  x = case x of y { I# x' -> if x' ==# 3 then x' else 8 }
		--	fw False x = 3

	-- Figure out whether the demand on the case binder is used, and use
	-- that to set the scrut_dmd.  This is utterly essential.
	-- Consider	f x = case x of y { (a,b) -> k y a }
	-- If we just take scrut_demand = U(L,A), then we won't pass x to the
	-- worker, so the worker will rebuild 
	--	x = (a, absent-error)
	-- and that'll crash.
	-- So at one stage I had:
	--	dead_case_bndr		 = isAbsentDmd (idDemandInfo case_bndr')
	--	keepity | dead_case_bndr = Drop
	--		| otherwise	 = Keep		
	--
	-- But then consider
	--	case x of y { (a,b) -> h y + a }
	-- where h : U(LL) -> T
	-- The above code would compute a Keep for x, since y is not Abs, which is silly
	-- The insight is, of course, that a demand on y is a demand on the
	-- scrutinee, so we need to `both` it with the scrut demand

	alt_dmd 	   = Eval (Prod [idDemandInfo b | b <- bndrs', isId b])
        scrut_dmd 	   = alt_dmd `both`
			     idDemandInfo case_bndr'

	(scrut_ty, scrut') = dmdAnal env scrut_dmd scrut
    in
    (alt_ty1 `bothType` scrut_ty, Case scrut' case_bndr' ty [alt'])

dmdAnal env dmd (Case scrut case_bndr ty alts)
  = let
	(alt_tys, alts')        = mapAndUnzip (dmdAnalAlt env dmd) alts
	(scrut_ty, scrut')      = dmdAnal env evalDmd scrut
	(alt_ty, case_bndr')	= annotateBndr (foldr1 lubType alt_tys) case_bndr
    in
--    pprTrace "dmdAnal:Case" (ppr alts $$ ppr alt_tys)
    (alt_ty `bothType` scrut_ty, Case scrut' case_bndr' ty alts')

dmdAnal env dmd (Let (NonRec id rhs) body)
  = let
	(sigs', lazy_fv, (id1, rhs')) = dmdAnalRhs NotTopLevel NonRecursive env (id, rhs)
	(body_ty, body') 	      = dmdAnal (updSigEnv env sigs') dmd body
	(body_ty1, id2)    	      = annotateBndr body_ty id1
	body_ty2		      = addLazyFVs body_ty1 lazy_fv
    in
	-- If the actual demand is better than the vanilla call
	-- demand, you might think that we might do better to re-analyse 
	-- the RHS with the stronger demand.
	-- But (a) That seldom happens, because it means that *every* path in 
	-- 	   the body of the let has to use that stronger demand
	-- (b) It often happens temporarily in when fixpointing, because
	--     the recursive function at first seems to place a massive demand.
	--     But we don't want to go to extra work when the function will
	--     probably iterate to something less demanding.  
	-- In practice, all the times the actual demand on id2 is more than
	-- the vanilla call demand seem to be due to (b).  So we don't
	-- bother to re-analyse the RHS.
    (body_ty2, Let (NonRec id2 rhs') body')    

dmdAnal env dmd (Let (Rec pairs) body)
  = let
	bndrs			 = map fst pairs
	(sigs', lazy_fv, pairs') = dmdFix NotTopLevel env pairs
	(body_ty, body')         = dmdAnal (updSigEnv env sigs') dmd body
	body_ty1		 = addLazyFVs body_ty lazy_fv
    in
    sigs' `seq` body_ty `seq`
    let
	(body_ty2, _) = annotateBndrs body_ty1 bndrs
		-- Don't bother to add demand info to recursive
		-- binders as annotateBndr does; 
		-- being recursive, we can't treat them strictly.
		-- But we do need to remove the binders from the result demand env
    in
    (body_ty2,  Let (Rec pairs') body')


dmdAnalAlt :: AnalEnv -> Demand -> Alt Var -> (DmdType, Alt Var)
dmdAnalAlt env dmd (con,bndrs,rhs)
  = let 
	(rhs_ty, rhs')   = dmdAnal env dmd rhs
        rhs_ty'          = addDataConPatDmds con bndrs rhs_ty
	(alt_ty, bndrs') = annotateBndrs rhs_ty' bndrs
	final_alt_ty | io_hack_reqd = alt_ty `lubType` topDmdType
		     | otherwise    = alt_ty

	-- There's a hack here for I/O operations.  Consider
	-- 	case foo x s of { (# s, r #) -> y }
	-- Is this strict in 'y'.  Normally yes, but what if 'foo' is an I/O
	-- operation that simply terminates the program (not in an erroneous way)?
	-- In that case we should not evaluate y before the call to 'foo'.
	-- Hackish solution: spot the IO-like situation and add a virtual branch,
	-- as if we had
	-- 	case foo x s of 
	--	   (# s, r #) -> y 
	--	   other      -> return ()
	-- So the 'y' isn't necessarily going to be evaluated
	--
	-- A more complete example where this shows up is:
	--	do { let len = <expensive> ;
	--	   ; when (...) (exitWith ExitSuccess)
	--	   ; print len }

	io_hack_reqd = con == DataAlt unboxedPairDataCon &&
		       idType (head bndrs) `coreEqType` realWorldStatePrimTy
    in	
    (final_alt_ty, (con, bndrs', rhs'))

addDataConPatDmds :: AltCon -> [Var] -> DmdType -> DmdType
-- See Note [Add demands for strict constructors]
addDataConPatDmds DEFAULT    _ dmd_ty = dmd_ty
addDataConPatDmds (LitAlt _) _ dmd_ty = dmd_ty
addDataConPatDmds (DataAlt con) bndrs dmd_ty
  = foldr add dmd_ty str_bndrs 
  where
    add bndr dmd_ty = addVarDmd dmd_ty bndr seqDmd
    str_bndrs = [ b | (b,s) <- zipEqual "addDataConPatBndrs"
                                   (filter isId bndrs)
                                   (dataConRepStrictness con)
                    , isMarkedStrict s ]
\end{code}

Note [Add demands for strict constructors]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Consider this program (due to Roman):

    data X a = X !a

    foo :: X Int -> Int -> Int
    foo (X a) n = go 0
     where
       go i | i < n     = a + go (i+1)
            | otherwise = 0

We want the worker for 'foo' too look like this:

    $wfoo :: Int# -> Int# -> Int#

with the first argument unboxed, so that it is not eval'd each time
around the loop (which would otherwise happen, since 'foo' is not
strict in 'a'.  It is sound for the wrapper to pass an unboxed arg
because X is strict, so its argument must be evaluated.  And if we
*don't* pass an unboxed argument, we can't even repair it by adding a
`seq` thus:

    foo (X a) n = a `seq` go 0

because the seq is discarded (very early) since X is strict!

There is the usual danger of reboxing, which as usual we ignore. But 
if X is monomorphic, and has an UNPACK pragma, then this optimisation
is even more important.  We don't want the wrapper to rebox an unboxed
argument, and pass an Int to $wfoo!


%************************************************************************
%*									*
                    Demand transformer
%*									*
%************************************************************************

\begin{code}
dmdTransform :: AnalEnv		-- The strictness environment
	     -> Id		-- The function
	     -> Demand		-- The demand on the function
	     -> DmdType		-- The demand type of the function in this context
	-- Returned DmdEnv includes the demand on 
	-- this function plus demand on its free variables

dmdTransform env var dmd

------ 	DATA CONSTRUCTOR
  | isDataConWorkId var		-- Data constructor
  = let 
	StrictSig dmd_ty    = idStrictness var	-- It must have a strictness sig
	DmdType _ _ con_res = dmd_ty
	arity		    = idArity var
    in
    if arity == call_depth then		-- Saturated, so unleash the demand
	let 
		-- Important!  If we Keep the constructor application, then
		-- we need the demands the constructor places (always lazy)
		-- If not, we don't need to.  For example:
		--	f p@(x,y) = (p,y)	-- S(AL)
		--	g a b     = f (a,b)
		-- It's vital that we don't calculate Absent for a!
	   dmd_ds = case res_dmd of
			Box (Eval ds) -> mapDmds box ds
			Eval ds	      -> ds
			_	      -> Poly Top

		-- ds can be empty, when we are just seq'ing the thing
		-- If so we must make up a suitable bunch of demands
	   arg_ds = case dmd_ds of
		      Poly d  -> replicate arity d
		      Prod ds -> ASSERT( ds `lengthIs` arity ) ds

	in
	mkDmdType emptyDmdEnv arg_ds con_res
		-- Must remember whether it's a product, hence con_res, not TopRes
    else
	topDmdType

------ 	IMPORTED FUNCTION
  | isGlobalId var,		-- Imported function
    let StrictSig dmd_ty = idStrictness var
  = -- pprTrace "strict-sig" (ppr var $$ ppr dmd_ty) $
    if dmdTypeDepth dmd_ty <= call_depth then	-- Saturated, so unleash the demand
	dmd_ty
    else
	topDmdType

------ 	LOCAL LET/REC BOUND THING
  | Just (StrictSig dmd_ty, top_lvl) <- lookupSigEnv env var
  = let
	fn_ty | dmdTypeDepth dmd_ty <= call_depth = dmd_ty 
	      | otherwise   		          = deferType dmd_ty
	-- NB: it's important to use deferType, and not just return topDmdType
	-- Consider	let { f x y = p + x } in f 1
	-- The application isn't saturated, but we must nevertheless propagate 
	--	a lazy demand for p!  
    in
    if isTopLevel top_lvl then fn_ty	-- Don't record top level things
    else addVarDmd fn_ty var dmd

------ 	LOCAL NON-LET/REC BOUND THING
  | otherwise	 		-- Default case
  = unitVarDmd var dmd

  where
    (call_depth, res_dmd) = splitCallDmd dmd
\end{code}

%************************************************************************
%*									*
\subsection{Bindings}
%*									*
%************************************************************************

\begin{code}
dmdFix :: TopLevelFlag
       -> AnalEnv 		-- Does not include bindings for this binding
       -> [(Id,CoreExpr)]
       -> (SigEnv, DmdEnv,
	   [(Id,CoreExpr)])	-- Binders annotated with stricness info

dmdFix top_lvl env orig_pairs
  = loop 1 initial_env orig_pairs
  where
    bndrs        = map fst orig_pairs
    initial_env = addInitialSigs top_lvl env bndrs
    
    loop :: Int
	 -> AnalEnv			-- Already contains the current sigs
	 -> [(Id,CoreExpr)] 		
	 -> (SigEnv, DmdEnv, [(Id,CoreExpr)])
    loop n env pairs
      = -- pprTrace "dmd loop" (ppr n <+> ppr bndrs $$ ppr env) $
        loop' n env pairs

    loop' n env pairs
      | found_fixpoint
      = (sigs', lazy_fv, pairs')
		-- Note: return pairs', not pairs.   pairs' is the result of 
		-- processing the RHSs with sigs (= sigs'), whereas pairs 
		-- is the result of processing the RHSs with the *previous* 
		-- iteration of sigs.

      | n >= 10  
      = pprTrace "dmdFix loop" (ppr n <+> (vcat 
			[ text "Sigs:" <+> ppr [ (id,lookupVarEnv sigs id, lookupVarEnv sigs' id) 
                                               | (id,_) <- pairs],
			  text "env:" <+> ppr env,
			  text "binds:" <+> pprCoreBinding (Rec pairs)]))
	(sigEnv env, lazy_fv, orig_pairs)	-- Safe output
		-- The lazy_fv part is really important!  orig_pairs has no strictness
		-- info, including nothing about free vars.  But if we have
		--	letrec f = ....y..... in ...f...
		-- where 'y' is free in f, we must record that y is mentioned, 
		-- otherwise y will get recorded as absent altogether

      | otherwise
      = loop (n+1) (nonVirgin sigs') pairs'
      where
        sigs = sigEnv env
	found_fixpoint = all (same_sig sigs sigs') bndrs 

	((sigs',lazy_fv), pairs') = mapAccumL my_downRhs (sigs, emptyDmdEnv) pairs
		-- mapAccumL: Use the new signature to do the next pair
		-- The occurrence analyser has arranged them in a good order
		-- so this can significantly reduce the number of iterations needed
	
        my_downRhs (sigs,lazy_fv) (id,rhs)
          = ((sigs', lazy_fv'), pair')
          where
	    (sigs', lazy_fv1, pair') = dmdAnalRhs top_lvl Recursive (updSigEnv env sigs) (id,rhs)
	    lazy_fv'		     = plusVarEnv_C both lazy_fv lazy_fv1
	   
    same_sig sigs sigs' var = lookup sigs var == lookup sigs' var
    lookup sigs var = case lookupVarEnv sigs var of
			Just (sig,_) -> sig
                        Nothing      -> pprPanic "dmdFix" (ppr var)

dmdAnalRhs :: TopLevelFlag -> RecFlag
	-> AnalEnv -> (Id, CoreExpr)
	-> (SigEnv,  DmdEnv, (Id, CoreExpr))
-- Process the RHS of the binding, add the strictness signature
-- to the Id, and augment the environment with the signature as well.

dmdAnalRhs top_lvl rec_flag env (id, rhs)
 = (sigs', lazy_fv, (id', rhs'))
 where
  arity		     = idArity id   -- The idArity should be up to date
				    -- The simplifier was run just beforehand
  (rhs_dmd_ty, rhs') = dmdAnal env (vanillaCall arity) rhs
  (lazy_fv, sig_ty)  = WARN( arity /= dmdTypeDepth rhs_dmd_ty && not (exprIsTrivial rhs), ppr id )
				-- The RHS can be eta-reduced to just a variable, 
				-- in which case we should not complain. 
		       mkSigTy top_lvl rec_flag id rhs rhs_dmd_ty
  id'		     = id `setIdStrictness` sig_ty
  sigs'		     = extendSigEnv top_lvl (sigEnv env) id sig_ty
\end{code}


%************************************************************************
%*									*
\subsection{Strictness signatures and types}
%*									*
%************************************************************************

\begin{code}
mkTopSigTy :: CoreExpr -> DmdType -> StrictSig
	-- Take a DmdType and turn it into a StrictSig
	-- NB: not used for never-inline things; hence False
mkTopSigTy rhs dmd_ty = snd (mk_sig_ty False False rhs dmd_ty)

mkSigTy :: TopLevelFlag -> RecFlag -> Id -> CoreExpr -> DmdType -> (DmdEnv, StrictSig)
mkSigTy top_lvl rec_flag id rhs dmd_ty 
  = mk_sig_ty never_inline thunk_cpr_ok rhs dmd_ty
  where
    never_inline = isNeverActive (idInlineActivation id)
    maybe_id_dmd = idDemandInfo_maybe id
	-- Is Nothing the first time round

    thunk_cpr_ok
	| isTopLevel top_lvl       = False	-- Top level things don't get
						-- their demandInfo set at all
	| isRec rec_flag	   = False	-- Ditto recursive things
	| Just dmd <- maybe_id_dmd = isStrictDmd dmd
	| otherwise 		   = True	-- Optimistic, first time round
						-- See notes below
\end{code}

The thunk_cpr_ok stuff [CPR-AND-STRICTNESS]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
If the rhs is a thunk, we usually forget the CPR info, because
it is presumably shared (else it would have been inlined, and 
so we'd lose sharing if w/w'd it into a function).  E.g.

	let r = case expensive of
		  (a,b) -> (b,a)
	in ...

If we marked r as having the CPR property, then we'd w/w into

	let $wr = \() -> case expensive of
			    (a,b) -> (# b, a #)
	    r = case $wr () of
		  (# b,a #) -> (b,a)
	in ...

But now r is a thunk, which won't be inlined, so we are no further ahead.
But consider

	f x = let r = case expensive of (a,b) -> (b,a)
	      in if foo r then r else (x,x)

Does f have the CPR property?  Well, no.

However, if the strictness analyser has figured out (in a previous 
iteration) that it's strict, then we DON'T need to forget the CPR info.
Instead we can retain the CPR info and do the thunk-splitting transform 
(see WorkWrap.splitThunk).

This made a big difference to PrelBase.modInt, which had something like
	modInt = \ x -> let r = ... -> I# v in
			...body strict in r...
r's RHS isn't a value yet; but modInt returns r in various branches, so
if r doesn't have the CPR property then neither does modInt
Another case I found in practice (in Complex.magnitude), looks like this:
		let k = if ... then I# a else I# b
		in ... body strict in k ....
(For this example, it doesn't matter whether k is returned as part of
the overall result; but it does matter that k's RHS has the CPR property.)  
Left to itself, the simplifier will make a join point thus:
		let $j k = ...body strict in k...
		if ... then $j (I# a) else $j (I# b)
With thunk-splitting, we get instead
		let $j x = let k = I#x in ...body strict in k...
		in if ... then $j a else $j b
This is much better; there's a good chance the I# won't get allocated.

The difficulty with this is that we need the strictness type to
look at the body... but we now need the body to calculate the demand
on the variable, so we can decide whether its strictness type should
have a CPR in it or not.  Simple solution: 
	a) use strictness info from the previous iteration
	b) make sure we do at least 2 iterations, by doing a second
	   round for top-level non-recs.  Top level recs will get at
	   least 2 iterations except for totally-bottom functions
	   which aren't very interesting anyway.

NB: strictly_demanded is never true of a top-level Id, or of a recursive Id.

The Nothing case in thunk_cpr_ok [CPR-AND-STRICTNESS]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Demand info now has a 'Nothing' state, just like strictness info.
The analysis works from 'dangerous' towards a 'safe' state; so we 
start with botSig for 'Nothing' strictness infos, and we start with
"yes, it's demanded" for 'Nothing' in the demand info.  The
fixpoint iteration will sort it all out.

We can't start with 'not-demanded' because then consider
	f x = let 
		  t = ... I# x
	      in
	      if ... then t else I# y else f x'

In the first iteration we'd have no demand info for x, so assume
not-demanded; then we'd get TopRes for f's CPR info.  Next iteration
we'd see that t was demanded, and so give it the CPR property, but by
now f has TopRes, so it will stay TopRes.  Instead, with the Nothing
setting the first time round, we say 'yes t is demanded' the first
time.

However, this does mean that for non-recursive bindings we must
iterate twice to be sure of not getting over-optimistic CPR info,
in the case where t turns out to be not-demanded.  This is handled
by dmdAnalTopBind.


Note [NOINLINE and strictness]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The strictness analyser used to have a HACK which ensured that NOINLNE
things were not strictness-analysed.  The reason was unsafePerformIO. 
Left to itself, the strictness analyser would discover this strictness 
for unsafePerformIO:
	unsafePerformIO:  C(U(AV))
But then consider this sub-expression
	unsafePerformIO (\s -> let r = f x in 
			       case writeIORef v r s of (# s1, _ #) ->
			       (# s1, r #)
The strictness analyser will now find that r is sure to be eval'd,
and may then hoist it out.  This makes tests/lib/should_run/memo002
deadlock.

Solving this by making all NOINLINE things have no strictness info is overkill.
In particular, it's overkill for runST, which is perfectly respectable.
Consider
	f x = runST (return x)
This should be strict in x.

So the new plan is to define unsafePerformIO using the 'lazy' combinator:

	unsafePerformIO (IO m) = lazy (case m realWorld# of (# _, r #) -> r)

Remember, 'lazy' is a wired-in identity-function Id, of type a->a, which is 
magically NON-STRICT, and is inlined after strictness analysis.  So
unsafePerformIO will look non-strict, and that's what we want.

Now we don't need the hack in the strictness analyser.  HOWEVER, this
decision does mean that even a NOINLINE function is not entirely
opaque: some aspect of its implementation leaks out, notably its
strictness.  For example, if you have a function implemented by an
error stub, but which has RULES, you may want it not to be eliminated
in favour of error!


\begin{code}
mk_sig_ty :: Bool -> Bool -> CoreExpr
          -> DmdType -> (DmdEnv, StrictSig)
mk_sig_ty _never_inline thunk_cpr_ok rhs (DmdType fv dmds res) 
  = (lazy_fv, mkStrictSig dmd_ty)
	-- Re unused never_inline, see Note [NOINLINE and strictness]
  where
    dmd_ty = DmdType strict_fv final_dmds res'

    lazy_fv   = filterUFM (not . isStrictDmd) fv
    strict_fv = filterUFM isStrictDmd         fv
	-- We put the strict FVs in the DmdType of the Id, so 
	-- that at its call sites we unleash demands on its strict fvs.
	-- An example is 'roll' in imaginary/wheel-sieve2
	-- Something like this:
	--	roll x = letrec 
	--		     go y = if ... then roll (x-1) else x+1
	--		 in 
	--		 go ms
	-- We want to see that roll is strict in x, which is because
	-- go is called.   So we put the DmdEnv for x in go's DmdType.
	--
	-- Another example:
	--	f :: Int -> Int -> Int
	--	f x y = let t = x+1
	--	    h z = if z==0 then t else 
	--		  if z==1 then x+1 else
	--		  x + h (z-1)
	--	in
	--	h y
	-- Calling h does indeed evaluate x, but we can only see
	-- that if we unleash a demand on x at the call site for t.
	--
	-- Incidentally, here's a place where lambda-lifting h would
	-- lose the cigar --- we couldn't see the joint strictness in t/x
	--
	--	ON THE OTHER HAND
	-- We don't want to put *all* the fv's from the RHS into the
	-- DmdType, because that makes fixpointing very slow --- the 
	-- DmdType gets full of lazy demands that are slow to converge.

    final_dmds = setUnpackStrategy dmds
	-- Set the unpacking strategy
	
    res' = case res of
		RetCPR | ignore_cpr_info -> TopRes
		_	 		 -> res
    ignore_cpr_info = not (exprIsHNF rhs || thunk_cpr_ok)
\end{code}

The unpack strategy determines whether we'll *really* unpack the argument,
or whether we'll just remember its strictness.  If unpacking would give
rise to a *lot* of worker args, we may decide not to unpack after all.

\begin{code}
setUnpackStrategy :: [Demand] -> [Demand]
setUnpackStrategy ds
  = snd (go (opt_MaxWorkerArgs - nonAbsentArgs ds) ds)
  where
    go :: Int 			-- Max number of args available for sub-components of [Demand]
       -> [Demand]
       -> (Int, [Demand])	-- Args remaining after subcomponents of [Demand] are unpacked

    go n (Eval (Prod cs) : ds) 
	| n' >= 0   = Eval (Prod cs') `cons` go n'' ds
        | otherwise = Box (Eval (Prod cs)) `cons` go n ds
	where
	  (n'',cs') = go n' cs
	  n' = n + 1 - non_abs_args
		-- Add one to the budget 'cos we drop the top-level arg
	  non_abs_args = nonAbsentArgs cs
		-- Delete # of non-absent args to which we'll now be committed
				
    go n (d:ds) = d `cons` go n ds
    go n []     = (n,[])

    cons d (n,ds) = (n, d:ds)

nonAbsentArgs :: [Demand] -> Int
nonAbsentArgs []	 = 0
nonAbsentArgs (Abs : ds) = nonAbsentArgs ds
nonAbsentArgs (_   : ds) = 1 + nonAbsentArgs ds
\end{code}


%************************************************************************
%*									*
\subsection{Strictness signatures and types}
%*									*
%************************************************************************

\begin{code}
unitVarDmd :: Var -> Demand -> DmdType
unitVarDmd var dmd = DmdType (unitVarEnv var dmd) [] TopRes

addVarDmd :: DmdType -> Var -> Demand -> DmdType
addVarDmd (DmdType fv ds res) var dmd
  = DmdType (extendVarEnv_C both fv var dmd) ds res

addLazyFVs :: DmdType -> DmdEnv -> DmdType
addLazyFVs (DmdType fv ds res) lazy_fvs
  = DmdType both_fv1 ds res
  where
    both_fv = plusVarEnv_C both fv lazy_fvs
    both_fv1 = modifyEnv (isBotRes res) (`both` Bot) lazy_fvs fv both_fv
	-- This modifyEnv is vital.  Consider
	--	let f = \x -> (x,y)
	--	in  error (f 3)
	-- Here, y is treated as a lazy-fv of f, but we must `both` that L
	-- demand with the bottom coming up from 'error'
	-- 
	-- I got a loop in the fixpointer without this, due to an interaction
	-- with the lazy_fv filtering in mkSigTy.  Roughly, it was
	--	letrec f n x 
	--	    = letrec g y = x `fatbar` 
	--			   letrec h z = z + ...g...
	--			   in h (f (n-1) x)
	-- 	in ...
	-- In the initial iteration for f, f=Bot
	-- Suppose h is found to be strict in z, but the occurrence of g in its RHS
	-- is lazy.  Now consider the fixpoint iteration for g, esp the demands it
	-- places on its free variables.  Suppose it places none.  Then the
	-- 	x `fatbar` ...call to h...
	-- will give a x->V demand for x.  That turns into a L demand for x,
	-- which floats out of the defn for h.  Without the modifyEnv, that
	-- L demand doesn't get both'd with the Bot coming up from the inner
	-- call to f.  So we just get an L demand for x for g.
	--
	-- A better way to say this is that the lazy-fv filtering should give the
	-- same answer as putting the lazy fv demands in the function's type.

annotateBndr :: DmdType -> Var -> (DmdType, Var)
-- The returned env has the var deleted
-- The returned var is annotated with demand info
-- No effect on the argument demands
annotateBndr dmd_ty@(DmdType fv ds res) var
  | isTyCoVar var = (dmd_ty, var)
  | otherwise   = (DmdType fv' ds res, setIdDemandInfo var dmd)
  where
    (fv', dmd) = removeFV fv var res

annotateBndrs :: DmdType -> [Var] -> (DmdType, [Var])
annotateBndrs = mapAccumR annotateBndr

annotateLamIdBndr :: AnalEnv
                  -> DmdType 	-- Demand type of body
		  -> Id 	-- Lambda binder
		  -> (DmdType, 	-- Demand type of lambda
		      Id)	-- and binder annotated with demand	

annotateLamIdBndr env (DmdType fv ds res) id
-- For lambdas we add the demand to the argument demands
-- Only called for Ids
  = ASSERT( isId id )
    (final_ty, setIdDemandInfo id hacked_dmd)
  where
      -- Watch out!  See note [Lambda-bound unfoldings]
    final_ty = case maybeUnfoldingTemplate (idUnfolding id) of
                 Nothing  -> main_ty
                 Just unf -> main_ty `bothType` unf_ty
                          where
                             (unf_ty, _) = dmdAnal env dmd unf
    
    main_ty = DmdType fv' (hacked_dmd:ds) res

    (fv', dmd) = removeFV fv id res
    hacked_dmd = argDemand dmd
	-- This call to argDemand is vital, because otherwise we label
	-- a lambda binder with demand 'B'.  But in terms of calling
	-- conventions that's Abs, because we don't pass it.  But
	-- when we do a w/w split we get
	--	fw x = (\x y:B -> ...) x (error "oops")
	-- And then the simplifier things the 'B' is a strict demand
	-- and evaluates the (error "oops").  Sigh

removeFV :: DmdEnv -> Var -> DmdResult -> (DmdEnv, Demand)
removeFV fv id res = (fv', zapUnlifted id dmd)
		where
		  fv' = fv `delVarEnv` id
		  dmd = lookupVarEnv fv id `orElse` deflt
	 	  deflt | isBotRes res = Bot
		        | otherwise    = Abs

zapUnlifted :: Id -> Demand -> Demand
-- For unlifted-type variables, we are only 
-- interested in Bot/Abs/Box Abs
zapUnlifted _  Bot = Bot
zapUnlifted _  Abs = Abs
zapUnlifted id dmd | isUnLiftedType (idType id) = lazyDmd
		   | otherwise			= dmd
\end{code}

Note [Lamba-bound unfoldings]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
We allow a lambda-bound variable to carry an unfolding, a facility that is used
exclusively for join points; see Note [Case binders and join points].  If so,
we must be careful to demand-analyse the RHS of the unfolding!  Example
   \x. \y{=Just x}. <body>
Then if <body> uses 'y', then transitively it uses 'x', and we must not
forget that fact, otherwise we might make 'x' absent when it isn't.


%************************************************************************
%*									*
\subsection{Strictness signatures}
%*									*
%************************************************************************

\begin{code}
data AnalEnv
  = AE { ae_sigs   :: SigEnv
       , ae_virgin :: Bool }  -- True on first iteration only
		              -- See Note [Initialising strictness]
	-- We use the se_env to tell us whether to
	-- record info about a variable in the DmdEnv
	-- We do so if it's a LocalId, but not top-level
	--
	-- The DmdEnv gives the demand on the free vars of the function
	-- when it is given enough args to satisfy the strictness signature

type SigEnv = VarEnv (StrictSig, TopLevelFlag)

instance Outputable AnalEnv where
  ppr (AE { ae_sigs = env, ae_virgin = virgin })
    = ptext (sLit "AE") <+> braces (vcat
         [ ptext (sLit "ae_virgin =") <+> ppr virgin
         , ptext (sLit "ae_sigs =") <+> ppr env ])

emptySigEnv :: SigEnv
emptySigEnv = emptyVarEnv

sigEnv :: AnalEnv -> SigEnv
sigEnv = ae_sigs

updSigEnv :: AnalEnv -> SigEnv -> AnalEnv
updSigEnv env sigs = env { ae_sigs = sigs }

extendAnalEnv :: TopLevelFlag -> AnalEnv -> Id -> StrictSig -> AnalEnv
extendAnalEnv top_lvl env var sig
  = env { ae_sigs = extendSigEnv top_lvl (ae_sigs env) var sig }

extendSigEnv :: TopLevelFlag -> SigEnv -> Id -> StrictSig -> SigEnv
extendSigEnv top_lvl sigs var sig = extendVarEnv sigs var (sig, top_lvl)

lookupSigEnv :: AnalEnv -> Id -> Maybe (StrictSig, TopLevelFlag)
lookupSigEnv env id = lookupVarEnv (ae_sigs env) id

addInitialSigs :: TopLevelFlag -> AnalEnv -> [Id] -> AnalEnv
-- See Note [Initialising strictness]
addInitialSigs top_lvl env@(AE { ae_sigs = sigs, ae_virgin = virgin }) ids
  = env { ae_sigs = extendVarEnvList sigs [ (id, (init_sig id, top_lvl))
                                          | id <- ids ] }
  where
    init_sig | virgin    = \_ -> botSig
             | otherwise = idStrictness

virgin, nonVirgin :: SigEnv -> AnalEnv
virgin    sigs = AE { ae_sigs = sigs, ae_virgin = True }
nonVirgin sigs = AE { ae_sigs = sigs, ae_virgin = False }

extendSigsWithLam :: AnalEnv -> Id -> AnalEnv
-- Extend the AnalEnv when we meet a lambda binder
-- If the binder is marked demanded with a product demand, then give it a CPR 
-- signature, because in the likely event that this is a lambda on a fn defn 
-- [we only use this when the lambda is being consumed with a call demand],
-- it'll be w/w'd and so it will be CPR-ish.  E.g.
--	f = \x::(Int,Int).  if ...strict in x... then
--				x
--			    else
--				(a,b)
-- We want f to have the CPR property because x does, by the time f has been w/w'd
--
-- Also note that we only want to do this for something that
-- definitely has product type, else we may get over-optimistic 
-- CPR results (e.g. from \x -> x!).

extendSigsWithLam env id
  = case idDemandInfo_maybe id of
	Nothing	             -> extendAnalEnv NotTopLevel env id cprSig
		-- Optimistic in the Nothing case;
		-- See notes [CPR-AND-STRICTNESS]
	Just (Eval (Prod _)) -> extendAnalEnv NotTopLevel env id cprSig
	_                    -> env
\end{code}

Note [Initialising strictness]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Our basic plan is to initialise the strictness of each Id in 
a recursive group to "bottom", and find a fixpoint from there.
However, this group A might be inside an *enclosing* recursive
group B, in which case we'll do the entire fixpoint shebang on A
for each iteration of B.

To speed things up, we initialise each iteration of B from the result
of the last one, which is neatly recorded in each binder.  That way we
make use of earlier iterations of the fixpoint algorithm.  (Cunning
plan.)  

But on the *first* iteration we want to *ignore* the current strictness
of the Id, and start from "bottom".  Nowadays the Id can have a current
strictness, because interface files record strictness for nested bindings.
To know when we are in the first iteration, we look at the ae_virgin
field of the AnalEnv.


%************************************************************************
%*									*
                   Demands
%*									*
%************************************************************************

\begin{code}
splitDmdTy :: DmdType -> (Demand, DmdType)
-- Split off one function argument
-- We already have a suitable demand on all
-- free vars, so no need to add more!
splitDmdTy (DmdType fv (dmd:dmds) res_ty) = (dmd, DmdType fv dmds res_ty)
splitDmdTy ty@(DmdType _ [] res_ty)       = (resTypeArgDmd res_ty, ty)

splitCallDmd :: Demand -> (Int, Demand)
splitCallDmd (Call d) = case splitCallDmd d of
			  (n, r) -> (n+1, r)
splitCallDmd d	      = (0, d)

vanillaCall :: Arity -> Demand
vanillaCall 0 = evalDmd
vanillaCall n = Call (vanillaCall (n-1))

deferType :: DmdType -> DmdType
deferType (DmdType fv _ _) = DmdType (deferEnv fv) [] TopRes
	-- Notice that we throw away info about both arguments and results
	-- For example,   f = let ... in \x -> x
	-- We don't want to get a stricness type V->T for f.

deferEnv :: DmdEnv -> DmdEnv
deferEnv fv = mapVarEnv defer fv


----------------
argDemand :: Demand -> Demand
-- The 'Defer' demands are just Lazy at function boundaries
-- Ugly!  Ask John how to improve it.
argDemand Top 	    = lazyDmd
argDemand (Defer _) = lazyDmd
argDemand (Eval ds) = Eval (mapDmds argDemand ds)
argDemand (Box Bot) = evalDmd
argDemand (Box d)   = box (argDemand d)
argDemand Bot	    = Abs	-- Don't pass args that are consumed (only) by bottom
argDemand d	    = d
\end{code}

\begin{code}
-------------------------
lubType :: DmdType -> DmdType -> DmdType
-- Consider (if x then y else []) with demand V
-- Then the first branch gives {y->V} and the second
--  *implicitly* has {y->A}.  So we must put {y->(V `lub` A)}
-- in the result env.
lubType (DmdType fv1 ds1 r1) (DmdType fv2 ds2 r2)
  = DmdType lub_fv2 (lub_ds ds1 ds2) (r1 `lubRes` r2)
  where
    lub_fv  = plusVarEnv_C lub fv1 fv2
    lub_fv1 = modifyEnv (not (isBotRes r1)) absLub fv2 fv1 lub_fv
    lub_fv2 = modifyEnv (not (isBotRes r2)) absLub fv1 fv2 lub_fv1
	-- lub is the identity for Bot

	-- Extend the shorter argument list to match the longer
    lub_ds (d1:ds1) (d2:ds2) = lub d1 d2 : lub_ds ds1 ds2
    lub_ds []	    []	     = []
    lub_ds ds1	    []	     = map (`lub` resTypeArgDmd r2) ds1
    lub_ds []	    ds2	     = map (resTypeArgDmd r1 `lub`) ds2

-----------------------------------
bothType :: DmdType -> DmdType -> DmdType
-- (t1 `bothType` t2) takes the argument/result info from t1,
-- using t2 just for its free-var info
-- NB: Don't forget about r2!  It might be BotRes, which is
--     a bottom demand on all the in-scope variables.
-- Peter: can this be done more neatly?
bothType (DmdType fv1 ds1 r1) (DmdType fv2 _ r2)
  = DmdType both_fv2 ds1 (r1 `bothRes` r2)
  where
    both_fv  = plusVarEnv_C both fv1 fv2
    both_fv1 = modifyEnv (isBotRes r1) (`both` Bot) fv2 fv1 both_fv
    both_fv2 = modifyEnv (isBotRes r2) (`both` Bot) fv1 fv2 both_fv1
	-- both is the identity for Abs
\end{code}


\begin{code}
lubRes :: DmdResult -> DmdResult -> DmdResult
lubRes BotRes r      = r
lubRes r      BotRes = r
lubRes RetCPR RetCPR = RetCPR
lubRes _      _      = TopRes

bothRes :: DmdResult -> DmdResult -> DmdResult
-- If either diverges, the whole thing does
-- Otherwise take CPR info from the first
bothRes _  BotRes = BotRes
bothRes r1 _      = r1
\end{code}

\begin{code}
modifyEnv :: Bool			-- No-op if False
	  -> (Demand -> Demand)		-- The zapper
	  -> DmdEnv -> DmdEnv		-- Env1 and Env2
	  -> DmdEnv -> DmdEnv		-- Transform this env
	-- Zap anything in Env1 but not in Env2
	-- Assume: dom(env) includes dom(Env1) and dom(Env2)

modifyEnv need_to_modify zapper env1 env2 env
  | need_to_modify = foldr zap env (varEnvKeys (env1 `minusUFM` env2))
  | otherwise	   = env
  where
    zap uniq env = addToUFM_Directly env uniq (zapper current_val)
		 where
		   current_val = expectJust "modifyEnv" (lookupUFM_Directly env uniq)
\end{code}


%************************************************************************
%*									*
\subsection{LUB and BOTH}
%*									*
%************************************************************************

\begin{code}
lub :: Demand -> Demand -> Demand

lub Bot  	d2 = d2
lub Abs  	d2 = absLub d2
lub Top 	_  = Top
lub (Defer ds1) d2 = defer (Eval ds1 `lub` d2)

lub (Call d1)   (Call d2)    = Call (d1 `lub` d2)
lub d1@(Call _) (Box d2)     = d1 `lub` d2	-- Just strip the box
lub    (Call _) d2@(Eval _)  = d2		-- Presumably seq or vanilla eval
lub d1@(Call _) d2	     = d2 `lub` d1	-- Bot, Abs, Top

-- For the Eval case, we use these approximation rules
-- Box Bot	 <= Eval (Box Bot ...)
-- Box Top	 <= Defer (Box Bot ...)
-- Box (Eval ds) <= Eval (map Box ds)
lub (Eval ds1)  (Eval ds2)  	  = Eval (ds1 `lubs` ds2)
lub (Eval ds1)  (Box Bot)   	  = Eval (mapDmds (`lub` Box Bot) ds1)
lub (Eval ds1)  (Box (Eval ds2)) = Eval (ds1 `lubs` mapDmds box ds2)
lub (Eval ds1)  (Box Abs)        = deferEval (mapDmds (`lub` Box Bot) ds1)
lub d1@(Eval _) d2	          = d2 `lub` d1	-- Bot,Abs,Top,Call,Defer

lub (Box d1)   (Box d2) = box (d1 `lub` d2)
lub d1@(Box _)  d2	= d2 `lub` d1

lubs :: Demands -> Demands -> Demands
lubs ds1 ds2 = zipWithDmds lub ds1 ds2

---------------------
box :: Demand -> Demand
-- box is the smart constructor for Box
-- It computes <B,bot> & d
-- INVARIANT: (Box d) => d = Bot, Abs, Eval
-- Seems to be no point in allowing (Box (Call d))
box (Call d)  = Call d	-- The odd man out.  Why?
box (Box d)   = Box d
box (Defer _) = lazyDmd
box Top       = lazyDmd	-- Box Abs and Box Top
box Abs       = lazyDmd	-- are the same <B,L>
box d 	      = Box d	-- Bot, Eval

---------------
defer :: Demand -> Demand

-- defer is the smart constructor for Defer
-- The idea is that (Defer ds) = <U(ds), L>
--
-- It specifies what happens at a lazy function argument
-- or a lambda; the L* operator
-- Set the strictness part to L, but leave
-- the boxity side unaffected
-- It also ensures that Defer (Eval [LLLL]) = L

defer Bot	 = Abs
defer Abs	 = Abs
defer Top	 = Top
defer (Call _)	 = lazyDmd	-- Approximation here?
defer (Box _)	 = lazyDmd
defer (Defer ds) = Defer ds
defer (Eval ds)  = deferEval ds

deferEval :: Demands -> Demand
-- deferEval ds = defer (Eval ds)
deferEval ds | allTop ds = Top
	     | otherwise  = Defer ds

---------------------
absLub :: Demand -> Demand
-- Computes (Abs `lub` d)
-- For the Bot case consider
--	f x y = if ... then x else error x
--   Then for y we get Abs `lub` Bot, and we really
--   want Abs overall
absLub Bot  	  = Abs
absLub Abs  	  = Abs
absLub Top 	  = Top
absLub (Call _)   = Top
absLub (Box _)    = Top
absLub (Eval ds)  = Defer (absLubs ds)	-- Or (Defer ds)?
absLub (Defer ds) = Defer (absLubs ds)	-- Or (Defer ds)?

absLubs :: Demands -> Demands
absLubs = mapDmds absLub

---------------
both :: Demand -> Demand -> Demand

both Abs d2 = d2

-- Note [Bottom demands]
both Bot Bot 	    = Bot
both Bot Abs 	    = Bot 
both Bot (Eval ds)  = Eval (mapDmds (`both` Bot) ds)
both Bot (Defer ds) = Eval (mapDmds (`both` Bot) ds)
both Bot _          = errDmd

both Top Bot 	    = errDmd
both Top Abs 	    = Top
both Top Top 	    = Top
both Top (Box d)    = Box d
both Top (Call d)   = Call d
both Top (Eval ds)  = Eval (mapDmds (`both` Top) ds)
both Top (Defer ds) 	-- = defer (Top `both` Eval ds)
			-- = defer (Eval (mapDmds (`both` Top) ds))
		     = deferEval (mapDmds (`both` Top) ds)


both (Box d1) 	(Box d2)    = box (d1 `both` d2)
both (Box d1) 	d2@(Call _) = box (d1 `both` d2)
both (Box d1) 	d2@(Eval _) = box (d1 `both` d2)
both (Box d1) 	(Defer _)   = Box d1
both d1@(Box _) d2	    = d2 `both` d1

both (Call d1) 	 (Call d2)   = Call (d1 `both` d2)
both (Call d1) 	 (Eval _)    = Call d1	-- Could do better for (Poly Bot)?
both (Call d1) 	 (Defer _)   = Call d1	-- Ditto
both d1@(Call _) d2	     = d2 `both` d1

both (Eval ds1)  (Eval  ds2) = Eval (ds1 `boths` ds2)
both (Eval ds1)  (Defer ds2) = Eval (ds1 `boths` mapDmds defer ds2)
both d1@(Eval _) d2	     = d2 `both` d1

both (Defer ds1)  (Defer ds2) = deferEval (ds1 `boths` ds2)
both d1@(Defer _) d2	      = d2 `both` d1
 
boths :: Demands -> Demands -> Demands
boths ds1 ds2 = zipWithDmds both ds1 ds2
\end{code}

Note [Bottom demands]
~~~~~~~~~~~~~~~~~~~~~
Consider
	f x = error x
From 'error' itself we get demand Bot on x
From the arg demand on x we get 
	x :-> evalDmd = Box (Eval (Poly Abs))
So we get  Bot `both` Box (Eval (Poly Abs))
	    = Seq Keep (Poly Bot)

Consider also
	f x = if ... then error (fst x) else fst x
Then we get (Eval (Box Bot, Bot) `lub` Eval (SA))
	= Eval (SA)
which is what we want.

Consider also
  f x = error [fst x]
Then we get 
     x :-> Bot `both` Defer [SA]
and we want the Bot demand to cancel out the Defer
so that we get Eval [SA].  Otherwise we'd have the odd
situation that
  f x = error (fst x)      -- Strictness U(SA)b
  g x = error ('y':fst x)  -- Strictness Tb