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

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

    {-# LANGUAGE CPP #-}

    module DmdAnal ( dmdAnalProgram ) where

    #include "HsVersions.h"

    import DynFlags
    import WwLib            ( findTypeShape, deepSplitProductType_maybe )
    import Demand   -- All of it
    import CoreSyn
    import Outputable
    import VarEnv
    import BasicTypes
    import FastString
    import Data.List
    import DataCon
    import Id
    import CoreUtils        ( exprIsHNF, exprType, exprIsTrivial )
    import TyCon
    import Type
    import FamInstEnv
    import Util
    import Maybes           ( isJust )
    import TysWiredIn       ( unboxedPairDataCon )
    import TysPrim          ( realWorldStatePrimTy )
    import ErrUtils         ( dumpIfSet_dyn )
    import Name             ( getName, stableNameCmp )
    import Data.Function    ( on )

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

    dmdAnalProgram :: DynFlags -> FamInstEnvs -> CoreProgram -> IO CoreProgram
    dmdAnalProgram dflags fam_envs binds
      = do {
            let { binds_plus_dmds = do_prog binds } ;
            dumpIfSet_dyn dflags Opt_D_dump_strsigs "Strictness signatures" $
                dumpStrSig binds_plus_dmds ;
            return binds_plus_dmds
        }
      where
        do_prog :: CoreProgram -> CoreProgram
        do_prog binds = snd $ mapAccumL dmdAnalTopBind (emptyAnalEnv dflags fam_envs) binds

    -- Analyse a (group of) top-level binding(s)
    dmdAnalTopBind :: AnalEnv
                   -> CoreBind
                   -> (AnalEnv, CoreBind)
    dmdAnalTopBind sigs (NonRec id rhs)
      = (extendAnalEnv TopLevel sigs id sig, NonRec id2 rhs2)
      where
        (  _, _, _,   rhs1) = dmdAnalRhs TopLevel Nothing sigs             id rhs
        (sig, _, id2, rhs2) = dmdAnalRhs TopLevel Nothing (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 sigs pairs
                    -- We get two iterations automatically
                    -- c.f. the NonRec case above

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

    Note [Ensure demand is strict]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    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!
    -}

    -- If e is complicated enough to become a thunk, its contents will be evaluated
    -- at most once, so oneify it.
    dmdTransformThunkDmd :: CoreExpr -> Demand -> Demand
    dmdTransformThunkDmd e
      | exprIsTrivial e = id
      | otherwise       = oneifyDmd

    -- Do not process absent demands
    -- Otherwise act like in a normal demand analysis
    -- See |-* relation in the companion paper
    dmdAnalStar :: AnalEnv
                -> Demand   -- This one takes a *Demand*
                -> CoreExpr -> (BothDmdArg, CoreExpr)
    dmdAnalStar env dmd e
      | (cd, defer_and_use) <- toCleanDmd dmd (exprType e)
      , (dmd_ty, e')        <- dmdAnal env cd e
      = (postProcessDmdTypeM defer_and_use dmd_ty, e')

    -- Main Demand Analsysis machinery
    dmdAnal, dmdAnal' :: AnalEnv
            -> CleanDemand         -- The main one takes a *CleanDemand*
            -> CoreExpr -> (DmdType, CoreExpr)

    -- The CleanDemand is always strict and not absent
    --    See Note [Ensure demand is strict]

    dmdAnal env d e = -- pprTrace "dmdAnal" (ppr d <+> ppr e) $
                      dmdAnal' env d e

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

    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

    {-       ----- I don't get this, so commenting out -------
        to_co        = pSnd (coercionKind co)
        dmd'
          | Just tc <- tyConAppTyCon_maybe to_co
          , isRecursiveTyCon tc = cleanEvalDmd
          | 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 (Tick t e)
      = (dmd_ty, Tick t 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)
      = -- This case handles value arguments (type args handled above)
        -- Crucially, coercions /are/ handled here, because they are
        -- value arguments (Trac #10288)
        let
            call_dmd          = mkCallDmd dmd
            (fun_ty, fun')    = dmdAnal env call_dmd fun
            (arg_dmd, res_ty) = splitDmdTy fun_ty
            (arg_ty, arg')    = dmdAnalStar env (dmdTransformThunkDmd arg arg_dmd) arg
        in
    --    pprTrace "dmdAnal:app" (vcat
    --         [ text "dmd =" <+> ppr dmd
    --         , text "expr =" <+> ppr (App fun arg)
    --         , text "fun dmd_ty =" <+> ppr fun_ty
    --         , text "arg dmd =" <+> ppr arg_dmd
    --         , text "arg dmd_ty =" <+> ppr arg_ty
    --         , text "res dmd_ty =" <+> ppr res_ty
    --         , text "overall res dmd_ty =" <+> ppr (res_ty `bothDmdType` arg_ty) ])
        (res_ty `bothDmdType` arg_ty, App fun' arg')

    -- this is an anonymous lambda, since @dmdAnalRhs@ uses @collectBinders@
    dmdAnal' env dmd (Lam var body)
      | isTyVar var
      = let
            (body_ty, body') = dmdAnal env dmd body
        in
        (body_ty, Lam var body')

      | otherwise
      = let (body_dmd, defer_and_use@(_,one_shot)) = peelCallDmd dmd
              -- body_dmd  - a demand to analyze the body
              -- one_shot  - one-shotness of the lambda
              --             hence, cardinality of its free vars

            env'             = extendSigsWithLam env var
            (body_ty, body') = dmdAnal env' body_dmd body
            (lam_ty, var')   = annotateLamIdBndr env notArgOfDfun body_ty one_shot var
        in
        (postProcessUnsat defer_and_use lam_ty, Lam var' body')

    dmdAnal' env dmd (Case scrut case_bndr ty [(DataAlt dc, bndrs, rhs)])
      -- Only one alternative with a product constructor
      | let tycon = dataConTyCon dc
      , isProductTyCon tycon
      , Just rec_tc' <- checkRecTc (ae_rec_tc env) tycon
      = let
            env_w_tc      = env { ae_rec_tc = rec_tc' }
            env_alt       = extendAnalEnv NotTopLevel env_w_tc case_bndr case_bndr_sig
            case_bndr_sig = cprProdSig (dataConRepArity dc)
                    -- cprProdSig: 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

            (rhs_ty, rhs')                  = dmdAnal env_alt dmd rhs
            (alt_ty1, dmds)                 = findBndrsDmds env rhs_ty bndrs
            (alt_ty2, case_bndr_dmd)        = findBndrDmd env False alt_ty1 case_bndr
            id_dmds                         = addCaseBndrDmd case_bndr_dmd dmds
            alt_ty3 | io_hack_reqd dc bndrs = deferAfterIO alt_ty2
                    | otherwise             = alt_ty2

            -- Compute demand on the scrutinee
            -- See Note [Demand on scrutinee of a product case]
            scrut_dmd          = mkProdDmd (addDataConStrictness dc id_dmds)
            (scrut_ty, scrut') = dmdAnal env scrut_dmd scrut
            res_ty             = alt_ty3 `bothDmdType` toBothDmdArg scrut_ty
            case_bndr'         = setIdDemandInfo case_bndr case_bndr_dmd
            bndrs'             = setBndrsDemandInfo bndrs id_dmds
        in
    --    pprTrace "dmdAnal:Case1" (vcat [ text "scrut" <+> ppr scrut
    --                                   , text "dmd" <+> ppr dmd
    --                                   , text "case_bndr_dmd" <+> ppr (idDemandInfo case_bndr')
    --                                   , text "scrut_dmd" <+> ppr scrut_dmd
    --                                   , text "scrut_ty" <+> ppr scrut_ty
    --                                   , text "alt_ty" <+> ppr alt_ty2
    --                                   , text "res_ty" <+> ppr res_ty ]) $
        (res_ty, Case scrut' case_bndr' ty [(DataAlt dc, bndrs', rhs')])

    dmdAnal' env dmd (Case scrut case_bndr ty alts)
      = let      -- Case expression with multiple alternatives
            (alt_tys, alts')     = mapAndUnzip (dmdAnalAlt env dmd case_bndr) alts
            (scrut_ty, scrut')   = dmdAnal env cleanEvalDmd scrut
            (alt_ty, case_bndr') = annotateBndr env (foldr lubDmdType botDmdType alt_tys) case_bndr
            res_ty               = alt_ty `bothDmdType` toBothDmdArg scrut_ty
        in
    --    pprTrace "dmdAnal:Case2" (vcat [ text "scrut" <+> ppr scrut
    --                                   , text "scrut_ty" <+> ppr scrut_ty
    --                                   , text "alt_tys" <+> ppr alt_tys
    --                                   , text "alt_ty" <+> ppr alt_ty
    --                                   , text "res_ty" <+> ppr res_ty ]) $
        (res_ty, Case scrut' case_bndr' ty alts')

    dmdAnal' env dmd (Let (NonRec id rhs) body)
      = (body_ty2, Let (NonRec id2 annotated_rhs) body')
      where
        (sig, lazy_fv, id1, rhs') = dmdAnalRhs NotTopLevel Nothing env id rhs
        (body_ty, body')          = dmdAnal (extendAnalEnv NotTopLevel env id sig) dmd body
        (body_ty1, id2)           = annotateBndr env body_ty id1
        body_ty2                  = addLazyFVs body_ty1 lazy_fv

        -- Annotate top-level lambdas at RHS basing on the aggregated demand info
        -- See Note [Annotating lambdas at right-hand side]
        annotated_rhs = annLamWithShotness (idDemandInfo id2) rhs'

            -- 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.

    dmdAnal' env dmd (Let (Rec pairs) body)
      = let
            (env', lazy_fv, pairs') = dmdFix NotTopLevel env pairs
            (body_ty, body')        = dmdAnal env' dmd body
            body_ty1                = deleteFVs body_ty (map fst pairs)
            body_ty2                = addLazyFVs body_ty1 lazy_fv
        in
        body_ty2 `seq`
        (body_ty2,  Let (Rec pairs') body')

    io_hack_reqd :: DataCon -> [Var] -> Bool
    -- Note [IO hack in the demand analyser]
    --
    -- 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 (Trac #148, #1592) where this shows up is:
    --      do { let len = <expensive> ;
    --         ; when (...) (exitWith ExitSuccess)
    --         ; print len }
    io_hack_reqd con bndrs
      | (bndr:_) <- bndrs
      = con == unboxedPairDataCon &&
        idType bndr `eqType` realWorldStatePrimTy
      | otherwise
      = False

    annLamWithShotness :: Demand -> CoreExpr -> CoreExpr
    annLamWithShotness d e
      | Just u <- cleanUseDmd_maybe d
      = go u e
      | otherwise = e
      where
        go u e
          | Just (c, u') <- peelUseCall u
          , Lam bndr body <- e
          = if isTyVar bndr
            then Lam bndr                    (go u  body)
            else Lam (setOneShotness c bndr) (go u' body)
          | otherwise
          = e

    setOneShotness :: Count -> Id -> Id
    setOneShotness One  bndr = setOneShotLambda bndr
    setOneShotness Many bndr = bndr

    dmdAnalAlt :: AnalEnv -> CleanDemand -> Id -> Alt Var -> (DmdType, Alt Var)
    dmdAnalAlt env dmd case_bndr (con,bndrs,rhs)
      | null bndrs    -- Literals, DEFAULT, and nullary constructors
      , (rhs_ty, rhs') <- dmdAnal env dmd rhs
      = (rhs_ty, (con, [], rhs'))

      | otherwise     -- Non-nullary data constructors
      , (rhs_ty, rhs') <- dmdAnal env dmd rhs
      , (alt_ty, dmds) <- findBndrsDmds env rhs_ty bndrs
      , let case_bndr_dmd = findIdDemand alt_ty case_bndr
            id_dmds       = addCaseBndrDmd case_bndr_dmd dmds
      = (alt_ty, (con, setBndrsDemandInfo bndrs id_dmds, rhs'))

    {- Note [Demand on the scrutinee of a product case]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    When figuring out the demand on the scrutinee of a product case,
    we use the demands of the case alternative, i.e. id_dmds.
    But note that these include the demand on the case binder;
    see Note [Demand on case-alternative binders] in Demand.hs.
    This is crucial. Example:
       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.

    Note [Aggregated demand for cardinality]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    We use different strategies for strictness and usage/cardinality to
    "unleash" demands captured on free variables by bindings. Let us
    consider the example:

    f1 y = let {-# NOINLINE h #-}
               h = y
           in  (h, h)

    We are interested in obtaining cardinality demand U1 on |y|, as it is
    used only in a thunk, and, therefore, is not going to be updated any
    more. Therefore, the demand on |y|, captured and unleashed by usage of
    |h| is U1. However, if we unleash this demand every time |h| is used,
    and then sum up the effects, the ultimate demand on |y| will be U1 +
    U1 = U. In order to avoid it, we *first* collect the aggregate demand
    on |h| in the body of let-expression, and only then apply the demand
    transformer:

    transf[x](U) = {y |-> U1}

    so the resulting demand on |y| is U1.

    The situation is, however, different for strictness, where this
    aggregating approach exhibits worse results because of the nature of
    |both| operation for strictness. Consider the example:

    f y c =
      let h x = y |seq| x
       in case of
            True  -> h True
            False -> y

    It is clear that |f| is strict in |y|, however, the suggested analysis
    will infer from the body of |let| that |h| is used lazily (as it is
    used in one branch only), therefore lazy demand will be put on its
    free variable |y|. Conversely, if the demand on |h| is unleashed right
    on the spot, we will get the desired result, namely, that |f| is
    strict in |y|.

    Note [Annotating lambdas at right-hand side]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Let us take a look at the following example:

    g f = let x = 100
              h = \y -> f x y
           in h 5

    One can see that |h| is called just once, therefore the RHS of h can
    be annotated as a one-shot lambda. This is done by the function
    annLamWithShotness *a posteriori*, i.e., basing on the aggregated
    usage demand on |h| from the body of |let|-expression, which is C1(U)
    in this case.

    In other words, for locally-bound lambdas we can infer
    one-shotness.
    -}

    {-
    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 'go' 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!

    We add these extra strict demands to the demand on the *scrutinee* of
    the case expression; hence the use of addDataConStrictness when
    forming scrut_dmd.  The case alternatives aren't strict in their
    sub-components, but simply evaluating the scrutinee to HNF does force
    those sub-components.

    ************************************************************************
    *                                                                      *
                        Demand transformer
    *                                                                      *
    ************************************************************************
    -}

    dmdTransform :: AnalEnv         -- The strictness environment
                 -> Id              -- The function
                 -> CleanDemand     -- 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
      | isDataConWorkId var                          -- Data constructor
      = dmdTransformDataConSig (idArity var) (idStrictness var) dmd

      | gopt Opt_DmdTxDictSel (ae_dflags env),
        Just _ <- isClassOpId_maybe var -- Dictionary component selector
      = dmdTransformDictSelSig (idStrictness var) dmd

      | isGlobalId var                               -- Imported function
      = let res = dmdTransformSig (idStrictness var) dmd in
    --    pprTrace "dmdTransform" (vcat [ppr var, ppr (idStrictness var), ppr dmd, ppr res])
        res

      | Just (sig, top_lvl) <- lookupSigEnv env var  -- Local letrec bound thing
      , let fn_ty = dmdTransformSig sig dmd
      = -- pprTrace "dmdTransform" (vcat [ppr var, ppr sig, ppr dmd, ppr fn_ty]) $
        if isTopLevel top_lvl
        then fn_ty   -- Don't record top level things
        else addVarDmd fn_ty var (mkOnceUsedDmd dmd)

      | otherwise                                    -- Local non-letrec-bound thing
      = unitVarDmd var (mkOnceUsedDmd dmd)

    {-
    ************************************************************************
    *                                                                      *
    \subsection{Bindings}
    *                                                                      *
    ************************************************************************
    -}

    -- Recursive bindings
    dmdFix :: TopLevelFlag
           -> AnalEnv               -- Does not include bindings for this binding
           -> [(Id,CoreExpr)]
           -> (AnalEnv, DmdEnv,
               [(Id,CoreExpr)])     -- Binders annotated with stricness info

    dmdFix top_lvl env orig_pairs
      = (updSigEnv env (sigEnv final_env), lazy_fv, pairs')
         -- Return to original virgin state, keeping new signatures
      where
        bndrs        = map fst orig_pairs
        initial_env = addInitialSigs top_lvl env bndrs
        (final_env, lazy_fv, pairs') = loop 1 initial_env orig_pairs

        loop :: Int
             -> AnalEnv                     -- Already contains the current sigs
             -> [(Id,CoreExpr)]
             -> (AnalEnv, 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
          = (env', 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 (sigEnv env) id,
            --                                              lookupVarEnv (sigEnv env') id)
            --                                          | (id,_) <- pairs],
            --                   text "env:" <+> ppr env,
            --                   text "binds:" <+> pprCoreBinding (Rec pairs)]))
            (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 env') pairs'
          where
            found_fixpoint = all (same_sig (sigEnv env) (sigEnv env')) bndrs

            ((env',lazy_fv), pairs') = mapAccumL my_downRhs (env, 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 (env, lazy_fv) (id,rhs)
              = ((env', lazy_fv'), (id', rhs'))
              where
                (sig, lazy_fv1, id', rhs') = dmdAnalRhs top_lvl (Just bndrs) env id rhs
                lazy_fv'                   = plusVarEnv_C bothDmd lazy_fv lazy_fv1
                env'                       = extendAnalEnv top_lvl env id sig

        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)

    -- Non-recursive bindings
    dmdAnalRhs :: TopLevelFlag
               -> Maybe [Id]   -- Just bs <=> recursive, Nothing <=> non-recursive
               -> AnalEnv -> Id -> CoreExpr
               -> (StrictSig,  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
      | Just fn <- unpackTrivial rhs   -- See Note [Demand analysis for trivial right-hand sides]
      , let fn_str = getStrictness env fn
            fn_fv | isLocalId fn = unitVarEnv fn topDmd
                  | otherwise    = emptyDmdEnv
            -- Note [Remember to demand the function itself]
            -- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
            -- fn_fv: don't forget to produce a demand for fn itself
            -- Lacking this caused Trac #9128
            -- The demand is very conservative (topDmd), but that doesn't
            -- matter; trivial bindings are usually inlined, so it only
            -- kicks in for top-level bindings and NOINLINE bindings
      = (fn_str, fn_fv, set_idStrictness env id fn_str, rhs)

      | otherwise
      = (sig_ty, lazy_fv, id', mkLams bndrs' body')
      where
        (bndrs, body)    = collectBinders rhs
        env_body         = foldl extendSigsWithLam env bndrs
        (body_ty, body') = dmdAnal env_body body_dmd body
        body_ty'         = removeDmdTyArgs body_ty -- zap possible deep CPR info
        (DmdType rhs_fv rhs_dmds rhs_res, bndrs')
                         = annotateLamBndrs env (isDFunId id) body_ty' bndrs
        sig_ty           = mkStrictSig (mkDmdType sig_fv rhs_dmds rhs_res')
        id'              = set_idStrictness env id sig_ty
            -- See Note [NOINLINE and strictness]

        -- See Note [Product demands for function body]
        body_dmd = case deepSplitProductType_maybe (ae_fam_envs env) (exprType body) of
                     Nothing            -> cleanEvalDmd
                     Just (dc, _, _, _) -> cleanEvalProdDmd (dataConRepArity dc)

        -- See Note [Lazy and unleashable free variables]
        -- See Note [Aggregated demand for cardinality]
        rhs_fv1 = case rec_flag of
                    Just bs -> reuseEnv (delVarEnvList rhs_fv bs)
                    Nothing -> rhs_fv

        (lazy_fv, sig_fv) = splitFVs is_thunk rhs_fv1

        rhs_res'  = trimCPRInfo trim_all trim_sums rhs_res
        trim_all  = is_thunk && not_strict
        trim_sums = not (isTopLevel top_lvl) -- See Note [CPR for sum types]

        -- See Note [CPR for thunks]
        is_thunk = not (exprIsHNF rhs)
        not_strict
           =  isTopLevel top_lvl  -- Top level and recursive things don't
           || isJust rec_flag     -- get their demandInfo set at all
           || not (isStrictDmd (idDemandInfo id) || ae_virgin env)
              -- See Note [Optimistic CPR in the "virgin" case]

    unpackTrivial :: CoreExpr -> Maybe Id
    -- Returns (Just v) if the arg is really equal to v, modulo
    -- casts, type applications etc
    -- See Note [Demand analysis for trivial right-hand sides]
    unpackTrivial (Var v)                 = Just v
    unpackTrivial (Cast e _)              = unpackTrivial e
    unpackTrivial (Lam v e) | isTyVar v   = unpackTrivial e
    unpackTrivial (App e a) | isTypeArg a = unpackTrivial e
    unpackTrivial _                       = Nothing

    {-
    Note [Demand analysis for trivial right-hand sides]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Consider
            foo = plusInt |> co
    where plusInt is an arity-2 function with known strictness.  Clearly
    we want plusInt's strictness to propagate to foo!  But because it has
    no manifest lambdas, it won't do so automatically, and indeed 'co' might
    have type (Int->Int->Int) ~ T, so we *can't* eta-expand.  So we have a
    special case for right-hand sides that are "trivial", namely variables,
    casts, type applications, and the like.

    Note that this can mean that 'foo' has an arity that is smaller than that
    indicated by its demand info.  e.g. if co :: (Int->Int->Int) ~ T, then
    foo's arity will be zero (see Note [exprArity invariant] in CoreArity),
    but its demand signature will be that of plusInt. A small example is the
    test case of Trac #8963.

    Note [Product demands for function body]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    This example comes from shootout/binary_trees:

        Main.check' = \ b z ds. case z of z' { I# ip ->
                                    case ds_d13s of
                                      Main.Nil -> z'
                                      Main.Node s14k s14l s14m ->
                                        Main.check' (not b)
                                          (Main.check' b
                                             (case b {
                                                False -> I# (-# s14h s14k);
                                                True  -> I# (+# s14h s14k)
                                              })
                                             s14l)
                                         s14m   }   }   }

    Here we *really* want to unbox z, even though it appears to be used boxed in
    the Nil case.  Partly the Nil case is not a hot path.  But more specifically,
    the whole function gets the CPR property if we do.

    So for the demand on the body of a RHS we use a product demand if it's
    a product type.

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

    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 bothDmd fv var dmd) ds res

    addLazyFVs :: DmdType -> DmdEnv -> DmdType
    addLazyFVs dmd_ty lazy_fvs
      = dmd_ty `bothDmdType` mkBothDmdArg lazy_fvs
            -- Using bothDmdType (rather than just both'ing the envs)
            -- 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 `bothDmd` 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 dmdAnalRhs.  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.

    {-
    Note [Do not strictify the argument dictionaries of a dfun]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    The typechecker can tie recursive knots involving dfuns, so we do the
    conservative thing and refrain from strictifying a dfun's argument
    dictionaries.
    -}

    setBndrsDemandInfo :: [Var] -> [Demand] -> [Var]
    setBndrsDemandInfo (b:bs) (d:ds)
      | isTyVar b = b : setBndrsDemandInfo bs (d:ds)
      | otherwise = setIdDemandInfo b d : setBndrsDemandInfo bs ds
    setBndrsDemandInfo [] ds = ASSERT( null ds ) []
    setBndrsDemandInfo bs _  = pprPanic "setBndrsDemandInfo" (ppr bs)

    annotateBndr :: AnalEnv -> DmdType -> Var -> (DmdType, Var)
    -- The returned env has the var deleted
    -- The returned var is annotated with demand info
    -- according to the result demand of the provided demand type
    -- No effect on the argument demands
    annotateBndr env dmd_ty var
      | isId var  = (dmd_ty', setIdDemandInfo var dmd)
      | otherwise = (dmd_ty, var)
      where
        (dmd_ty', dmd) = findBndrDmd env False dmd_ty var

    annotateLamBndrs :: AnalEnv -> DFunFlag -> DmdType -> [Var] -> (DmdType, [Var])
    annotateLamBndrs env args_of_dfun ty bndrs = mapAccumR annotate ty bndrs
      where
        annotate dmd_ty bndr
          | isId bndr = annotateLamIdBndr env args_of_dfun dmd_ty Many bndr
          | otherwise = (dmd_ty, bndr)

    annotateLamIdBndr :: AnalEnv
                      -> DFunFlag   -- is this lambda at the top of the RHS of a dfun?
                      -> DmdType    -- Demand type of body
                      -> Count      -- One-shot-ness of the lambda
                      -> Id         -- Lambda binder
                      -> (DmdType,  -- Demand type of lambda
                          Id)       -- and binder annotated with demand

    annotateLamIdBndr env arg_of_dfun dmd_ty one_shot id
    -- For lambdas we add the demand to the argument demands
    -- Only called for Ids
      = ASSERT( isId id )
        -- pprTrace "annLamBndr" (vcat [ppr id, ppr _dmd_ty]) $
        (final_ty, setOneShotness one_shot (setIdDemandInfo id dmd))
      where
          -- Watch out!  See note [Lambda-bound unfoldings]
        final_ty = case maybeUnfoldingTemplate (idUnfolding id) of
                     Nothing  -> main_ty
                     Just unf -> main_ty `bothDmdType` unf_ty
                              where
                                 (unf_ty, _) = dmdAnalStar env dmd unf

        main_ty = addDemand dmd dmd_ty'
        (dmd_ty', dmd) = findBndrDmd env arg_of_dfun dmd_ty id

    deleteFVs :: DmdType -> [Var] -> DmdType
    deleteFVs (DmdType fvs dmds res) bndrs
      = DmdType (delVarEnvList fvs bndrs) dmds res

    {-
    Note [CPR for sum types]
    ~~~~~~~~~~~~~~~~~~~~~~~~
    At the moment we do not do CPR for let-bindings that
       * non-top level
       * bind a sum type
    Reason: I found that in some benchmarks we were losing let-no-escapes,
    which messed it all up.  Example
       let j = \x. ....
       in case y of
            True  -> j False
            False -> j True
    If we w/w this we get
       let j' = \x. ....
       in case y of
            True  -> case j' False of { (# a #) -> Just a }
            False -> case j' True of { (# a #) -> Just a }
    Notice that j' is not a let-no-escape any more.

    However this means in turn that the *enclosing* function
    may be CPR'd (via the returned Justs).  But in the case of
    sums, there may be Nothing alternatives; and that messes
    up the sum-type CPR.

    Conclusion: only do this for products.  It's still not
    guaranteed OK for products, but sums definitely lose sometimes.

    Note [CPR for thunks]
    ~~~~~~~~~~~~~~~~~~~~~
    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.

    Note [Optimistic CPR in the "virgin" case]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Demand and strictness info are initialized by top elements. However,
    this prevents from inferring a CPR property in the first pass of the
    analyser, so we keep an explicit flag ae_virgin in the AnalEnv
    datatype.

    We can't start with 'not-demanded' (i.e., top) 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, by checking the
    ae_virgin flag at 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!

    Note [Lazy and unleasheable free variables]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    We put the strict and once-used 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.

    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}
    *                                                                      *
    ************************************************************************
    -}

    type DFunFlag = Bool  -- indicates if the lambda being considered is in the
                          -- sequence of lambdas at the top of the RHS of a dfun
    notArgOfDfun :: DFunFlag
    notArgOfDfun = False

    data AnalEnv
      = AE { ae_dflags :: DynFlags
           , ae_sigs   :: SigEnv
           , ae_virgin :: Bool    -- True on first iteration only
                                  -- See Note [Initialising strictness]
           , ae_rec_tc :: RecTcChecker
           , ae_fam_envs :: FamInstEnvs
     }

            -- 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 ])

    emptyAnalEnv :: DynFlags -> FamInstEnvs -> AnalEnv
    emptyAnalEnv dflags fam_envs
        = AE { ae_dflags = dflags
             , ae_sigs = emptySigEnv
             , ae_virgin = True
             , ae_rec_tc = initRecTc
             , ae_fam_envs = fam_envs
             }

    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

    getStrictness :: AnalEnv -> Id -> StrictSig
    getStrictness env fn
      | isGlobalId fn                        = idStrictness fn
      | Just (sig, _) <- lookupSigEnv env fn = sig
      | otherwise                            = nopSig

    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

    nonVirgin :: AnalEnv -> AnalEnv
    nonVirgin env = env { ae_virgin = False }

    extendSigsWithLam :: AnalEnv -> Id -> AnalEnv
    -- Extend the AnalEnv when we meet a lambda binder
    extendSigsWithLam env id
      | isId id
      , isStrictDmd (idDemandInfo id) || ae_virgin env
           -- See Note [Optimistic CPR in the "virgin" case]
           -- See Note [Initial CPR for strict binders]
      , Just (dc,_,_,_) <- deepSplitProductType_maybe (ae_fam_envs env) $ idType id
      = extendAnalEnv NotTopLevel env id (cprProdSig (dataConRepArity dc))

      | otherwise
      = env

    addDataConStrictness :: DataCon -> [Demand] -> [Demand]
    -- See Note [Add demands for strict constructors]
    addDataConStrictness con ds
      = ASSERT2( equalLength strs ds, ppr con $$ ppr strs $$ ppr ds )
        zipWith add ds strs
      where
        strs = dataConRepStrictness con
        add dmd str | isMarkedStrict str = dmd `bothDmd` seqDmd
                    | otherwise          = dmd
        -- Yes, even if 'dmd' is Absent!

    findBndrsDmds :: AnalEnv -> DmdType -> [Var] -> (DmdType, [Demand])
    -- Return the demands on the Ids in the [Var]
    findBndrsDmds env dmd_ty bndrs
      = go dmd_ty bndrs
      where
        go dmd_ty []  = (dmd_ty, [])
        go dmd_ty (b:bs)
          | isId b    = let (dmd_ty1, dmds) = go dmd_ty bs
                            (dmd_ty2, dmd)  = findBndrDmd env False dmd_ty1 b
                        in (dmd_ty2, dmd : dmds)
          | otherwise = go dmd_ty bs

    findBndrDmd :: AnalEnv -> Bool -> DmdType -> Id -> (DmdType, Demand)
    -- See Note [Trimming a demand to a type] in Demand.lhs
    findBndrDmd env arg_of_dfun dmd_ty id
      = (dmd_ty', dmd')
      where
        dmd' = killUsageDemand (ae_dflags env) $
               strictify $
               trimToType starting_dmd (findTypeShape fam_envs id_ty)

        (dmd_ty', starting_dmd) = peelFV dmd_ty id

        id_ty = idType id

        strictify dmd
          | gopt Opt_DictsStrict (ae_dflags env)
                 -- We never want to strictify a recursive let. At the moment
                 -- annotateBndr is only call for non-recursive lets; if that
                 -- changes, we need a RecFlag parameter and another guard here.
          , not arg_of_dfun -- See Note [Do not strictify the argument dictionaries of a dfun]
          = strictifyDictDmd id_ty dmd
          | otherwise
          = dmd

        fam_envs = ae_fam_envs env

    set_idStrictness :: AnalEnv -> Id -> StrictSig -> Id
    set_idStrictness env id sig
      = setIdStrictness id (killUsageSig (ae_dflags env) sig)

    dumpStrSig :: CoreProgram -> SDoc
    dumpStrSig binds = vcat (map printId ids)
      where
      ids = sortBy (stableNameCmp `on` getName) (concatMap getIds binds)
      getIds (NonRec i _) = [ i ]
      getIds (Rec bs)     = map fst bs
      printId id | isExportedId id = ppr id <> colon <+> pprIfaceStrictSig (idStrictness id)
                 | otherwise       = empty

    {-
    Note [Initial CPR for strict binders]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    CPR is initialized for a lambda binder in an optimistic manner, i.e,
    if the binder is used strictly and at least some of its components as
    a product are used, which is checked by the value of the absence
    demand.

    If the binder is marked demanded with a strict 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!).

    Note [Initialising strictness]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    See section 9.2 (Finding fixpoints) of the paper.

    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 B might be inside an *enclosing* recursiveb group A, in
    which case we'll do the entire fixpoint shebang on for each iteration
    of A. This can be illustrated by the following example:

    Example:

      f [] = []
      f (x:xs) = let g []     = f xs
                     g (y:ys) = y+1 : g ys
                  in g (h x)

    At each iteration of the fixpoint for f, the analyser has to find a
    fixpoint for the enclosed function g. In the meantime, the demand
    values for g at each iteration for f are *greater* than those we
    encountered in the previous iteration for f. Therefore, we can begin
    the fixpoint for g not with the bottom value but rather with the
    result of the previous analysis. I.e., when beginning the fixpoint
    process for g, we can start from the demand signature computed for g
    previously and attached to the binding occurrence of g.

    To speed things up, we initialise each iteration of A (the enclosing
    one) 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.
    -}