basicTypes/Demand.hs

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

    \section[Demand]{@Demand@: A decoupled implementation of a demand domain}
    -}

    {-# LANGUAGE CPP, FlexibleInstances, TypeSynonymInstances, RecordWildCards #-}

    module Demand (
            StrDmd, UseDmd(..), Count,

            Demand, CleanDemand, getStrDmd, getUseDmd,
            mkProdDmd, mkOnceUsedDmd, mkManyUsedDmd, mkHeadStrict, oneifyDmd,
            toCleanDmd,
            absDmd, topDmd, botDmd, seqDmd,
            lubDmd, bothDmd,
            lazyApply1Dmd, lazyApply2Dmd, strictApply1Dmd,
            catchArgDmd,
            isTopDmd, isAbsDmd, isSeqDmd,
            peelUseCall, cleanUseDmd_maybe, strictenDmd, bothCleanDmd,
            addCaseBndrDmd,

            DmdType(..), dmdTypeDepth, lubDmdType, bothDmdType,
            nopDmdType, botDmdType, mkDmdType,
            addDemand, removeDmdTyArgs,
            BothDmdArg, mkBothDmdArg, toBothDmdArg,

            DmdEnv, emptyDmdEnv,
            peelFV, findIdDemand,

            DmdResult, CPRResult,
            isBotRes, isTopRes,
            topRes, botRes, exnRes, cprProdRes,
            vanillaCprProdRes, cprSumRes,
            appIsBottom, isBottomingSig, pprIfaceStrictSig,
            trimCPRInfo, returnsCPR_maybe,
            StrictSig(..), mkStrictSig, mkClosedStrictSig,
            nopSig, botSig, exnSig, cprProdSig,
            isTopSig, hasDemandEnvSig,
            splitStrictSig, strictSigDmdEnv,
            increaseStrictSigArity,

            seqDemand, seqDemandList, seqDmdType, seqStrictSig,

            evalDmd, cleanEvalDmd, cleanEvalProdDmd, isStrictDmd,
            splitDmdTy, splitFVs,
            deferAfterIO,
            postProcessUnsat, postProcessDmdType,

            splitProdDmd_maybe, peelCallDmd, mkCallDmd, mkWorkerDemand,
            dmdTransformSig, dmdTransformDataConSig, dmdTransformDictSelSig,
            argOneShots, argsOneShots, saturatedByOneShots,
            trimToType, TypeShape(..),

            useCount, isUsedOnce, reuseEnv,
            killUsageDemand, killUsageSig, zapUsageDemand, zapUsageEnvSig,
            zapUsedOnceDemand, zapUsedOnceSig,
            strictifyDictDmd

         ) where

    #include "HsVersions.h"

    import DynFlags
    import Outputable
    import Var ( Var )
    import VarEnv
    import UniqFM
    import Util
    import BasicTypes
    import Binary
    import Maybes           ( orElse )

    import Type            ( Type, isUnliftedType )
    import TyCon           ( isNewTyCon, isClassTyCon )
    import DataCon         ( splitDataProductType_maybe )

    {-
    ************************************************************************
    *                                                                      *
            Joint domain for Strictness and Absence
    *                                                                      *
    ************************************************************************
    -}

    data JointDmd s u = JD { sd :: s, ud :: u }
      deriving ( Eq, Show )

    getStrDmd :: JointDmd s u -> s
    getStrDmd = sd

    getUseDmd :: JointDmd s u -> u
    getUseDmd = ud

    -- Pretty-printing
    instance (Outputable s, Outputable u) => Outputable (JointDmd s u) where
      ppr (JD {sd = s, ud = u}) = angleBrackets (ppr s <> char ',' <> ppr u)

    -- Well-formedness preserving constructors for the joint domain
    mkJointDmd :: s -> u -> JointDmd s u
    mkJointDmd s u = JD { sd = s, ud = u }

    mkJointDmds :: [s] -> [u] -> [JointDmd s u]
    mkJointDmds ss as = zipWithEqual "mkJointDmds" mkJointDmd ss as

    {-
    ************************************************************************
    *                                                                      *
                Strictness domain
    *                                                                      *
    ************************************************************************

            Lazy
             |
      ExnStr x -
               |
            HeadStr
            /     \
        SCall      SProd
            \      /
            HyperStr

    Note [Exceptions and strictness]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Exceptions need rather careful treatment, especially because of 'catch'
    ('catch#'), 'catchSTM' ('catchSTM#'), and 'orElse' ('catchRetry#').
    See Trac #11555, #10712 and #13330, and for some more background, #11222.

    There are three main pieces.

    * The Termination type includes ThrowsExn, meaning "under the given
      demand this expression either diverges or throws an exception".

      This is relatively uncontroversial. The primops raise# and
      raiseIO# both return ThrowsExn; nothing else does.

    * An ArgStr has an ExnStr flag to say how to process the Termination
      result of the argument.  If the ExnStr flag is ExnStr, we squash
      ThrowsExn to topRes.  (This is done in postProcessDmdResult.)

    Here is the key example

        catchRetry# (\s -> retry# s) blah

    We analyse the argument (\s -> retry# s) with demand
        Str ExnStr (SCall HeadStr)
    i.e. with the ExnStr flag set.
      - First we analyse the argument with the "clean-demand" (SCall
        HeadStr), getting a DmdResult of ThrowsExn from the saturated
        application of retry#.
      - Then we apply the post-processing for the shell, squashing the
        ThrowsExn to topRes.

    This also applies uniformly to free variables.  Consider

        let r = \st -> retry# st
        in catchRetry# (\s -> ...(r s')..) handler st

    If we give the first argument of catch a strict signature, we'll get a demand
    'C(S)' for 'r'; that is, 'r' is definitely called with one argument, which
    indeed it is.  But when we post-process the free-var demands on catchRetry#'s
    argument (in postProcessDmdEnv), we'll give 'r' a demand of (Str ExnStr (SCall
    HeadStr)); and if we feed that into r's RHS (which would be reasonable) we'll
    squash the retry just as if we'd inlined 'r'.

    * We don't try to get clever about 'catch#' and 'catchSTM#' at the moment. We
    previously (#11222) tried to take advantage of the fact that 'catch#' calls its
    first argument eagerly. See especially commit
    9915b6564403a6d17651e9969e9ea5d7d7e78e7f. We analyzed that first argument with
    a strict demand, and then performed a post-processing step at the end to change
    ThrowsExn to TopRes.  The trouble, I believe, is that to use this approach
    correctly, we'd need somewhat different information about that argument.
    Diverges, ThrowsExn (i.e., diverges or throws an exception), and Dunno are the
    wrong split here.  In order to evaluate part of the argument speculatively,
    we'd need to know that it *does not throw an exception*. That is, that it
    either diverges or succeeds. But we don't currently have a way to talk about
    that. Abstractly and approximately,

    catch# m f s = case ORACLE m s of
      DivergesOrSucceeds -> m s
      Fails exc -> f exc s

    where the magical ORACLE determines whether or not (m s) throws an exception
    when run, and if so which one. If we want, we can safely consider (catch# m f s)
    strict in anything that both branches are strict in (by performing demand
    analysis for 'catch#' in the same way we do for case). We could also safely
    consider it strict in anything demanded by (m s) that is guaranteed not to
    throw an exception under that demand, but I don't know if we have the means
    to express that.

    My mind keeps turning to this model (not as an actual change to the type, but
    as a way to think about what's going on in the analysis):

    newtype IO a = IO {unIO :: State# s -> (# s, (# SomeException | a #) #)}
    instance Monad IO where
      return a = IO $ \s -> (# s, (# | a #) #)
      IO m >>= f = IO $ \s -> case m s of
        (# s', (# e | #) #) -> (# s', e #)
        (# s', (# | a #) #) -> unIO (f a) s
    raiseIO# e s = (# s, (# e | #) #)
    catch# m f s = case m s of
      (# s', (# e | #) #) -> f e s'
      res -> res

    Thinking about it this way seems likely to be productive for analyzing IO
    exception behavior, but imprecise exceptions and asynchronous exceptions remain
    quite slippery beasts. Can we incorporate them? I think we can. We can imagine
    applying 'seq#' to evaluate @m s@, determining whether it throws an imprecise
    or asynchronous exception or whether it succeeds or throws an IO exception.
    This confines the peculiarities to 'seq#', which is indeed rather essentially
    peculiar.
    -}

    -- Vanilla strictness domain
    data StrDmd
      = HyperStr             -- Hyper-strict
                             -- Bottom of the lattice
                             -- Note [HyperStr and Use demands]

      | SCall StrDmd         -- Call demand
                             -- Used only for values of function type

      | SProd [ArgStr]     -- Product
                             -- Used only for values of product type
                             -- Invariant: not all components are HyperStr (use HyperStr)
                             --            not all components are Lazy     (use HeadStr)

      | HeadStr              -- Head-Strict
                             -- A polymorphic demand: used for values of all types,
                             --                       including a type variable

      deriving ( Eq, Show )

    type ArgStr = Str StrDmd

    data Str s = Lazy         -- Lazy
                              -- Top of the lattice
               | Str ExnStr s
      deriving ( Eq, Show )

    data ExnStr  -- See Note [Exceptions and strictness]
      = VanStr   -- "Vanilla" case, ordinary strictness

      | ExnStr   -- (Str ExnStr d) means be strict like 'd' but then degrade
                 --                the Termination info ThrowsExn to Dunno
      deriving( Eq, Show )

    -- Well-formedness preserving constructors for the Strictness domain
    strBot, strTop :: ArgStr
    strBot = Str VanStr HyperStr
    strTop = Lazy

    mkSCall :: StrDmd -> StrDmd
    mkSCall HyperStr = HyperStr
    mkSCall s        = SCall s

    mkSProd :: [ArgStr] -> StrDmd
    mkSProd sx
      | any isHyperStr sx = HyperStr
      | all isLazy     sx = HeadStr
      | otherwise         = SProd sx

    isLazy :: ArgStr -> Bool
    isLazy Lazy     = True
    isLazy (Str {}) = False

    isHyperStr :: ArgStr -> Bool
    isHyperStr (Str _ HyperStr) = True
    isHyperStr _                = False

    -- Pretty-printing
    instance Outputable StrDmd where
      ppr HyperStr      = char 'B'
      ppr (SCall s)     = char 'C' <> parens (ppr s)
      ppr HeadStr       = char 'S'
      ppr (SProd sx)    = char 'S' <> parens (hcat (map ppr sx))

    instance Outputable ArgStr where
      ppr (Str x s)     = (case x of VanStr -> empty; ExnStr -> char 'x')
                          <> ppr s
      ppr Lazy          = char 'L'

    lubArgStr :: ArgStr -> ArgStr -> ArgStr
    lubArgStr Lazy        _           = Lazy
    lubArgStr _           Lazy        = Lazy
    lubArgStr (Str x1 s1) (Str x2 s2) = Str (x1 `lubExnStr` x2) (s1 `lubStr` s2)

    lubExnStr :: ExnStr -> ExnStr -> ExnStr
    lubExnStr VanStr VanStr = VanStr
    lubExnStr _      _      = ExnStr   -- ExnStr is lazier

    lubStr :: StrDmd -> StrDmd -> StrDmd
    lubStr HyperStr s              = s
    lubStr (SCall s1) HyperStr     = SCall s1
    lubStr (SCall _)  HeadStr      = HeadStr
    lubStr (SCall s1) (SCall s2)   = SCall (s1 `lubStr` s2)
    lubStr (SCall _)  (SProd _)    = HeadStr
    lubStr (SProd sx) HyperStr     = SProd sx
    lubStr (SProd _)  HeadStr      = HeadStr
    lubStr (SProd s1) (SProd s2)
        | length s1 == length s2   = mkSProd (zipWith lubArgStr s1 s2)
        | otherwise                = HeadStr
    lubStr (SProd _) (SCall _)     = HeadStr
    lubStr HeadStr   _             = HeadStr

    bothArgStr :: ArgStr -> ArgStr -> ArgStr
    bothArgStr Lazy        s           = s
    bothArgStr s           Lazy        = s
    bothArgStr (Str x1 s1) (Str x2 s2) = Str (x1 `bothExnStr` x2) (s1 `bothStr` s2)

    bothExnStr :: ExnStr -> ExnStr -> ExnStr
    bothExnStr ExnStr ExnStr = ExnStr
    bothExnStr _      _      = VanStr

    bothStr :: StrDmd -> StrDmd -> StrDmd
    bothStr HyperStr _             = HyperStr
    bothStr HeadStr s              = s
    bothStr (SCall _)  HyperStr    = HyperStr
    bothStr (SCall s1) HeadStr     = SCall s1
    bothStr (SCall s1) (SCall s2)  = SCall (s1 `bothStr` s2)
    bothStr (SCall _)  (SProd _)   = HyperStr  -- Weird

    bothStr (SProd _)  HyperStr    = HyperStr
    bothStr (SProd s1) HeadStr     = SProd s1
    bothStr (SProd s1) (SProd s2)
        | length s1 == length s2   = mkSProd (zipWith bothArgStr s1 s2)
        | otherwise                = HyperStr  -- Weird
    bothStr (SProd _) (SCall _)    = HyperStr

    -- utility functions to deal with memory leaks
    seqStrDmd :: StrDmd -> ()
    seqStrDmd (SProd ds)   = seqStrDmdList ds
    seqStrDmd (SCall s)    = seqStrDmd s
    seqStrDmd _            = ()

    seqStrDmdList :: [ArgStr] -> ()
    seqStrDmdList [] = ()
    seqStrDmdList (d:ds) = seqArgStr d `seq` seqStrDmdList ds

    seqArgStr :: ArgStr -> ()
    seqArgStr Lazy      = ()
    seqArgStr (Str x s) = x `seq` seqStrDmd s

    -- Splitting polymorphic demands
    splitArgStrProdDmd :: Int -> ArgStr -> Maybe [ArgStr]
    splitArgStrProdDmd n Lazy      = Just (replicate n Lazy)
    splitArgStrProdDmd n (Str _ s) = splitStrProdDmd n s

    splitStrProdDmd :: Int -> StrDmd -> Maybe [ArgStr]
    splitStrProdDmd n HyperStr   = Just (replicate n strBot)
    splitStrProdDmd n HeadStr    = Just (replicate n strTop)
    splitStrProdDmd n (SProd ds) = WARN( not (ds `lengthIs` n),
                                         text "splitStrProdDmd" $$ ppr n $$ ppr ds )
                                   Just ds
    splitStrProdDmd _ (SCall {}) = Nothing
          -- This can happen when the programmer uses unsafeCoerce,
          -- and we don't then want to crash the compiler (Trac #9208)

    {-
    ************************************************************************
    *                                                                      *
                Absence domain
    *                                                                      *
    ************************************************************************

             Used
             /   \
         UCall   UProd
             \   /
             UHead
              |
      Count x -
            |
           Abs
    -}

    -- Domain for genuine usage
    data UseDmd
      = UCall Count UseDmd   -- Call demand for absence
                             -- Used only for values of function type

      | UProd [ArgUse]     -- Product
                             -- Used only for values of product type
                             -- See Note [Don't optimise UProd(Used) to Used]
                             -- [Invariant] Not all components are Abs
                             --             (in that case, use UHead)

      | UHead                -- May be used; but its sub-components are
                             -- definitely *not* used.  Roughly U(AAA)
                             -- Eg the usage of x in x `seq` e
                             -- A polymorphic demand: used for values of all types,
                             --                       including a type variable
                             -- Since (UCall _ Abs) is ill-typed, UHead doesn't
                             -- make sense for lambdas

      | Used                 -- May be used; and its sub-components may be used
                             -- Top of the lattice
      deriving ( Eq, Show )

    -- Extended usage demand for absence and counting
    type ArgUse = Use UseDmd

    data Use u
      = Abs             -- Definitely unused
                        -- Bottom of the lattice

      | Use Count u     -- May be used with some cardinality
      deriving ( Eq, Show )

    -- Abstract counting of usages
    data Count = One | Many
      deriving ( Eq, Show )

    -- Pretty-printing
    instance Outputable ArgUse where
      ppr Abs           = char 'A'
      ppr (Use Many a)   = ppr a
      ppr (Use One  a)   = char '1' <> char '*' <> ppr a

    instance Outputable UseDmd where
      ppr Used           = char 'U'
      ppr (UCall c a)    = char 'C' <> ppr c <> parens (ppr a)
      ppr UHead          = char 'H'
      ppr (UProd as)     = char 'U' <> parens (hcat (punctuate (char ',') (map ppr as)))

    instance Outputable Count where
      ppr One  = char '1'
      ppr Many = text ""

    useBot, useTop :: ArgUse
    useBot     = Abs
    useTop     = Use Many Used

    mkUCall :: Count -> UseDmd -> UseDmd
    --mkUCall c Used = Used c
    mkUCall c a  = UCall c a

    mkUProd :: [ArgUse] -> UseDmd
    mkUProd ux
      | all (== Abs) ux    = UHead
      | otherwise          = UProd ux

    lubCount :: Count -> Count -> Count
    lubCount _ Many = Many
    lubCount Many _ = Many
    lubCount x _    = x

    lubArgUse :: ArgUse -> ArgUse -> ArgUse
    lubArgUse Abs x                   = x
    lubArgUse x Abs                   = x
    lubArgUse (Use c1 a1) (Use c2 a2) = Use (lubCount c1 c2) (lubUse a1 a2)

    lubUse :: UseDmd -> UseDmd -> UseDmd
    lubUse UHead       u               = u
    lubUse (UCall c u) UHead           = UCall c u
    lubUse (UCall c1 u1) (UCall c2 u2) = UCall (lubCount c1 c2) (lubUse u1 u2)
    lubUse (UCall _ _) _               = Used
    lubUse (UProd ux) UHead            = UProd ux
    lubUse (UProd ux1) (UProd ux2)
         | length ux1 == length ux2    = UProd $ zipWith lubArgUse ux1 ux2
         | otherwise                   = Used
    lubUse (UProd {}) (UCall {})       = Used
    -- lubUse (UProd {}) Used             = Used
    lubUse (UProd ux) Used             = UProd (map (`lubArgUse` useTop) ux)
    lubUse Used       (UProd ux)       = UProd (map (`lubArgUse` useTop) ux)
    lubUse Used _                      = Used  -- Note [Used should win]

    -- `both` is different from `lub` in its treatment of counting; if
    -- `both` is computed for two used, the result always has
    --  cardinality `Many` (except for the inner demands of UCall demand -- [TODO] explain).
    --  Also,  x `bothUse` x /= x (for anything but Abs).

    bothArgUse :: ArgUse -> ArgUse -> ArgUse
    bothArgUse Abs x                   = x
    bothArgUse x Abs                   = x
    bothArgUse (Use _ a1) (Use _ a2)   = Use Many (bothUse a1 a2)

    bothUse :: UseDmd -> UseDmd -> UseDmd
    bothUse UHead       u               = u
    bothUse (UCall c u) UHead           = UCall c u

    -- Exciting special treatment of inner demand for call demands:
    --    use `lubUse` instead of `bothUse`!
    bothUse (UCall _ u1) (UCall _ u2)   = UCall Many (u1 `lubUse` u2)

    bothUse (UCall {}) _                = Used
    bothUse (UProd ux) UHead            = UProd ux
    bothUse (UProd ux1) (UProd ux2)
          | length ux1 == length ux2    = UProd $ zipWith bothArgUse ux1 ux2
          | otherwise                   = Used
    bothUse (UProd {}) (UCall {})       = Used
    -- bothUse (UProd {}) Used             = Used  -- Note [Used should win]
    bothUse Used (UProd ux)             = UProd (map (`bothArgUse` useTop) ux)
    bothUse (UProd ux) Used             = UProd (map (`bothArgUse` useTop) ux)
    bothUse Used _                      = Used  -- Note [Used should win]

    peelUseCall :: UseDmd -> Maybe (Count, UseDmd)
    peelUseCall (UCall c u)   = Just (c,u)
    peelUseCall _             = Nothing

    addCaseBndrDmd :: Demand    -- On the case binder
                   -> [Demand]  -- On the components of the constructor
                   -> [Demand]  -- Final demands for the components of the constructor
    -- See Note [Demand on case-alternative binders]
    addCaseBndrDmd (JD { sd = ms, ud = mu }) alt_dmds
      = case mu of
         Abs     -> alt_dmds
         Use _ u -> zipWith bothDmd alt_dmds (mkJointDmds ss us)
                 where
                    Just ss = splitArgStrProdDmd arity ms  -- Guaranteed not to be a call
                    Just us = splitUseProdDmd      arity u   -- Ditto
      where
        arity = length alt_dmds

    {- Note [Demand on case-alternative binders]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    The demand on a binder in a case alternative comes
      (a) From the demand on the binder itself
      (b) From the demand on the case binder
    Forgetting (b) led directly to Trac #10148.

    Example. Source code:
      f x@(p,_) = if p then foo x else True

      foo (p,True) = True
      foo (p,q)    = foo (q,p)

    After strictness analysis:
      f = \ (x_an1 [Dmd=<S(SL),1*U(U,1*U)>] :: (Bool, Bool)) ->
          case x_an1
          of wild_X7 [Dmd=<L,1*U(1*U,1*U)>]
          { (p_an2 [Dmd=<S,1*U>], ds_dnz [Dmd=<L,A>]) ->
          case p_an2 of _ {
            False -> GHC.Types.True;
            True -> foo wild_X7 }

    It's true that ds_dnz is *itself* absent, but the use of wild_X7 means
    that it is very much alive and demanded.  See Trac #10148 for how the
    consequences play out.

    This is needed even for non-product types, in case the case-binder
    is used but the components of the case alternative are not.

    Note [Don't optimise UProd(Used) to Used]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    These two UseDmds:
       UProd [Used, Used]   and    Used
    are semantically equivalent, but we do not turn the former into
    the latter, for a regrettable-subtle reason.  Suppose we did.
    then
      f (x,y) = (y,x)
    would get
      StrDmd = Str  = SProd [Lazy, Lazy]
      UseDmd = Used = UProd [Used, Used]
    But with the joint demand of <Str, Used> doesn't convey any clue
    that there is a product involved, and so the worthSplittingFun
    will not fire.  (We'd need to use the type as well to make it fire.)
    Moreover, consider
      g h p@(_,_) = h p
    This too would get <Str, Used>, but this time there really isn't any
    point in w/w since the components of the pair are not used at all.

    So the solution is: don't aggressively collapse UProd [Used,Used] to
    Used; intead leave it as-is. In effect we are using the UseDmd to do a
    little bit of boxity analysis.  Not very nice.

    Note [Used should win]
    ~~~~~~~~~~~~~~~~~~~~~~
    Both in lubUse and bothUse we want (Used `both` UProd us) to be Used.
    Why?  Because Used carries the implication the whole thing is used,
    box and all, so we don't want to w/w it.  If we use it both boxed and
    unboxed, then we are definitely using the box, and so we are quite
    likely to pay a reboxing cost.  So we make Used win here.

    Example is in the Buffer argument of GHC.IO.Handle.Internals.writeCharBuffer

    Baseline: (A) Not making Used win (UProd wins)
    Compare with: (B) making Used win for lub and both

                Min          -0.3%     -5.6%    -10.7%    -11.0%    -33.3%
                Max          +0.3%    +45.6%    +11.5%    +11.5%     +6.9%
     Geometric Mean          -0.0%     +0.5%     +0.3%     +0.2%     -0.8%

    Baseline: (B) Making Used win for both lub and both
    Compare with: (C) making Used win for both, but UProd win for lub

                Min          -0.1%     -0.3%     -7.9%     -8.0%     -6.5%
                Max          +0.1%     +1.0%    +21.0%    +21.0%     +0.5%
     Geometric Mean          +0.0%     +0.0%     -0.0%     -0.1%     -0.1%
    -}

    -- If a demand is used multiple times (i.e. reused), than any use-once
    -- mentioned there, that is not protected by a UCall, can happen many times.
    markReusedDmd :: ArgUse -> ArgUse
    markReusedDmd Abs         = Abs
    markReusedDmd (Use _ a)   = Use Many (markReused a)

    markReused :: UseDmd -> UseDmd
    markReused (UCall _ u)      = UCall Many u   -- No need to recurse here
    markReused (UProd ux)       = UProd (map markReusedDmd ux)
    markReused u                = u

    isUsedMU :: ArgUse -> Bool
    -- True <=> markReusedDmd d = d
    isUsedMU Abs          = True
    isUsedMU (Use One _)  = False
    isUsedMU (Use Many u) = isUsedU u

    isUsedU :: UseDmd -> Bool
    -- True <=> markReused d = d
    isUsedU Used           = True
    isUsedU UHead          = True
    isUsedU (UProd us)     = all isUsedMU us
    isUsedU (UCall One _)  = False
    isUsedU (UCall Many _) = True  -- No need to recurse

    -- Squashing usage demand demands
    seqUseDmd :: UseDmd -> ()
    seqUseDmd (UProd ds)   = seqArgUseList ds
    seqUseDmd (UCall c d)  = c `seq` seqUseDmd d
    seqUseDmd _            = ()

    seqArgUseList :: [ArgUse] -> ()
    seqArgUseList []     = ()
    seqArgUseList (d:ds) = seqArgUse d `seq` seqArgUseList ds

    seqArgUse :: ArgUse -> ()
    seqArgUse (Use c u)  = c `seq` seqUseDmd u
    seqArgUse _          = ()

    -- Splitting polymorphic Maybe-Used demands
    splitUseProdDmd :: Int -> UseDmd -> Maybe [ArgUse]
    splitUseProdDmd n Used        = Just (replicate n useTop)
    splitUseProdDmd n UHead       = Just (replicate n Abs)
    splitUseProdDmd n (UProd ds)  = WARN( not (ds `lengthIs` n),
                                          text "splitUseProdDmd" $$ ppr n
                                                                 $$ ppr ds )
                                    Just ds
    splitUseProdDmd _ (UCall _ _) = Nothing
          -- This can happen when the programmer uses unsafeCoerce,
          -- and we don't then want to crash the compiler (Trac #9208)

    useCount :: Use u -> Count
    useCount Abs         = One
    useCount (Use One _) = One
    useCount _           = Many

    {-
    ************************************************************************
    *                                                                      *
             Clean demand for Strictness and Usage
    *                                                                      *
    ************************************************************************

    This domain differst from JointDemand in the sence that pure absence
    is taken away, i.e., we deal *only* with non-absent demands.

    Note [Strict demands]
    ~~~~~~~~~~~~~~~~~~~~~
    isStrictDmd returns true only of demands that are
       both strict
       and  used
    In particular, it is False for <HyperStr, Abs>, which can and does
    arise in, say (Trac #7319)
       f x = raise# <some exception>
    Then 'x' is not used, so f gets strictness <HyperStr,Abs> -> .
    Now the w/w generates
       fx = let x <HyperStr,Abs> = absentError "unused"
            in raise <some exception>
    At this point we really don't want to convert to
       fx = case absentError "unused" of x -> raise <some exception>
    Since the program is going to diverge, this swaps one error for another,
    but it's really a bad idea to *ever* evaluate an absent argument.
    In Trac #7319 we get
       T7319.exe: Oops!  Entered absent arg w_s1Hd{v} [lid] [base:GHC.Base.String{tc 36u}]

    Note [Dealing with call demands]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Call demands are constructed and deconstructed coherently for
    strictness and absence. For instance, the strictness signature for the
    following function

    f :: (Int -> (Int, Int)) -> (Int, Bool)
    f g = (snd (g 3), True)

    should be: <L,C(U(AU))>m
    -}

    type CleanDemand = JointDmd StrDmd UseDmd
         -- A demand that is at least head-strict

    bothCleanDmd :: CleanDemand -> CleanDemand -> CleanDemand
    bothCleanDmd (JD { sd = s1, ud = a1}) (JD { sd = s2, ud = a2})
      = JD { sd = s1 `bothStr` s2, ud = a1 `bothUse` a2 }

    mkHeadStrict :: CleanDemand -> CleanDemand
    mkHeadStrict cd = cd { sd = HeadStr }

    mkOnceUsedDmd, mkManyUsedDmd :: CleanDemand -> Demand
    mkOnceUsedDmd (JD {sd = s,ud = a}) = JD { sd = Str VanStr s, ud = Use One a }
    mkManyUsedDmd (JD {sd = s,ud = a}) = JD { sd = Str VanStr s, ud = Use Many a }

    evalDmd :: Demand
    -- Evaluated strictly, and used arbitrarily deeply
    evalDmd = JD { sd = Str VanStr HeadStr, ud = useTop }

    mkProdDmd :: [Demand] -> CleanDemand
    mkProdDmd dx
      = JD { sd = mkSProd $ map getStrDmd dx
           , ud = mkUProd $ map getUseDmd dx }

    mkCallDmd :: CleanDemand -> CleanDemand
    mkCallDmd (JD {sd = d, ud = u})
      = JD { sd = mkSCall d, ud = mkUCall One u }

    -- See Note [Demand on the worker] in WorkWrap
    mkWorkerDemand :: Int -> Demand
    mkWorkerDemand n = JD { sd = Lazy, ud = Use One (go n) }
      where go 0 = Used
            go n = mkUCall One $ go (n-1)

    cleanEvalDmd :: CleanDemand
    cleanEvalDmd = JD { sd = HeadStr, ud = Used }

    cleanEvalProdDmd :: Arity -> CleanDemand
    cleanEvalProdDmd n = JD { sd = HeadStr, ud = UProd (replicate n useTop) }

    {-
    ************************************************************************
    *                                                                      *
               Demand: combining stricness and usage
    *                                                                      *
    ************************************************************************
    -}

    type Demand = JointDmd ArgStr ArgUse

    lubDmd :: Demand -> Demand -> Demand
    lubDmd (JD {sd = s1, ud = a1}) (JD {sd = s2, ud = a2})
     = JD { sd = s1 `lubArgStr` s2
          , ud = a1 `lubArgUse` a2 }

    bothDmd :: Demand -> Demand -> Demand
    bothDmd (JD {sd = s1, ud = a1}) (JD {sd = s2, ud = a2})
     = JD { sd = s1 `bothArgStr` s2
          , ud = a1 `bothArgUse` a2 }

    lazyApply1Dmd, lazyApply2Dmd, strictApply1Dmd, catchArgDmd :: Demand

    strictApply1Dmd = JD { sd = Str VanStr (SCall HeadStr)
                         , ud = Use Many (UCall One Used) }

    -- First argument of catchRetry# and catchSTM#:
    --    uses its arg once, applies it once
    --    and catches exceptions (the ExnStr) part
    catchArgDmd = JD { sd = Str ExnStr (SCall HeadStr)
                     , ud = Use One (UCall One Used) }

    lazyApply1Dmd = JD { sd = Lazy
                       , ud = Use One (UCall One Used) }

    -- Second argument of catch#:
    --    uses its arg at most once, applies it once
    --    but is lazy (might not be called at all)
    lazyApply2Dmd = JD { sd = Lazy
                       , ud = Use One (UCall One (UCall One Used)) }

    absDmd :: Demand
    absDmd = JD { sd = Lazy, ud = Abs }

    topDmd :: Demand
    topDmd = JD { sd = Lazy, ud = useTop }

    botDmd :: Demand
    botDmd = JD { sd = strBot, ud = useBot }

    seqDmd :: Demand
    seqDmd = JD { sd = Str VanStr HeadStr, ud = Use One UHead }

    oneifyDmd :: Demand -> Demand
    oneifyDmd (JD { sd = s, ud = Use _ a }) = JD { sd = s, ud = Use One a }
    oneifyDmd jd                            = jd

    isTopDmd :: Demand -> Bool
    -- Used to suppress pretty-printing of an uninformative demand
    isTopDmd (JD {sd = Lazy, ud = Use Many Used}) = True
    isTopDmd _                                    = False

    isAbsDmd :: Demand -> Bool
    isAbsDmd (JD {ud = Abs}) = True   -- The strictness part can be HyperStr
    isAbsDmd _               = False  -- for a bottom demand

    isSeqDmd :: Demand -> Bool
    isSeqDmd (JD {sd = Str VanStr HeadStr, ud = Use _ UHead}) = True
    isSeqDmd _                                                = False

    isUsedOnce :: Demand -> Bool
    isUsedOnce (JD { ud = a }) = case useCount a of
                                   One  -> True
                                   Many -> False

    -- More utility functions for strictness
    seqDemand :: Demand -> ()
    seqDemand (JD {sd = s, ud = u}) = seqArgStr s `seq` seqArgUse u

    seqDemandList :: [Demand] -> ()
    seqDemandList [] = ()
    seqDemandList (d:ds) = seqDemand d `seq` seqDemandList ds

    isStrictDmd :: Demand -> Bool
    -- See Note [Strict demands]
    isStrictDmd (JD {ud = Abs})  = False
    isStrictDmd (JD {sd = Lazy}) = False
    isStrictDmd _                = True

    isWeakDmd :: Demand -> Bool
    isWeakDmd (JD {sd = s, ud = a}) = isLazy s && isUsedMU a

    cleanUseDmd_maybe :: Demand -> Maybe UseDmd
    cleanUseDmd_maybe (JD { ud = Use _ u }) = Just u
    cleanUseDmd_maybe _                     = Nothing

    splitFVs :: Bool   -- Thunk
             -> DmdEnv -> (DmdEnv, DmdEnv)
    splitFVs is_thunk rhs_fvs
      | is_thunk  = nonDetFoldUFM_Directly add (emptyVarEnv, emptyVarEnv) rhs_fvs
                    -- It's OK to use nonDetFoldUFM_Directly because we
                    -- immediately forget the ordering by putting the elements
                    -- in the envs again
      | otherwise = partitionVarEnv isWeakDmd rhs_fvs
      where
        add uniq dmd@(JD { sd = s, ud = u }) (lazy_fv, sig_fv)
          | Lazy <- s = (addToUFM_Directly lazy_fv uniq dmd, sig_fv)
          | otherwise = ( addToUFM_Directly lazy_fv uniq (JD { sd = Lazy, ud = u })
                        , addToUFM_Directly sig_fv  uniq (JD { sd = s,    ud = Abs }) )

    data TypeShape = TsFun TypeShape
                   | TsProd [TypeShape]
                   | TsUnk

    instance Outputable TypeShape where
      ppr TsUnk        = text "TsUnk"
      ppr (TsFun ts)   = text "TsFun" <> parens (ppr ts)
      ppr (TsProd tss) = parens (hsep $ punctuate comma $ map ppr tss)

    trimToType :: Demand -> TypeShape -> Demand
    -- See Note [Trimming a demand to a type]
    trimToType (JD { sd = ms, ud = mu }) ts
      = JD (go_ms ms ts) (go_mu mu ts)
      where
        go_ms :: ArgStr -> TypeShape -> ArgStr
        go_ms Lazy      _  = Lazy
        go_ms (Str x s) ts = Str x (go_s s ts)

        go_s :: StrDmd -> TypeShape -> StrDmd
        go_s HyperStr    _            = HyperStr
        go_s (SCall s)   (TsFun ts)   = SCall (go_s s ts)
        go_s (SProd mss) (TsProd tss)
          | equalLength mss tss       = SProd (zipWith go_ms mss tss)
        go_s _           _            = HeadStr

        go_mu :: ArgUse -> TypeShape -> ArgUse
        go_mu Abs _ = Abs
        go_mu (Use c u) ts = Use c (go_u u ts)

        go_u :: UseDmd -> TypeShape -> UseDmd
        go_u UHead       _          = UHead
        go_u (UCall c u) (TsFun ts) = UCall c (go_u u ts)
        go_u (UProd mus) (TsProd tss)
          | equalLength mus tss      = UProd (zipWith go_mu mus tss)
        go_u _           _           = Used

    {-
    Note [Trimming a demand to a type]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Consider this:

      f :: a -> Bool
      f x = case ... of
              A g1 -> case (x |> g1) of (p,q) -> ...
              B    -> error "urk"

    where A,B are the constructors of a GADT.  We'll get a U(U,U) demand
    on x from the A branch, but that's a stupid demand for x itself, which
    has type 'a'. Indeed we get ASSERTs going off (notably in
    splitUseProdDmd, Trac #8569).

    Bottom line: we really don't want to have a binder whose demand is more
    deeply-nested than its type.  There are various ways to tackle this.
    When processing (x |> g1), we could "trim" the incoming demand U(U,U)
    to match x's type.  But I'm currently doing so just at the moment when
    we pin a demand on a binder, in DmdAnal.findBndrDmd.

    Note [Threshold demands]
    ~~~~~~~~~~~~~~~~~~~~~~~~
    Threshold usage demand is generated to figure out if
    cardinality-instrumented demands of a binding's free variables should
    be unleashed. See also [Aggregated demand for cardinality].

    Note [Replicating polymorphic demands]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Some demands can be considered as polymorphic. Generally, it is
    applicable to such beasts as tops, bottoms as well as Head-Used and
    Head-stricts demands. For instance,

    S ~ S(L, ..., L)

    Also, when top or bottom is occurred as a result demand, it in fact
    can be expanded to saturate a callee's arity.
    -}

    splitProdDmd_maybe :: Demand -> Maybe [Demand]
    -- Split a product into its components, iff there is any
    -- useful information to be extracted thereby
    -- The demand is not necessarily strict!
    splitProdDmd_maybe (JD { sd = s, ud = u })
      = case (s,u) of
          (Str _ (SProd sx), Use _ u) | Just ux <- splitUseProdDmd (length sx) u
                                      -> Just (mkJointDmds sx ux)
          (Str _ s, Use _ (UProd ux)) | Just sx <- splitStrProdDmd (length ux) s
                                      -> Just (mkJointDmds sx ux)
          (Lazy,    Use _ (UProd ux)) -> Just (mkJointDmds (replicate (length ux) Lazy) ux)
          _ -> Nothing

    {-
    ************************************************************************
    *                                                                      *
                       Demand results
    *                                                                      *
    ************************************************************************

    DmdResult:     Dunno CPRResult
                   /
               ThrowsExn
                 /
            Diverges

    CPRResult:         NoCPR
                       /    \
                RetProd    RetSum ConTag

    Product constructors return (Dunno (RetProd rs))
    In a fixpoint iteration, start from Diverges
    We have lubs, but not glbs; but that is ok.
    -}

    ------------------------------------------------------------------------
    -- Constructed Product Result
    ------------------------------------------------------------------------

    data Termination r
      = Diverges    -- Definitely diverges
      | ThrowsExn   -- Definitely throws an exception or diverges
      | Dunno r     -- Might diverge or converge
      deriving( Eq, Show )

    type DmdResult = Termination CPRResult

    data CPRResult = NoCPR          -- Top of the lattice
                   | RetProd        -- Returns a constructor from a product type
                   | RetSum ConTag  -- Returns a constructor from a data type
                   deriving( Eq, Show )

    lubCPR :: CPRResult -> CPRResult -> CPRResult
    lubCPR (RetSum t1) (RetSum t2)
      | t1 == t2                       = RetSum t1
    lubCPR RetProd     RetProd     = RetProd
    lubCPR _ _                     = NoCPR

    lubDmdResult :: DmdResult -> DmdResult -> DmdResult
    lubDmdResult Diverges       r              = r
    lubDmdResult ThrowsExn      Diverges       = ThrowsExn
    lubDmdResult ThrowsExn      r              = r
    lubDmdResult (Dunno c1)     Diverges       = Dunno c1
    lubDmdResult (Dunno c1)     ThrowsExn      = Dunno c1
    lubDmdResult (Dunno c1)     (Dunno c2)     = Dunno (c1 `lubCPR` c2)
    -- This needs to commute with defaultDmd, i.e.
    -- defaultDmd (r1 `lubDmdResult` r2) = defaultDmd r1 `lubDmd` defaultDmd r2
    -- (See Note [Default demand on free variables] for why)

    bothDmdResult :: DmdResult -> Termination () -> DmdResult
    -- See Note [Asymmetry of 'both' for DmdType and DmdResult]
    bothDmdResult _ Diverges   = Diverges
    bothDmdResult r ThrowsExn  = case r of { Diverges -> r; _ -> ThrowsExn }
    bothDmdResult r (Dunno {}) = r
    -- This needs to commute with defaultDmd, i.e.
    -- defaultDmd (r1 `bothDmdResult` r2) = defaultDmd r1 `bothDmd` defaultDmd r2
    -- (See Note [Default demand on free variables] for why)

    instance Outputable r => Outputable (Termination r) where
      ppr Diverges      = char 'b'
      ppr ThrowsExn     = char 'x'
      ppr (Dunno c)     = ppr c

    instance Outputable CPRResult where
      ppr NoCPR        = empty
      ppr (RetSum n)   = char 'm' <> int n
      ppr RetProd      = char 'm'

    seqDmdResult :: DmdResult -> ()
    seqDmdResult Diverges  = ()
    seqDmdResult ThrowsExn = ()
    seqDmdResult (Dunno c) = seqCPRResult c

    seqCPRResult :: CPRResult -> ()
    seqCPRResult NoCPR        = ()
    seqCPRResult (RetSum n)   = n `seq` ()
    seqCPRResult RetProd      = ()

    ------------------------------------------------------------------------
    -- Combined demand result                                             --
    ------------------------------------------------------------------------

    -- [cprRes] lets us switch off CPR analysis
    -- by making sure that everything uses TopRes
    topRes, exnRes, botRes :: DmdResult
    topRes = Dunno NoCPR
    exnRes = ThrowsExn
    botRes = Diverges

    cprSumRes :: ConTag -> DmdResult
    cprSumRes tag = Dunno $ RetSum tag

    cprProdRes :: [DmdType] -> DmdResult
    cprProdRes _arg_tys = Dunno $ RetProd

    vanillaCprProdRes :: Arity -> DmdResult
    vanillaCprProdRes _arity = Dunno $ RetProd

    isTopRes :: DmdResult -> Bool
    isTopRes (Dunno NoCPR) = True
    isTopRes _             = False

    isBotRes :: DmdResult -> Bool
    -- True if the result diverges or throws an exception
    isBotRes Diverges   = True
    isBotRes ThrowsExn  = True
    isBotRes (Dunno {}) = False

    trimCPRInfo :: Bool -> Bool -> DmdResult -> DmdResult
    trimCPRInfo trim_all trim_sums res
      = trimR res
      where
        trimR (Dunno c) = Dunno (trimC c)
        trimR res       = res

        trimC (RetSum n)   | trim_all || trim_sums = NoCPR
                           | otherwise             = RetSum n
        trimC RetProd      | trim_all  = NoCPR
                           | otherwise = RetProd
        trimC NoCPR = NoCPR

    returnsCPR_maybe :: DmdResult -> Maybe ConTag
    returnsCPR_maybe (Dunno c) = retCPR_maybe c
    returnsCPR_maybe _         = Nothing

    retCPR_maybe :: CPRResult -> Maybe ConTag
    retCPR_maybe (RetSum t)  = Just t
    retCPR_maybe RetProd     = Just fIRST_TAG
    retCPR_maybe NoCPR       = Nothing

    -- See Notes [Default demand on free variables]
    -- and [defaultDmd vs. resTypeArgDmd]
    defaultDmd :: Termination r -> Demand
    defaultDmd (Dunno {}) = absDmd
    defaultDmd _          = botDmd  -- Diverges or ThrowsExn

    resTypeArgDmd :: Termination r -> Demand
    -- TopRes and BotRes are polymorphic, so that
    --      BotRes === (Bot -> BotRes) === ...
    --      TopRes === (Top -> TopRes) === ...
    -- This function makes that concrete
    -- Also see Note [defaultDmd vs. resTypeArgDmd]
    resTypeArgDmd (Dunno _) = topDmd
    resTypeArgDmd _         = botDmd   -- Diverges or ThrowsExn

    {-
    Note [defaultDmd and resTypeArgDmd]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    These functions are similar: They express the demand on something not
    explicitly mentioned in the environment resp. the argument list. Yet they are
    different:
     * Variables not mentioned in the free variables environment are definitely
       unused, so we can use absDmd there.
     * Further arguments *can* be used, of course. Hence topDmd is used.

    Note [Worthy functions for Worker-Wrapper split]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    For non-bottoming functions a worker-wrapper transformation takes into
    account several possibilities to decide if the function is worthy for
    splitting:

    1. The result is of product type and the function is strict in some
    (or even all) of its arguments. The check that the argument is used is
    more of sanity nature, since strictness implies usage. Example:

    f :: (Int, Int) -> Int
    f p = (case p of (a,b) -> a) + 1

    should be splitted to

    f :: (Int, Int) -> Int
    f p = case p of (a,b) -> $wf a

    $wf :: Int -> Int
    $wf a = a + 1

    2. Sometimes it also makes sense to perform a WW split if the
    strictness analysis cannot say for sure if the function is strict in
    components of its argument. Then we reason according to the inferred
    usage information: if the function uses its product argument's
    components, the WW split can be beneficial. Example:

    g :: Bool -> (Int, Int) -> Int
    g c p = case p of (a,b) ->
              if c then a else b

    The function g is strict in is argument p and lazy in its
    components. However, both components are used in the RHS. The idea is
    since some of the components (both in this case) are used in the
    right-hand side, the product must presumable be taken apart.

    Therefore, the WW transform splits the function g to

    g :: Bool -> (Int, Int) -> Int
    g c p = case p of (a,b) -> $wg c a b

    $wg :: Bool -> Int -> Int -> Int
    $wg c a b = if c then a else b

    3. If an argument is absent, it would be silly to pass it to a
    function, hence the worker with reduced arity is generated.

    Note [Worker-wrapper for bottoming functions]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    We used not to split if the result is bottom.
    [Justification:  there's no efficiency to be gained.]

    But it's sometimes bad not to make a wrapper.  Consider
            fw = \x# -> let x = I# x# in case e of
                                            p1 -> error_fn x
                                            p2 -> error_fn x
                                            p3 -> the real stuff
    The re-boxing code won't go away unless error_fn gets a wrapper too.
    [We don't do reboxing now, but in general it's better to pass an
    unboxed thing to f, and have it reboxed in the error cases....]

    However we *don't* want to do this when the argument is not actually
    taken apart in the function at all.  Otherwise we risk decomposing a
    massive tuple which is barely used.  Example:

            f :: ((Int,Int) -> String) -> (Int,Int) -> a
            f g pr = error (g pr)

            main = print (f fst (1, error "no"))

    Here, f does not take 'pr' apart, and it's stupid to do so.
    Imagine that it had millions of fields. This actually happened
    in GHC itself where the tuple was DynFlags

    ************************************************************************
    *                                                                      *
               Demand environments and types
    *                                                                      *
    ************************************************************************
    -}

    type DmdEnv = VarEnv Demand   -- See Note [Default demand on free variables]

    data DmdType = DmdType
                      DmdEnv        -- Demand on explicitly-mentioned
                                    --      free variables
                      [Demand]      -- Demand on arguments
                      DmdResult     -- See [Nature of result demand]

    {-
    Note [Nature of result demand]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    A DmdResult contains information about termination (currently distinguishing
    definite divergence and no information; it is possible to include definite
    convergence here), and CPR information about the result.

    The semantics of this depends on whether we are looking at a DmdType, i.e. the
    demand put on by an expression _under a specific incoming demand_ on its
    environment, or at a StrictSig describing a demand transformer.

    For a
     * DmdType, the termination information is true given the demand it was
       generated with, while for
     * a StrictSig it holds after applying enough arguments.

    The CPR information, though, is valid after the number of arguments mentioned
    in the type is given. Therefore, when forgetting the demand on arguments, as in
    dmdAnalRhs, this needs to be considere (via removeDmdTyArgs).

    Consider
      b2 x y = x `seq` y `seq` error (show x)
    this has a strictness signature of
      <S><S>b
    meaning that "b2 `seq` ()" and "b2 1 `seq` ()" might well terminate, but
    for "b2 1 2 `seq` ()" we get definite divergence.

    For comparison,
      b1 x = x `seq` error (show x)
    has a strictness signature of
      <S>b
    and "b1 1 `seq` ()" is known to terminate.

    Now consider a function h with signature "<C(S)>", and the expression
      e1 = h b1
    now h puts a demand of <C(S)> onto its argument, and the demand transformer
    turns it into
      <S>b
    Now the DmdResult "b" does apply to us, even though "b1 `seq` ()" does not
    diverge, and we do not anything being passed to b.

    Note [Asymmetry of 'both' for DmdType and DmdResult]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    'both' for DmdTypes is *asymmetrical*, because there is only one
    result!  For example, given (e1 e2), we get a DmdType dt1 for e1, use
    its arg demand to analyse e2 giving dt2, and then do (dt1 `bothType` dt2).
    Similarly with
      case e of { p -> rhs }
    we get dt_scrut from the scrutinee and dt_rhs from the RHS, and then
    compute (dt_rhs `bothType` dt_scrut).

    We
     1. combine the information on the free variables,
     2. take the demand on arguments from the first argument
     3. combine the termination results, but
     4. take CPR info from the first argument.

    3 and 4 are implementd in bothDmdResult.
    -}

    -- Equality needed for fixpoints in DmdAnal
    instance Eq DmdType where
      (==) (DmdType fv1 ds1 res1)
           (DmdType fv2 ds2 res2) = nonDetUFMToList fv1 == nonDetUFMToList fv2
             -- It's OK to use nonDetUFMToList here because we're testing for
             -- equality and even though the lists will be in some arbitrary
             -- Unique order, it is the same order for both
                                  && ds1 == ds2 && res1 == res2

    lubDmdType :: DmdType -> DmdType -> DmdType
    lubDmdType d1 d2
      = DmdType lub_fv lub_ds lub_res
      where
        n = max (dmdTypeDepth d1) (dmdTypeDepth d2)
        (DmdType fv1 ds1 r1) = ensureArgs n d1
        (DmdType fv2 ds2 r2) = ensureArgs n d2

        lub_fv  = plusVarEnv_CD lubDmd fv1 (defaultDmd r1) fv2 (defaultDmd r2)
        lub_ds  = zipWithEqual "lubDmdType" lubDmd ds1 ds2
        lub_res = lubDmdResult r1 r2

    {-
    Note [The need for BothDmdArg]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Previously, the right argument to bothDmdType, as well as the return value of
    dmdAnalStar via postProcessDmdType, was a DmdType. But bothDmdType only needs
    to know about the free variables and termination information, but nothing about
    the demand put on arguments, nor cpr information. So we make that explicit by
    only passing the relevant information.
    -}

    type BothDmdArg = (DmdEnv, Termination ())

    mkBothDmdArg :: DmdEnv -> BothDmdArg
    mkBothDmdArg env = (env, Dunno ())

    toBothDmdArg :: DmdType -> BothDmdArg
    toBothDmdArg (DmdType fv _ r) = (fv, go r)
      where
        go (Dunno {}) = Dunno ()
        go ThrowsExn  = ThrowsExn
        go Diverges   = Diverges

    bothDmdType :: DmdType -> BothDmdArg -> DmdType
    bothDmdType (DmdType fv1 ds1 r1) (fv2, t2)
        -- See Note [Asymmetry of 'both' for DmdType and DmdResult]
        -- 'both' takes the argument/result info from its *first* arg,
        -- using its second arg just for its free-var info.
      = DmdType (plusVarEnv_CD bothDmd fv1 (defaultDmd r1) fv2 (defaultDmd t2))
                ds1
                (r1 `bothDmdResult` t2)

    instance Outputable DmdType where
      ppr (DmdType fv ds res)
        = hsep [hcat (map ppr ds) <> ppr res,
                if null fv_elts then empty
                else braces (fsep (map pp_elt fv_elts))]
        where
          pp_elt (uniq, dmd) = ppr uniq <> text "->" <> ppr dmd
          fv_elts = nonDetUFMToList fv
            -- It's OK to use nonDetUFMToList here because we only do it for
            -- pretty printing

    emptyDmdEnv :: VarEnv Demand
    emptyDmdEnv = emptyVarEnv

    -- nopDmdType is the demand of doing nothing
    -- (lazy, absent, no CPR information, no termination information).
    -- Note that it is ''not'' the top of the lattice (which would be "may use everything"),
    -- so it is (no longer) called topDmd
    nopDmdType, botDmdType, exnDmdType :: DmdType
    nopDmdType = DmdType emptyDmdEnv [] topRes
    botDmdType = DmdType emptyDmdEnv [] botRes
    exnDmdType = DmdType emptyDmdEnv [] exnRes

    cprProdDmdType :: Arity -> DmdType
    cprProdDmdType arity
      = DmdType emptyDmdEnv [] (vanillaCprProdRes arity)

    isTopDmdType :: DmdType -> Bool
    isTopDmdType (DmdType env [] res)
      | isTopRes res && isEmptyVarEnv env = True
    isTopDmdType _                        = False

    mkDmdType :: DmdEnv -> [Demand] -> DmdResult -> DmdType
    mkDmdType fv ds res = DmdType fv ds res

    dmdTypeDepth :: DmdType -> Arity
    dmdTypeDepth (DmdType _ ds _) = length ds

    -- Remove any demand on arguments. This is used in dmdAnalRhs on the body
    removeDmdTyArgs :: DmdType -> DmdType
    removeDmdTyArgs = ensureArgs 0

    -- This makes sure we can use the demand type with n arguments,
    -- It extends the argument list with the correct resTypeArgDmd
    -- It also adjusts the DmdResult: Divergence survives additional arguments,
    -- CPR information does not (and definite converge also would not).
    ensureArgs :: Arity -> DmdType -> DmdType
    ensureArgs n d | n == depth = d
                   | otherwise  = DmdType fv ds' r'
      where depth = dmdTypeDepth d
            DmdType fv ds r = d

            ds' = take n (ds ++ repeat (resTypeArgDmd r))
            r' = case r of    -- See [Nature of result demand]
                  Dunno _ -> topRes
                  _       -> r

    seqDmdType :: DmdType -> ()
    seqDmdType (DmdType env ds res) =
      seqDmdEnv env `seq` seqDemandList ds `seq` seqDmdResult res `seq` ()

    seqDmdEnv :: DmdEnv -> ()
    seqDmdEnv env = seqEltsUFM seqDemandList env

    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)

    -- When e is evaluated after executing an IO action, and d is e's demand, then
    -- what of this demand should we consider, given that the IO action can cleanly
    -- exit?
    -- * We have to kill all strictness demands (i.e. lub with a lazy demand)
    -- * We can keep usage information (i.e. lub with an absent demand)
    -- * We have to kill definite divergence
    -- * We can keep CPR information.
    -- See Note [IO hack in the demand analyser] in DmdAnal
    deferAfterIO :: DmdType -> DmdType
    deferAfterIO d@(DmdType _ _ res) =
        case d `lubDmdType` nopDmdType of
            DmdType fv ds _ -> DmdType fv ds (defer_res res)
      where
      defer_res r@(Dunno {}) = r
      defer_res _            = topRes  -- Diverges and ThrowsExn

    strictenDmd :: Demand -> CleanDemand
    strictenDmd (JD { sd = s, ud = u})
      = JD { sd = poke_s s, ud = poke_u u }
      where
        poke_s Lazy      = HeadStr
        poke_s (Str _ s) = s
        poke_u Abs       = UHead
        poke_u (Use _ u) = u

    -- Deferring and peeling

    type DmdShell   -- Describes the "outer shell"
                    -- of a Demand
       = JointDmd (Str ()) (Use ())

    toCleanDmd :: Demand -> Type -> (DmdShell, CleanDemand)
    -- Splits a Demand into its "shell" and the inner "clean demand"
    toCleanDmd (JD { sd = s, ud = u }) expr_ty
      = (JD { sd = ss, ud = us }, JD { sd = s', ud = u' })
        -- See Note [Analyzing with lazy demand and lambdas]
      where
        (ss, s') = case s of
                    Str x s'           -> (Str x      (), s')
                    Lazy | is_unlifted -> (Str VanStr (), HeadStr)
                         | otherwise   -> (Lazy,          HeadStr)

        (us, u') = case u of
                     Use c u'          -> (Use c   (), u')
                     Abs | is_unlifted -> (Use One (), Used)
                         | otherwise   -> (Abs,        Used)

        is_unlifted = isUnliftedType expr_ty
        -- See Note [Analysing with absent demand]

    -- This is used in dmdAnalStar when post-processing
    -- a function's argument demand. So we only care about what
    -- does to free variables, and whether it terminates.
    -- see Note [The need for BothDmdArg]
    postProcessDmdType :: DmdShell -> DmdType -> BothDmdArg
    postProcessDmdType du@(JD { sd = ss }) (DmdType fv _ res_ty)
        = (postProcessDmdEnv du fv, term_info)
        where
           term_info = case postProcessDmdResult ss res_ty of
                         Dunno _   -> Dunno ()
                         ThrowsExn -> ThrowsExn
                         Diverges  -> Diverges

    postProcessDmdResult :: Str () -> DmdResult -> DmdResult
    postProcessDmdResult Lazy           _         = topRes
    postProcessDmdResult (Str ExnStr _) ThrowsExn = topRes  -- Key point!
    -- Note that only ThrowsExn results can be caught, not Diverges
    postProcessDmdResult _              res       = res

    postProcessDmdEnv :: DmdShell -> DmdEnv -> DmdEnv
    postProcessDmdEnv ds@(JD { sd = ss, ud = us }) env
      | Abs <- us       = emptyDmdEnv
        -- In this case (postProcessDmd ds) == id; avoid a redundant rebuild
        -- of the environment. Be careful, bad things will happen if this doesn't
        -- match postProcessDmd (see #13977).
      | Str VanStr _ <- ss
      , Use One _ <- us = env
      | otherwise       = mapVarEnv (postProcessDmd ds) env
      -- For the Absent case just discard all usage information
      -- We only processed the thing at all to analyse the body
      -- See Note [Always analyse in virgin pass]

    reuseEnv :: DmdEnv -> DmdEnv
    reuseEnv = mapVarEnv (postProcessDmd
                            (JD { sd = Str VanStr (), ud = Use Many () }))

    postProcessUnsat :: DmdShell -> DmdType -> DmdType
    postProcessUnsat ds@(JD { sd = ss }) (DmdType fv args res_ty)
      = DmdType (postProcessDmdEnv ds fv)
                (map (postProcessDmd ds) args)
                (postProcessDmdResult ss res_ty)

    postProcessDmd :: DmdShell -> Demand -> Demand
    postProcessDmd (JD { sd = ss, ud = us }) (JD { sd = s, ud = a})
      = JD { sd = s', ud = a' }
      where
        s' = case ss of
               Lazy         -> Lazy
               Str ExnStr _ -> markExnStr s
               Str VanStr _ -> s
        a' = case us of
               Abs        -> Abs
               Use Many _ -> markReusedDmd a
               Use One  _ -> a

    markExnStr :: ArgStr -> ArgStr
    markExnStr (Str VanStr s) = Str ExnStr s
    markExnStr s              = s

    -- Peels one call level from the demand, and also returns
    -- whether it was unsaturated (separately for strictness and usage)
    peelCallDmd :: CleanDemand -> (CleanDemand, DmdShell)
    -- Exploiting the fact that
    -- on the strictness side      C(B) = B
    -- and on the usage side       C(U) = U
    peelCallDmd (JD {sd = s, ud = u})
      = (JD { sd = s', ud = u' }, JD { sd = ss, ud = us })
      where
        (s', ss) = case s of
                     SCall s' -> (s',       Str VanStr ())
                     HyperStr -> (HyperStr, Str VanStr ())
                     _        -> (HeadStr,  Lazy)
        (u', us) = case u of
                     UCall c u' -> (u',   Use c    ())
                     _          -> (Used, Use Many ())
           -- The _ cases for usage includes UHead which seems a bit wrong
           -- because the body isn't used at all!
           -- c.f. the Abs case in toCleanDmd

    -- Peels that multiple nestings of calls clean demand and also returns
    -- whether it was unsaturated (separately for strictness and usage
    -- see Note [Demands from unsaturated function calls]
    peelManyCalls :: Int -> CleanDemand -> DmdShell
    peelManyCalls n (JD { sd = str, ud = abs })
      = JD { sd = go_str n str, ud = go_abs n abs }
      where
        go_str :: Int -> StrDmd -> Str ()  -- True <=> unsaturated, defer
        go_str 0 _          = Str VanStr ()
        go_str _ HyperStr   = Str VanStr () -- == go_str (n-1) HyperStr, as HyperStr = Call(HyperStr)
        go_str n (SCall d') = go_str (n-1) d'
        go_str _ _          = Lazy

        go_abs :: Int -> UseDmd -> Use ()      -- Many <=> unsaturated, or at least
        go_abs 0 _              = Use One ()   --          one UCall Many in the demand
        go_abs n (UCall One d') = go_abs (n-1) d'
        go_abs _ _              = Use Many ()

    {-
    Note [Demands from unsaturated function calls]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    Consider a demand transformer d1 -> d2 -> r for f.
    If a sufficiently detailed demand is fed into this transformer,
    e.g <C(C(S)), C1(C1(S))> arising from "f x1 x2" in a strict, use-once context,
    then d1 and d2 is precisely the demand unleashed onto x1 and x2 (similar for
    the free variable environment) and furthermore the result information r is the
    one we want to use.

    An anonymous lambda is also an unsaturated function all (needs one argument,
    none given), so this applies to that case as well.

    But the demand fed into f might be less than <C(C(S)), C1(C1(S))>. There are a few cases:
     * Not enough demand on the strictness side:
       - In that case, we need to zap all strictness in the demand on arguments and
         free variables.
       - Furthermore, we remove CPR information. It could be left, but given the incoming
         demand is not enough to evaluate so far we just do not bother.
       - And finally termination information: If r says that f diverges for sure,
         then this holds when the demand guarantees that two arguments are going to
         be passed. If the demand is lower, we may just as well converge.
         If we were tracking definite convegence, than that would still hold under
         a weaker demand than expected by the demand transformer.
     * Not enough demand from the usage side: The missing usage can be expanded
       using UCall Many, therefore this is subsumed by the third case:
     * At least one of the uses has a cardinality of Many.
       - Even if f puts a One demand on any of its argument or free variables, if
         we call f multiple times, we may evaluate this argument or free variable
         multiple times. So forget about any occurrence of "One" in the demand.

    In dmdTransformSig, we call peelManyCalls to find out if we are in any of these
    cases, and then call postProcessUnsat to reduce the demand appropriately.

    Similarly, dmdTransformDictSelSig and dmdAnal, when analyzing a Lambda, use
    peelCallDmd, which peels only one level, but also returns the demand put on the
    body of the function.
    -}

    peelFV :: DmdType -> Var -> (DmdType, Demand)
    peelFV (DmdType fv ds res) id = -- pprTrace "rfv" (ppr id <+> ppr dmd $$ ppr fv)
                                   (DmdType fv' ds res, dmd)
      where
      fv' = fv `delVarEnv` id
      -- See Note [Default demand on free variables]
      dmd  = lookupVarEnv fv id `orElse` defaultDmd res

    addDemand :: Demand -> DmdType -> DmdType
    addDemand dmd (DmdType fv ds res) = DmdType fv (dmd:ds) res

    findIdDemand :: DmdType -> Var -> Demand
    findIdDemand (DmdType fv _ res) id
      = lookupVarEnv fv id `orElse` defaultDmd res

    {-
    Note [Default demand on free variables]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    If the variable is not mentioned in the environment of a demand type,
    its demand is taken to be a result demand of the type.
        For the stricness component,
         if the result demand is a Diverges, then we use HyperStr
                                             else we use Lazy
        For the usage component, we use Absent.
    So we use either absDmd or botDmd.

    Also note the equations for lubDmdResult (resp. bothDmdResult) noted there.

    Note [Always analyse in virgin pass]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Tricky point: make sure that we analyse in the 'virgin' pass. Consider
       rec { f acc x True  = f (...rec { g y = ...g... }...)
             f acc x False = acc }
    In the virgin pass for 'f' we'll give 'f' a very strict (bottom) type.
    That might mean that we analyse the sub-expression containing the
    E = "...rec g..." stuff in a bottom demand.  Suppose we *didn't analyse*
    E, but just returned botType.

    Then in the *next* (non-virgin) iteration for 'f', we might analyse E
    in a weaker demand, and that will trigger doing a fixpoint iteration
    for g.  But *because it's not the virgin pass* we won't start g's
    iteration at bottom.  Disaster.  (This happened in $sfibToList' of
    nofib/spectral/fibheaps.)

    So in the virgin pass we make sure that we do analyse the expression
    at least once, to initialise its signatures.

    Note [Analyzing with lazy demand and lambdas]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    The insight for analyzing lambdas follows from the fact that for
    strictness S = C(L). This polymorphic expansion is critical for
    cardinality analysis of the following example:

    {-# NOINLINE build #-}
    build g = (g (:) [], g (:) [])

    h c z = build (\x ->
                    let z1 = z ++ z
                     in if c
                        then \y -> x (y ++ z1)
                        else \y -> x (z1 ++ y))

    One can see that `build` assigns to `g` demand <L,C(C1(U))>.
    Therefore, when analyzing the lambda `(\x -> ...)`, we
    expect each lambda \y -> ... to be annotated as "one-shot"
    one. Therefore (\x -> \y -> x (y ++ z)) should be analyzed with a
    demand <C(C(..), C(C1(U))>.

    This is achieved by, first, converting the lazy demand L into the
    strict S by the second clause of the analysis.

    Note [Analysing with absent demand]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Suppose we analyse an expression with demand <L,A>.  The "A" means
    "absent", so this expression will never be needed.  What should happen?
    There are several wrinkles:

    * We *do* want to analyse the expression regardless.
      Reason: Note [Always analyse in virgin pass]

      But we can post-process the results to ignore all the usage
      demands coming back. This is done by postProcessDmdType.

    * But in the case of an *unlifted type* we must be extra careful,
      because unlifted values are evaluated even if they are not used.
      Example (see Trac #9254):
         f :: (() -> (# Int#, () #)) -> ()
              -- Strictness signature is
              --    <C(S(LS)), 1*C1(U(A,1*U()))>
              -- I.e. calls k, but discards first component of result
         f k = case k () of (# _, r #) -> r

         g :: Int -> ()
         g y = f (\n -> (# case y of I# y2 -> y2, n #))

      Here f's strictness signature says (correctly) that it calls its
      argument function and ignores the first component of its result.
      This is correct in the sense that it'd be fine to (say) modify the
      function so that always returned 0# in the first component.

      But in function g, we *will* evaluate the 'case y of ...', because
      it has type Int#.  So 'y' will be evaluated.  So we must record this
      usage of 'y', else 'g' will say 'y' is absent, and will w/w so that
      'y' is bound to an aBSENT_ERROR thunk.

      An alternative would be to replace the 'case y of ...' with (say) 0#,
      but I have not tried that. It's not a common situation, but it is
      not theoretical: unsafePerformIO's implementation is very very like
      'f' above.

    ************************************************************************
    *                                                                      *
                         Demand signatures
    *                                                                      *
    ************************************************************************

    In a let-bound Id we record its strictness info.
    In principle, this strictness info is a demand transformer, mapping
    a demand on the Id into a DmdType, which gives
            a) the free vars of the Id's value
            b) the Id's arguments
            c) an indication of the result of applying
               the Id to its arguments

    However, in fact we store in the Id an extremely emascuated demand
    transfomer, namely

                    a single DmdType
    (Nevertheless we dignify StrictSig as a distinct type.)

    This DmdType gives the demands unleashed by the Id when it is applied
    to as many arguments as are given in by the arg demands in the DmdType.
    Also see Note [Nature of result demand] for the meaning of a DmdResult in a
    strictness signature.

    If an Id is applied to less arguments than its arity, it means that
    the demand on the function at a call site is weaker than the vanilla
    call demand, used for signature inference. Therefore we place a top
    demand on all arguments. Otherwise, the demand is specified by Id's
    signature.

    For example, the demand transformer described by the demand signature
            StrictSig (DmdType {x -> <S,1*U>} <L,A><L,U(U,U)>m)
    says that when the function is applied to two arguments, it
    unleashes demand <S,1*U> on the free var x, <L,A> on the first arg,
    and <L,U(U,U)> on the second, then returning a constructor.

    If this same function is applied to one arg, all we can say is that it
    uses x with <L,U>, and its arg with demand <L,U>.
    -}

    newtype StrictSig = StrictSig DmdType
                      deriving( Eq )

    instance Outputable StrictSig where
       ppr (StrictSig ty) = ppr ty

    -- Used for printing top-level strictness pragmas in interface files
    pprIfaceStrictSig :: StrictSig -> SDoc
    pprIfaceStrictSig (StrictSig (DmdType _ dmds res))
      = hcat (map ppr dmds) <> ppr res

    mkStrictSig :: DmdType -> StrictSig
    mkStrictSig dmd_ty = StrictSig dmd_ty

    mkClosedStrictSig :: [Demand] -> DmdResult -> StrictSig
    mkClosedStrictSig ds res = mkStrictSig (DmdType emptyDmdEnv ds res)

    splitStrictSig :: StrictSig -> ([Demand], DmdResult)
    splitStrictSig (StrictSig (DmdType _ dmds res)) = (dmds, res)

    increaseStrictSigArity :: Int -> StrictSig -> StrictSig
    -- Add extra arguments to a strictness signature
    increaseStrictSigArity arity_increase (StrictSig (DmdType env dmds res))
      = StrictSig (DmdType env (replicate arity_increase topDmd ++ dmds) res)

    isTopSig :: StrictSig -> Bool
    isTopSig (StrictSig ty) = isTopDmdType ty

    hasDemandEnvSig :: StrictSig -> Bool
    hasDemandEnvSig (StrictSig (DmdType env _ _)) = not (isEmptyVarEnv env)

    strictSigDmdEnv :: StrictSig -> DmdEnv
    strictSigDmdEnv (StrictSig (DmdType env _ _)) = env

    isBottomingSig :: StrictSig -> Bool
    -- True if the signature diverges or throws an exception
    isBottomingSig (StrictSig (DmdType _ _ res)) = isBotRes res

    nopSig, botSig, exnSig :: StrictSig
    nopSig = StrictSig nopDmdType
    botSig = StrictSig botDmdType
    exnSig = StrictSig exnDmdType

    cprProdSig :: Arity -> StrictSig
    cprProdSig arity = StrictSig (cprProdDmdType arity)

    seqStrictSig :: StrictSig -> ()
    seqStrictSig (StrictSig ty) = seqDmdType ty

    dmdTransformSig :: StrictSig -> CleanDemand -> DmdType
    -- (dmdTransformSig fun_sig dmd) considers a call to a function whose
    -- signature is fun_sig, with demand dmd.  We return the demand
    -- that the function places on its context (eg its args)
    dmdTransformSig (StrictSig dmd_ty@(DmdType _ arg_ds _)) cd
      = postProcessUnsat (peelManyCalls (length arg_ds) cd) dmd_ty
        -- see Note [Demands from unsaturated function calls]

    dmdTransformDataConSig :: Arity -> StrictSig -> CleanDemand -> DmdType
    -- Same as dmdTransformSig but for a data constructor (worker),
    -- which has a special kind of demand transformer.
    -- If the constructor is saturated, we feed the demand on
    -- the result into the constructor arguments.
    dmdTransformDataConSig arity (StrictSig (DmdType _ _ con_res))
                                 (JD { sd = str, ud = abs })
      | Just str_dmds <- go_str arity str
      , Just abs_dmds <- go_abs arity abs
      = DmdType emptyDmdEnv (mkJointDmds str_dmds abs_dmds) con_res
                    -- Must remember whether it's a product, hence con_res, not TopRes

      | otherwise   -- Not saturated
      = nopDmdType
      where
        go_str 0 dmd        = splitStrProdDmd arity dmd
        go_str n (SCall s') = go_str (n-1) s'
        go_str n HyperStr   = go_str (n-1) HyperStr
        go_str _ _          = Nothing

        go_abs 0 dmd            = splitUseProdDmd arity dmd
        go_abs n (UCall One u') = go_abs (n-1) u'
        go_abs _ _              = Nothing

    dmdTransformDictSelSig :: StrictSig -> CleanDemand -> DmdType
    -- Like dmdTransformDataConSig, we have a special demand transformer
    -- for dictionary selectors.  If the selector is saturated (ie has one
    -- argument: the dictionary), we feed the demand on the result into
    -- the indicated dictionary component.
    dmdTransformDictSelSig (StrictSig (DmdType _ [dict_dmd] _)) cd
       | (cd',defer_use) <- peelCallDmd cd
       , Just jds <- splitProdDmd_maybe dict_dmd
       = postProcessUnsat defer_use $
         DmdType emptyDmdEnv [mkOnceUsedDmd $ mkProdDmd $ map (enhance cd') jds] topRes
       | otherwise
       = nopDmdType              -- See Note [Demand transformer for a dictionary selector]
      where
        enhance cd old | isAbsDmd old = old
                       | otherwise    = mkOnceUsedDmd cd  -- This is the one!

    dmdTransformDictSelSig _ _ = panic "dmdTransformDictSelSig: no args"

    {-
    Note [Demand transformer for a dictionary selector]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    If we evaluate (op dict-expr) under demand 'd', then we can push the demand 'd'
    into the appropriate field of the dictionary. What *is* the appropriate field?
    We just look at the strictness signature of the class op, which will be
    something like: U(AAASAAAAA).  Then replace the 'S' by the demand 'd'.

    For single-method classes, which are represented by newtypes the signature
    of 'op' won't look like U(...), so the splitProdDmd_maybe will fail.
    That's fine: if we are doing strictness analysis we are also doing inlining,
    so we'll have inlined 'op' into a cast.  So we can bale out in a conservative
    way, returning nopDmdType.

    It is (just.. Trac #8329) possible to be running strictness analysis *without*
    having inlined class ops from single-method classes.  Suppose you are using
    ghc --make; and the first module has a local -O0 flag.  So you may load a class
    without interface pragmas, ie (currently) without an unfolding for the class
    ops.   Now if a subsequent module in the --make sweep has a local -O flag
    you might do strictness analysis, but there is no inlining for the class op.
    This is weird, so I'm not worried about whether this optimises brilliantly; but
    it should not fall over.
    -}

    argsOneShots :: StrictSig -> Arity -> [[OneShotInfo]]
    -- See Note [Computing one-shot info]
    argsOneShots (StrictSig (DmdType _ arg_ds _)) n_val_args
      | unsaturated_call = []
      | otherwise = go arg_ds
      where
        unsaturated_call = arg_ds `lengthExceeds` n_val_args

        go []               = []
        go (arg_d : arg_ds) = argOneShots arg_d `cons` go arg_ds

        -- Avoid list tail like [ [], [], [] ]
        cons [] [] = []
        cons a  as = a:as

    -- saturatedByOneShots n C1(C1(...)) = True,
    --   <=>
    -- there are at least n nested C1(..) calls
    -- See Note [Demand on the worker] in WorkWrap
    saturatedByOneShots :: Int -> Demand -> Bool
    saturatedByOneShots n (JD { ud = usg })
      = case usg of
          Use _ arg_usg -> go n arg_usg
          _             -> False
      where
        go 0 _             = True
        go n (UCall One u) = go (n-1) u
        go _ _             = False

    argOneShots :: Demand          -- depending on saturation
                -> [OneShotInfo]
    argOneShots (JD { ud = usg })
      = case usg of
          Use _ arg_usg -> go arg_usg
          _             -> []
      where
        go (UCall One  u) = OneShotLam : go u
        go (UCall Many u) = NoOneShotInfo : go u
        go _              = []

    {- Note [Computing one-shot info]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    Consider a call
        f (\pqr. e1) (\xyz. e2) e3
    where f has usage signature
        C1(C(C1(U))) C1(U) U
    Then argsOneShots returns a [[OneShotInfo]] of
        [[OneShot,NoOneShotInfo,OneShot],  [OneShot]]
    The occurrence analyser propagates this one-shot infor to the
    binders \pqr and \xyz; see Note [Use one-shot information] in OccurAnal.
    -}

    -- appIsBottom returns true if an application to n args
    -- would diverge or throw an exception
    -- See Note [Unsaturated applications]
    appIsBottom :: StrictSig -> Int -> Bool
    appIsBottom (StrictSig (DmdType _ ds res)) n
                | isBotRes res                   = not $ lengthExceeds ds n
    appIsBottom _                              _ = False

    {-
    Note [Unsaturated applications]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    If a function having bottom as its demand result is applied to a less
    number of arguments than its syntactic arity, we cannot say for sure
    that it is going to diverge. This is the reason why we use the
    function appIsBottom, which, given a strictness signature and a number
    of arguments, says conservatively if the function is going to diverge
    or not.

    Zap absence or one-shot information, under control of flags

    Note [Killing usage information]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    The flags -fkill-one-shot and -fkill-absence let you switch off the generation
    of absence or one-shot information altogether.  This is only used for performance
    tests, to see how important they are.
    -}

    zapUsageEnvSig :: StrictSig -> StrictSig
    -- Remove the usage environment from the demand
    zapUsageEnvSig (StrictSig (DmdType _ ds r)) = mkClosedStrictSig ds r

    zapUsageDemand :: Demand -> Demand
    -- Remove the usage info, but not the strictness info, from the demand
    zapUsageDemand = kill_usage $ KillFlags
        { kf_abs         = True
        , kf_used_once   = True
        , kf_called_once = True
        }

    -- | Remove all 1* information (but not C1 information) from the demand
    zapUsedOnceDemand :: Demand -> Demand
    zapUsedOnceDemand = kill_usage $ KillFlags
        { kf_abs         = False
        , kf_used_once   = True
        , kf_called_once = False
        }

    -- | Remove all 1* information (but not C1 information) from the strictness
    --   signature
    zapUsedOnceSig :: StrictSig -> StrictSig
    zapUsedOnceSig (StrictSig (DmdType env ds r))
        = StrictSig (DmdType env (map zapUsedOnceDemand ds) r)

    killUsageDemand :: DynFlags -> Demand -> Demand
    -- See Note [Killing usage information]
    killUsageDemand dflags dmd
      | Just kfs <- killFlags dflags = kill_usage kfs dmd
      | otherwise                    = dmd

    killUsageSig :: DynFlags -> StrictSig -> StrictSig
    -- See Note [Killing usage information]
    killUsageSig dflags sig@(StrictSig (DmdType env ds r))
      | Just kfs <- killFlags dflags = StrictSig (DmdType env (map (kill_usage kfs) ds) r)
      | otherwise                    = sig

    data KillFlags = KillFlags
        { kf_abs         :: Bool
        , kf_used_once   :: Bool
        , kf_called_once :: Bool
        }

    killFlags :: DynFlags -> Maybe KillFlags
    -- See Note [Killing usage information]
    killFlags dflags
      | not kf_abs && not kf_used_once = Nothing
      | otherwise                      = Just (KillFlags {..})
      where
        kf_abs         = gopt Opt_KillAbsence dflags
        kf_used_once   = gopt Opt_KillOneShot dflags
        kf_called_once = kf_used_once

    kill_usage :: KillFlags -> Demand -> Demand
    kill_usage kfs (JD {sd = s, ud = u}) = JD {sd = s, ud = zap_musg kfs u}

    zap_musg :: KillFlags -> ArgUse -> ArgUse
    zap_musg kfs Abs
      | kf_abs kfs = useTop
      | otherwise  = Abs
    zap_musg kfs (Use c u)
      | kf_used_once kfs = Use Many (zap_usg kfs u)
      | otherwise        = Use c    (zap_usg kfs u)

    zap_usg :: KillFlags -> UseDmd -> UseDmd
    zap_usg kfs (UCall c u)
        | kf_called_once kfs = UCall Many (zap_usg kfs u)
        | otherwise          = UCall c    (zap_usg kfs u)
    zap_usg kfs (UProd us)   = UProd (map (zap_musg kfs) us)
    zap_usg _   u            = u

    -- If the argument is a used non-newtype dictionary, give it strict
    -- demand. Also split the product type & demand and recur in order to
    -- similarly strictify the argument's contained used non-newtype
    -- superclass dictionaries. We use the demand as our recursive measure
    -- to guarantee termination.
    strictifyDictDmd :: Type -> Demand -> Demand
    strictifyDictDmd ty dmd = case getUseDmd dmd of
      Use n _ |
        Just (tycon, _arg_tys, _data_con, inst_con_arg_tys)
          <- splitDataProductType_maybe ty,
        not (isNewTyCon tycon), isClassTyCon tycon -- is a non-newtype dictionary
        -> seqDmd `bothDmd` -- main idea: ensure it's strict
           case splitProdDmd_maybe dmd of
             -- superclass cycles should not be a problem, since the demand we are
             -- consuming would also have to be infinite in order for us to diverge
             Nothing -> dmd -- no components have interesting demand, so stop
                            -- looking for superclass dicts
             Just dmds
               | all (not . isAbsDmd) dmds -> evalDmd
                 -- abstract to strict w/ arbitrary component use, since this
                 -- smells like reboxing; results in CBV boxed
                 --
                 -- TODO revisit this if we ever do boxity analysis
               | otherwise -> case mkProdDmd $ zipWith strictifyDictDmd inst_con_arg_tys dmds of
                   JD {sd = s,ud = a} -> JD (Str VanStr s) (Use n a)
                 -- TODO could optimize with an aborting variant of zipWith since
                 -- the superclass dicts are always a prefix
      _ -> dmd -- unused or not a dictionary

    {-
    Note [HyperStr and Use demands]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    The information "HyperStr" needs to be in the strictness signature, and not in
    the demand signature, because we still want to know about the demand on things. Consider

        f (x,y) True  = error (show x)
        f (x,y) False = x+1

    The signature of f should be <S(SL),1*U(1*U(U),A)><S,1*U>m. If we were not
    distinguishing the uses on x and y in the True case, we could either not figure
    out how deeply we can unpack x, or that we do not have to pass y.

    ************************************************************************
    *                                                                      *
                         Serialisation
    *                                                                      *
    ************************************************************************
    -}

    instance Binary StrDmd where
      put_ bh HyperStr     = do putByte bh 0
      put_ bh HeadStr      = do putByte bh 1
      put_ bh (SCall s)    = do putByte bh 2
                                put_ bh s
      put_ bh (SProd sx)   = do putByte bh 3
                                put_ bh sx
      get bh = do
             h <- getByte bh
             case h of
               0 -> do return HyperStr
               1 -> do return HeadStr
               2 -> do s  <- get bh
                       return (SCall s)
               _ -> do sx <- get bh
                       return (SProd sx)

    instance Binary ExnStr where
      put_ bh VanStr = putByte bh 0
      put_ bh ExnStr = putByte bh 1

      get bh = do h <- getByte bh
                  return (case h of
                            0 -> VanStr
                            _ -> ExnStr)

    instance Binary ArgStr where
        put_ bh Lazy         = do
                putByte bh 0
        put_ bh (Str x s)    = do
                putByte bh 1
                put_ bh x
                put_ bh s

        get  bh = do
                h <- getByte bh
                case h of
                  0 -> return Lazy
                  _ -> do x <- get bh
                          s  <- get bh
                          return $ Str x s

    instance Binary Count where
        put_ bh One  = do putByte bh 0
        put_ bh Many = do putByte bh 1

        get  bh = do h <- getByte bh
                     case h of
                       0 -> return One
                       _ -> return Many

    instance Binary ArgUse where
        put_ bh Abs          = do
                putByte bh 0
        put_ bh (Use c u)    = do
                putByte bh 1
                put_ bh c
                put_ bh u

        get  bh = do
                h <- getByte bh
                case h of
                  0 -> return Abs
                  _ -> do c  <- get bh
                          u  <- get bh
                          return $ Use c u

    instance Binary UseDmd where
        put_ bh Used         = do
                putByte bh 0
        put_ bh UHead        = do
                putByte bh 1
        put_ bh (UCall c u)    = do
                putByte bh 2
                put_ bh c
                put_ bh u
        put_ bh (UProd ux)   = do
                putByte bh 3
                put_ bh ux

        get  bh = do
                h <- getByte bh
                case h of
                  0 -> return $ Used
                  1 -> return $ UHead
                  2 -> do c <- get bh
                          u <- get bh
                          return (UCall c u)
                  _ -> do ux <- get bh
                          return (UProd ux)

    instance (Binary s, Binary u) => Binary (JointDmd s u) where
        put_ bh (JD { sd = x, ud = y }) = do put_ bh x; put_ bh y
        get  bh = do
                  x <- get bh
                  y <- get bh
                  return $ JD { sd = x, ud = y }

    instance Binary StrictSig where
        put_ bh (StrictSig aa) = do
                put_ bh aa
        get bh = do
              aa <- get bh
              return (StrictSig aa)

    instance Binary DmdType where
      -- Ignore DmdEnv when spitting out the DmdType
      put_ bh (DmdType _ ds dr)
           = do put_ bh ds
                put_ bh dr
      get bh
          = do ds <- get bh
               dr <- get bh
               return (DmdType emptyDmdEnv ds dr)

    instance Binary DmdResult where
      put_ bh (Dunno c)     = do { putByte bh 0; put_ bh c }
      put_ bh ThrowsExn     = putByte bh 1
      put_ bh Diverges      = putByte bh 2

      get bh = do { h <- getByte bh
                  ; case h of
                      0 -> do { c <- get bh; return (Dunno c) }
                      1 -> return ThrowsExn
                      _ -> return Diverges }

    instance Binary CPRResult where
        put_ bh (RetSum n)   = do { putByte bh 0; put_ bh n }
        put_ bh RetProd      = putByte bh 1
        put_ bh NoCPR        = putByte bh 2

        get  bh = do
                h <- getByte bh
                case h of
                  0 -> do { n <- get bh; return (RetSum n) }
                  1 -> return RetProd
                  _ -> return NoCPR