%
-% (c) The GRASP/AQUA Project, Glasgow University, 1993-1995
+% (c) The GRASP/AQUA Project, Glasgow University, 1993-1996
%
\section[SaAbsInt]{Abstract interpreter for strictness analysis}
isBot
) where
-IMPORT_Trace -- ToDo: rm
-import Pretty
---import FiniteMap
-import Outputable
-
-import AbsPrel ( PrimOp(..), PrimKind )
-import AbsUniType ( isPrimType, getUniDataTyCon_maybe,
- maybeSingleConstructorTyCon,
- returnsRealWorld,
- isEnumerationTyCon, TyVarTemplate, TyCon
- IF_ATTACK_PRAGMAS(COMMA cmpTyCon)
+IMP_Ubiq(){-uitous-}
+
+import CoreSyn
+import CoreUnfold ( UnfoldingDetails(..), FormSummary )
+import CoreUtils ( unTagBinders )
+import Id ( idType, getIdStrictness, getIdUnfolding,
+ dataConTyCon, dataConArgTys
)
-import Id ( getIdStrictness, getIdUniType, getIdUnfolding,
- getDataConSig, getInstantiatedDataConSig,
- DataCon(..), isBottomingId
+import IdInfo ( StrictnessInfo(..), Demand(..),
+ wwPrim, wwStrict, wwEnum, wwUnpack
)
-
-import IdInfo -- various bits
-import IdEnv
-import CoreFuns ( unTagBinders )
-import Maybes ( maybeToBool, Maybe(..) )
-import PlainCore
+import MagicUFs ( MagicUnfoldingFun )
+import Maybes ( maybeToBool )
+import Outputable ( Outputable(..){-instance * []-} )
+import PprStyle ( PprStyle(..) )
+import Pretty ( ppStr )
+import PrimOp ( PrimOp(..) )
import SaLib
-import SimplEnv ( FormSummary(..) ) -- nice data abstraction, huh? (WDP 95/03)
-import Util
+import TyCon ( maybeTyConSingleCon, isEnumerationTyCon,
+ TyCon{-instance Eq-}
+ )
+import Type ( maybeAppDataTyConExpandingDicts, isPrimType )
+import TysWiredIn ( intTyCon, integerTyCon, doubleTyCon,
+ floatTyCon, wordTyCon, addrTyCon
+ )
+import Util ( isIn, isn'tIn, nOfThem, zipWithEqual,
+ pprTrace, panic, pprPanic, assertPanic
+ )
+
+returnsRealWorld = panic "SaAbsInt.returnsRealWorld (ToDo)"
\end{code}
%************************************************************************
-- always returns bottom, such as \y.x,
-- when x is bound to bottom.
-lub (AbsProd xs) (AbsProd ys) = ASSERT (length xs == length ys)
- AbsProd (zipWith lub xs ys)
+lub (AbsProd xs) (AbsProd ys) = AbsProd (zipWithEqual "lub" lub xs ys)
lub _ _ = AbsTop -- Crude, but conservative
- -- The crudity only shows up if there
+ -- The crudity only shows up if there
-- are functions involved
-- Slightly funny glb; for absence analysis only;
--
-- f = \a b -> ...
--
--- g = \x y z -> case x of
+-- g = \x y z -> case x of
-- [] -> f x
-- (p:ps) -> f p
--
-- Deal with functions specially, because AbsTop isn't the
-- top of their domain.
-glb v1 v2
+glb v1 v2
| is_fun v1 || is_fun v2
- = if not (anyBot v1) && not (anyBot v2)
+ = if not (anyBot v1) && not (anyBot v2)
then
AbsTop
else
-- The non-functional cases are quite straightforward
-glb (AbsProd xs) (AbsProd ys) = ASSERT (length xs == length ys)
- AbsProd (zipWith glb xs ys)
+glb (AbsProd xs) (AbsProd ys) = AbsProd (zipWithEqual "glb" glb xs ys)
glb AbsTop v2 = v2
glb v1 AbsTop = v1
-combineCaseValues
+combineCaseValues
:: AnalysisKind
-> AbsVal -- Value of scrutinee
-> [AbsVal] -- Value of branches (at least one)
AbsTop -> True; -- i.e., cool
AbsProd _ -> True; -- ditto
_ -> False -- party over
- }
+ }
-- For absence analysis, check if the scrutinee is all poison (isBot)
-- If so, return poison (AbsBot); otherwise, any nested poison will come
isBot AbsBot = True
isBot (AbsFun args body env) = isBot (absEval StrAnal body env)
- -- Don't bother to extend the envt because
- -- unbound variables default to AbsTop anyway
+ -- Don't bother to extend the envt because
+ -- unbound variables default to AbsTop anyway
isBot other = False
\end{code}
\begin{code}
widen :: AnalysisKind -> AbsVal -> AbsVal
-widen StrAnal (AbsFun args body env)
+widen StrAnal (AbsFun args body env)
| isBot (absEval StrAnal body env) = AbsBot
| otherwise
= ASSERT (not (null args))
-- alternative here would be to bind g to its exact abstract
-- value, but that entails lots of potential re-computation, at
-- every application of g.)
-
+
widen StrAnal (AbsProd vals) = AbsProd (map (widen StrAnal) vals)
widen StrAnal other_val = other_val
-widen AbsAnal (AbsFun args body env)
+widen AbsAnal (AbsFun args body env)
| anyBot (absEval AbsAnal body env) = AbsBot
-- In the absence-analysis case it's *essential* to check
-- that the function has no poison in its body. If it does,
| otherwise
= ASSERT (not (null args))
AbsApproxFun (map (findDemandAbsOnly env body) args)
-
+
widen AbsAnal (AbsProd vals) = AbsProd (map (widen AbsAnal) vals)
-- It's desirable to do a good job of widening for product
widen AbsAnal other_val = other_val
--- OLD if anyBot val then AbsBot else AbsTop
+-- WAS: if anyBot val then AbsBot else AbsTop
-- Nowadays widen is doing a better job on functions for absence analysis.
\end{code}
sameVal AbsTop AbsTop = True
sameVal AbsTop other = False -- Right?
-sameVal (AbsProd vals1) (AbsProd vals2) = ASSERT (length vals1 == length vals2)
- and (zipWith sameVal vals1 vals2)
+sameVal (AbsProd vals1) (AbsProd vals2) = and (zipWithEqual "sameVal" sameVal vals1 vals2)
sameVal (AbsProd _) AbsTop = False
sameVal (AbsProd _) AbsBot = False
(@True@ is the exciting answer; @False@ is always safe.)
\begin{code}
-evalStrictness :: Demand
- -> AbsVal
- -> Bool -- True iff the value is sure
+evalStrictness :: Demand
+ -> AbsVal
+ -> Bool -- True iff the value is sure
-- to be less defined than the Demand
evalStrictness (WwLazy _) _ = False
= case val of
AbsTop -> False
AbsBot -> True
- AbsProd vals -> ASSERT (length vals == length demand_info)
- or (zipWith evalStrictness demand_info vals)
+ AbsProd vals -> or (zipWithEqual "evalStrictness" evalStrictness demand_info vals)
_ -> trace "evalStrictness?" False
evalStrictness WwPrim val
= case val of
- AbsTop -> False
+ AbsTop -> False
- other -> -- A primitive value should be defined, never bottom;
+ other -> -- A primitive value should be defined, never bottom;
-- hence this paranoia check
pprPanic "evalStrictness: WwPrim:" (ppr PprDebug other)
\end{code}
possibly} hit poison.
\begin{code}
-evalAbsence (WwLazy True) _ = False -- Can't possibly hit poison
+evalAbsence (WwLazy True) _ = False -- Can't possibly hit poison
-- with Absent demand
evalAbsence (WwUnpack demand_info) val
= case val of
AbsTop -> False -- No poison in here
AbsBot -> True -- Pure poison
- AbsProd vals -> ASSERT (length demand_info == length vals)
- or (zipWith evalAbsence demand_info vals)
+ AbsProd vals -> or (zipWithEqual "evalAbsence" evalAbsence demand_info vals)
_ -> panic "evalAbsence: other"
evalAbsence other val = anyBot val
result =
case (lookupAbsValEnv env var, getIdStrictness var, getIdUnfolding var) of
- (Just abs_val, _, _) ->
+ (Just abs_val, _, _) ->
abs_val -- Bound in the environment
- (Nothing, NoStrictnessInfo, LiteralForm _) ->
- AbsTop -- Literals all terminate, and have no poison
-
- (Nothing, NoStrictnessInfo, ConstructorForm _ _ _) ->
- AbsTop -- An imported constructor won't have
- -- bottom components, nor poison!
-
- (Nothing, NoStrictnessInfo, GeneralForm _ _ unfolding _) ->
+ (Nothing, NoStrictnessInfo, GenForm _ unfolding _) ->
-- We have an unfolding for the expr
-- Assume the unfolding has no free variables since it
-- came from inside the Id
-- "U(U(U(U(SL)LLLLLLLLL)LL)LLLLLSLLLLL)" _N_ _N_ #-}
- (Nothing, strictness_info, _) ->
+ (Nothing, strictness_info, _) ->
-- Includes MagicForm, IWantToBeINLINEd, NoUnfoldingDetails
-- Try the strictness info
absValFromStrictness anal strictness_info
-
-
- -- Done via strictness now
- -- GeneralForm _ BottomForm _ _ -> AbsBot
in
- -- pprTrace "absId:" (ppBesides [ppr PprDebug var, ppStr "=:", pp_anal anal, ppStr ":=",ppr PprDebug result]) (
+ -- pprTrace "absId:" (ppBesides [ppr PprDebug var, ppStr "=:", pp_anal anal, ppStr ":=",ppr PprDebug result]) $
result
- -- )
where
pp_anal StrAnal = ppStr "STR"
pp_anal AbsAnal = ppStr "ABS"
-absEvalAtom anal (CoVarAtom v) env = absId anal v env
-absEvalAtom anal (CoLitAtom _) env = AbsTop
+absEvalAtom anal (VarArg v) env = absId anal v env
+absEvalAtom anal (LitArg _) env = AbsTop
\end{code}
\begin{code}
-absEval :: AnalysisKind -> PlainCoreExpr -> AbsValEnv -> AbsVal
+absEval :: AnalysisKind -> CoreExpr -> AbsValEnv -> AbsVal
-absEval anal (CoVar var) env = absId anal var env
+absEval anal (Var var) env = absId anal var env
-absEval anal (CoLit _) env = AbsTop
+absEval anal (Lit _) env = AbsTop
-- What if an unboxed literal? That's OK: it terminates, so its
-- abstract value is AbsTop.
to make sure that any poison (?????)
\begin{code}
-absEval StrAnal (CoPrim SeqOp [t] [e]) env
- = if isBot (absEvalAtom StrAnal e env) then AbsBot else AbsTop
+absEval StrAnal (Prim SeqOp [TyArg _, e]) env
+ = ASSERT(isValArg e)
+ if isBot (absEvalAtom StrAnal e env) then AbsBot else AbsTop
-- This is a special case to ensure that seq# is strict in its argument.
-- The comments below (for most normal PrimOps) do not apply.
-absEval StrAnal (CoPrim op ts es) env = AbsTop
+absEval StrAnal (Prim op es) env = AbsTop
-- The arguments are all of unboxed type, so they will already
-- have been eval'd. If the boxed version was bottom, we'll
-- already have returned bottom.
-- uses boxed args and we don't know whether or not it's
-- strict, so we assume laziness. (JSM)
-absEval AbsAnal (CoPrim op ts as) env
- = if any anyBot [absEvalAtom AbsAnal a env | a <- as]
+absEval AbsAnal (Prim op as) env
+ = if any anyBot [absEvalAtom AbsAnal a env | a <- as, isValArg a]
then AbsBot
else AbsTop
-- For absence analysis, we want to see if the poison shows up...
-absEval anal (CoCon con ts as) env
+absEval anal (Con con as) env
| has_single_con
- = AbsProd [absEvalAtom anal a env | a <- as]
+ = AbsProd [absEvalAtom anal a env | a <- as, isValArg a]
| otherwise -- Not single-constructor
= case anal of
StrAnal -> -- Strictness case: it's easy: it certainly terminates
- AbsTop
- AbsAnal -> -- In the absence case we need to be more
+ AbsTop
+ AbsAnal -> -- In the absence case we need to be more
-- careful: look to see if there's any
-- poison in the components
- if any anyBot [absEvalAtom AbsAnal a env | a <- as]
+ if any anyBot [absEvalAtom AbsAnal a env | a <- as, isValArg a]
then AbsBot
else AbsTop
where
- (_,_,_, tycon) = getDataConSig con
- has_single_con = maybeToBool (maybeSingleConstructorTyCon tycon)
+ has_single_con = maybeToBool (maybeTyConSingleCon (dataConTyCon con))
\end{code}
\begin{code}
-absEval anal (CoLam [] body) env = absEval anal body env -- paranoia
-absEval anal (CoLam binders body) env = AbsFun binders body env
-absEval anal (CoTyLam ty expr) env = absEval anal expr env
-absEval anal (CoApp e1 e2) env = absApply anal (absEval anal e1 env)
- (absEvalAtom anal e2 env)
-absEval anal (CoTyApp expr ty) env = absEval anal expr env
+absEval anal (Lam (ValBinder binder) body) env
+ = AbsFun [binder] body env
+absEval anal (Lam other_binder expr) env
+ = absEval anal expr env
+absEval anal (App f a) env | isValArg a
+ = absApply anal (absEval anal f env) (absEvalAtom anal a env)
+absEval anal (App expr _) env
+ = absEval anal expr env
\end{code}
For primitive cases, just GLB the branches, then LUB with the expr part.
\begin{code}
-absEval anal (CoCase expr (CoPrimAlts alts deflt)) env
+absEval anal (Case expr (PrimAlts alts deflt)) env
= let
expr_val = absEval anal expr env
abs_alts = [ absEval anal rhs env | (_, rhs) <- alts ]
combineCaseValues anal expr_val
(abs_deflt ++ abs_alts)
-absEval anal (CoCase expr (CoAlgAlts alts deflt)) env
+absEval anal (Case expr (AlgAlts alts deflt)) env
= let
- expr_val = absEval anal expr env
+ expr_val = absEval anal expr env
abs_alts = [ absEvalAlgAlt anal expr_val alt env | alt <- alts ]
abs_deflt = absEvalDefault anal expr_val deflt env
in
result
\end{code}
-For @CoLets@ we widen the value we get. This is nothing to
+For @Lets@ we widen the value we get. This is nothing to
do with fixpointing. The reason is so that we don't get an explosion
in the amount of computation. For example, consider:
\begin{verbatim}
f x = case x of
p1 -> ...g r...
p2 -> ...g s...
- in
+ in
f e
\end{verbatim}
If we bind @f@ and @g@ to their exact abstract value, then we'll
and be prepared to bale out.
\begin{code}
-absEval anal (CoLet (CoNonRec binder e1) e2) env
+absEval anal (Let (NonRec binder e1) e2) env
= let
new_env = addOneToAbsValEnv env binder (widen anal (absEval anal e1 env))
in
- -- The binder of a CoNonRec should *not* be of unboxed type,
+ -- The binder of a NonRec should *not* be of unboxed type,
-- hence no need to strictly evaluate the Rhs.
absEval anal e2 new_env
-absEval anal (CoLet (CoRec pairs) body) env
+absEval anal (Let (Rec pairs) body) env
= let
(binders,rhss) = unzip pairs
rhs_vals = cheapFixpoint anal binders rhss env -- Returns widened values
new_env = growAbsValEnvList env (binders `zip` rhs_vals)
in
absEval anal body new_env
-\end{code}
-\begin{code}
-absEval anal (CoSCC cc expr) env = absEval anal expr env
-
--- ToDo: add DPH stuff here
+absEval anal (SCC cc expr) env = absEval anal expr env
+absEval anal (Coerce c ty expr) env = absEval anal expr env
\end{code}
\begin{code}
-absEvalAlgAlt :: AnalysisKind -> AbsVal -> (Id,[Id],PlainCoreExpr) -> AbsValEnv -> AbsVal
+absEvalAlgAlt :: AnalysisKind -> AbsVal -> (Id,[Id],CoreExpr) -> AbsValEnv -> AbsVal
absEvalAlgAlt anal (AbsProd arg_vals) (con, args, rhs) env
= -- The scrutinee is a product value, so it must be of a single-constr
_ -> False -- party over
}
-
-absEvalDefault :: AnalysisKind
+
+absEvalDefault :: AnalysisKind
-> AbsVal -- Value of scrutinee
- -> PlainCoreCaseDefault
- -> AbsValEnv
+ -> CoreCaseDefault
+ -> AbsValEnv
-> [AbsVal] -- Empty or singleton
-absEvalDefault anal scrut_val CoNoDefault env = []
-absEvalDefault anal scrut_val (CoBindDefault binder expr) env
+absEvalDefault anal scrut_val NoDefault env = []
+absEvalDefault anal scrut_val (BindDefault binder expr) env
= [absEval anal expr (addOneToAbsValEnv env binder scrut_val)]
\end{code}
-- AbsBot represents the abstract bottom *function* too
absApply StrAnal AbsTop arg = AbsTop
-absApply AbsAnal AbsTop arg = if anyBot arg
+absApply AbsAnal AbsTop arg = if anyBot arg
then AbsBot
else AbsTop
-- To be conservative, we have to assume that a function about
\end{code}
An @AbsFun@ with only one more argument needed---bind it and eval the
-result. A @CoLam@ with two or more args: return another @AbsFun@ with
+result. A @Lam@ with two or more args: return another @AbsFun@ with
an augmented environment.
\begin{code}
See notes on @addStrictnessInfoToId@.
\begin{code}
-findStrictness :: [UniType] -- Types of args in which strictness is wanted
- -> AbsVal -- Abstract strictness value of function
+findStrictness :: StrAnalFlags
+ -> [Type] -- Types of args in which strictness is wanted
+ -> AbsVal -- Abstract strictness value of function
-> AbsVal -- Abstract absence value of function
-> [Demand] -- Resulting strictness annotation
-findStrictness [] str_val abs_val = []
+findStrictness strflags [] str_val abs_val = []
-findStrictness (ty:tys) str_val abs_val
+findStrictness strflags (ty:tys) str_val abs_val
= let
- demand = findRecDemand [] str_fn abs_fn ty
+ demand = findRecDemand strflags [] str_fn abs_fn ty
str_fn val = absApply StrAnal str_val val
abs_fn val = absApply AbsAnal abs_val val
- demands = findStrictness tys (absApply StrAnal str_val AbsTop)
- (absApply AbsAnal abs_val AbsTop)
+ demands = findStrictness strflags tys
+ (absApply StrAnal str_val AbsTop)
+ (absApply AbsAnal abs_val AbsTop)
in
- -- pprTrace "findRecDemand:" (ppCat [ppr PprDebug demand, ppr PprDebug ty]) (
demand : demands
- -- )
\end{code}
\begin{code}
findDemandStrOnly str_env expr binder -- Only strictness environment available
- = findRecDemand [] str_fn abs_fn (getIdUniType binder)
+ = findRecDemand strflags [] str_fn abs_fn (idType binder)
where
str_fn val = absEval StrAnal expr (addOneToAbsValEnv str_env binder val)
abs_fn val = AbsBot -- Always says poison; so it looks as if
-- nothing is absent; safe
-
+ strflags = getStrAnalFlags str_env
findDemandAbsOnly abs_env expr binder -- Only absence environment available
- = findRecDemand [] str_fn abs_fn (getIdUniType binder)
+ = findRecDemand strflags [] str_fn abs_fn (idType binder)
where
str_fn val = AbsBot -- Always says non-termination;
-- that'll make findRecDemand peer into the
-- structure of the value.
abs_fn val = absEval AbsAnal expr (addOneToAbsValEnv abs_env binder val)
-
+ strflags = getStrAnalFlags abs_env
+
findDemand str_env abs_env expr binder
- = findRecDemand [] str_fn abs_fn (getIdUniType binder)
+ = findRecDemand strflags [] str_fn abs_fn (idType binder)
where
str_fn val = absEval StrAnal expr (addOneToAbsValEnv str_env binder val)
abs_fn val = absEval AbsAnal expr (addOneToAbsValEnv abs_env binder val)
+ strflags = getStrAnalFlags str_env
\end{code}
@findRecDemand@ is where we finally convert strictness/absence info
Ho hum.
\begin{code}
-findRecDemand :: [TyCon] -- TyCons already seen; used to avoid
+findRecDemand :: StrAnalFlags
+ -> [TyCon] -- TyCons already seen; used to avoid
-- zooming into recursive types
-> (AbsVal -> AbsVal) -- The strictness function
-> (AbsVal -> AbsVal) -- The absence function
- -> UniType -- The type of the argument
+ -> Type -- The type of the argument
-> Demand
-findRecDemand seen str_fn abs_fn ty
+findRecDemand strflags seen str_fn abs_fn ty
= if isPrimType ty then -- It's a primitive type!
wwPrim
-- We prefer absence over strictness: see NOTE above.
WwLazy True
- else if not (isBot (str_fn AbsBot)) then -- It's not strict
- WwLazy False
+ else if not (all_strict ||
+ (num_strict && is_numeric_type ty) ||
+ (isBot (str_fn AbsBot))) then
+ WwLazy False -- It's not strict and we're not pretending
- else -- It's strict!
+ else -- It's strict (or we're pretending it is)!
- case getUniDataTyCon_maybe ty of
+ case (maybeAppDataTyConExpandingDicts ty) of
Nothing -> wwStrict
Just (tycon,tycon_arg_tys,[data_con]) | tycon `not_elem` seen ->
-- Single constructor case, tycon not already seen higher up
let
- (_,cmpnt_tys,_) = getInstantiatedDataConSig data_con tycon_arg_tys
+ cmpnt_tys = dataConArgTys data_con tycon_arg_tys
prod_len = length cmpnt_tys
compt_strict_infos
- = [ findRecDemand (tycon:seen)
+ = [ findRecDemand strflags (tycon:seen)
(\ cmpnt_val ->
str_fn (mkMainlyTopProd prod_len i cmpnt_val)
)
else
wwStrict
where
+ (all_strict, num_strict) = strflags
+
+ is_numeric_type ty
+ = case (maybeAppDataTyConExpandingDicts ty) of -- NB: duplicates stuff done above
+ Nothing -> False
+ Just (tycon, _, _)
+ | tycon `is_elem`
+ [intTyCon, integerTyCon,
+ doubleTyCon, floatTyCon,
+ wordTyCon, addrTyCon]
+ -> True
+ _{-something else-} -> False
+ where
+ is_elem = isIn "is_numeric_type"
+
-- mkMainlyTopProd: make an AbsProd that is all AbsTops ("n"-1 of
-- them) except for a given value in the "i"th position.
approximation.
\begin{code}
-cheapFixpoint :: AnalysisKind -> [Id] -> [PlainCoreExpr] -> AbsValEnv -> [AbsVal]
+cheapFixpoint :: AnalysisKind -> [Id] -> [CoreExpr] -> AbsValEnv -> [AbsVal]
cheapFixpoint AbsAnal [id] [rhs] env
= [crudeAbsWiden (absEval AbsAnal rhs new_env)]
= [widen anal (absEval anal rhs new_env) | rhs <- rhss]
-- We do just one iteration, starting from a safe
-- approximation. This won't do a good job in situations
- -- like:
+ -- like:
-- \x -> letrec f = ...g...
-- g = ...f...x...
-- in
\end{verbatim}
\begin{code}
-fixpoint :: AnalysisKind -> [Id] -> [PlainCoreExpr] -> AbsValEnv -> [AbsVal]
+fixpoint :: AnalysisKind -> [Id] -> [CoreExpr] -> AbsValEnv -> [AbsVal]
fixpoint anal [] _ env = []
-fixpoint anal ids rhss env
+fixpoint anal ids rhss env
= fix_loop initial_vals
where
initial_val id
= case anal of -- The (unsafe) starting point
- StrAnal -> if (returnsRealWorld (getIdUniType id))
+ StrAnal -> if (returnsRealWorld (idType id))
then AbsTop -- this is a massively horrible hack (SLPJ 95/05)
else AbsBot
AbsAnal -> AbsTop
fix_loop :: [AbsVal] -> [AbsVal]
- fix_loop current_widened_vals
+ fix_loop current_widened_vals
= let
new_env = growAbsValEnvList env (ids `zip` current_widened_vals)
new_vals = [ absEval anal rhs new_env | rhs <- rhss ]
new_widened_vals = map (widen anal) new_vals
- in
+ in
if (and (zipWith sameVal current_widened_vals new_widened_vals)) then
current_widened_vals
+ -- NB: I was too chicken to make that a zipWithEqual,
+ -- lest I jump into a black hole. WDP 96/02
+
-- Return the widened values. We might get a slightly
-- better value by returning new_vals (which we used to
-- do, see below), but alas that means that whenever the
letrec
x = ...p..d...
d = (x,y)
- in
+ in
...
\end{verbatim}
Here, if p is @AbsBot@, then we'd better {\em not} end up with a ``fixed