[ghc-hetmet.git] / compiler / coreSyn / CoreUtils.lhs

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

Utility functions on @Core@ syntax

\begin{code}
{-# OPTIONS -fno-warn-incomplete-patterns #-}
-- The above warning supression flag is a temporary kludge.
-- While working on this module you are encouraged to remove it and fix
-- any warnings in the module. See
--     http://hackage.haskell.org/trac/ghc/wiki/Commentary/CodingStyle#Warnings
-- for details

-- | Commonly useful utilites for manipulating the Core language
module CoreUtils (
        -- * Constructing expressions
        mkInlineMe, mkSCC, mkCoerce, mkCoerceI,
        bindNonRec, needsCaseBinding,
        mkAltExpr, mkPiType, mkPiTypes,

        -- * Taking expressions apart
        findDefault, findAlt, isDefaultAlt, mergeAlts, trimConArgs,

        -- * Properties of expressions
        exprType, coreAltType, coreAltsType,
        exprIsDupable, exprIsTrivial, exprIsCheap, exprIsExpandable,
        exprIsHNF,exprOkForSpeculation, exprIsBig, 
        exprIsConApp_maybe, exprIsBottom,
        rhsIsStatic,

        -- * Expression and bindings size
        coreBindsSize, exprSize,

        -- * Hashing
        hashExpr,

        -- * Equality
        cheapEqExpr, 

        -- * Manipulating data constructors and types
        applyTypeToArgs, applyTypeToArg,
        dataConOrigInstPat, dataConRepInstPat, dataConRepFSInstPat
    ) where

#include "HsVersions.h"

import CoreSyn
import PprCore
import Var
import SrcLoc
import VarEnv
import VarSet
import Name
import Module
#if mingw32_TARGET_OS
import Packages
#endif
import Literal
import DataCon
import PrimOp
import Id
import IdInfo
import NewDemand
import Type
import Coercion
import TyCon
import CostCentre
import Unique
import Outputable
import TysPrim
import FastString
import Maybes
import Util
import Data.Word
import Data.Bits
\end{code}


%************************************************************************
%*                                                                      *
\subsection{Find the type of a Core atom/expression}
%*                                                                      *
%************************************************************************

\begin{code}
exprType :: CoreExpr -> Type
-- ^ Recover the type of a well-typed Core expression. Fails when
-- applied to the actual 'CoreSyn.Type' expression as it cannot
-- really be said to have a type
exprType (Var var)           = idType var
exprType (Lit lit)           = literalType lit
exprType (Let _ body)        = exprType body
exprType (Case _ _ ty _)     = ty
exprType (Cast _ co)         = snd (coercionKind co)
exprType (Note _ e)          = exprType e
exprType (Lam binder expr)   = mkPiType binder (exprType expr)
exprType e@(App _ _)
  = case collectArgs e of
        (fun, args) -> applyTypeToArgs e (exprType fun) args

exprType other = pprTrace "exprType" (pprCoreExpr other) alphaTy

coreAltType :: CoreAlt -> Type
-- ^ Returns the type of the alternatives right hand side
coreAltType (_,bs,rhs) 
  | any bad_binder bs = expandTypeSynonyms ty
  | otherwise         = ty    -- Note [Existential variables and silly type synonyms]
  where
    ty           = exprType rhs
    free_tvs     = tyVarsOfType ty
    bad_binder b = isTyVar b && b `elemVarSet` free_tvs

coreAltsType :: [CoreAlt] -> Type
-- ^ Returns the type of the first alternative, which should be the same as for all alternatives
coreAltsType (alt:_) = coreAltType alt
coreAltsType []      = panic "corAltsType"
\end{code}

Note [Existential variables and silly type synonyms]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Consider
        data T = forall a. T (Funny a)
        type Funny a = Bool
        f :: T -> Bool
        f (T x) = x

Now, the type of 'x' is (Funny a), where 'a' is existentially quantified.
That means that 'exprType' and 'coreAltsType' may give a result that *appears*
to mention an out-of-scope type variable.  See Trac #3409 for a more real-world
example.

Various possibilities suggest themselves:

 - Ignore the problem, and make Lint not complain about such variables

 - Expand all type synonyms (or at least all those that discard arguments)
      This is tricky, because at least for top-level things we want to
      retain the type the user originally specified.

 - Expand synonyms on the fly, when the problem arises. That is what
   we are doing here.  It's not too expensive, I think.

\begin{code}
mkPiType  :: Var   -> Type -> Type
-- ^ Makes a @(->)@ type or a forall type, depending
-- on whether it is given a type variable or a term variable.
mkPiTypes :: [Var] -> Type -> Type
-- ^ 'mkPiType' for multiple type or value arguments

mkPiType v ty
   | isId v    = mkFunTy (idType v) ty
   | otherwise = mkForAllTy v ty

mkPiTypes vs ty = foldr mkPiType ty vs
\end{code}

\begin{code}
applyTypeToArg :: Type -> CoreExpr -> Type
-- ^ Determines the type resulting from applying an expression to a function with the given type
applyTypeToArg fun_ty (Type arg_ty) = applyTy fun_ty arg_ty
applyTypeToArg fun_ty _             = funResultTy fun_ty

applyTypeToArgs :: CoreExpr -> Type -> [CoreExpr] -> Type
-- ^ A more efficient version of 'applyTypeToArg' when we have several arguments.
-- The first argument is just for debugging, and gives some context
applyTypeToArgs _ op_ty [] = op_ty

applyTypeToArgs e op_ty (Type ty : args)
  =     -- Accumulate type arguments so we can instantiate all at once
    go [ty] args
  where
    go rev_tys (Type ty : args) = go (ty:rev_tys) args
    go rev_tys rest_args        = applyTypeToArgs e op_ty' rest_args
                                where
                                  op_ty' = applyTysD msg op_ty (reverse rev_tys)
                                  msg = ptext (sLit "applyTypeToArgs") <+> 
                                        panic_msg e op_ty

applyTypeToArgs e op_ty (_ : args)
  = case (splitFunTy_maybe op_ty) of
        Just (_, res_ty) -> applyTypeToArgs e res_ty args
        Nothing -> pprPanic "applyTypeToArgs" (panic_msg e op_ty)

panic_msg :: CoreExpr -> Type -> SDoc
panic_msg e op_ty = pprCoreExpr e $$ ppr op_ty
\end{code}

%************************************************************************
%*                                                                      *
\subsection{Attaching notes}
%*                                                                      *
%************************************************************************

mkNote removes redundant coercions, and SCCs where possible

\begin{code}
#ifdef UNUSED
mkNote :: Note -> CoreExpr -> CoreExpr
mkNote (SCC cc) expr               = mkSCC cc expr
mkNote InlineMe expr               = mkInlineMe expr
mkNote note     expr               = Note note expr
#endif
\end{code}

Drop trivial InlineMe's.  This is somewhat important, because if we have an unfolding
that looks like (Note InlineMe (Var v)), the InlineMe doesn't go away because it may
not be *applied* to anything.

We don't use exprIsTrivial here, though, because we sometimes generate worker/wrapper
bindings like
        fw = ...
        f  = inline_me (coerce t fw)
As usual, the inline_me prevents the worker from getting inlined back into the wrapper.
We want the split, so that the coerces can cancel at the call site.  

However, we can get left with tiresome type applications.  Notably, consider
        f = /\ a -> let t = e in (t, w)
Then lifting the let out of the big lambda gives
        t' = /\a -> e
        f = /\ a -> let t = inline_me (t' a) in (t, w)
The inline_me is to stop the simplifier inlining t' right back
into t's RHS.  In the next phase we'll substitute for t (since
its rhs is trivial) and *then* we could get rid of the inline_me.
But it hardly seems worth it, so I don't bother.

\begin{code}
-- | Wraps the given expression in an inlining hint unless the expression
-- is trivial in some sense, so that doing so would usually hurt us
mkInlineMe :: CoreExpr -> CoreExpr
mkInlineMe e@(Var _)           = e
mkInlineMe e@(Note InlineMe _) = e
mkInlineMe e                   = Note InlineMe e
\end{code}

\begin{code}
-- | Wrap the given expression in the coercion, dropping identity coercions and coalescing nested coercions
mkCoerceI :: CoercionI -> CoreExpr -> CoreExpr
mkCoerceI IdCo e = e
mkCoerceI (ACo co) e = mkCoerce co e

-- | Wrap the given expression in the coercion safely, coalescing nested coercions
mkCoerce :: Coercion -> CoreExpr -> CoreExpr
mkCoerce co (Cast expr co2)
  = ASSERT(let { (from_ty, _to_ty) = coercionKind co; 
                 (_from_ty2, to_ty2) = coercionKind co2} in
           from_ty `coreEqType` to_ty2 )
    mkCoerce (mkTransCoercion co2 co) expr

mkCoerce co expr 
  = let (from_ty, _to_ty) = coercionKind co in
--    if to_ty `coreEqType` from_ty
--    then expr
--    else 
        ASSERT2(from_ty `coreEqType` (exprType expr), text "Trying to coerce" <+> text "(" <> ppr expr $$ text "::" <+> ppr (exprType expr) <> text ")" $$ ppr co $$ ppr (coercionKindPredTy co))
         (Cast expr co)
\end{code}

\begin{code}
-- | Wraps the given expression in the cost centre unless
-- in a way that maximises their utility to the user
mkSCC :: CostCentre -> Expr b -> Expr b
        -- Note: Nested SCC's *are* preserved for the benefit of
        --       cost centre stack profiling
mkSCC _  (Lit lit)          = Lit lit
mkSCC cc (Lam x e)          = Lam x (mkSCC cc e)  -- Move _scc_ inside lambda
mkSCC cc (Note (SCC cc') e) = Note (SCC cc) (Note (SCC cc') e)
mkSCC cc (Note n e)         = Note n (mkSCC cc e) -- Move _scc_ inside notes
mkSCC cc (Cast e co)        = Cast (mkSCC cc e) co -- Move _scc_ inside cast
mkSCC cc expr               = Note (SCC cc) expr
\end{code}


%************************************************************************
%*                                                                      *
\subsection{Other expression construction}
%*                                                                      *
%************************************************************************

\begin{code}
bindNonRec :: Id -> CoreExpr -> CoreExpr -> CoreExpr
-- ^ @bindNonRec x r b@ produces either:
--
-- > let x = r in b
--
-- or:
--
-- > case r of x { _DEFAULT_ -> b }
--
-- depending on whether we have to use a @case@ or @let@
-- binding for the expression (see 'needsCaseBinding').
-- It's used by the desugarer to avoid building bindings
-- that give Core Lint a heart attack, although actually
-- the simplifier deals with them perfectly well. See
-- also 'MkCore.mkCoreLet'
bindNonRec bndr rhs body 
  | needsCaseBinding (idType bndr) rhs = Case rhs bndr (exprType body) [(DEFAULT, [], body)]
  | otherwise                          = Let (NonRec bndr rhs) body

-- | Tests whether we have to use a @case@ rather than @let@ binding for this expression
-- as per the invariants of 'CoreExpr': see "CoreSyn#let_app_invariant"
needsCaseBinding :: Type -> CoreExpr -> Bool
needsCaseBinding ty rhs = isUnLiftedType ty && not (exprOkForSpeculation rhs)
        -- Make a case expression instead of a let
        -- These can arise either from the desugarer,
        -- or from beta reductions: (\x.e) (x +# y)
\end{code}

\begin{code}
mkAltExpr :: AltCon     -- ^ Case alternative constructor
          -> [CoreBndr] -- ^ Things bound by the pattern match
          -> [Type]     -- ^ The type arguments to the case alternative
          -> CoreExpr
-- ^ This guy constructs the value that the scrutinee must have
-- given that you are in one particular branch of a case
mkAltExpr (DataAlt con) args inst_tys
  = mkConApp con (map Type inst_tys ++ varsToCoreExprs args)
mkAltExpr (LitAlt lit) [] []
  = Lit lit
mkAltExpr (LitAlt _) _ _ = panic "mkAltExpr LitAlt"
mkAltExpr DEFAULT _ _ = panic "mkAltExpr DEFAULT"
\end{code}


%************************************************************************
%*                                                                      *
\subsection{Taking expressions apart}
%*                                                                      *
%************************************************************************

The default alternative must be first, if it exists at all.
This makes it easy to find, though it makes matching marginally harder.

\begin{code}
-- | Extract the default case alternative
findDefault :: [CoreAlt] -> ([CoreAlt], Maybe CoreExpr)
findDefault ((DEFAULT,args,rhs) : alts) = ASSERT( null args ) (alts, Just rhs)
findDefault alts                        =                     (alts, Nothing)

isDefaultAlt :: CoreAlt -> Bool
isDefaultAlt (DEFAULT, _, _) = True
isDefaultAlt _               = False


-- | Find the case alternative corresponding to a particular 
-- constructor: panics if no such constructor exists
findAlt :: AltCon -> [CoreAlt] -> Maybe CoreAlt
    -- A "Nothing" result *is* legitmiate
    -- See Note [Unreachable code]
findAlt con alts
  = case alts of
        (deflt@(DEFAULT,_,_):alts) -> go alts (Just deflt)
        _                          -> go alts Nothing
  where
    go []                     deflt = deflt
    go (alt@(con1,_,_) : alts) deflt
      = case con `cmpAltCon` con1 of
          LT -> deflt   -- Missed it already; the alts are in increasing order
          EQ -> Just alt
          GT -> ASSERT( not (con1 == DEFAULT) ) go alts deflt

---------------------------------
mergeAlts :: [CoreAlt] -> [CoreAlt] -> [CoreAlt]
-- ^ Merge alternatives preserving order; alternatives in
-- the first argument shadow ones in the second
mergeAlts [] as2 = as2
mergeAlts as1 [] = as1
mergeAlts (a1:as1) (a2:as2)
  = case a1 `cmpAlt` a2 of
        LT -> a1 : mergeAlts as1      (a2:as2)
        EQ -> a1 : mergeAlts as1      as2       -- Discard a2
        GT -> a2 : mergeAlts (a1:as1) as2


---------------------------------
trimConArgs :: AltCon -> [CoreArg] -> [CoreArg]
-- ^ Given:
--
-- > case (C a b x y) of
-- >        C b x y -> ...
--
-- We want to drop the leading type argument of the scrutinee
-- leaving the arguments to match agains the pattern

trimConArgs DEFAULT      args = ASSERT( null args ) []
trimConArgs (LitAlt _)   args = ASSERT( null args ) []
trimConArgs (DataAlt dc) args = dropList (dataConUnivTyVars dc) args
\end{code}

Note [Unreachable code]
~~~~~~~~~~~~~~~~~~~~~~~
It is possible (although unusual) for GHC to find a case expression
that cannot match.  For example: 

     data Col = Red | Green | Blue
     x = Red
     f v = case x of 
              Red -> ...
              _ -> ...(case x of { Green -> e1; Blue -> e2 })...

Suppose that for some silly reason, x isn't substituted in the case
expression.  (Perhaps there's a NOINLINE on it, or profiling SCC stuff
gets in the way; cf Trac #3118.)  Then the full-lazines pass might produce
this

     x = Red
     lvl = case x of { Green -> e1; Blue -> e2 })
     f v = case x of 
             Red -> ...
             _ -> ...lvl...

Now if x gets inlined, we won't be able to find a matching alternative
for 'Red'.  That's because 'lvl' is unreachable.  So rather than crashing
we generate (error "Inaccessible alternative").

Similar things can happen (augmented by GADTs) when the Simplifier
filters down the matching alternatives in Simplify.rebuildCase.



%************************************************************************
%*                                                                      *
\subsection{Figuring out things about expressions}
%*                                                                      *
%************************************************************************

@exprIsTrivial@ is true of expressions we are unconditionally happy to
                duplicate; simple variables and constants, and type
                applications.  Note that primop Ids aren't considered
                trivial unless 

Note [Variable are trivial]
~~~~~~~~~~~~~~~~~~~~~~~~~~~
There used to be a gruesome test for (hasNoBinding v) in the
Var case:
        exprIsTrivial (Var v) | hasNoBinding v = idArity v == 0
The idea here is that a constructor worker, like \$wJust, is
really short for (\x -> \$wJust x), becuase \$wJust has no binding.
So it should be treated like a lambda.  Ditto unsaturated primops.
But now constructor workers are not "have-no-binding" Ids.  And
completely un-applied primops and foreign-call Ids are sufficiently
rare that I plan to allow them to be duplicated and put up with
saturating them.

Note [SCCs are trivial]
~~~~~~~~~~~~~~~~~~~~~~~
We used not to treat (_scc_ "foo" x) as trivial, because it really
generates code, (and a heap object when it's a function arg) to
capture the cost centre.  However, the profiling system discounts the
allocation costs for such "boxing thunks" whereas the extra costs of
*not* inlining otherwise-trivial bindings can be high, and are hard to
discount.

\begin{code}
exprIsTrivial :: CoreExpr -> Bool
exprIsTrivial (Var _)          = True        -- See Note [Variables are trivial]
exprIsTrivial (Type _)         = True
exprIsTrivial (Lit lit)        = litIsTrivial lit
exprIsTrivial (App e arg)      = not (isRuntimeArg arg) && exprIsTrivial e
exprIsTrivial (Note _       e) = exprIsTrivial e  -- See Note [SCCs are trivial]
exprIsTrivial (Cast e _)       = exprIsTrivial e
exprIsTrivial (Lam b body)     = not (isRuntimeVar b) && exprIsTrivial body
exprIsTrivial _                = False
\end{code}


@exprIsDupable@ is true of expressions that can be duplicated at a modest
                cost in code size.  This will only happen in different case
                branches, so there's no issue about duplicating work.

                That is, exprIsDupable returns True of (f x) even if
                f is very very expensive to call.

                Its only purpose is to avoid fruitless let-binding
                and then inlining of case join points


\begin{code}
exprIsDupable :: CoreExpr -> Bool
exprIsDupable (Type _)          = True
exprIsDupable (Var _)           = True
exprIsDupable (Lit lit)         = litIsDupable lit
exprIsDupable (Note InlineMe _) = True
exprIsDupable (Note _ e)        = exprIsDupable e
exprIsDupable (Cast e _)        = exprIsDupable e
exprIsDupable expr
  = go expr 0
  where
    go (Var _)   _      = True
    go (App f a) n_args =  n_args < dupAppSize
                        && exprIsDupable a
                        && go f (n_args+1)
    go _         _      = False

dupAppSize :: Int
dupAppSize = 4          -- Size of application we are prepared to duplicate
\end{code}

@exprIsCheap@ looks at a Core expression and returns \tr{True} if
it is obviously in weak head normal form, or is cheap to get to WHNF.
[Note that that's not the same as exprIsDupable; an expression might be
big, and hence not dupable, but still cheap.]

By ``cheap'' we mean a computation we're willing to:
        push inside a lambda, or
        inline at more than one place
That might mean it gets evaluated more than once, instead of being
shared.  The main examples of things which aren't WHNF but are
``cheap'' are:

  *     case e of
          pi -> ei
        (where e, and all the ei are cheap)

  *     let x = e in b
        (where e and b are cheap)

  *     op x1 ... xn
        (where op is a cheap primitive operator)

  *     error "foo"
        (because we are happy to substitute it inside a lambda)

Notice that a variable is considered 'cheap': we can push it inside a lambda,
because sharing will make sure it is only evaluated once.

\begin{code}
exprIsCheap' :: (Id -> Bool) -> CoreExpr -> Bool
exprIsCheap' _          (Lit _)           = True
exprIsCheap' _          (Type _)          = True
exprIsCheap' _          (Var _)           = True
exprIsCheap' _          (Note InlineMe _) = True
exprIsCheap' is_conlike (Note _ e)        = exprIsCheap' is_conlike e
exprIsCheap' is_conlike (Cast e _)        = exprIsCheap' is_conlike e
exprIsCheap' is_conlike (Lam x e)         = isRuntimeVar x
                                            || exprIsCheap' is_conlike e
exprIsCheap' is_conlike (Case e _ _ alts) = exprIsCheap' is_conlike e && 
                                and [exprIsCheap' is_conlike rhs | (_,_,rhs) <- alts]
        -- Experimentally, treat (case x of ...) as cheap
        -- (and case __coerce x etc.)
        -- This improves arities of overloaded functions where
        -- there is only dictionary selection (no construction) involved
exprIsCheap' is_conlike (Let (NonRec x _) e)  
      | isUnLiftedType (idType x) = exprIsCheap' is_conlike e
      | otherwise                 = False
        -- strict lets always have cheap right hand sides,
        -- and do no allocation.

exprIsCheap' is_conlike other_expr      -- Applications and variables
  = go other_expr []
  where
        -- Accumulate value arguments, then decide
    go (App f a) val_args | isRuntimeArg a = go f (a:val_args)
                          | otherwise      = go f val_args

    go (Var _) [] = True        -- Just a type application of a variable
                                -- (f t1 t2 t3) counts as WHNF
    go (Var f) args
        = case idDetails f of
                RecSelId {}  -> go_sel args
                ClassOpId _  -> go_sel args
                PrimOpId op  -> go_primop op args

                _ | is_conlike f -> go_pap args
                  | length args < idArity f -> go_pap args

                _ -> isBottomingId f
                        -- Application of a function which
                        -- always gives bottom; we treat this as cheap
                        -- because it certainly doesn't need to be shared!
        
    go _ _ = False
 
    --------------
    go_pap args = all exprIsTrivial args
        -- For constructor applications and primops, check that all
        -- the args are trivial.  We don't want to treat as cheap, say,
        --      (1:2:3:4:5:[])
        -- We'll put up with one constructor application, but not dozens
        
    --------------
    go_primop op args = primOpIsCheap op && all (exprIsCheap' is_conlike) args
        -- In principle we should worry about primops
        -- that return a type variable, since the result
        -- might be applied to something, but I'm not going
        -- to bother to check the number of args
 
    --------------
    go_sel [arg] = exprIsCheap' is_conlike arg  -- I'm experimenting with making record selection
    go_sel _     = False                -- look cheap, so we will substitute it inside a
                                        -- lambda.  Particularly for dictionary field selection.
                -- BUT: Take care with (sel d x)!  The (sel d) might be cheap, but
                --      there's no guarantee that (sel d x) will be too.  Hence (n_val_args == 1)

exprIsCheap :: CoreExpr -> Bool
exprIsCheap = exprIsCheap' isDataConWorkId

exprIsExpandable :: CoreExpr -> Bool
exprIsExpandable = exprIsCheap' isConLikeId
\end{code}

\begin{code}
-- | 'exprOkForSpeculation' returns True of an expression that is:
--
--  * Safe to evaluate even if normal order eval might not 
--    evaluate the expression at all, or
--
--  * Safe /not/ to evaluate even if normal order would do so
--
-- Precisely, it returns @True@ iff:
--
--  * The expression guarantees to terminate, 
--
--  * soon, 
--
--  * without raising an exception,
--
--  * without causing a side effect (e.g. writing a mutable variable)
--
-- Note that if @exprIsHNF e@, then @exprOkForSpecuation e@.
-- As an example of the considerations in this test, consider:
--
-- > let x = case y# +# 1# of { r# -> I# r# }
-- > in E
--
-- being translated to:
--
-- > case y# +# 1# of { r# -> 
-- >    let x = I# r#
-- >    in E 
-- > }
-- 
-- We can only do this if the @y + 1@ is ok for speculation: it has no
-- side effects, and can't diverge or raise an exception.
exprOkForSpeculation :: CoreExpr -> Bool
exprOkForSpeculation (Lit _)     = True
exprOkForSpeculation (Type _)    = True
    -- Tick boxes are *not* suitable for speculation
exprOkForSpeculation (Var v)     = isUnLiftedType (idType v)
                                 && not (isTickBoxOp v)
exprOkForSpeculation (Note _ e)  = exprOkForSpeculation e
exprOkForSpeculation (Cast e _)  = exprOkForSpeculation e
exprOkForSpeculation other_expr
  = case collectArgs other_expr of
        (Var f, args) -> spec_ok (idDetails f) args
        _             -> False
 
  where
    spec_ok (DataConWorkId _) _
      = True    -- The strictness of the constructor has already
                -- been expressed by its "wrapper", so we don't need
                -- to take the arguments into account

    spec_ok (PrimOpId op) args
      | isDivOp op,             -- Special case for dividing operations that fail
        [arg1, Lit lit] <- args -- only if the divisor is zero
      = not (isZeroLit lit) && exprOkForSpeculation arg1
                -- Often there is a literal divisor, and this 
                -- can get rid of a thunk in an inner looop

      | otherwise
      = primOpOkForSpeculation op && 
        all exprOkForSpeculation args
                                -- A bit conservative: we don't really need
                                -- to care about lazy arguments, but this is easy

    spec_ok _ _ = False

-- | True of dyadic operators that can fail only if the second arg is zero!
isDivOp :: PrimOp -> Bool
-- This function probably belongs in PrimOp, or even in 
-- an automagically generated file.. but it's such a 
-- special case I thought I'd leave it here for now.
isDivOp IntQuotOp        = True
isDivOp IntRemOp         = True
isDivOp WordQuotOp       = True
isDivOp WordRemOp        = True
isDivOp FloatDivOp       = True
isDivOp DoubleDivOp      = True
isDivOp _                = False
\end{code}

\begin{code}
-- | True of expressions that are guaranteed to diverge upon execution
exprIsBottom :: CoreExpr -> Bool
exprIsBottom e = go 0 e
               where
                -- n is the number of args
                 go n (Note _ e)     = go n e
                 go n (Cast e _)     = go n e
                 go n (Let _ e)      = go n e
                 go _ (Case e _ _ _) = go 0 e   -- Just check the scrut
                 go n (App e _)      = go (n+1) e
                 go n (Var v)        = idAppIsBottom v n
                 go _ (Lit _)        = False
                 go _ (Lam _ _)      = False
                 go _ (Type _)       = False

idAppIsBottom :: Id -> Int -> Bool
idAppIsBottom id n_val_args = appIsBottom (idNewStrictness id) n_val_args
\end{code}

\begin{code}

-- | This returns true for expressions that are certainly /already/ 
-- evaluated to /head/ normal form.  This is used to decide whether it's ok 
-- to change:
--
-- > case x of _ -> e
--
-- into:
--
-- > e
--
-- and to decide whether it's safe to discard a 'seq'.
-- So, it does /not/ treat variables as evaluated, unless they say they are.
-- However, it /does/ treat partial applications and constructor applications
-- as values, even if their arguments are non-trivial, provided the argument
-- type is lifted. For example, both of these are values:
--
-- > (:) (f x) (map f xs)
-- > map (...redex...)
--
-- Because 'seq' on such things completes immediately.
--
-- For unlifted argument types, we have to be careful:
--
-- > C (f x :: Int#)
--
-- Suppose @f x@ diverges; then @C (f x)@ is not a value. However this can't 
-- happen: see "CoreSyn#let_app_invariant". This invariant states that arguments of
-- unboxed type must be ok-for-speculation (or trivial).
exprIsHNF :: CoreExpr -> Bool           -- True => Value-lambda, constructor, PAP
exprIsHNF (Var v)       -- NB: There are no value args at this point
  =  isDataConWorkId v  -- Catches nullary constructors, 
                        --      so that [] and () are values, for example
  || idArity v > 0      -- Catches (e.g.) primops that don't have unfoldings
  || isEvaldUnfolding (idUnfolding v)
        -- Check the thing's unfolding; it might be bound to a value
        -- A worry: what if an Id's unfolding is just itself: 
        -- then we could get an infinite loop...

exprIsHNF (Lit _)          = True
exprIsHNF (Type _)         = True       -- Types are honorary Values;
                                        -- we don't mind copying them
exprIsHNF (Lam b e)        = isRuntimeVar b || exprIsHNF e
exprIsHNF (Note _ e)       = exprIsHNF e
exprIsHNF (Cast e _)       = exprIsHNF e
exprIsHNF (App e (Type _)) = exprIsHNF e
exprIsHNF (App e a)        = app_is_value e [a]
exprIsHNF _                = False

-- There is at least one value argument
app_is_value :: CoreExpr -> [CoreArg] -> Bool
app_is_value (Var fun) args
  = idArity fun > valArgCount args      -- Under-applied function
    ||  isDataConWorkId fun             --  or data constructor
app_is_value (Note _ f) as = app_is_value f as
app_is_value (Cast f _) as = app_is_value f as
app_is_value (App f a)  as = app_is_value f (a:as)
app_is_value _          _  = False
\end{code}

These InstPat functions go here to avoid circularity between DataCon and Id

\begin{code}
dataConRepInstPat, dataConOrigInstPat :: [Unique] -> DataCon -> [Type] -> ([TyVar], [CoVar], [Id])
dataConRepFSInstPat :: [FastString] -> [Unique] -> DataCon -> [Type] -> ([TyVar], [CoVar], [Id])

dataConRepInstPat   = dataConInstPat dataConRepArgTys (repeat ((fsLit "ipv")))
dataConRepFSInstPat = dataConInstPat dataConRepArgTys
dataConOrigInstPat  = dataConInstPat dc_arg_tys       (repeat ((fsLit "ipv")))
  where 
    dc_arg_tys dc = map mkPredTy (dataConEqTheta dc) ++ map mkPredTy (dataConDictTheta dc) ++ dataConOrigArgTys dc
        -- Remember to include the existential dictionaries

dataConInstPat :: (DataCon -> [Type])      -- function used to find arg tys
                  -> [FastString]          -- A long enough list of FSs to use for names
                  -> [Unique]              -- An equally long list of uniques, at least one for each binder
                  -> DataCon
                  -> [Type]                -- Types to instantiate the universally quantified tyvars
               -> ([TyVar], [CoVar], [Id]) -- Return instantiated variables
-- dataConInstPat arg_fun fss us con inst_tys returns a triple 
-- (ex_tvs, co_tvs, arg_ids),
--
--   ex_tvs are intended to be used as binders for existential type args
--
--   co_tvs are intended to be used as binders for coercion args and the kinds
--     of these vars have been instantiated by the inst_tys and the ex_tys
--     The co_tvs include both GADT equalities (dcEqSpec) and 
--     programmer-specified equalities (dcEqTheta)
--
--   arg_ids are indended to be used as binders for value arguments, 
--     and their types have been instantiated with inst_tys and ex_tys
--     The arg_ids include both dicts (dcDictTheta) and
--     programmer-specified arguments (after rep-ing) (deRepArgTys)
--
-- Example.
--  The following constructor T1
--
--  data T a where
--    T1 :: forall b. Int -> b -> T(a,b)
--    ...
--
--  has representation type 
--   forall a. forall a1. forall b. (a ~ (a1,b)) => 
--     Int -> b -> T a
--
--  dataConInstPat fss us T1 (a1',b') will return
--
--  ([a1'', b''], [c :: (a1', b')~(a1'', b'')], [x :: Int, y :: b''])
--
--  where the double-primed variables are created with the FastStrings and
--  Uniques given as fss and us
dataConInstPat arg_fun fss uniqs con inst_tys 
  = (ex_bndrs, co_bndrs, arg_ids)
  where 
    univ_tvs = dataConUnivTyVars con
    ex_tvs   = dataConExTyVars con
    arg_tys  = arg_fun con
    eq_spec  = dataConEqSpec con
    eq_theta = dataConEqTheta con
    eq_preds = eqSpecPreds eq_spec ++ eq_theta

    n_ex = length ex_tvs
    n_co = length eq_preds

      -- split the Uniques and FastStrings
    (ex_uniqs, uniqs')   = splitAt n_ex uniqs
    (co_uniqs, id_uniqs) = splitAt n_co uniqs'

    (ex_fss, fss')     = splitAt n_ex fss
    (co_fss, id_fss)   = splitAt n_co fss'

      -- Make existential type variables
    ex_bndrs = zipWith3 mk_ex_var ex_uniqs ex_fss ex_tvs
    mk_ex_var uniq fs var = mkTyVar new_name kind
      where
        new_name = mkSysTvName uniq fs
        kind     = tyVarKind var

      -- Make the instantiating substitution
    subst = zipOpenTvSubst (univ_tvs ++ ex_tvs) (inst_tys ++ map mkTyVarTy ex_bndrs)

      -- Make new coercion vars, instantiating kind
    co_bndrs = zipWith3 mk_co_var co_uniqs co_fss eq_preds
    mk_co_var uniq fs eq_pred = mkCoVar new_name co_kind
       where
         new_name = mkSysTvName uniq fs
         co_kind  = substTy subst (mkPredTy eq_pred)

      -- make value vars, instantiating types
    mk_id_var uniq fs ty = mkUserLocal (mkVarOccFS fs) uniq (substTy subst ty) noSrcSpan
    arg_ids = zipWith3 mk_id_var id_uniqs id_fss arg_tys

-- | Returns @Just (dc, [x1..xn])@ if the argument expression is 
-- a constructor application of the form @dc x1 .. xn@
exprIsConApp_maybe :: CoreExpr -> Maybe (DataCon, [CoreExpr])
exprIsConApp_maybe (Cast expr co)
  =     -- Here we do the KPush reduction rule as described in the FC paper
    case exprIsConApp_maybe expr of {
        Nothing            -> Nothing ;
        Just (dc, dc_args) -> 

        -- The transformation applies iff we have
        --      (C e1 ... en) `cast` co
        -- where co :: (T t1 .. tn) ~ (T s1 ..sn)
        -- That is, with a T at the top of both sides
        -- The left-hand one must be a T, because exprIsConApp returned True
        -- but the right-hand one might not be.  (Though it usually will.)

    let (from_ty, to_ty)           = coercionKind co
        (from_tc, from_tc_arg_tys) = splitTyConApp from_ty
                -- The inner one must be a TyConApp
    in
    case splitTyConApp_maybe to_ty of {
        Nothing -> Nothing ;
        Just (to_tc, to_tc_arg_tys) 
                | from_tc /= to_tc -> Nothing
                -- These two Nothing cases are possible; we might see 
                --      (C x y) `cast` (g :: T a ~ S [a]),
                -- where S is a type function.  In fact, exprIsConApp
                -- will probably not be called in such circumstances,
                -- but there't nothing wrong with it 

                | otherwise  ->
    let
        tc_arity = tyConArity from_tc

        (univ_args, rest1)        = splitAt tc_arity dc_args
        (ex_args, rest2)          = splitAt n_ex_tvs rest1
        (co_args_spec, rest3)     = splitAt n_cos_spec rest2
        (co_args_theta, val_args) = splitAt n_cos_theta rest3

        arg_tys             = dataConRepArgTys dc
        dc_univ_tyvars      = dataConUnivTyVars dc
        dc_ex_tyvars        = dataConExTyVars dc
        dc_eq_spec          = dataConEqSpec dc
        dc_eq_theta         = dataConEqTheta dc
        dc_tyvars           = dc_univ_tyvars ++ dc_ex_tyvars
        n_ex_tvs            = length dc_ex_tyvars
        n_cos_spec          = length dc_eq_spec
        n_cos_theta         = length dc_eq_theta

        -- Make the "theta" from Fig 3 of the paper
        gammas              = decomposeCo tc_arity co
        new_tys             = gammas ++ map (\ (Type t) -> t) ex_args
        theta               = zipOpenTvSubst dc_tyvars new_tys

          -- First we cast the existential coercion arguments
        cast_co_spec (tv, ty) co 
          = cast_co_theta (mkEqPred (mkTyVarTy tv, ty)) co
        cast_co_theta eqPred (Type co) 
          | (ty1, ty2) <- getEqPredTys eqPred
          = Type $ mkSymCoercion (substTy theta ty1)
                   `mkTransCoercion` co
                   `mkTransCoercion` (substTy theta ty2)
        new_co_args = zipWith cast_co_spec  dc_eq_spec  co_args_spec ++
                      zipWith cast_co_theta dc_eq_theta co_args_theta
  
          -- ...and now value arguments
        new_val_args = zipWith cast_arg arg_tys val_args
        cast_arg arg_ty arg = mkCoerce (substTy theta arg_ty) arg

    in
    ASSERT( length univ_args == tc_arity )
    ASSERT( from_tc == dataConTyCon dc )
    ASSERT( and (zipWith coreEqType [t | Type t <- univ_args] from_tc_arg_tys) )
    ASSERT( all isTypeArg (univ_args ++ ex_args) )
    ASSERT2( equalLength val_args arg_tys, ppr dc $$ ppr dc_tyvars $$ ppr dc_ex_tyvars $$ ppr arg_tys $$ ppr dc_args $$ ppr univ_args $$ ppr ex_args $$ ppr val_args $$ ppr arg_tys  )

    Just (dc, map Type to_tc_arg_tys ++ ex_args ++ new_co_args ++ new_val_args)
    }}

{-
-- We do not want to tell the world that we have a
-- Cons, to *stop* Case of Known Cons, which removes
-- the TickBox.
exprIsConApp_maybe (Note (TickBox {}) expr)
  = Nothing
exprIsConApp_maybe (Note (BinaryTickBox {}) expr)
  = Nothing
-}

exprIsConApp_maybe (Note _ expr)
  = exprIsConApp_maybe expr
    -- We ignore InlineMe notes in case we have
    --  x = __inline_me__ (a,b)
    -- All part of making sure that INLINE pragmas never hurt
    -- Marcin tripped on this one when making dictionaries more inlinable
    --
    -- In fact, we ignore all notes.  For example,
    --          case _scc_ "foo" (C a b) of
    --                  C a b -> e
    -- should be optimised away, but it will be only if we look
    -- through the SCC note.

exprIsConApp_maybe expr = analyse (collectArgs expr)
  where
    analyse (Var fun, args)
        | Just con <- isDataConWorkId_maybe fun,
          args `lengthAtLeast` dataConRepArity con
                -- Might be > because the arity excludes type args
        = Just (con,args)

        -- Look through unfoldings, but only cheap ones, because
        -- we are effectively duplicating the unfolding
    analyse (Var fun, [])
        | let unf = idUnfolding fun,
          isExpandableUnfolding unf
        = exprIsConApp_maybe (unfoldingTemplate unf)

    analyse _ = Nothing
\end{code}



%************************************************************************
%*                                                                      *
\subsection{Equality}
%*                                                                      *
%************************************************************************

\begin{code}
-- | A cheap equality test which bales out fast!
--      If it returns @True@ the arguments are definitely equal,
--      otherwise, they may or may not be equal.
--
-- See also 'exprIsBig'
cheapEqExpr :: Expr b -> Expr b -> Bool

cheapEqExpr (Var v1)   (Var v2)   = v1==v2
cheapEqExpr (Lit lit1) (Lit lit2) = lit1 == lit2
cheapEqExpr (Type t1)  (Type t2)  = t1 `coreEqType` t2

cheapEqExpr (App f1 a1) (App f2 a2)
  = f1 `cheapEqExpr` f2 && a1 `cheapEqExpr` a2

cheapEqExpr (Cast e1 t1) (Cast e2 t2)
  = e1 `cheapEqExpr` e2 && t1 `coreEqCoercion` t2

cheapEqExpr _ _ = False

exprIsBig :: Expr b -> Bool
-- ^ Returns @True@ of expressions that are too big to be compared by 'cheapEqExpr'
exprIsBig (Lit _)      = False
exprIsBig (Var _)      = False
exprIsBig (Type _)     = False
exprIsBig (App f a)    = exprIsBig f || exprIsBig a
exprIsBig (Cast e _)   = exprIsBig e    -- Hopefully coercions are not too big!
exprIsBig _            = True
\end{code}



%************************************************************************
%*                                                                      *
\subsection{The size of an expression}
%*                                                                      *
%************************************************************************

\begin{code}
coreBindsSize :: [CoreBind] -> Int
coreBindsSize bs = foldr ((+) . bindSize) 0 bs

exprSize :: CoreExpr -> Int
-- ^ A measure of the size of the expressions, strictly greater than 0
-- It also forces the expression pretty drastically as a side effect
exprSize (Var v)         = v `seq` 1
exprSize (Lit lit)       = lit `seq` 1
exprSize (App f a)       = exprSize f + exprSize a
exprSize (Lam b e)       = varSize b + exprSize e
exprSize (Let b e)       = bindSize b + exprSize e
exprSize (Case e b t as) = seqType t `seq` exprSize e + varSize b + 1 + foldr ((+) . altSize) 0 as
exprSize (Cast e co)     = (seqType co `seq` 1) + exprSize e
exprSize (Note n e)      = noteSize n + exprSize e
exprSize (Type t)        = seqType t `seq` 1

noteSize :: Note -> Int
noteSize (SCC cc)       = cc `seq` 1
noteSize InlineMe       = 1
noteSize (CoreNote s)   = s `seq` 1  -- hdaume: core annotations
 
varSize :: Var -> Int
varSize b  | isTyVar b = 1
           | otherwise = seqType (idType b)             `seq`
                         megaSeqIdInfo (idInfo b)       `seq`
                         1

varsSize :: [Var] -> Int
varsSize = sum . map varSize

bindSize :: CoreBind -> Int
bindSize (NonRec b e) = varSize b + exprSize e
bindSize (Rec prs)    = foldr ((+) . pairSize) 0 prs

pairSize :: (Var, CoreExpr) -> Int
pairSize (b,e) = varSize b + exprSize e

altSize :: CoreAlt -> Int
altSize (c,bs,e) = c `seq` varsSize bs + exprSize e
\end{code}


%************************************************************************
%*                                                                      *
\subsection{Hashing}
%*                                                                      *
%************************************************************************

\begin{code}
hashExpr :: CoreExpr -> Int
-- ^ Two expressions that hash to the same @Int@ may be equal (but may not be)
-- Two expressions that hash to the different Ints are definitely unequal.
--
-- The emphasis is on a crude, fast hash, rather than on high precision.
-- 
-- But unequal here means \"not identical\"; two alpha-equivalent 
-- expressions may hash to the different Ints.
--
-- We must be careful that @\\x.x@ and @\\y.y@ map to the same hash code,
-- (at least if we want the above invariant to be true).

hashExpr e = fromIntegral (hash_expr (1,emptyVarEnv) e .&. 0x7fffffff)
             -- UniqFM doesn't like negative Ints

type HashEnv = (Int, VarEnv Int)  -- Hash code for bound variables

hash_expr :: HashEnv -> CoreExpr -> Word32
-- Word32, because we're expecting overflows here, and overflowing
-- signed types just isn't cool.  In C it's even undefined.
hash_expr env (Note _ e)              = hash_expr env e
hash_expr env (Cast e _)              = hash_expr env e
hash_expr env (Var v)                 = hashVar env v
hash_expr _   (Lit lit)               = fromIntegral (hashLiteral lit)
hash_expr env (App f e)               = hash_expr env f * fast_hash_expr env e
hash_expr env (Let (NonRec b r) e)    = hash_expr (extend_env env b) e * fast_hash_expr env r
hash_expr env (Let (Rec ((b,_):_)) e) = hash_expr (extend_env env b) e
hash_expr env (Case e _ _ _)          = hash_expr env e
hash_expr env (Lam b e)               = hash_expr (extend_env env b) e
hash_expr _   (Type _)                = WARN(True, text "hash_expr: type") 1
-- Shouldn't happen.  Better to use WARN than trace, because trace
-- prevents the CPR optimisation kicking in for hash_expr.

fast_hash_expr :: HashEnv -> CoreExpr -> Word32
fast_hash_expr env (Var v)      = hashVar env v
fast_hash_expr env (Type t)     = fast_hash_type env t
fast_hash_expr _   (Lit lit)    = fromIntegral (hashLiteral lit)
fast_hash_expr env (Cast e _)   = fast_hash_expr env e
fast_hash_expr env (Note _ e)   = fast_hash_expr env e
fast_hash_expr env (App _ a)    = fast_hash_expr env a  -- A bit idiosyncratic ('a' not 'f')!
fast_hash_expr _   _            = 1

fast_hash_type :: HashEnv -> Type -> Word32
fast_hash_type env ty 
  | Just tv <- getTyVar_maybe ty            = hashVar env tv
  | Just (tc,tys) <- splitTyConApp_maybe ty = let hash_tc = fromIntegral (hashName (tyConName tc))
                                              in foldr (\t n -> fast_hash_type env t + n) hash_tc tys
  | otherwise                               = 1

extend_env :: HashEnv -> Var -> (Int, VarEnv Int)
extend_env (n,env) b = (n+1, extendVarEnv env b n)

hashVar :: HashEnv -> Var -> Word32
hashVar (_,env) v
 = fromIntegral (lookupVarEnv env v `orElse` hashName (idName v))
\end{code}

%************************************************************************
%*                                                                      *
\subsection{Determining non-updatable right-hand-sides}
%*                                                                      *
%************************************************************************

Top-level constructor applications can usually be allocated
statically, but they can't if the constructor, or any of the
arguments, come from another DLL (because we can't refer to static
labels in other DLLs).

If this happens we simply make the RHS into an updatable thunk, 
and 'execute' it rather than allocating it statically.

\begin{code}
-- | This function is called only on *top-level* right-hand sides.
-- Returns @True@ if the RHS can be allocated statically in the output,
-- with no thunks involved at all.
rhsIsStatic :: PackageId -> CoreExpr -> Bool
-- It's called (i) in TidyPgm.hasCafRefs to decide if the rhs is, or
-- refers to, CAFs; (ii) in CoreToStg to decide whether to put an
-- update flag on it and (iii) in DsExpr to decide how to expand
-- list literals
--
-- The basic idea is that rhsIsStatic returns True only if the RHS is
--      (a) a value lambda
--      (b) a saturated constructor application with static args
--
-- BUT watch out for
--  (i) Any cross-DLL references kill static-ness completely
--      because they must be 'executed' not statically allocated
--      ("DLL" here really only refers to Windows DLLs, on other platforms,
--      this is not necessary)
--
-- (ii) We treat partial applications as redexes, because in fact we 
--      make a thunk for them that runs and builds a PAP
--      at run-time.  The only appliations that are treated as 
--      static are *saturated* applications of constructors.

-- We used to try to be clever with nested structures like this:
--              ys = (:) w ((:) w [])
-- on the grounds that CorePrep will flatten ANF-ise it later.
-- But supporting this special case made the function much more 
-- complicated, because the special case only applies if there are no 
-- enclosing type lambdas:
--              ys = /\ a -> Foo (Baz ([] a))
-- Here the nested (Baz []) won't float out to top level in CorePrep.
--
-- But in fact, even without -O, nested structures at top level are 
-- flattened by the simplifier, so we don't need to be super-clever here.
--
-- Examples
--
--      f = \x::Int. x+7        TRUE
--      p = (True,False)        TRUE
--
--      d = (fst p, False)      FALSE because there's a redex inside
--                              (this particular one doesn't happen but...)
--
--      h = D# (1.0## /## 2.0##)        FALSE (redex again)
--      n = /\a. Nil a                  TRUE
--
--      t = /\a. (:) (case w a of ...) (Nil a)  FALSE (redex)
--
--
-- This is a bit like CoreUtils.exprIsHNF, with the following differences:
--    a) scc "foo" (\x -> ...) is updatable (so we catch the right SCC)
--
--    b) (C x xs), where C is a contructors is updatable if the application is
--         dynamic
-- 
--    c) don't look through unfolding of f in (f x).

rhsIsStatic _this_pkg rhs = is_static False rhs
  where
  is_static :: Bool     -- True <=> in a constructor argument; must be atomic
          -> CoreExpr -> Bool
  
  is_static False (Lam b e) = isRuntimeVar b || is_static False e
  
  is_static _      (Note (SCC _) _) = False
  is_static in_arg (Note _ e)       = is_static in_arg e
  is_static in_arg (Cast e _)       = is_static in_arg e
  
  is_static _      (Lit lit)
    = case lit of
        MachLabel _ _ _ -> False
        _             -> True
        -- A MachLabel (foreign import "&foo") in an argument
        -- prevents a constructor application from being static.  The
        -- reason is that it might give rise to unresolvable symbols
        -- in the object file: under Linux, references to "weak"
        -- symbols from the data segment give rise to "unresolvable
        -- relocation" errors at link time This might be due to a bug
        -- in the linker, but we'll work around it here anyway. 
        -- SDM 24/2/2004
  
  is_static in_arg other_expr = go other_expr 0
   where
    go (Var f) n_val_args
#if mingw32_TARGET_OS
        | not (isDllName _this_pkg (idName f))
#endif
        =  saturated_data_con f n_val_args
        || (in_arg && n_val_args == 0)  
                -- A naked un-applied variable is *not* deemed a static RHS
                -- E.g.         f = g
                -- Reason: better to update so that the indirection gets shorted
                --         out, and the true value will be seen
                -- NB: if you change this, you'll break the invariant that THUNK_STATICs
                --     are always updatable.  If you do so, make sure that non-updatable
                --     ones have enough space for their static link field!

    go (App f a) n_val_args
        | isTypeArg a                    = go f n_val_args
        | not in_arg && is_static True a = go f (n_val_args + 1)
        -- The (not in_arg) checks that we aren't in a constructor argument;
        -- if we are, we don't allow (value) applications of any sort
        -- 
        -- NB. In case you wonder, args are sometimes not atomic.  eg.
        --   x = D# (1.0## /## 2.0##)
        -- can't float because /## can fail.

    go (Note (SCC _) _) _          = False
    go (Note _ f)       n_val_args = go f n_val_args
    go (Cast e _)       n_val_args = go e n_val_args

    go _                _          = False

    saturated_data_con f n_val_args
        = case isDataConWorkId_maybe f of
            Just dc -> n_val_args == dataConRepArity dc
            Nothing -> False
\end{code}