{-# LANGUAGE BangPatterns #-} {-# LANGUAGE CPP #-} {-# LANGUAGE FlexibleContexts #-} {-# LANGUAGE FlexibleInstances #-} {-# LANGUAGE TupleSections #-} {-# LANGUAGE UndecidableInstances #-} -- | Check that a datatype is strictly positive. module Agda.TypeChecking.Positivity where import Prelude hiding (null) import Control.Applicative hiding (empty) import Control.DeepSeq import Control.Monad.Reader import Data.Either import qualified Data.Foldable as Fold import Data.Function import Data.Graph (SCC(..), flattenSCC) import Data.List as List hiding (null) import Data.Map (Map) import qualified Data.Map as Map import qualified Data.Sequence as DS import Data.Set (Set) import qualified Data.Set as Set import Debug.Trace import Test.QuickCheck import Agda.Syntax.Common import qualified Agda.Syntax.Info as Info import Agda.Syntax.Internal import Agda.Syntax.Internal.Pattern import Agda.TypeChecking.Datatypes (isDataOrRecordType, DataOrRecord(..)) import Agda.TypeChecking.Records (unguardedRecord, recursiveRecord) import Agda.TypeChecking.Monad import Agda.TypeChecking.Monad.Builtin (primInf, CoinductionKit(..), coinductionKit) import Agda.TypeChecking.Reduce import Agda.TypeChecking.Positivity.Occurrence import Agda.TypeChecking.Pretty import Agda.TypeChecking.Substitute import Agda.TypeChecking.Telescope import qualified Agda.Utils.Graph.AdjacencyMap.Unidirectional as Graph import Agda.Utils.Functor import Agda.Utils.List import Agda.Utils.Maybe import Agda.Utils.Monad import Agda.Utils.Null import qualified Agda.Utils.Permutation as Perm import Agda.Utils.SemiRing import Agda.Utils.Size #include "undefined.h" import Agda.Utils.Impossible type Graph n e = Graph.Graph n n e -- | Check that the datatypes in the mutual block containing the given -- declarations are strictly positive. -- -- Also add information about positivity and recursivity of records -- to the signature. checkStrictlyPositive :: Info.MutualInfo -> Set QName -> TCM () checkStrictlyPositive mi qset = disableDestructiveUpdate $ do -- compute the occurrence graph for qs let qs = Set.toList qset reportSDoc "tc.pos.tick" 100 $ text "positivity of" <+> prettyTCM qs g <- buildOccurrenceGraph qset let (gstar, sccs') = Graph.gaussJordanFloydWarshallMcNaughtonYamada $ fmap occ g reportSDoc "tc.pos.tick" 100 $ text "constructed graph" reportSLn "tc.pos.graph" 5 $ "Positivity graph: N=" ++ show (size $ Graph.nodes g) ++ " E=" ++ show (length $ Graph.edges g) reportSDoc "tc.pos.graph" 10 $ vcat [ text "positivity graph for" <+> (fsep $ map prettyTCM qs) , nest 2 $ prettyTCM g ] reportSLn "tc.pos.graph" 5 $ "Positivity graph (completed): E=" ++ show (length $ Graph.edges gstar) reportSDoc "tc.pos.graph" 50 $ vcat [ text "transitive closure of positivity graph for" <+> prettyTCM qs , nest 2 $ prettyTCM gstar ] -- remember argument occurrences for qs in the signature setArgOccs qset qs gstar reportSDoc "tc.pos.tick" 100 $ text "set args" -- check positivity for all strongly connected components of the graph for qs let sccs = map flattenSCC sccs' reportSDoc "tc.pos.graph.sccs" 10 $ do let (triv, others) = partitionEithers $ for sccs' $ \ scc -> case scc of AcyclicSCC v -> Left v CyclicSCC vs -> Right vs sep [ text $ show (length triv) ++ " trivial sccs" , text $ show (length others) ++ " non-trivial sccs with lengths " ++ show (map length others) ] reportSDoc "tc.pos.graph.sccs" 15 $ text $ " sccs = " ++ show sccs forM_ sccs $ \ scc -> setMut [ q | DefNode q <- scc ] mapM_ (checkPos g gstar) $ qs reportSDoc "tc.pos.tick" 100 $ text "checked positivity" where checkPos :: Graph Node Edge -> Graph Node Occurrence -> QName -> TCM () checkPos g gstar q = inConcreteOrAbstractMode q $ do -- we check positivity only for data or record definitions whenJustM (isDatatype q) $ \ dr -> do reportSDoc "tc.pos.check" 10 $ text "Checking positivity of" <+> prettyTCM q let loop :: Maybe Occurrence loop = Graph.lookup (DefNode q) (DefNode q) gstar -- Note the property -- Agda.Utils.Graph.AdjacencyMap.Unidirectional.Tests.prop_productOfEdgesInBoundedWalk, -- which relates productOfEdgesInBoundedWalk to -- gaussJordanFloydWarshallMcNaughtonYamada. how :: String -> Occurrence -> TCM Doc how msg bound = case productOfEdgesInBoundedWalk occ g (DefNode q) (DefNode q) bound of Just (Edge _ how) -> fsep $ [prettyTCM q] ++ pwords "is" ++ pwords (msg ++ ", because it occurs") ++ [prettyTCM how] Nothing -> __IMPOSSIBLE__ -- if we have a negative loop, raise error -- ASR (23 December 2015). We don't raise a strictly positive -- error if the NO_POSITIVITY_CHECK pragma was set on in the -- mutual block. See Issue 1614. when (Info.mutualPositivityCheck mi) $ whenM positivityCheckEnabled $ case loop of Just o | o <= JustPos -> do err <- how "not strictly positive" JustPos setCurrentRange q $ typeError $ GenericDocError err _ -> return () -- if we find an unguarded record, mark it as such when (dr == IsRecord) $ case loop of Just o | o <= StrictPos -> do reportSDoc "tc.pos.record" 5 $ how "not guarded" StrictPos unguardedRecord q checkInduction q -- otherwise, if the record is recursive, mark it as well Just o | o <= GuardPos -> do reportSDoc "tc.pos.record" 5 $ how "recursive" GuardPos recursiveRecord q checkInduction q _ -> return () checkInduction :: QName -> TCM () checkInduction q = -- ASR (01 January 2016). We don't raise this error if the -- NO_POSITIVITY_CHECK pragma was set on in the record. See -- Issue 1760. when (Info.mutualPositivityCheck mi) $ whenM positivityCheckEnabled $ do -- Check whether the recursive record has been declared as -- 'Inductive' or 'Coinductive'. Otherwise, error. unlessM (isJust . recInduction . theDef <$> getConstInfo q) $ setCurrentRange (nameBindingSite $ qnameName q) $ typeError . GenericDocError =<< text "Recursive record" <+> prettyTCM q <+> text "needs to be declared as either inductive or coinductive" occ (Edge o _) = o isDatatype :: QName -> TCM (Maybe DataOrRecord) isDatatype q = do def <- theDef <$> getConstInfo q return $ case def of Datatype{dataClause = Nothing} -> Just IsData Record {recClause = Nothing} -> Just IsRecord _ -> Nothing -- Set the mutually recursive identifiers for a SCC. setMut :: [QName] -> TCM () setMut [] = return () -- nothing to do setMut [q] = return () -- no mutual recursion setMut qs = forM_ qs $ \ q -> setMutual q (delete q qs) -- TODO: The previous line is at least quadratic in the length -- of qs (assuming that the expression "delete q qs" is always -- forced, for instance due to serialisation). Presumably qs is -- usually short, but in some cases (for instance for generated -- code) it may be long. Wouldn't it be better to assign a -- unique identifier to each SCC, and avoid storing lists? -- Set the polarity of the arguments to a couple of definitions setArgOccs :: Set QName -> [QName] -> Graph Node Occurrence -> TCM () setArgOccs qset qs g = do -- Compute a map from each name in q to the maximal argument index let maxs = Map.fromListWith max [ (q, i) | ArgNode q i <- Set.toList $ Graph.sourceNodes g, q `Set.member` qset ] forM_ qs $ \ q -> inConcreteOrAbstractMode q $ do reportSDoc "tc.pos.args" 10 $ text "checking args of" <+> prettyTCM q n <- getDefArity =<< getConstInfo q -- If there is no outgoing edge @ArgNode q i@, all @n@ arguments are @Unused@. -- Otherwise, we obtain the occurrences from the Graph. let findOcc i = fromMaybe Unused $ Graph.lookup (ArgNode q i) (DefNode q) g args = caseMaybe (Map.lookup q maxs) (replicate n Unused) $ \ m -> map findOcc [0 .. max m (n - 1)] reportSDoc "tc.pos.args" 10 $ sep [ text "args of" <+> prettyTCM q <+> text "=" , nest 2 $ prettyList $ map (text . show) args ] -- The list args can take a long time to compute, but contains -- small elements, and is stored in the interface (right?), so -- it is computed deep-strictly. setArgOccurrences q $!! args getDefArity :: Definition -> TCM Int getDefArity def = case theDef def of defn@Function{} -> do let dropped = projectionArgs defn -- TODO: instantiateFull followed by arity could perhaps be -- optimised, presumably the instantiation can be performed -- lazily. subtract dropped . arity <$> instantiateFull (defType def) Datatype{ dataPars = n } -> return n Record{ recPars = n } -> return n _ -> return 0 -- Specification of occurrences ------------------------------------------- -- See also Agda.TypeChecking.Positivity.Occurrence. -- | Description of an occurrence. data OccursWhere = Unknown -- ^ an unknown position (treated as negative) | Known (DS.Seq Where) -- ^ The elements of the sequence, from left to right, explain how -- to get to the occurrence. deriving (Show, Eq, Ord) -- | One part of the description of an occurrence. data Where = LeftOfArrow | DefArg QName Nat -- ^ in the nth argument of a define constant | UnderInf -- ^ in the principal argument of built-in ∞ | VarArg -- ^ as an argument to a bound variable | MetaArg -- ^ as an argument of a metavariable | ConArgType QName -- ^ in the type of a constructor | IndArgType QName -- ^ in a datatype index of a constructor | InClause Nat -- ^ in the nth clause of a defined function | Matched -- ^ matched against in a clause of a defined function | InDefOf QName -- ^ in the definition of a constant deriving (Show, Eq, Ord) (>*<) :: OccursWhere -> OccursWhere -> OccursWhere Unknown >*< _ = Unknown Known _ >*< Unknown = Unknown Known os1 >*< Known os2 = Known (os1 DS.>< os2) instance PrettyTCM OccursWhere where prettyTCM o = prettyOs $ map maxOneLeftOfArrow $ uniq $ splitOnDef o where nth 0 = pwords "first" nth 1 = pwords "second" nth 2 = pwords "third" nth n = pwords $ show (n + 1) ++ "th" -- remove consecutive duplicates uniq = map head . group prettyOs [] = __IMPOSSIBLE__ prettyOs [o] = prettyO o <> text "." prettyOs (o:os) = prettyO o <> text ", which occurs" $$ prettyOs os prettyO Unknown = empty prettyO (Known ws) = Fold.foldrM (\w d -> return d $$ fsep (prettyW w)) empty ws prettyW w = case w of LeftOfArrow -> pwords "to the left of an arrow" DefArg q i -> pwords "in the" ++ nth i ++ pwords "argument to" ++ [prettyTCM q] UnderInf -> pwords "under" ++ [do -- this cannot fail if an 'UnderInf' has been generated Def inf _ <- ignoreSharing <$> primInf prettyTCM inf] VarArg -> pwords "in an argument to a bound variable" MetaArg -> pwords "in an argument to a metavariable" ConArgType c -> pwords "in the type of the constructor" ++ [prettyTCM c] IndArgType c -> pwords "in an index of the target type of the constructor" ++ [prettyTCM c] InClause i -> pwords "in the" ++ nth i ++ pwords "clause" Matched -> pwords "as matched against" InDefOf d -> pwords "in the definition of" ++ [prettyTCM d] maxOneLeftOfArrow Unknown = Unknown maxOneLeftOfArrow (Known ws) = Known $ noArrows DS.>< case DS.viewl startsWithArrow of DS.EmptyL -> DS.empty w DS.:< ws -> w DS.<| DS.filter (not . isArrow) ws where (noArrows, startsWithArrow) = DS.breakl isArrow ws isArrow LeftOfArrow{} = True isArrow _ = False splitOnDef Unknown = [Unknown] splitOnDef (Known ws) = split ws DS.empty where split ws acc = case DS.viewl ws of w@InDefOf{} DS.:< ws -> let rest = split ws (DS.singleton w) in if DS.null acc then rest else Known acc : rest w DS.:< ws -> split ws (acc DS.|> w) DS.EmptyL -> [Known acc] instance Sized OccursWhere where size Unknown = 1 size (Known ws) = 1 + size ws -- Computing occurrences -------------------------------------------------- data Item = AnArg Nat | ADef QName deriving (Eq, Ord, Show) type Occurrences = Map Item [OccursWhere] -- | Used to build 'Occurrences' and occurrence graphs. data OccurrencesBuilder = Concat [OccurrencesBuilder] | OccursAs Where OccurrencesBuilder | OccursHere Item | OnlyVarsUpTo Nat OccurrencesBuilder -- ^ @OnlyVarsUpTo n occs@ discards occurrences of de Bruijn index -- @>= n@. -- | Used to build 'Occurrences' and occurrence graphs. data OccurrencesBuilder' = Concat' [OccurrencesBuilder'] | OccursAs' Where OccurrencesBuilder' | OccursHere' Item OccursWhere emptyOB :: OccurrencesBuilder emptyOB = Concat [] (>+<) :: OccurrencesBuilder -> OccurrencesBuilder -> OccurrencesBuilder occs1 >+< occs2 = Concat [occs1, occs2] -- | Removes 'OnlyVarsUpTo' entries and adds 'OccursWhere' entries. -- -- WARNING: There can be lots of sharing between the generated -- 'OccursWhere' entries. Traversing all of these entries could be -- expensive. (See 'computeEdges' for an example.) preprocess :: OccurrencesBuilder -> OccurrencesBuilder' preprocess ob = case pp Nothing DS.empty ob of Nothing -> Concat' [] Just ob -> ob where pp :: Maybe Nat -- ^ Variables larger than or equal to this number, if any, -- are not retained. -> DS.Seq Where -> OccurrencesBuilder -> Maybe OccurrencesBuilder' pp !m ws (Concat obs) = case catMaybes $ map (pp m ws) obs of [] -> Nothing obs -> return (Concat' obs) pp m ws (OccursAs w ob) = OccursAs' w <$> pp m (ws DS.|> w) ob pp m ws (OnlyVarsUpTo n ob) = pp (Just $! maybe n (min n) m) ws ob pp m ws (OccursHere i) = do guard keep return (OccursHere' i (Known ws)) where keep = case (m, i) of (Nothing, _) -> True (_, ADef _) -> True (Just m, AnArg i) -> i < m -- | A type used locally in 'flatten'. data OccursWheres = OccursWheres :++ OccursWheres | Occurs OccursWhere -- | An interpreter for 'OccurrencesBuilder'. -- -- WARNING: There can be lots of sharing between the generated -- 'OccursWhere' entries. Traversing all of these entries could be -- expensive. (See 'computeEdges' for an example.) flatten :: OccurrencesBuilder -> Occurrences flatten = fmap (flip flatten'' []) . Map.fromListWith (:++) . flip flatten' [] . preprocess where flatten' :: OccurrencesBuilder' -> [(Item, OccursWheres)] -> [(Item, OccursWheres)] flatten' (Concat' obs) = foldr (\occs f -> flatten' occs . f) id obs flatten' (OccursAs' _ ob) = flatten' ob flatten' (OccursHere' i o) = ((i, Occurs o) :) flatten'' (os1 :++ os2) = flatten'' os1 . flatten'' os2 flatten'' (Occurs o) = (o :) -- | Context for computing occurrences. data OccEnv = OccEnv { vars :: [Maybe Item] -- ^ Items corresponding to the free variables. -- -- Potential invariant: It seems as if the list has the form -- @'genericReplicate' n 'Nothing' ++ 'map' ('Just' . 'AnArg') is@, -- for some @n@ and @is@, where @is@ is decreasing -- (non-strictly). , inf :: Maybe QName -- ^ Name for ∞ builtin. } -- | Monad for computing occurrences. type OccM = Reader OccEnv withExtendedOccEnv :: Maybe Item -> OccM a -> OccM a withExtendedOccEnv i = local $ \ e -> e { vars = i : vars e } -- | Running the monad getOccurrences :: (Show a, PrettyTCM a, ComputeOccurrences a) => [Maybe Item] -> a -> TCM OccurrencesBuilder getOccurrences vars a = do reportSDoc "tc.pos.occ" 70 $ text "computing occurrences in " <+> text (show a) reportSDoc "tc.pos.occ" 20 $ text "computing occurrences in " <+> prettyTCM a kit <- coinductionKit return $ runReader (occurrences a) $ OccEnv vars $ fmap nameOfInf kit class ComputeOccurrences a where occurrences :: a -> OccM OccurrencesBuilder instance ComputeOccurrences Clause where occurrences cl = do let ps = unnumberPatVars $ clausePats cl (Concat (mapMaybe matching (zip [0..] ps)) >+<) <$> walk (patItems ps) (clauseBody cl) where matching (i, p) | properlyMatching (unArg p) = Just $ OccursAs Matched $ OccursHere $ AnArg i | otherwise = Nothing walk _ NoBody = return emptyOB walk [] (Body v) = occurrences v walk (i : pis) (Bind b) = withExtendedOccEnv i $ walk pis $ absBody b walk [] Bind{} = __IMPOSSIBLE__ walk (_ : _) Body{} = __IMPOSSIBLE__ -- @patItems ps@ creates a map from the pattern variables of @ps@ -- to the index of the argument they are bound in. -- This map is given as a list. patItems ps = concat $ zipWith patItem [0..] ps -- @patItem i p@ replicates index @i@ as often as there are -- pattern variables in @p@ (dot patterns count as variable) patItem :: Int -> Arg Pattern -> [Maybe Item] patItem i p = map (const $ Just $ AnArg i) $ patternVars p instance ComputeOccurrences Term where occurrences v = case unSpine v of Var i args -> do vars <- asks vars occs <- occurrences args -- Apparently some development version of GHC chokes if the -- following line is replaced by vars ! i. let mi | i < length vars = vars !! i | otherwise = flip trace __IMPOSSIBLE__ $ "impossible: occurrence of de Bruijn index " ++ show i ++ " in vars " ++ show vars ++ " is unbound" return $ maybe emptyOB OccursHere mi >+< OccursAs VarArg occs Def d args -> do inf <- asks inf let occsAs = if Just d /= inf then OccursAs . DefArg d else \ n -> -- the principal argument of builtin INF (∞) is the second (n==1) -- the first is a level argument (n==0, counting from 0!) if n == 1 then OccursAs UnderInf else OccursAs (DefArg d n) occs <- mapM occurrences args return $ OccursHere (ADef d) >+< Concat (zipWith occsAs [0..] occs) Con c args -> occurrences args MetaV _ args -> OccursAs MetaArg <$> occurrences args Pi a b -> do oa <- occurrences a ob <- occurrences b return $ OccursAs LeftOfArrow oa >+< ob Lam _ b -> occurrences b Level l -> occurrences l Lit{} -> return emptyOB Sort{} -> return emptyOB DontCare _ -> return emptyOB -- Andreas, 2011-09-09: do we need to check for negative occurrences in irrelevant positions? Shared p -> occurrences $ derefPtr p instance ComputeOccurrences Level where occurrences (Max as) = occurrences as instance ComputeOccurrences PlusLevel where occurrences ClosedLevel{} = return emptyOB occurrences (Plus _ l) = occurrences l instance ComputeOccurrences LevelAtom where occurrences l = case l of MetaLevel _ vs -> OccursAs MetaArg <$> occurrences vs BlockedLevel _ v -> occurrences v NeutralLevel _ v -> occurrences v UnreducedLevel v -> occurrences v instance ComputeOccurrences Type where occurrences (El _ v) = occurrences v instance ComputeOccurrences a => ComputeOccurrences (Tele a) where occurrences EmptyTel = return emptyOB occurrences (ExtendTel a b) = occurrences (a, b) instance ComputeOccurrences a => ComputeOccurrences (Abs a) where occurrences (Abs _ b) = withExtendedOccEnv Nothing $ occurrences b occurrences (NoAbs _ b) = occurrences b instance ComputeOccurrences a => ComputeOccurrences (Elim' a) where occurrences Proj{} = __IMPOSSIBLE__ occurrences (Apply a) = occurrences a instance ComputeOccurrences a => ComputeOccurrences (Arg a) where occurrences = occurrences . unArg instance ComputeOccurrences a => ComputeOccurrences (Dom a) where occurrences = occurrences . unDom instance ComputeOccurrences a => ComputeOccurrences [a] where occurrences vs = Concat <$> mapM occurrences vs instance (ComputeOccurrences a, ComputeOccurrences b) => ComputeOccurrences (a, b) where occurrences (x, y) = do ox <- occurrences x oy <- occurrences y return $ ox >+< oy -- | Computes the occurrences in the given definition. -- -- WARNING: There can be lots of sharing between the 'OccursWhere' -- entries. Traversing all of these entries could be expensive. (See -- 'computeEdges' for an example.) computeOccurrences :: QName -> TCM Occurrences computeOccurrences q = flatten <$> computeOccurrences' q -- | Computes the occurrences in the given definition. computeOccurrences' :: QName -> TCM OccurrencesBuilder computeOccurrences' q = inConcreteOrAbstractMode q $ do reportSDoc "tc.pos" 25 $ do a <- defAbstract <$> getConstInfo q m <- asks envAbstractMode text "computeOccurrences" <+> prettyTCM q <+> text (show a) <+> text (show m) def <- getConstInfo q OccursAs (InDefOf q) <$> case theDef def of Function{funClauses = cs} -> do n <- getDefArity def cs <- map (etaExpandClause n) <$> instantiateFull cs Concat . zipWith (OccursAs . InClause) [0..] <$> mapM (getOccurrences []) cs Datatype{dataClause = Just c} -> getOccurrences [] =<< instantiateFull c Datatype{dataPars = np, dataCons = cs} -> do -- Andreas, 2013-02-27: first, each data index occurs as matched on. TelV tel t <- telView $ defType def let xs = [np .. size tel - 1] -- argument positions corresponding to indices ioccs = Concat $ map (OccursAs Matched . OccursHere . AnArg) xs -- Then, we compute the occurrences in the constructor types. let conOcc c = do a <- defType <$> getConstInfo c TelV tel t <- telView' <$> normalise a -- normalization needed e.g. for test/succeed/Bush.agda let indices = case unEl t of Def _ vs -> genericDrop np vs _ -> __IMPOSSIBLE__ let tel' = telFromList $ genericDrop np $ telToList tel vars np = map (Just . AnArg) $ downFrom np (>+<) <$> (OccursAs (ConArgType c) <$> getOccurrences (vars np) tel') <*> (OccursAs (IndArgType c) . OnlyVarsUpTo np <$> getOccurrences (vars $ size tel) indices) (>+<) ioccs <$> (Concat <$> mapM conOcc cs) Record{recClause = Just c} -> getOccurrences [] =<< instantiateFull c Record{recPars = np, recTel = tel} -> do let tel' = telFromList $ genericDrop np $ telToList tel vars = map (Just . AnArg) $ downFrom np getOccurrences vars =<< instantiateFull tel' -- Arguments to other kinds of definitions are hard-wired. Constructor{} -> return emptyOB Axiom{} -> return emptyOB Primitive{} -> return emptyOB -- | Eta expand a clause to have the given number of variables. -- Warning: doesn't put correct types in telescope! -- This is used instead of special treatment of lambdas -- (which was unsound: issue 121) etaExpandClause :: Nat -> Clause -> Clause etaExpandClause n c@Clause{ clauseTel = tel, namedClausePats = ps, clauseBody = b } | m <= 0 = c | otherwise = c { namedClausePats = raise m ps ++ map (defaultArg . unnamed . VarP . (,underscore)) (downFrom m) , clauseBody = liftBody m b , clauseTel = telFromList $ telToList tel ++ (replicate m $ (underscore,) <$> dummyDom) -- dummyDom, not __IMPOSSIBLE__, because of debug printing. } where m = n - genericLength ps bind 0 = id bind n = Bind . Abs underscore . bind (n - 1) vars = map (defaultArg . var) $ downFrom m -- vars = reverse [ defaultArg $ var i | i <- [0..m - 1] ] liftBody m (Bind b) = Bind $ fmap (liftBody m) b liftBody m NoBody = bind m NoBody liftBody m (Body v) = bind m $ Body $ raise m v `apply` vars -- Building the occurrence graph ------------------------------------------ data Node = DefNode !QName | ArgNode !QName !Nat deriving (Eq, Ord) instance Show Node where show (DefNode q) = show q show (ArgNode q i) = show q ++ "." ++ show i instance PrettyTCM Node where prettyTCM (DefNode q) = prettyTCM q prettyTCM (ArgNode q i) = prettyTCM q <> text ("." ++ show i) instance PrettyTCM n => PrettyTCM (WithNode n Edge) where prettyTCM (WithNode n (Edge o w)) = prettyTCM o <+> prettyTCM n <+> fsep (pwords $ show w) -- | Edge labels for the positivity graph. data Edge = Edge !Occurrence OccursWhere deriving (Eq, Ord, Show) instance Null Edge where null (Edge o _) = null o empty = Edge empty Unknown -- | These operations form a semiring if we quotient by the relation -- \"the 'Occurrence' components are equal\". instance SemiRing Edge where ozero = Edge ozero Unknown oone = Edge oone Unknown oplus _ e@(Edge Mixed _) = e -- dominant oplus e@(Edge Mixed _) _ = e oplus (Edge Unused _) e = e -- neutral oplus e (Edge Unused _) = e oplus (Edge JustNeg _) e@(Edge JustNeg _) = e oplus _ e@(Edge JustNeg w) = Edge Mixed w oplus e@(Edge JustNeg w) _ = Edge Mixed w oplus _ e@(Edge JustPos _) = e -- dominates strict pos. oplus e@(Edge JustPos _) _ = e oplus _ e@(Edge StrictPos _) = e -- dominates 'GuardPos' oplus e@(Edge StrictPos _) _ = e oplus (Edge GuardPos _) e@(Edge GuardPos _) = e otimes (Edge o1 w1) (Edge o2 w2) = Edge (otimes o1 o2) (w1 >*< w2) -- | As 'OccursWhere' does not have an 'oplus' we cannot do something meaningful -- for the @OccursWhere@ here. -- -- E.g. @ostar (Edge JustNeg w) = Edge Mixed (w `oplus` (w >*< w))@ -- would probably more sense, if we could do it. instance StarSemiRing Edge where ostar (Edge o w) = Edge (ostar o) w -- | WARNING: There can be lots of sharing between the 'OccursWhere' -- entries in the edges. Traversing all of these entries could be -- expensive. (See 'computeEdges' for an example.) buildOccurrenceGraph :: Set QName -> TCM (Graph Node Edge) buildOccurrenceGraph qs = Graph.fromListWith oplus . concat <$> mapM defGraph (Set.toList qs) where defGraph :: QName -> TCM [Graph.Edge Node Node Edge] defGraph q = do occs <- computeOccurrences' q es <- computeEdges qs q occs reportSDoc "tc.pos.occs" 40 $ (text "Occurrences in" <+> prettyTCM q <> text ":") $+$ (nest 2 $ vcat $ map (\(i, (n, s)) -> text (show i) <> text ":" <+> text (show n) <+> text "occurrences, of total size" <+> text (show s)) $ sortBy (compare `on` fst . snd) $ map (\(i, os) -> (i, (length os, sum $ map size os))) $ Map.toList (flatten occs)) reportSDoc "tc.pos.occs" 50 $ (nest 2 $ vcat $ map (\(i, os) -> (text (show i) <> text ":") $+$ (nest 2 $ vcat $ map (text . show) os)) (Map.toList (flatten occs))) reportSDoc "tc.pos.occs.edges" 60 $ text "Edges:" $+$ (nest 2 $ vcat $ map (\e -> let Edge o w = Graph.label e in prettyTCM (Graph.source e) <+> text "-[" <+> text (show o) <> text "," <+> text (show w) <+> text "]->" <+> prettyTCM (Graph.target e)) es) return es -- | Computes all non-'ozero' occurrence graph edges represented by -- the given 'OccurrencesBuilder'. -- -- WARNING: There can be lots of sharing between the 'OccursWhere' -- entries in the edges. Traversing all of these entries could be -- expensive. For instance, for the function @F@ in -- @benchmark/misc/SlowOccurrences.agda@ a large number of edges from -- the argument @X@ to the function @F@ are computed. These edges have -- polarity 'StrictPos', 'JustNeg' or 'JustPos', and contain the -- following 'OccursWhere' elements: -- -- * @'Known' ('DS.fromList' ['InDefOf' "F", 'InClause' 0])@, -- -- * @'Known' ('DS.fromList' ['InDefOf' "F", 'InClause' 0, 'LeftOfArrow'])@, -- -- * @'Known' ('DS.fromList' ['InDefOf' "F", 'InClause' 0, 'LeftOfArrow', 'LeftOfArrow'])@, -- -- * @'Known' ('DS.fromList' ['InDefOf' "F", 'InClause' 0, 'LeftOfArrow', 'LeftOfArrow', 'LeftOfArrow'])@, -- -- * and so on. computeEdges :: Set QName -- ^ The names in the current mutual block. -> QName -- ^ The current name. -> OccurrencesBuilder -> TCM [Graph.Edge Node Node Edge] computeEdges muts q ob = ($ []) <$> mkEdge __IMPOSSIBLE__ StrictPos (preprocess ob) where mkEdge to !pol ob = case ob of Concat' obs -> foldr (liftM2 (.)) (return id) [ mkEdge to pol ob | ob <- obs ] OccursAs' w ob -> do (to, pol) <- mkEdge' to pol w mkEdge to pol ob OccursHere' i o -> return $ if null pol then id else (Graph.Edge { Graph.source = case i of AnArg i -> ArgNode q i ADef q -> DefNode q , Graph.target = to , Graph.label = Edge pol o } :) mkEdge' to !pol w = case w of VarArg -> mixed MetaArg -> mixed LeftOfArrow -> negative DefArg d i -> do pol' <- isGuarding d if Set.member d muts then return (ArgNode d i, pol') else addPol =<< otimes pol' <$> getArgOccurrence d i UnderInf -> addPol GuardPos -- Andreas, 2012-06-09: ∞ is guarding ConArgType _ -> keepGoing IndArgType _ -> mixed InClause _ -> keepGoing Matched -> mixed -- consider arguments matched against as used InDefOf d -> do pol' <- isGuarding d return (DefNode d, pol') where keepGoing = return (to, pol) mixed = return (to, Mixed) negative = return (to, otimes pol JustNeg) addPol pol' = return (to, otimes pol pol') isGuarding d = do isDR <- isDataOrRecordType d return $ case isDR of Just IsData -> GuardPos -- a datatype is guarding _ -> StrictPos ------------------------------------------------------------------------ -- * Generators and tests ------------------------------------------------------------------------ instance Arbitrary OccursWhere where arbitrary = oneof [return Unknown, Known <$> arbitrary] shrink Unknown = [] shrink (Known ws) = Unknown : [ Known ws | ws <- shrink ws ] instance Arbitrary Where where arbitrary = oneof [ return LeftOfArrow , DefArg <$> arbitrary <*> arbitrary , return UnderInf , return VarArg , return MetaArg , ConArgType <$> arbitrary , IndArgType <$> arbitrary , InClause <$> arbitrary , return Matched , InDefOf <$> arbitrary ] instance CoArbitrary OccursWhere where coarbitrary (Known ws) = variant 0 . coarbitrary ws coarbitrary Unknown = variant 1 instance CoArbitrary Where where coarbitrary LeftOfArrow = variant 0 coarbitrary (DefArg a b) = variant 1 . coarbitrary (a, b) coarbitrary UnderInf = variant 2 coarbitrary VarArg = variant 3 coarbitrary MetaArg = variant 4 coarbitrary (ConArgType a) = variant 5 . coarbitrary a coarbitrary (IndArgType a) = variant 6 . coarbitrary a coarbitrary (InClause a) = variant 7 . coarbitrary a coarbitrary Matched = variant 8 coarbitrary (InDefOf a) = variant 9 . coarbitrary a instance Arbitrary Edge where arbitrary = Edge <$> arbitrary <*> arbitrary shrink (Edge o w) = [ Edge o w | o <- shrink o ] ++ [ Edge o w | w <- shrink w ] instance CoArbitrary Edge where coarbitrary (Edge o w) = coarbitrary (o, w) -- properties moved to Agda.TypeChecking.Positivity.Tests