target 0.1.3.0 → 0.2.0.0
raw patch · 18 files changed
+3851/−90 lines, 18 filesdep +hintdep ~QuickCheckdep ~basedep ~liquid-fixpointnew-component:exe:target
Dependencies added: hint
Dependency ranges changed: QuickCheck, base, liquid-fixpoint, liquidhaskell, template-haskell
Files
- bench/Main.hs +1/−1
- bin/Target.hs +29/−0
- src/Test/Target.hs +2/−2
- src/Test/Target/Eval.hs +4/−3
- src/Test/Target/Monad.hs +1/−1
- src/Test/Target/TH.hs +0/−52
- src/Test/Target/Targetable/Function.hs +1/−1
- src/Test/Target/Testable.hs +3/−2
- src/Test/Target/Types.hs +2/−2
- src/Test/Target/Util.hs +1/−1
- target.cabal +16/−25
- test/Expr.hs +124/−0
- test/HOFs.hs +22/−0
- test/List.hs +70/−0
- test/Map.hs +3131/−0
- test/MapTest.hs +158/−0
- test/RBTree.hs +239/−0
- test/RBTreeTest.hs +47/−0
bench/Main.hs view
@@ -20,7 +20,7 @@ import qualified Data.ByteString as B import qualified Data.ByteString.Char8 as B8 import qualified Data.ByteString.Lazy as LB-import Data.Csv+import Data.Csv hiding (header) import qualified Data.List as L import Data.IORef import Data.Monoid
+ bin/Target.hs view
@@ -0,0 +1,29 @@+{-# LANGUAGE LambdaCase #-}+module Main where++import Language.Haskell.Interpreter+import System.Environment+import System.Exit+import System.IO+import Test.Target+import Text.Printf++main :: IO ()+main = do+ [src, binder] <- getArgs+ r <- runInterpreter $ do+ loadModules [src]+ mods <- getLoadedModules+ -- liftIO $ print mods+ setImportsQ $ map (\m -> (m,Nothing)) mods+ ++ [("Test.Target", Nothing), ("Prelude", Nothing)]+ set [languageExtensions := [TemplateHaskell]]+ let expr = printf "$(targetResultTH '%s \"%s\")" binder src+ -- liftIO $ putStrLn expr+ interpret expr (as :: IO Result)+ case r of+ Left e -> hPrint stderr e >> exitWith (ExitFailure 2)+ Right x -> x >>= \case+ Errored e -> hPutStrLn stderr e >> exitWith (ExitFailure 2)+ Failed s -> printf "Found counter-example: %s\n" s >> exitWith (ExitFailure 1)+ Passed n -> printf "OK! Passed %d tests.\n" n >> exitSuccess
src/Test/Target.hs view
@@ -18,16 +18,16 @@ import Control.Monad.State import qualified Language.Haskell.TH as TH import System.Process (terminateProcess)+import Test.QuickCheck.All (monomorphic) import Text.Printf (printf) import Language.Fixpoint.Names-import Language.Fixpoint.SmtLib2 hiding (verbose)+import Language.Fixpoint.Smt.Interface hiding (verbose) import Test.Target.Monad import Test.Target.Targetable (Targetable(..)) import Test.Target.Targetable.Function () import Test.Target.Testable-import Test.Target.TH import Test.Target.Types import Test.Target.Util
src/Test/Target/Eval.hs view
@@ -12,7 +12,8 @@ import Text.Printf import qualified GHC-import Language.Fixpoint.SmtLib2+import Language.Fixpoint.Smt.Interface+import Language.Fixpoint.Smt.Theories (theorySymbols) import Language.Fixpoint.Types hiding (R) import Language.Haskell.Liquid.Types hiding (var) @@ -80,7 +81,7 @@ Nothing -> throwM $ EvalError $ printf "applyMeasure(%s): no equation for %s" name (show ct) Just x -> return x -applyMeasure n m e _+applyMeasure n _ e _ = throwM $ EvalError $ printf "applyMeasure(%s, %s)" n (showpp e) setSym :: Symbol@@ -148,7 +149,7 @@ evalExpr (ESym s) _ = return $ ESym s evalExpr (EBin b e1 e2) m = evalBop b <$> evalExpr e1 m <*> evalExpr e2 m evalExpr (EApp f es) m- | val f == "Set_emp" || val f == "Set_sng" || val f `M.member` smt_set_funs+ | val f == "Set_emp" || val f == "Set_sng" || val f `M.member` theorySymbols = mapM (`evalExpr` m) es >>= \es' -> evalSet (val f) es' | otherwise = filter ((==f) . name) <$> gets measEnv >>= \case
src/Test/Target/Monad.hs view
@@ -42,7 +42,7 @@ import Language.Fixpoint.Config (SMTSolver (..)) import Language.Fixpoint.Names-import Language.Fixpoint.SmtLib2 hiding (verbose)+import Language.Fixpoint.Smt.Interface hiding (verbose) import Language.Fixpoint.Types import Language.Haskell.Liquid.PredType import Language.Haskell.Liquid.RefType
− src/Test/Target/TH.hs
@@ -1,52 +0,0 @@-{-# LANGUAGE RankNTypes #-}-{-# LANGUAGE TemplateHaskell #-}--module Test.Target.TH where--import Control.Monad-import qualified Language.Haskell.TH as TH--------------------------------------------------------------------------- Testing Polymorphic Functions (courtesy of QuickCheck)-------------------------------------------------------------------------type Error = forall a. String -> a---- | Monomorphise an arbitrary property by defaulting all type variables to 'Integer'.------ For example, if @f@ has type @'Ord' a => [a] -> [a]@--- then @$('monomorphic' 'f)@ has type @['Integer'] -> ['Integer']@.------ If you want to use 'monomorphic' in the same file where you defined the--- property, the same scoping problems pop up as in 'quickCheckAll':--- see the note there about @return []@.-monomorphic :: TH.Name -> TH.ExpQ-monomorphic t = do- ty0 <- fmap infoType (TH.reify t)- let err msg = error $ msg ++ ": " ++ TH.pprint ty0- (polys, ctx, ty) <- deconstructType err ty0- case polys of- [] -> return (TH.VarE t)- _ -> do- integer <- [t| Integer |]- ty' <- monomorphiseType err integer ty- return (TH.SigE (TH.VarE t) ty')--infoType :: TH.Info -> TH.Type-infoType (TH.ClassOpI _ ty _ _) = ty-infoType (TH.DataConI _ ty _ _) = ty-infoType (TH.VarI _ ty _ _) = ty--deconstructType :: Error -> TH.Type -> TH.Q ([TH.Name], TH.Cxt, TH.Type)-deconstructType err ty0@(TH.ForallT xs ctx ty) = do- let plain (TH.PlainTV _) = True- plain _ = False- unless (all plain xs) $ err "Higher-kinded type variables in type"- return (map (\(TH.PlainTV x) -> x) xs, ctx, ty)-deconstructType _ ty = return ([], [], ty)--monomorphiseType :: Error -> TH.Type -> TH.Type -> TH.TypeQ-monomorphiseType err mono ty@(TH.VarT n) = return mono-monomorphiseType err mono (TH.AppT t1 t2) = liftM2 TH.AppT (monomorphiseType err mono t1) (monomorphiseType err mono t2)-monomorphiseType err mono ty@(TH.ForallT _ _ _) = err $ "Higher-ranked type"-monomorphiseType err mono ty = return ty
src/Test/Target/Targetable/Function.hs view
@@ -21,7 +21,7 @@ import Data.Proxy import qualified Data.Text as T import qualified GHC-import Language.Fixpoint.SmtLib2+import Language.Fixpoint.Smt.Interface import Language.Fixpoint.Types hiding (ofReft, reft) import Language.Haskell.Liquid.GhcMisc (qualifiedNameSymbol) import Language.Haskell.Liquid.RefType (addTyConInfo, rTypeSort)
src/Test/Target/Testable.hs view
@@ -28,7 +28,8 @@ import qualified Data.Text.Lazy as LT import Text.Printf -import Language.Fixpoint.SmtLib2+import Language.Fixpoint.Smt.Interface+import Language.Fixpoint.Smt.Theories (theorySymbols) import Language.Fixpoint.Types hiding (Result) import Language.Haskell.Liquid.RefType import Language.Haskell.Liquid.Types hiding (Result (..), env, var)@@ -179,7 +180,7 @@ let defFun x t = io $ smtWrite ctx (makeDecl x t) forM_ ms $ \m -> do let x = val (name m)- if x `M.member` smt_set_funs+ if x `M.member` theorySymbols then return () else defFun x (rTypeSort emb (sort m)) -- assert constraints
src/Test/Target/Types.hs view
@@ -8,7 +8,7 @@ import qualified Data.Text as T import Data.Typeable -import Language.Fixpoint.SmtLib2+import Language.Fixpoint.Smt.Interface import Language.Fixpoint.Types import Language.Haskell.Liquid.Types @@ -66,6 +66,6 @@ data Result = Passed !Int | Failed !String | Errored !String- deriving (Show)+ deriving (Show, Typeable) -- resultPassed (Passed i) = i
src/Test/Target/Util.hs view
@@ -26,7 +26,7 @@ import qualified GHC.Paths import qualified HscTypes as GHC -import Language.Fixpoint.SmtLib2+import Language.Fixpoint.Smt.Interface import Language.Fixpoint.Types hiding (prop) import Language.Haskell.Liquid.CmdLine import Language.Haskell.Liquid.GhcInterface
target.cabal view
@@ -1,5 +1,5 @@ name: target-version: 0.1.3.0+version: 0.2.0.0 synopsis: Generate test-suites from refinement types. description: Target is a library for testing Haskell functions based on@@ -23,7 +23,7 @@ with refinement types, we have a series of <http://goto.ucsd.edu/~rjhala/liquid/haskell/blog/blog/categories/basic/ blog posts> as well as an- <http://github.com/ucsd-progsys/liquidhaskell/tree/master/docs/tutorial evolving tutorial>.+ <http://ucsd-progsys.github.io/liquidhaskell-tutorial/ evolving tutorial>. Target uses the same specification language as LiquidHaskell, so the examples should carry over. .@@ -46,8 +46,7 @@ library default-language: Haskell2010 hs-source-dirs: src- ghc-options: -Wall -fno-warn-name-shadowing -fno-warn-orphans- ghc-prof-options: -fprof-auto+ ghc-options: -Wall -fno-warn-name-shadowing -fno-warn-orphans -fno-warn-unused-imports -fno-warn-dodgy-imports -fno-warn-deprecated-flags -fno-warn-deprecations exposed-modules: Test.Target, Test.Target.Eval, Test.Target.Expr,@@ -55,7 +54,6 @@ Test.Target.Targetable, Test.Target.Targetable.Function, Test.Target.Testable,- Test.Target.TH, Test.Target.Types, Test.Target.Util @@ -66,27 +64,28 @@ , filepath >= 1.3.0.1 , ghc >= 7.8.3 , ghc-paths- , liquid-fixpoint >= 0.3.0.1- , liquidhaskell >= 0.4.1.1+ , liquid-fixpoint >= 0.4+ , liquidhaskell >= 0.5 , mtl >= 2.1.2 , pretty , process , syb >= 0.4.2 , tagged >= 0.7- , template-haskell+ , template-haskell >= 2.8 , text >= 1.0 , text-format , th-lift , transformers >= 0.3 , unordered-containers >= 0.2.3.0 , vector+ -- only for the `monomorphic` TH splice+ , QuickCheck >= 2.7 benchmark bench type: exitcode-stdio-1.0 default-language: Haskell2010 hs-source-dirs: bench ghc-options: -O2- ghc-prof-options: -fprof-auto main-is: Main.hs build-depends: base, aeson,@@ -124,6 +123,7 @@ hs-source-dirs: test ghc-options: -O2 main-is: Main.hs+ other-modules: Expr, HOFs, List, Map, MapTest, RBTree, RBTreeTest --c-sources: cbits/fpstring.c --include-dirs: include --includes: fpstring.h@@ -145,19 +145,10 @@ template-haskell, unordered-containers --- executable liquid-check--- default-language: Haskell2010--- hs-source-dirs: bin--- main-is: Target.hs--- build-depends: base,--- Target,--- liquid-fixpoint,--- data-timeout >= 0.3,--- ghc,--- ghc-paths,--- directory,--- filepath,--- process,--- text,--- time,--- transformers+executable target+ default-language: Haskell2010+ hs-source-dirs: bin+ main-is: Target.hs+ build-depends: base,+ hint,+ target
+ test/Expr.hs view
@@ -0,0 +1,124 @@+{-# LANGUAGE DeriveGeneric #-}+{-# LANGUAGE FlexibleInstances #-}+{-# LANGUAGE MultiParamTypeClasses #-}+{-# LANGUAGE OverloadedStrings #-}+{-# LANGUAGE ScopedTypeVariables #-}+{-@ LIQUID "--idirs=../src" @-}+module Expr where++import Data.Set (Set, (\\))+import qualified Data.Set as Set+import GHC.Generics++import Test.LiquidCheck+import Test.LiquidCheck.Eval (setSym)+import Test.LiquidCheck.Expr (app)++import BasicTypes (TupleSort (..))+import Control.Applicative+import Control.Monad.State+import qualified Data.HashMap.Strict as HM+import qualified Data.Map as M+import Data.Monoid+import Data.Proxy+import Language.Fixpoint.Types (Sort (..))+import Language.Haskell.Liquid.PredType+import Language.Haskell.Liquid.Types (RType (..))+import Test.LiquidCheck+import Test.LiquidCheck.Gen (GenState (..))+import Test.LiquidCheck.Util+import TysWiredIn (listTyCon, tupleTyCon)++data Expr = Var Char+ | Lam Char Expr+ | App Expr Expr+ deriving (Generic, Show)++hasDepth d (Var c) = d == 1+hasDepth d (Lam c e) = hasDepth (d-1) e+hasDepth d (App f e) = hasDepth (d-1) f || hasDepth (d-1) e++{-@ measure freeVars :: Expr -> (Set Char)+ freeVars (Var v) = (Set_sng v)+ freeVars (Lam v e) = (Set_dif (freeVars e) (Set_sng v))+ freeVars (App x y) = (Set_cup (freeVars x) (freeVars y))+ @-}++{-@ measure isLam :: Expr -> Prop+ isLam (Var v) = false+ isLam (Lam v e) = true+ isLam (App x y) = false+ @-}++{-@ data Expr = Var (x1 :: Char)+ | Lam (x2 :: Char) (x3 :: Expr)+ | App (x4 :: Expr) (x5 :: Expr)+ @-}++{-@ type Closed = {v:Expr | (Set_emp (freeVars v))} @-}++instance Constrain Expr++{-@ measure prop :: Bool -> Prop+ prop (True) = true+ prop (False) = false+ @-}+{-@ type Valid = {v:Bool | (prop v)} @-}++freeVars (Var v) = Set.singleton v+freeVars (Lam v e) = freeVars e \\ Set.singleton v+freeVars (App x y) = freeVars x `Set.union` freeVars y++{-@ inv :: Closed -> Valid @-}+inv e = Set.null $ freeVars e ++closed = inv++{- subst :: e1:Closed -> n:Char -> e2:Closed+ -> {v:Closed | (if (Set_mem n (freeVars e2))+ then (freeVars v) = (Set_cup (Set_dif (freeVars e2)+ (Set_sng n))+ (freeVars e1))+ else (freeVars v) = (freeVars e2))}+ @-}+{-@ subst :: e1:Closed -> n:Char -> e2:Closed -> Closed @-}+subst :: Expr -> Char -> Expr -> Expr+subst e1 v e2@(Var v')+ = if v == v' then e1 else e2+subst e1 v e2@(Lam v' e')+ | v == v' = e2+ | v' `Set.member` fvs = subst e1 v (freshen e2)+ | otherwise = Lam v' (subst e1 v e')+ where+ fvs = freeVars e1+subst e v (App e1 e2)+ = App e1' e2'+ where+ e1' = subst e v e1+ e2' = subst e v e2++{-@ freshen :: e:{Expr | (isLam e)} -> {v:Expr | (freeVars v) = (freeVars e)} @-}+freshen (Lam v e) = Lam v' (subst (Var v') v e)+ where+ v' = fresh v (freeVars e)++{-@ fresh :: n:Char -> ns:Set Char -> {v:Char | not (v == n || (Set_mem v ns))} @-}+fresh :: Char -> Set Char -> Char+fresh v vs = succ $ Set.findMax (Set.insert v vs)++--instance (Ord a, Constrain a) => Constrain (Set a) where+-- getType _ = FObj "Data.Set.Base.Set"+-- gen p d (RApp c ts ps r)+-- = do tyi <- gets tyconInfo+-- let listRTyCon = tyi HM.! listTyCon+-- gen (Proxy :: Proxy [a]) d (RApp listRTyCon ts [] mempty)+-- stitch d t = stitch d t >>= \(xs :: [a]) -> return $ Set.fromList xs+-- toExpr s = app setSym [toExpr x | x <- Set.toList s]++ encode v t = undefined++liquidTests :: [(String, Test)]+liquidTests = [ ("inv", T inv)+ , ("freshen", T freshen)+-- , ("fresh", T fresh)+ , ("subst", T subst)]
+ test/HOFs.hs view
@@ -0,0 +1,22 @@+{-# LANGUAGE TemplateHaskell #-}+module HOFs where++import Test.Target+import Text.Show.Functions ()++{-@ foo :: (x:Int -> {v:Int | v > x}) -> {v:Int | v > 0} @-}+foo :: (Int -> Int) -> Int+foo f = f 0++foo_bad :: (Int -> Int) -> Int+foo_bad f = f (-1)++{-@ list_foo :: xs:{[Int] | len xs > 0} -> (xs:[Int] -> {v:[Int] | len v < len xs})+ -> {v:[Int] | len v < len xs}+ @-}+list_foo :: [Int] -> ([Int] -> [Int]) -> [Int]+list_foo xs f = f xs++list_foo_bad :: [Int] -> ([Int] -> [Int]) -> [Int]+list_foo_bad xs f = f []+
+ test/List.hs view
@@ -0,0 +1,70 @@+{-# LANGUAGE FlexibleInstances #-}+{-# LANGUAGE MultiParamTypeClasses #-}+{-# LANGUAGE OverloadedStrings #-}+{-# LANGUAGE DeriveGeneric #-}+{-# LANGUAGE ScopedTypeVariables #-}+module List where++import GHC.Generics+import Test.Target+import Test.Target.Targetable++--------------------------------------------------------------------------------+--- Code+--------------------------------------------------------------------------------+data List a = Nil | Cons a (List a) deriving (Generic, Show)+infixr `Cons`++insert :: Int -> List Int -> List Int+insert x ys = insert' x ys++insert' x Nil+ = Cons x Nil+insert' x (y `Cons` ys)+ | x < y+ = x `Cons` y `Cons` ys+ | x == y+ = y `Cons` ys+ | otherwise+ = y `Cons` insert' x ys++insert_bad :: Int -> List Int -> List Int+insert_bad x Nil+ = Cons x Nil+insert_bad x (y `Cons` ys)+ | x < y+ = x `Cons` y `Cons` ys+ | otherwise+ = y `Cons` insert_bad x ys++mytake :: Int -> List Int -> List Int+mytake 0 xs = Nil+mytake _ Nil = Nil+mytake n (Cons x xs) = x `Cons` mytake (n-1) xs++{-@ mymap :: (Int -> Int) -> x:List Int -> {v:List Int | (llen v) = (llen x)} @-}+mymap :: (Int -> Int) -> List Int -> List Int+mymap f Nil = Nil+mymap f (Cons x xs) = Cons (f x) (mymap f xs)++--------------------------------------------------------------------------------+--- Target+--------------------------------------------------------------------------------+instance Targetable a => Targetable (List a)++{-@ data List a <p:: a -> a -> Prop> =+ Nil | Cons (zoo::a) (zoog::List <p> (a<p zoo>))+ @-}++{-@ measure llen :: List a -> Int+ llen(Nil) = 0+ llen(Cons x xs) = 1 + llen(xs)+ @-}++{-@ type SortedList a = List <{\x y -> x < y}> a @-}++{-@ mytake :: n:Nat -> xs:SortedList Nat+ -> {v:SortedList Nat | (Min (llen v) n (llen xs))} @-}++{-@ insert :: n:Int -> xs:SortedList Int -> SortedList Int @-}+
+ test/Map.hs view
@@ -0,0 +1,3131 @@+{-# LANGUAGE DeriveGeneric #-}+{-# LANGUAGE RankNTypes #-}+{-# LANGUAGE TemplateHaskell #-}+{-@ LIQUID "--totality" @-}+{-# LANGUAGE CPP #-}+#if __GLASGOW_HASKELL__+-- LIQUID {- LANGUAGE DeriveDataTypeable, StandaloneDeriving -}+#endif+#if !defined(TESTING) && __GLASGOW_HASKELL__ >= 703+{-# LANGUAGE Trustworthy #-}+#endif+-----------------------------------------------------------------------------+-- |+-- Module : Data.Map.Base+-- Copyright : (c) Daan Leijen 2002+-- (c) Andriy Palamarchuk 2008+-- License : BSD-style+-- Maintainer : libraries@haskell.org+-- Stability : provisional+-- Portability : portable+--+-- An efficient implementation of maps from keys to values (dictionaries).+--+-- Since many function names (but not the type name) clash with+-- "Prelude" names, this module is usually imported @qualified@, e.g.+--+-- > import Data.Map (Map)+-- > import qualified Data.Map as Map+--+-- The implementation of 'Map' is based on /size balanced/ binary trees (or+-- trees of /bounded balance/) as described by:+--+-- * Stephen Adams, \"/Efficient sets: a balancing act/\",+-- Journal of Functional Programming 3(4):553-562, October 1993,+-- <http://www.swiss.ai.mit.edu/~adams/BB/>.+--+-- * J. Nievergelt and E.M. Reingold,+-- \"/Binary search trees of bounded balance/\",+-- SIAM journal of computing 2(1), March 1973.+--+-- Note that the implementation is /left-biased/ -- the elements of a+-- first argument are always preferred to the second, for example in+-- 'union' or 'insert'.+--+-- Operation comments contain the operation time complexity in+-- the Big-O notation <http://en.wikipedia.org/wiki/Big_O_notation>.+-----------------------------------------------------------------------------++-- [Note: Using INLINABLE]+-- ~~~~~~~~~~~~~~~~~~~~~~~+-- It is crucial to the performance that the functions specialize on the Ord+-- type when possible. GHC 7.0 and higher does this by itself when it sees th+-- unfolding of a function -- that is why all public functions are marked+-- INLINABLE (that exposes the unfolding).+++-- [Note: Using INLINE]+-- ~~~~~~~~~~~~~~~~~~~~+-- For other compilers and GHC pre 7.0, we mark some of the functions INLINE.+-- We mark the functions that just navigate down the tree (lookup, insert,+-- delete and similar). That navigation code gets inlined and thus specialized+-- when possible. There is a price to pay -- code growth. The code INLINED is+-- therefore only the tree navigation, all the real work (rebalancing) is not+-- INLINED by using a NOINLINE.+--+-- All methods marked INLINE have to be nonrecursive -- a 'go' function doing+-- the real work is provided.+++-- [Note: Type of local 'go' function]+-- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+-- If the local 'go' function uses an Ord class, it sometimes heap-allocates+-- the Ord dictionary when the 'go' function does not have explicit type.+-- In that case we give 'go' explicit type. But this slightly decrease+-- performance, as the resulting 'go' function can float out to top level.+++-- [Note: Local 'go' functions and capturing]+-- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+-- As opposed to IntMap, when 'go' function captures an argument, increased+-- heap-allocation can occur: sometimes in a polymorphic function, the 'go'+-- floats out of its enclosing function and then it heap-allocates the+-- dictionary and the argument. Maybe it floats out too late and strictness+-- analyzer cannot see that these could be passed on stack.+--+-- For example, change 'member' so that its local 'go' function is not passing+-- argument k and then look at the resulting code for hedgeInt.+++-- [Note: Order of constructors]+-- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+-- The order of constructors of Map matters when considering performance.+-- Currently in GHC 7.0, when type has 2 constructors, a forward conditional+-- jump is made when successfully matching second constructor. Successful match+-- of first constructor results in the forward jump not taken.+-- On GHC 7.0, reordering constructors from Tip | Bin to Bin | Tip+-- improves the benchmark by up to 10% on x86.++module Map (+ -- * Map type+ Map(..) -- instance Eq,Show,Read++ -- * Operators+ , (!), (\\)++ -- * Query+ , null+ , size+ , member+ , notMember+ , lookup+ , findWithDefault+ , lookupLT+ , lookupGT+ , lookupLE+ , lookupGE++ -- * Construction+ , empty+ , singleton++ -- ** Insertion+ , insert+ , insertWith+ , insertWithKey+ , insertLookupWithKey++ -- ** Delete\/Update+ , delete+ , adjust+ , adjustWithKey+ , update+ , updateWithKey+ , updateLookupWithKey+ , alter++ -- * Combine++ -- ** Union+ , union+ , unionWith+ , unionWithKey+ , unions+ , unionsWith++ -- ** Difference+ , difference+ , differenceWith+ , differenceWithKey++ -- ** Intersection+ , intersection+ , intersectionWith+ , intersectionWithKey++ -- ** Universal combining function+ , mergeWithKey++ -- * Traversal+ -- ** Map+ , map+ , mapWithKey+ -- LIQUID, traverseWithKey+ , mapAccum+ , mapAccumWithKey+ , mapAccumRWithKey+ , mapKeys+ , mapKeysWith+ , mapKeysMonotonic++ -- * Folds+ , foldr+ , foldl+ , foldrWithKey+ , foldlWithKey+ -- ** Strict folds+ , foldr'+ , foldl'+ , foldrWithKey'+ , foldlWithKey'++ -- * Conversion+ , elems+ , keys+ , assocs+ -- LIQUID, keysSet+ -- LIQUID, fromSet++ -- ** Lists+ , toList+ , fromList+ , fromListWith+ , fromListWithKey++ -- ** Ordered lists+ , toAscList+ , toDescList+ , fromAscList+ , fromAscListWith+ , fromAscListWithKey+ , fromDistinctAscList++ -- * Filter+ , filter+ , filterWithKey+ , partition+ , partitionWithKey++ , mapMaybe+ , mapMaybeWithKey+ , mapEither+ , mapEitherWithKey++ , split+ , splitLookup++ -- * Submap+ , isSubmapOf, isSubmapOfBy+ , isProperSubmapOf, isProperSubmapOfBy++ -- * Indexed+ , lookupIndex+ , findIndex+ , elemAt+ , updateAt+ , deleteAt++ -- * Min\/Max+ , findMin+ , findMax+ , deleteMin+ , deleteMax+ , deleteFindMin+ , deleteFindMax+ , updateMin+ , updateMax+ , updateMinWithKey+ , updateMaxWithKey+ , minView+ , maxView+ , minViewWithKey+ , maxViewWithKey++ -- * Debugging+ -- , showTree+ -- , showTreeWith+ -- , valid++ -- -- Used by the strict version+ -- , bin+ -- , balance+ -- , balanced+ -- , balanceL+ -- , balanceR+ -- , delta+ -- , join'+ -- , merge+ -- , glue+ -- , trim, zoo1, zoo2+ -- , trimLookupLo+ -- , foldlStrict+ -- , MaybeS(..)+ -- , filterGt+ -- , filterLt++ -- LIQUID+ , Maybe, Char, Bool, Int, Either, MaybeS(..), delta, balanceL,+ balanceR, filterGt, filterLt, insertR, trim, insertMin,+ insertMax, bin, glue, join', merge+ ) where++import Prelude hiding (filter, foldl, foldr, lookup, map,+ null)+-- LIQUID import qualified Data.Set.Base as Set+-- LIQUID import Data.StrictPair+import Data.Monoid (Monoid (..))+-- LIQUID import Control.Applicative (Applicative(..), (<$>))+import qualified Data.Foldable as Foldable+import Data.Traversable (Traversable (traverse))+-- import Data.Typeable+import Control.DeepSeq (NFData (rnf))++#if __GLASGOW_HASKELL__+import Data.Data+import GHC.Exts (build)+import Text.Read++import Data.Set (Set)+import GHC.Generics+import Test.Target+import Test.Target.Targetable+#endif++-- Use macros to define strictness of functions.+-- STRICT_x_OF_y denotes an y-ary function strict in the x-th parameter.+-- We do not use BangPatterns, because they are not in any standard and we+-- want the compilers to be compiled by as many compilers as possible.+--LIQUID #define STRICT_1_OF_2(fn) fn arg _ | arg `seq` False = undefined+--LIQUID #define STRICT_1_OF_3(fn) fn arg _ _ | arg `seq` False = undefined+--LIQUID #define STRICT_2_OF_3(fn) fn _ arg _ | arg `seq` False = undefined+--LIQUID #define STRICT_1_OF_4(fn) fn arg _ _ _ | arg `seq` False = undefined+--LIQUID #define STRICT_2_OF_4(fn) fn _ arg _ _ | arg `seq` False = undefined++{--------------------------------------------------------------------+ Operators+--------------------------------------------------------------------}+infixl 9 !,\\ --++-- | /O(log n)/. Find the value at a key.+-- Calls 'error' when the element can not be found.+--+-- > fromList [(5,'a'), (3,'b')] ! 1 Error: element not in the map+-- > fromList [(5,'a'), (3,'b')] ! 5 == 'a'++{-@ (!) :: (Ord k) => OMap k a -> k -> a @-}+(!) :: Ord k => Map k a -> k -> a+m ! k = find k m+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE (!) #-}+#endif++-- | Same as 'difference'.+{-@ (\\) :: Ord k => OMap k a -> OMap k b -> OMap k a @-}+(\\) :: Ord k => Map k a -> Map k b -> Map k a+m1 \\ m2 = difference m1 m2+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE (\\) #-}+#endif++-- LiquidCheck tests+instance (Targetable k, Targetable a) => Targetable (Map k a)++{--------------------------------------------------------------------+ Size balanced trees.+--------------------------------------------------------------------}+-- | A Map from keys @k@ to values @a@.++-- See Note: Order of constructors+data Map k a = Bin Size k a (Map k a) (Map k a)+ | Tip+ deriving (Generic, Show)++type Size = Int++{-@ data Map [mlen] k a <l :: root:k -> k -> Prop, r :: root:k -> k -> Prop>+ = Bin (ssz :: Nat)+ (key :: k)+ (value :: a)+ (left :: Map <l, r> (k <l key>) a)+ (right :: {v:Map <l, r> (k <r key>) a | ssz = 1 + (mlen left) + (mlen v)})+ | Tip+ @-}++{-@ measure mlen :: (Map k a) -> Int+ mlen(Tip) = 0+ mlen(Bin s k v l r) = 1 + (mlen l) + (mlen r)+ @-}++{-@ type SumMLen A B = {v:Nat | v = (mlen A) + (mlen B)} @-}++{-@ invariant {v:Map k a | (mlen v) >= 0} @-}+++{-@ type OMap kk aa = {v:Map <{\root v -> v < root}, {\root v -> v > root}> kk aa | (isBalanced v)} @-}++{-@ measure isJustS :: forall a. MaybeS a -> Prop+ isJustS (JustS x) = true+ isJustS (NothingS) = false+@-}++{-@ measure fromJustS :: forall a. MaybeS a -> a+ fromJustS (JustS x) = x+ @-}++{-@ measure isBin :: Map k a -> Prop+ isBin (Bin sz kx x l r) = true+ isBin (Tip) = false+ @-}++{-@ measure mapKeys :: Map k a -> (Set k)+ mapKeys (Tip) = {v | (Set_emp v)}+ mapKeys (Bin s k v l r) = (Set_cup (Set_sng k) (Set_cup (mapKeys l) (mapKeys r)))+ @-}++{-@ invariant {v0: MaybeS {v: a | ((isJustS v0) && (v = (fromJustS v0)))} | true} @-}++{-@ predicate IfDefLe X Y = ((isJustS X) => ((fromJustS X) < Y)) @-}+{-@ predicate IfDefLt X Y = ((isJustS X) => ((fromJustS X) < Y)) @-}+{-@ predicate IfDefGt X Y = ((isJustS X) => (Y < (fromJustS X))) @-}+{-@ predicate RootLt Lo V = ((isBin V) => (IfDefLt Lo (key V))) @-}+{-@ predicate RootGt Hi V = ((isBin V) => (IfDefGt Hi (key V))) @-}+{-@ predicate RootBetween Lo Hi V = ((RootLt Lo V) && (RootGt Hi V)) @-}+{-@ predicate KeyBetween Lo Hi V = ((IfDefLt Lo V) && (IfDefGt Hi V)) @-}+++-- LIQUID instance (Ord k) => Monoid (Map k v) where+-- mempty = empty+-- mappend = union+-- mconcat = unions++#if __GLASGOW_HASKELL__++{--------------------------------------------------------------------+ A Data instance+--------------------------------------------------------------------}++-- This instance preserves data abstraction at the cost of inefficiency.+-- We omit reflection services for the sake of data abstraction.+-- LIQUID instance (Data k, Data a, Ord k) => Data (Map k a) where+-- LIQUID gfoldl f z m = z fromList `f` toList m+-- LIQUID toConstr _ = error "toConstr"+-- LIQUID gunfold _ _ = error "gunfold"+-- LIQUID dataTypeOf _ = mkNoRepType "Data.Map.Map"+-- LIQUID dataCast2 f = gcast2 f+#endif++{--------------------------------------------------------------------+ Query+--------------------------------------------------------------------}+-- | /O(1)/. Is the map empty?+--+-- > Data.Map.null (empty) == True+-- > Data.Map.null (singleton 1 'a') == False++{-@ null :: OMap k a -> Bool @-}+null :: Map k a -> Bool+null Tip = True+null (Bin {}) = False+{-# INLINE null #-}++-- | /O(1)/. The number of elements in the map.+--+-- > size empty == 0+-- > size (singleton 1 'a') == 1+-- > size (fromList([(1,'a'), (2,'c'), (3,'b')])) == 3++size :: Map k a -> Int+size Tip = 0+size (Bin sz _ _ _ _) = sz+{-# INLINE size #-}+++-- | /O(log n)/. Lookup the value at a key in the map.+--+-- The function will return the corresponding value as @('Just' value)@,+-- or 'Nothing' if the key isn't in the map.+--+-- An example of using @lookup@:+--+-- > import Prelude hiding (lookup)+-- > import Data.Map+-- >+-- > employeeDept = fromList([("John","Sales"), ("Bob","IT")])+-- > deptCountry = fromList([("IT","USA"), ("Sales","France")])+-- > countryCurrency = fromList([("USA", "Dollar"), ("France", "Euro")])+-- >+-- > employeeCurrency :: String -> Maybe String+-- > employeeCurrency name = do+-- > dept <- lookup name employeeDept+-- > country <- lookup dept deptCountry+-- > lookup country countryCurrency+-- >+-- > main = do+-- > putStrLn $ "John's currency: " ++ (show (employeeCurrency "John"))+-- > putStrLn $ "Pete's currency: " ++ (show (employeeCurrency "Pete"))+--+-- The output of this program:+--+-- > John's currency: Just "Euro"+-- > Pete's currency: Nothing++{-@ lookup :: (Ord k) => k -> OMap k a -> Maybe a @-}+lookup :: Ord k => k -> Map k a -> Maybe a+lookup = go+ where+ go _ Tip = Nothing+ go k (Bin _ kx x l r) = case compare k kx of+ LT -> go k l+ GT -> go k r+ EQ -> Just x+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE lookup #-}+#else+{-# INLINE lookup #-}+#endif++-- | /O(log n)/. Is the key a member of the map? See also 'notMember'.+--+-- > member 5 (fromList [(5,'a'), (3,'b')]) == True+-- > member 1 (fromList [(5,'a'), (3,'b')]) == False++{-@ member :: (Ord k) => k -> OMap k a -> Bool @-}+member :: Ord k => k -> Map k a -> Bool+member = go+ where+ go _ Tip = False+ go k (Bin _ kx _ l r) = case compare k kx of+ LT -> go k l+ GT -> go k r+ EQ -> True+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE member #-}+#else+{-# INLINE member #-}+#endif++-- | /O(log n)/. Is the key not a member of the map? See also 'member'.+--+-- > notMember 5 (fromList [(5,'a'), (3,'b')]) == False+-- > notMember 1 (fromList [(5,'a'), (3,'b')]) == True++{-@ notMember :: (Ord k) => k -> OMap k a -> Bool @-}+notMember :: Ord k => k -> Map k a -> Bool+notMember k m = not $ member k m+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE notMember #-}+#else+{-# INLINE notMember #-}+#endif++-- | /O(log n)/. Find the value at a key.+-- Calls 'error' when the element can not be found.++{-@ find :: (Ord k) => k -> OMap k a -> a @-}+find :: Ord k => k -> Map k a -> a+find = go+ where+ go _ Tip = error "Map.!: given key is not an element in the map"+ go k (Bin _ kx x l r) = case compare k kx of+ LT -> go k l+ GT -> go k r+ EQ -> x+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE find #-}+#else+{-# INLINE find #-}+#endif++-- | /O(log n)/. The expression @('findWithDefault' def k map)@ returns+-- the value at key @k@ or returns default value @def@+-- when the key is not in the map.+--+-- > findWithDefault 'x' 1 (fromList [(5,'a'), (3,'b')]) == 'x'+-- > findWithDefault 'x' 5 (fromList [(5,'a'), (3,'b')]) == 'a'++{-@ findWithDefault :: (Ord k) => a -> k -> OMap k a -> a @-}+findWithDefault :: Ord k => a -> k -> Map k a -> a+findWithDefault = go+ where+ go def _ Tip = def+ go def k (Bin _ kx x l r) = case compare k kx of+ LT -> go def k l+ GT -> go def k r+ EQ -> x+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE findWithDefault #-}+#else+{-# INLINE findWithDefault #-}+#endif++-- | /O(log n)/. Find largest key smaller than the given one and return the+-- corresponding (key, value) pair.+--+-- > lookupLT 3 (fromList [(3,'a'), (5,'b')]) == Nothing+-- > lookupLT 4 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')+{-@ lookupLT :: (Ord k) => k -> OMap k v -> Maybe (k, v) @-}+lookupLT :: Ord k => k -> Map k v -> Maybe (k, v)+lookupLT = goNothing+ where+ goNothing _ Tip = Nothing+ goNothing k (Bin _ kx x l r) | k <= kx = goNothing k l+ | otherwise = goJust k kx x r++ goJust _ kx' x' Tip = Just (kx', x')+ goJust k kx' x' (Bin _ kx x l r) | k <= kx = goJust k kx' x' l+ | otherwise = goJust k kx x r+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE lookupLT #-}+#else+{-# INLINE lookupLT #-}+#endif++-- | /O(log n)/. Find smallest key greater than the given one and return the+-- corresponding (key, value) pair.+--+-- > lookupGT 4 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')+-- > lookupGT 5 (fromList [(3,'a'), (5,'b')]) == Nothing+{-@ lookupGT :: (Ord k) => k -> OMap k v -> Maybe (k, v) @-}+lookupGT :: Ord k => k -> Map k v -> Maybe (k, v)+lookupGT = goNothing+ where+ goNothing _ Tip = Nothing+ goNothing k (Bin _ kx x l r) | k < kx = goJust k kx x l+ | otherwise = goNothing k r++ goJust _ kx' x' Tip = Just (kx', x')+ goJust k kx' x' (Bin _ kx x l r) | k < kx = goJust k kx x l+ | otherwise = goJust k kx' x' r+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE lookupGT #-}+#else+{-# INLINE lookupGT #-}+#endif++-- | /O(log n)/. Find largest key smaller or equal to the given one and return+-- the corresponding (key, value) pair.+--+-- > lookupLE 2 (fromList [(3,'a'), (5,'b')]) == Nothing+-- > lookupLE 4 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')+-- > lookupLE 5 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')+{-@ lookupLE :: (Ord k) => k -> OMap k v -> Maybe (k, v) @-}+lookupLE :: Ord k => k -> Map k v -> Maybe (k, v)+lookupLE = goNothing+ where+ goNothing _ Tip = Nothing+ goNothing k (Bin _ kx x l r) = case compare k kx of LT -> goNothing k l+ EQ -> Just (kx, x)+ GT -> goJust k kx x r++ goJust _ kx' x' Tip = Just (kx', x')+ goJust k kx' x' (Bin _ kx x l r) = case compare k kx of LT -> goJust k kx' x' l+ EQ -> Just (kx, x)+ GT -> goJust k kx x r+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE lookupLE #-}+#else+{-# INLINE lookupLE #-}+#endif++-- | /O(log n)/. Find smallest key greater or equal to the given one and return+-- the corresponding (key, value) pair.+--+-- > lookupGE 3 (fromList [(3,'a'), (5,'b')]) == Just (3, 'a')+-- > lookupGE 4 (fromList [(3,'a'), (5,'b')]) == Just (5, 'b')+-- > lookupGE 6 (fromList [(3,'a'), (5,'b')]) == Nothing+{-@ lookupGE :: (Ord k) => k -> OMap k v -> Maybe (k, v) @-}+lookupGE :: Ord k => k -> Map k v -> Maybe (k, v)+lookupGE = goNothing+ where+ goNothing _ Tip = Nothing+ goNothing k (Bin _ kx x l r) = case compare k kx of LT -> goJust k kx x l+ EQ -> Just (kx, x)+ GT -> goNothing k r++ goJust _ kx' x' Tip = Just (kx', x')+ goJust k kx' x' (Bin _ kx x l r) = case compare k kx of LT -> goJust k kx x l+ EQ -> Just (kx, x)+ GT -> goJust k kx' x' r+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE lookupGE #-}+#else+{-# INLINE lookupGE #-}+#endif++{--------------------------------------------------------------------+ Construction+--------------------------------------------------------------------}+-- | /O(1)/. The empty map.+--+-- > empty == fromList []+-- > size empty == 0+{-@ empty :: OMap k a @-}+empty :: Map k a+empty = Tip+{-# INLINE empty #-}++-- | /O(1)/. A map with a single element.+--+-- > singleton 1 'a' == fromList [(1, 'a')]+-- > size (singleton 1 'a') == 1++{-@ singleton :: k -> a -> OMap k a @-}+singleton :: k -> a -> Map k a+singleton k x = Bin 1 k x Tip Tip+{-# INLINE singleton #-}++{--------------------------------------------------------------------+ Insertion+--------------------------------------------------------------------}+-- | /O(log n)/. Insert a new key and value in the map.+-- If the key is already present in the map, the associated value is+-- replaced with the supplied value. 'insert' is equivalent to+-- @'insertWith' 'const'@.+--+-- > insert 5 'x' (fromList [(5,'a'), (3,'b')]) == fromList [(3, 'b'), (5, 'x')]+-- > insert 7 'x' (fromList [(5,'a'), (3,'b')]) == fromList [(3, 'b'), (5, 'a'), (7, 'x')]+-- > insert 5 'x' empty == singleton 5 'x'++-- See Note: Type of local 'go' function+{-@ insert :: (Ord k) => k:k -> a -> x:OMap k a+ -> {v:OMap k a | (mapKeys v) = (Set_cup (Set_sng k) (mapKeys x))}+ @-}+insert :: Ord k => k -> a -> Map k a -> Map k a+insert = insert_go+--LIQUID insert = go+--LIQUID where+--LIQUID go :: Ord k => k -> a -> Map k a -> Map k a+--LIQUID go kx x Tip = singleton kx x+--LIQUID go kx x (Bin sz ky y l r) =+--LIQUID case compare kx ky of+--LIQUID -- Bin ky y (go kx x l) r+--LIQUID LT -> balanceL ky y (go kx x l) r+--LIQUID GT -> balanceR ky y l (go kx x r)+--LIQUID EQ -> Bin sz kx x l r++{-@ insert_go :: (Ord k) => k -> a -> OMap k a -> OMap k a @-}+insert_go :: Ord k => k -> a -> Map k a -> Map k a+insert_go kx x Tip = singleton kx x+insert_go kx x (Bin sz ky y l r) =+ case compare kx ky of+ -- Bin ky y (insert_go kx x l) r+ LT -> balanceL ky y (insert_go kx x l) r+ GT -> balanceR ky y l (insert_go kx x r)+ EQ -> Bin sz kx x l r+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE insert #-}+#else+{-# INLINE insert #-}+#endif++-- Insert a new key and value in the map if it is not already present.+-- Used by `union`.++-- See Note: Type of local 'go' function+insertR :: Ord k => k -> a -> Map k a -> Map k a+insertR = insertR_go+--LIQUID insertR = go+--LIQUID where+--LIQUID go :: Ord k => k -> a -> Map k a -> Map k a+--LIQUID go kx x Tip = singleton kx x+--LIQUID go kx x t@(Bin _ ky y l r) =+--LIQUID case compare kx ky of+--LIQUID LT -> balanceL ky y (go kx x l) r+--LIQUID GT -> balanceR ky y l (go kx x r)+--LIQUID EQ -> t++insertR_go :: Ord k => k -> a -> Map k a -> Map k a+insertR_go kx x Tip = singleton kx x+insertR_go kx x t@(Bin _ ky y l r) =+ case compare kx ky of+ LT -> balanceL ky y (insertR_go kx x l) r+ GT -> balanceR ky y l (insertR_go kx x r)+ EQ -> t+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE insertR #-}+#else+{-# INLINE insertR #-}+#endif++-- | /O(log n)/. Insert with a function, combining new value and old value.+-- @'insertWith' f key value mp@+-- will insert the pair (key, value) into @mp@ if key does+-- not exist in the map. If the key does exist, the function will+-- insert the pair @(key, f new_value old_value)@.+--+-- > insertWith (++) 5 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "xxxa")]+-- > insertWith (++) 7 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "xxx")]+-- > insertWith (++) 5 "xxx" empty == singleton 5 "xxx"++{-@ insertWith :: (Ord k) => (a -> a -> a) -> k -> a -> OMap k a -> OMap k a @-}+insertWith :: Ord k => (a -> a -> a) -> k -> a -> Map k a -> Map k a+insertWith f = insertWithKey (\_ x' y' -> f x' y')+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE insertWith #-}+#else+{-# INLINE insertWith #-}+#endif++-- | /O(log n)/. Insert with a function, combining key, new value and old value.+-- @'insertWithKey' f key value mp@+-- will insert the pair (key, value) into @mp@ if key does+-- not exist in the map. If the key does exist, the function will+-- insert the pair @(key,f key new_value old_value)@.+-- Note that the key passed to f is the same key passed to 'insertWithKey'.+--+-- > let f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value+-- > insertWithKey f 5 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:xxx|a")]+-- > insertWithKey f 7 "xxx" (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "xxx")]+-- > insertWithKey f 5 "xxx" empty == singleton 5 "xxx"++-- See Note: Type of local 'go' function++{-@ insertWithKey :: (Ord k) => (k -> a -> a -> a) -> k -> a -> OMap k a -> OMap k a @-}+insertWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> Map k a+insertWithKey = insertWithKey_go+--LIQUID insertWithKey = go+--LIQUID where+--LIQUID go :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> Map k a+--LIQUID go _ kx x Tip = singleton kx x+--LIQUID go f kx x (Bin sy ky y l r) =+--LIQUID case compare kx ky of+--LIQUID LT -> balanceL ky y (go f kx x l) r+--LIQUID GT -> balanceR ky y l (go f kx x r)+--LIQUID EQ -> Bin sy kx (f kx x y) l r++{-@ insertWithKey_go :: (Ord k) => (k -> a -> a -> a) -> k -> a -> OMap k a -> OMap k a @-}+insertWithKey_go :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> Map k a+insertWithKey_go _ kx x Tip = singleton kx x+insertWithKey_go f kx x (Bin sy ky y l r) =+ case compare kx ky of+ LT -> balanceL ky y (insertWithKey_go f kx x l) r+ GT -> balanceR ky y l (insertWithKey_go f kx x r)+ EQ -> Bin sy kx (f kx x y) l r+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE insertWithKey #-}+#else+{-# INLINE insertWithKey #-}+#endif++-- | /O(log n)/. Combines insert operation with old value retrieval.+-- The expression (@'insertLookupWithKey' f k x map@)+-- is a pair where the first element is equal to (@'lookup' k map@)+-- and the second element equal to (@'insertWithKey' f k x map@).+--+-- > let f key new_value old_value = (show key) ++ ":" ++ new_value ++ "|" ++ old_value+-- > insertLookupWithKey f 5 "xxx" (fromList [(5,"a"), (3,"b")]) == (Just "a", fromList [(3, "b"), (5, "5:xxx|a")])+-- > insertLookupWithKey f 7 "xxx" (fromList [(5,"a"), (3,"b")]) == (Nothing, fromList [(3, "b"), (5, "a"), (7, "xxx")])+-- > insertLookupWithKey f 5 "xxx" empty == (Nothing, singleton 5 "xxx")+--+-- This is how to define @insertLookup@ using @insertLookupWithKey@:+--+-- > let insertLookup kx x t = insertLookupWithKey (\_ a _ -> a) kx x t+-- > insertLookup 5 "x" (fromList [(5,"a"), (3,"b")]) == (Just "a", fromList [(3, "b"), (5, "x")])+-- > insertLookup 7 "x" (fromList [(5,"a"), (3,"b")]) == (Nothing, fromList [(3, "b"), (5, "a"), (7, "x")])++-- See Note: Type of local 'go' function++{-@ insertLookupWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> OMap k a -> (Maybe a, OMap k a) @-}+insertLookupWithKey :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> (Maybe a, Map k a)+insertLookupWithKey = insertLookupWithKey_go+--LIQUID insertLookupWithKey = go+--LIQUID where+--LIQUID go :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> (Maybe a, Map k a)+--LIQUID go _ kx x Tip = (Nothing, singleton kx x)+--LIQUID go f kx x (Bin sy ky y l r) =+--LIQUID case compare kx ky of+--LIQUID LT -> let (found, l') = go f kx x l+--LIQUID in (found, balanceL ky y l' r)+--LIQUID GT -> let (found, r') = go f kx x r+--LIQUID in (found, balanceR ky y l r')+--LIQUID EQ -> (Just y, Bin sy kx (f kx x y) l r)++{-@ insertLookupWithKey_go :: Ord k => (k -> a -> a -> a) -> k -> a -> OMap k a -> (Maybe a, OMap k a) @-}+insertLookupWithKey_go :: Ord k => (k -> a -> a -> a) -> k -> a -> Map k a -> (Maybe a, Map k a)+insertLookupWithKey_go _ kx x Tip = (Nothing, singleton kx x)+insertLookupWithKey_go f kx x (Bin sy ky y l r) =+ case compare kx ky of+ LT -> let (found, l') = insertLookupWithKey_go f kx x l+ in (found, balanceL ky y l' r)+ GT -> let (found, r') = insertLookupWithKey_go f kx x r+ in (found, balanceR ky y l r')+ EQ -> (Just y, Bin sy kx (f kx x y) l r)+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE insertLookupWithKey #-}+#else+{-# INLINE insertLookupWithKey #-}+#endif++{--------------------------------------------------------------------+ Deletion+--------------------------------------------------------------------}+-- | /O(log n)/. Delete a key and its value from the map. When the key is not+-- a member of the map, the original map is returned.+--+-- > delete 5 (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"+-- > delete 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]+-- > delete 5 empty == empty++-- See Note: Type of local 'go' function+{-@ delete :: (Ord k) => k:k -> x:OMap k a+ -> {v:OMap k a | (mapKeys v) = (Set_dif (mapKeys x) (Set_sng k))} @-}+delete :: Ord k => k -> Map k a -> Map k a+delete = delete_go+--LIQUID delete = go+--LIQUID where+--LIQUID go :: Ord k => k -> Map k a -> Map k a+--LIQUID go _ Tip = Tip+--LIQUID go k (Bin _ kx x l r) =+--LIQUID case compare k kx of+--LIQUID LT -> balanceR kx x (go k l) r+--LIQUID GT -> balanceL kx x l (go k r)+--LIQUID EQ -> glue kx l r++{-@ delete_go :: (Ord k) => k -> OMap k a -> OMap k a @-}+delete_go :: Ord k => k -> Map k a -> Map k a+delete_go _ Tip = Tip+delete_go k (Bin _ kx x l r) =+ case compare k kx of+ LT -> balanceR kx x (delete_go k l) r+ GT -> balanceL kx x l (delete_go k r)+ EQ -> glue kx l r+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE delete #-}+#else+{-# INLINE delete #-}+#endif++-- | /O(log n)/. Update a value at a specific key with the result of the provided function.+-- When the key is not+-- a member of the map, the original map is returned.+--+-- > adjust ("new " ++) 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "new a")]+-- > adjust ("new " ++) 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]+-- > adjust ("new " ++) 7 empty == empty++{-@ adjust :: (Ord k) => (a -> a) -> k -> OMap k a -> OMap k a @-}+adjust :: Ord k => (a -> a) -> k -> Map k a -> Map k a+adjust f = adjustWithKey (\_ x -> f x)+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE adjust #-}+#else+{-# INLINE adjust #-}+#endif++-- | /O(log n)/. Adjust a value at a specific key. When the key is not+-- a member of the map, the original map is returned.+--+-- > let f key x = (show key) ++ ":new " ++ x+-- > adjustWithKey f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:new a")]+-- > adjustWithKey f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]+-- > adjustWithKey f 7 empty == empty++{-@ adjustWithKey :: (Ord k) => (k -> a -> a) -> k -> OMap k a -> OMap k a @-}+adjustWithKey :: Ord k => (k -> a -> a) -> k -> Map k a -> Map k a+adjustWithKey f = updateWithKey (\k' x' -> Just (f k' x'))+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE adjustWithKey #-}+#else+{-# INLINE adjustWithKey #-}+#endif++-- | /O(log n)/. The expression (@'update' f k map@) updates the value @x@+-- at @k@ (if it is in the map). If (@f x@) is 'Nothing', the element is+-- deleted. If it is (@'Just' y@), the key @k@ is bound to the new value @y@.+--+-- > let f x = if x == "a" then Just "new a" else Nothing+-- > update f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "new a")]+-- > update f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]+-- > update f 3 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"++{-@ update :: (Ord k) => (a -> Maybe a) -> k -> OMap k a -> OMap k a @-}+update :: Ord k => (a -> Maybe a) -> k -> Map k a -> Map k a+update f = updateWithKey (\_ x -> f x)+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE update #-}+#else+{-# INLINE update #-}+#endif++-- | /O(log n)/. The expression (@'updateWithKey' f k map@) updates the+-- value @x@ at @k@ (if it is in the map). If (@f k x@) is 'Nothing',+-- the element is deleted. If it is (@'Just' y@), the key @k@ is bound+-- to the new value @y@.+--+-- > let f k x = if x == "a" then Just ((show k) ++ ":new a") else Nothing+-- > updateWithKey f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "5:new a")]+-- > updateWithKey f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]+-- > updateWithKey f 3 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"++-- See Note: Type of local 'go' function++{-@ updateWithKey :: (Ord k) => (k -> a -> Maybe a) -> k -> OMap k a -> OMap k a @-}+updateWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> Map k a+updateWithKey = updateWithKey_go+--LIQUID updateWithKey = go+--LIQUID where+--LIQUID go :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> Map k a+--LIQUID go _ _ Tip = Tip+--LIQUID go f k(Bin sx kx x l r) =+--LIQUID case compare k kx of+--LIQUID LT -> balanceR kx x (go f k l) r+--LIQUID GT -> balanceL kx x l (go f k r)+--LIQUID EQ -> case f kx x of+--LIQUID Just x' -> Bin sx kx x' l r+--LIQUID Nothing -> glue kx l r+{-@ updateWithKey_go :: (Ord k) => (k -> a -> Maybe a) -> k -> OMap k a -> OMap k a @-}+updateWithKey_go :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> Map k a+updateWithKey_go _ _ Tip = Tip+updateWithKey_go f k(Bin sx kx x l r) =+ case compare k kx of+ LT -> balanceR kx x (updateWithKey_go f k l) r+ GT -> balanceL kx x l (updateWithKey_go f k r)+ EQ -> case f kx x of+ Just x' -> Bin sx kx x' l r+ Nothing -> glue kx l r++#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE updateWithKey #-}+#else+{-# INLINE updateWithKey #-}+#endif++-- | /O(log n)/. Lookup and update. See also 'updateWithKey'.+-- The function returns changed value, if it is updated.+-- Returns the original key value if the map entry is deleted.+--+-- > let f k x = if x == "a" then Just ((show k) ++ ":new a") else Nothing+-- > updateLookupWithKey f 5 (fromList [(5,"a"), (3,"b")]) == (Just "5:new a", fromList [(3, "b"), (5, "5:new a")])+-- > updateLookupWithKey f 7 (fromList [(5,"a"), (3,"b")]) == (Nothing, fromList [(3, "b"), (5, "a")])+-- > updateLookupWithKey f 3 (fromList [(5,"a"), (3,"b")]) == (Just "b", singleton 5 "a")++-- See Note: Type of local 'go' function++{-@ updateLookupWithKey :: (Ord k) => (k -> a -> Maybe a) -> k -> OMap k a -> (Maybe a, OMap k a) @-}+updateLookupWithKey :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> (Maybe a,Map k a)+updateLookupWithKey = updateLookupWithKey_go+--LIQUID updateLookupWithKey = go+--LIQUID where+--LIQUID go :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> (Maybe a,Map k a)+--LIQUID go _ _ Tip = (Nothing,Tip)+--LIQUID go f k (Bin sx kx x l r) =+--LIQUID case compare k kx of+--LIQUID LT -> let (found,l') = go f k l in (found,balanceR kx x l' r)+--LIQUID GT -> let (found,r') = go f k r in (found,balanceL kx x l r')+--LIQUID EQ -> case f kx x of+--LIQUID Just x' -> (Just x',Bin sx kx x' l r)+--LIQUID Nothing -> (Just x,glue kx l r)++{-@ updateLookupWithKey_go :: (Ord k) => (k -> a -> Maybe a) -> k -> OMap k a -> (Maybe a, OMap k a) @-}+updateLookupWithKey_go :: Ord k => (k -> a -> Maybe a) -> k -> Map k a -> (Maybe a,Map k a)+updateLookupWithKey_go _ _ Tip = (Nothing,Tip)+updateLookupWithKey_go f k (Bin sx kx x l r) =+ case compare k kx of+ LT -> let (found,l') = updateLookupWithKey_go f k l in (found,balanceR kx x l' r)+ GT -> let (found,r') = updateLookupWithKey_go f k r in (found,balanceL kx x l r')+ EQ -> case f kx x of+ Just x' -> (Just x',Bin sx kx x' l r)+ Nothing -> (Just x,glue kx l r)++#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE updateLookupWithKey #-}+#else+{-# INLINE updateLookupWithKey #-}+#endif++-- | /O(log n)/. The expression (@'alter' f k map@) alters the value @x@ at @k@, or absence thereof.+-- 'alter' can be used to insert, delete, or update a value in a 'Map'.+-- In short : @'lookup' k ('alter' f k m) = f ('lookup' k m)@.+--+-- > let f _ = Nothing+-- > alter f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a")]+-- > alter f 5 (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"+-- >+-- > let f _ = Just "c"+-- > alter f 7 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "a"), (7, "c")]+-- > alter f 5 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "c")]++-- See Note: Type of local 'go' function++{-@ alter :: (Ord k) => (Maybe a -> Maybe a) -> k -> OMap k a -> OMap k a @-}+alter :: Ord k => (Maybe a -> Maybe a) -> k -> Map k a -> Map k a+alter = alter_go+--LIQUID alter = go+--LIQUID where+--LIQUID go :: Ord k => (Maybe a -> Maybe a) -> k -> Map k a -> Map k a+--LIQUID go f k Tip = case f Nothing of+--LIQUID Nothing -> Tip+--LIQUID Just x -> singleton k x+--LIQUID+--LIQUID go f k (Bin sx kx x l r) = case compare k kx of+--LIQUID LT -> balance kx x (go f k l) r+--LIQUID GT -> balance kx x l (go f k r)+--LIQUID EQ -> case f (Just x) of+--LIQUID Just x' -> Bin sx kx x' l r+--LIQUID Nothing -> glue kx l r++alter_go :: Ord k => (Maybe a -> Maybe a) -> k -> Map k a -> Map k a+alter_go f k Tip = case f Nothing of+ Nothing -> Tip+ Just x -> singleton k x++alter_go f k (Bin sx kx x l r) = case compare k kx of+ LT -> balance kx x (alter_go f k l) r+ GT -> balance kx x l (alter_go f k r)+ EQ -> case f (Just x) of+ Just x' -> Bin sx kx x' l r+ Nothing -> glue kx l r++#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE alter #-}+#else+{-# INLINE alter #-}+#endif++{--------------------------------------------------------------------+ Indexing+--------------------------------------------------------------------}+-- | /O(log n)/. Return the /index/ of a key. The index is a number from+-- /0/ up to, but not including, the 'size' of the map. Calls 'error' when+-- the key is not a 'member' of the map.+--+-- > findIndex 2 (fromList [(5,"a"), (3,"b")]) Error: element is not in the map+-- > findIndex 3 (fromList [(5,"a"), (3,"b")]) == 0+-- > findIndex 5 (fromList [(5,"a"), (3,"b")]) == 1+-- > findIndex 6 (fromList [(5,"a"), (3,"b")]) Error: element is not in the map++-- See Note: Type of local 'go' function++{-@ findIndex :: (Ord k) => k -> OMap k a -> GHC.Types.Int @-}+findIndex :: Ord k => k -> Map k a -> Int+findIndex = findIndex_go 0+--LIQUID findIndex = go 0+--LIQUID where+--LIQUID go :: Ord k => Int -> k -> Map k a -> Int+--LIQUID go _ _ Tip = error "Map.findIndex: element is not in the map"+--LIQUID go idx k (Bin _ kx _ l r) = case compare k kx of+--LIQUID LT -> go idx k l+--LIQUID GT -> go (idx + size l + 1) k r+--LIQUID EQ -> idx + size l++{-@ findIndex_go :: (Ord k) => Int -> k -> OMap k a -> GHC.Types.Int @-}+{-@ Decrease findIndex_go 4 @-}+findIndex_go :: Ord k => Int -> k -> Map k a -> Int+findIndex_go _ _ Tip = error "Map.findIndex: element is not in the map"+findIndex_go idx k (Bin _ kx _ l r) = case compare k kx of+ LT -> findIndex_go idx k l+ GT -> findIndex_go (idx + size l + 1) k r+ EQ -> idx + size l+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE findIndex #-}+#endif++-- | /O(log n)/. Lookup the /index/ of a key. The index is a number from+-- /0/ up to, but not including, the 'size' of the map.+--+-- > isJust (lookupIndex 2 (fromList [(5,"a"), (3,"b")])) == False+-- > fromJust (lookupIndex 3 (fromList [(5,"a"), (3,"b")])) == 0+-- > fromJust (lookupIndex 5 (fromList [(5,"a"), (3,"b")])) == 1+-- > isJust (lookupIndex 6 (fromList [(5,"a"), (3,"b")])) == False++-- See Note: Type of local 'go' function+{-@ lookupIndex :: (Ord k) => k -> OMap k a -> Maybe GHC.Types.Int @-}+lookupIndex :: Ord k => k -> Map k a -> Maybe Int+lookupIndex = lookupIndex_go 0+--LIQUID lookupIndex = go 0+--LIQUID where+--LIQUID go :: Ord k => Int -> k -> Map k a -> Maybe Int+--LIQUID go _ _ Tip = Nothing+--LIQUID go idx k (Bin _ kx _ l r) = case compare k kx of+--LIQUID LT -> go idx k l+--LIQUID GT -> go (idx + size l + 1) k r+--LIQUID EQ -> Just $! idx + size l++{-@ lookupIndex_go :: (Ord k) => Int -> k -> OMap k a -> Maybe GHC.Types.Int @-}+{-@ Decrease lookupIndex_go 4 @-}+lookupIndex_go :: Ord k => Int -> k -> Map k a -> Maybe Int+lookupIndex_go _ _ Tip = Nothing+lookupIndex_go idx k (Bin _ kx _ l r) = case compare k kx of+ LT -> lookupIndex_go idx k l+ GT -> lookupIndex_go (idx + size l + 1) k r+ EQ -> Just $! idx + size l+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE lookupIndex #-}+#endif++-- | /O(log n)/. Retrieve an element by /index/. Calls 'error' when an+-- invalid index is used.+--+-- > elemAt 0 (fromList [(5,"a"), (3,"b")]) == (3,"b")+-- > elemAt 1 (fromList [(5,"a"), (3,"b")]) == (5, "a")+-- > elemAt 2 (fromList [(5,"a"), (3,"b")]) Error: index out of range+++{-@ elemAt :: GHC.Types.Int -> OMap k a -> (k, a) @-}+{-@ Decrease elemAt 2 @-}+elemAt :: Int -> Map k a -> (k,a)+elemAt _ Tip = error "Map.elemAt: index out of range"+elemAt i (Bin _ kx x l r)+ = case compare i sizeL of+ LT -> elemAt i l+ GT -> elemAt (i-sizeL-1) r+ EQ -> (kx,x)+ where+ sizeL = size l++-- | /O(log n)/. Update the element at /index/. Calls 'error' when an+-- invalid index is used.+--+-- > updateAt (\ _ _ -> Just "x") 0 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "x"), (5, "a")]+-- > updateAt (\ _ _ -> Just "x") 1 (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "x")]+-- > updateAt (\ _ _ -> Just "x") 2 (fromList [(5,"a"), (3,"b")]) Error: index out of range+-- > updateAt (\ _ _ -> Just "x") (-1) (fromList [(5,"a"), (3,"b")]) Error: index out of range+-- > updateAt (\_ _ -> Nothing) 0 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"+-- > updateAt (\_ _ -> Nothing) 1 (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"+-- > updateAt (\_ _ -> Nothing) 2 (fromList [(5,"a"), (3,"b")]) Error: index out of range+-- > updateAt (\_ _ -> Nothing) (-1) (fromList [(5,"a"), (3,"b")]) Error: index out of range++{-@ updateAt :: (k -> a -> Maybe a) -> GHC.Types.Int -> OMap k a -> OMap k a @-}+{-@ Decrease updateAt 3 @-}+updateAt :: (k -> a -> Maybe a) -> Int -> Map k a -> Map k a+updateAt f i t = i `seq`+ case t of+ Tip -> error "Map.updateAt: index out of range"+ Bin sx kx x l r -> case compare i sizeL of+ LT -> balanceR kx x (updateAt f i l) r+ GT -> balanceL kx x l (updateAt f (i-sizeL-1) r)+ EQ -> case f kx x of+ Just x' -> Bin sx kx x' l r+ Nothing -> glue kx l r+ where+ sizeL = size l++-- | /O(log n)/. Delete the element at /index/.+-- Defined as (@'deleteAt' i map = 'updateAt' (\k x -> 'Nothing') i map@).+--+-- > deleteAt 0 (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"+-- > deleteAt 1 (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"+-- > deleteAt 2 (fromList [(5,"a"), (3,"b")]) Error: index out of range+-- > deleteAt (-1) (fromList [(5,"a"), (3,"b")]) Error: index out of range++{-@ deleteAt :: GHC.Types.Int -> OMap k a -> OMap k a @-}+{-@ Decrease deleteAt 2 @-}+deleteAt :: Int -> Map k a -> Map k a+deleteAt i t = i `seq`+ case t of+ Tip -> error "Map.deleteAt: index out of range"+ Bin _ kx x l r -> case compare i sizeL of+ LT -> balanceR kx x (deleteAt i l) r+ GT -> balanceL kx x l (deleteAt (i-sizeL-1) r)+ EQ -> glue kx l r+ where+ sizeL = size l+++{--------------------------------------------------------------------+ Minimal, Maximal+--------------------------------------------------------------------}+-- | /O(log n)/. The minimal key of the map. Calls 'error' if the map is empty.+--+-- > findMin (fromList [(5,"a"), (3,"b")]) == (3,"b")+-- > findMin empty Error: empty map has no minimal element++{-@ findMin :: OMap k a -> (k, a) @-}+findMin :: Map k a -> (k,a)+findMin (Bin _ kx x Tip _) = (kx,x)+findMin (Bin _ _ _ l _) = findMin l+findMin Tip = error "Map.findMin: empty map has no minimal element"++-- | /O(log n)/. The maximal key of the map. Calls 'error' if the map is empty.+--+-- > findMax (fromList [(5,"a"), (3,"b")]) == (5,"a")+-- > findMax empty Error: empty map has no maximal element++{-@ findMax :: OMap k a -> (k, a) @-}+findMax :: Map k a -> (k,a)+findMax (Bin _ kx x _ Tip) = (kx,x)+findMax (Bin _ _ _ _ r) = findMax r+findMax Tip = error "Map.findMax: empty map has no maximal element"++-- | /O(log n)/. Delete the minimal key. Returns an empty map if the map is empty.+--+-- > deleteMin (fromList [(5,"a"), (3,"b"), (7,"c")]) == fromList [(5,"a"), (7,"c")]+-- > deleteMin empty == empty+++{-@ deleteMin :: OMap k a -> OMap k a @-}+deleteMin :: Map k a -> Map k a+deleteMin (Bin _ _ _ Tip r) = r+deleteMin (Bin _ kx x l r) = balanceR kx x (deleteMin l) r+deleteMin Tip = Tip++-- | /O(log n)/. Delete the maximal key. Returns an empty map if the map is empty.+--+-- > deleteMax (fromList [(5,"a"), (3,"b"), (7,"c")]) == fromList [(3,"b"), (5,"a")]+-- > deleteMax empty == empty++{-@ deleteMax :: OMap k a -> OMap k a @-}+deleteMax :: Map k a -> Map k a+deleteMax (Bin _ _ _ l Tip) = l+deleteMax (Bin _ kx x l r) = balanceL kx x l (deleteMax r)+deleteMax Tip = Tip++-- | /O(log n)/. Update the value at the minimal key.+--+-- > updateMin (\ a -> Just ("X" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3, "Xb"), (5, "a")]+-- > updateMin (\ _ -> Nothing) (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"++{-@ updateMin :: (a -> Maybe a) -> OMap k a -> OMap k a @-}+updateMin :: (a -> Maybe a) -> Map k a -> Map k a+updateMin f m+ = updateMinWithKey (\_ x -> f x) m++-- | /O(log n)/. Update the value at the maximal key.+--+-- > updateMax (\ a -> Just ("X" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3, "b"), (5, "Xa")]+-- > updateMax (\ _ -> Nothing) (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"++{-@ updateMax :: (a -> Maybe a) -> OMap k a -> OMap k a @-}+updateMax :: (a -> Maybe a) -> Map k a -> Map k a+updateMax f m+ = updateMaxWithKey (\_ x -> f x) m+++-- | /O(log n)/. Update the value at the minimal key.+--+-- > updateMinWithKey (\ k a -> Just ((show k) ++ ":" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3,"3:b"), (5,"a")]+-- > updateMinWithKey (\ _ _ -> Nothing) (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"++{-@ updateMinWithKey :: (k -> a -> Maybe a) -> OMap k a -> OMap k a @-}+updateMinWithKey :: (k -> a -> Maybe a) -> Map k a -> Map k a+updateMinWithKey _ Tip = Tip+updateMinWithKey f (Bin sx kx x Tip r) = case f kx x of+ Nothing -> r+ Just x' -> Bin sx kx x' Tip r+updateMinWithKey f (Bin _ kx x l r) = balanceR kx x (updateMinWithKey f l) r++-- | /O(log n)/. Update the value at the maximal key.+--+-- > updateMaxWithKey (\ k a -> Just ((show k) ++ ":" ++ a)) (fromList [(5,"a"), (3,"b")]) == fromList [(3,"b"), (5,"5:a")]+-- > updateMaxWithKey (\ _ _ -> Nothing) (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"++{-@ updateMaxWithKey :: (k -> a -> Maybe a) -> OMap k a -> OMap k a @-}+updateMaxWithKey :: (k -> a -> Maybe a) -> Map k a -> Map k a+updateMaxWithKey _ Tip = Tip+updateMaxWithKey f (Bin sx kx x l Tip) = case f kx x of+ Nothing -> l+ Just x' -> Bin sx kx x' l Tip+updateMaxWithKey f (Bin _ kx x l r) = balanceL kx x l (updateMaxWithKey f r)++-- | /O(log n)/. Retrieves the minimal (key,value) pair of the map, and+-- the map stripped of that element, or 'Nothing' if passed an empty map.+--+-- > minViewWithKey (fromList [(5,"a"), (3,"b")]) == Just ((3,"b"), singleton 5 "a")+-- > minViewWithKey empty == Nothing++{-@ minViewWithKey :: OMap k a -> Maybe (k, a, OMap k a) @-}+minViewWithKey :: Map k a -> Maybe (k, a, Map k a)+minViewWithKey Tip = Nothing+minViewWithKey x = Just (deleteFindMin x)++-- | /O(log n)/. Retrieves the maximal (key,value) pair of the map, and+-- the map stripped of that element, or 'Nothing' if passed an empty map.+--+-- > maxViewWithKey (fromList [(5,"a"), (3,"b")]) == Just ((5,"a"), singleton 3 "b")+-- > maxViewWithKey empty == Nothing++{-@ maxViewWithKey :: OMap k a -> Maybe (k, a, OMap k a) @-}+maxViewWithKey :: Map k a -> Maybe (k, a, Map k a)+maxViewWithKey Tip = Nothing+maxViewWithKey x = Just (deleteFindMax x)++-- | /O(log n)/. Retrieves the value associated with minimal key of the+-- map, and the map stripped of that element, or 'Nothing' if passed an+-- empty map.+--+-- > minView (fromList [(5,"a"), (3,"b")]) == Just ("b", singleton 5 "a")+-- > minView empty == Nothing++{-@ minView :: OMap k a -> Maybe (a, OMap k a) @-}+minView :: Map k a -> Maybe (a, Map k a)+minView Tip = Nothing+minView x = let (_, m, t) = deleteFindMin x in Just (m ,t) -- (first snd $ deleteFindMin x)++-- | /O(log n)/. Retrieves the value associated with maximal key of the+-- map, and the map stripped of that element, or 'Nothing' if passed an+--+-- > maxView (fromList [(5,"a"), (3,"b")]) == Just ("a", singleton 3 "b")+-- > maxView empty == Nothing++{-@ maxView :: OMap k a -> Maybe (a, OMap k a) @-}+maxView :: Map k a -> Maybe (a, Map k a)+maxView Tip = Nothing+maxView x = let (_, m, t) = deleteFindMax x in Just (m, t)++-- Update the 1st component of a tuple (special case of Control.Arrow.first)+-- first :: (a -> b) -> (a, c) -> (b, c)+-- first f (x, y) = (f x, y)++{--------------------------------------------------------------------+ Union.+--------------------------------------------------------------------}+-- | The union of a list of maps:+-- (@'unions' == 'Prelude.foldl' 'union' 'empty'@).+--+-- > unions [(fromList [(5, "a"), (3, "b")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "A3"), (3, "B3")])]+-- > == fromList [(3, "b"), (5, "a"), (7, "C")]+-- > unions [(fromList [(5, "A3"), (3, "B3")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "a"), (3, "b")])]+-- > == fromList [(3, "B3"), (5, "A3"), (7, "C")]++{-@ unions :: (Ord k) => [OMap k a] -> OMap k a @-}+unions :: Ord k => [Map k a] -> Map k a+unions ts+ = foldlStrict union empty ts+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE unions #-}+#endif++-- | The union of a list of maps, with a combining operation:+-- (@'unionsWith' f == 'Prelude.foldl' ('unionWith' f) 'empty'@).+--+-- > unionsWith (++) [(fromList [(5, "a"), (3, "b")]), (fromList [(5, "A"), (7, "C")]), (fromList [(5, "A3"), (3, "B3")])]+-- > == fromList [(3, "bB3"), (5, "aAA3"), (7, "C")]++{-@ unionsWith :: (Ord k) => (a->a->a) -> [OMap k a] -> OMap k a @-}+unionsWith :: Ord k => (a->a->a) -> [Map k a] -> Map k a+unionsWith f ts+ = foldlStrict (unionWith f) empty ts+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE unionsWith #-}+#endif++-- | /O(n+m)/.+-- The expression (@'union' t1 t2@) takes the left-biased union of @t1@ and @t2@.+-- It prefers @t1@ when duplicate keys are encountered,+-- i.e. (@'union' == 'unionWith' 'const'@).+-- The implementation uses the efficient /hedge-union/ algorithm.+-- Hedge-union is more efficient on (bigset \``union`\` smallset).+--+-- > union (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "a"), (7, "C")]++{-@ union :: (Ord k) => x:OMap k a -> y:OMap k a+ -> {v:OMap k a | (mapKeys v) = (Set_cup (mapKeys x) (mapKeys y))}+ @-}+union :: Ord k => Map k a -> Map k a -> Map k a+union Tip t2 = t2+union t1 Tip = t1+union t1 t2 = hedgeUnion NothingS NothingS t1 t2+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE union #-}+#endif++-- left-biased hedge union+{-@ hedgeUnion :: (Ord k) => lo: MaybeS k+ -> hi: MaybeS {v: k | (IfDefLt lo v) }+ -> OMap {v: k | (KeyBetween lo hi v) } a+ -> {v: OMap k a | (RootBetween lo hi v) }+ -> OMap {v: k | (KeyBetween lo hi v)} a @-}+hedgeUnion :: Ord a => MaybeS a -> MaybeS a -> Map a b -> Map a b -> Map a b+hedgeUnion _ _ t1 Tip = t1+hedgeUnion blo bhi Tip (Bin _ kx x l r) = join' kx x (filterGt blo l) (filterLt bhi r)+hedgeUnion _ _ t1 (Bin _ kx x Tip Tip) = insertR kx x t1 -- According to benchmarks, this special case increases+ -- performance up to 30%. It does not help in difference or intersection.+hedgeUnion blo bhi (Bin _ kx x l r) t2 = join' kx x (hedgeUnion blo bmi l (trim blo bmi t2))+ (hedgeUnion bmi bhi r (trim bmi bhi t2))+ where bmi = JustS kx+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE hedgeUnion #-}+#endif++{--------------------------------------------------------------------+ Union with a combining function+--------------------------------------------------------------------}+-- | /O(n+m)/. Union with a combining function. The implementation uses the efficient /hedge-union/ algorithm.+--+-- > unionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "aA"), (7, "C")]++{-@ unionWith :: (Ord k) => (a -> a -> a) -> OMap k a -> OMap k a -> OMap k a @-}+unionWith :: Ord k => (a -> a -> a) -> Map k a -> Map k a -> Map k a+unionWith f m1 m2+ = unionWithKey (\_ x y -> f x y) m1 m2+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE unionWith #-}+#endif++-- | /O(n+m)/.+-- Union with a combining function. The implementation uses the efficient /hedge-union/ algorithm.+-- Hedge-union is more efficient on (bigset \``union`\` smallset).+--+-- > let f key left_value right_value = (show key) ++ ":" ++ left_value ++ "|" ++ right_value+-- > unionWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == fromList [(3, "b"), (5, "5:a|A"), (7, "C")]++{-@ unionWithKey :: (Ord k) => (k -> a -> a -> a) -> OMap k a -> OMap k a -> OMap k a @-}+unionWithKey :: Ord k => (k -> a -> a -> a) -> Map k a -> Map k a -> Map k a+unionWithKey f t1 t2 = mergeWithKey (\k x1 x2 -> Just $ f k x1 x2) (\ _ _ x -> x) (\ _ _ x -> x) t1 t2+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE unionWithKey #-}+#endif++{--------------------------------------------------------------------+ Difference+--------------------------------------------------------------------}+-- | /O(n+m)/. Difference of two maps.+-- Return elements of the first map not existing in the second map.+-- The implementation uses an efficient /hedge/ algorithm comparable with /hedge-union/.+--+-- > difference (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 3 "b"++{-@ difference :: (Ord k) => x:OMap k a -> y:OMap k b+ -> {v:OMap k a | (mapKeys v) = (Set_dif (mapKeys x) (mapKeys y))}+ @-}+difference :: Ord k => Map k a -> Map k b -> Map k a+difference Tip _ = Tip+difference t1 Tip = t1+difference t1 t2 = hedgeDiff NothingS NothingS t1 t2+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE difference #-}+#endif++{-@ hedgeDiff :: (Ord k) => lo: MaybeS k+ -> hi: MaybeS {v: k | (IfDefLt lo v) }+ -> {v: OMap k a | (RootBetween lo hi v) }+ -> OMap {v: k | (KeyBetween lo hi v) } b+ -> OMap {v: k | (KeyBetween lo hi v) } a @-}+{-@ Decrease hedgeDiff 5 @-}+hedgeDiff :: Ord a => MaybeS a -> MaybeS a -> Map a b -> Map a c -> Map a b+hedgeDiff _ _ Tip _ = Tip+hedgeDiff blo bhi (Bin _ kx x l r) Tip = join' kx x (filterGt blo l) (filterLt bhi r)+hedgeDiff blo bhi t (Bin _ kx _ l r) = merge kx (hedgeDiff blo bmi (trim blo bmi t) l)+ (hedgeDiff bmi bhi (trim bmi bhi t) r)+ where bmi = JustS kx+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE hedgeDiff #-}+#endif++-- | /O(n+m)/. Difference with a combining function.+-- When two equal keys are+-- encountered, the combining function is applied to the values of these keys.+-- If it returns 'Nothing', the element is discarded (proper set difference). If+-- it returns (@'Just' y@), the element is updated with a new value @y@.+-- The implementation uses an efficient /hedge/ algorithm comparable with /hedge-union/.+--+-- > let f al ar = if al == "b" then Just (al ++ ":" ++ ar) else Nothing+-- > differenceWith f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (3, "B"), (7, "C")])+-- > == singleton 3 "b:B"++{-@ differenceWith :: (Ord k) => (a -> b -> Maybe a) -> OMap k a -> OMap k b -> OMap k a @-}+differenceWith :: Ord k => (a -> b -> Maybe a) -> Map k a -> Map k b -> Map k a+differenceWith f m1 m2+ = differenceWithKey (\_ x y -> f x y) m1 m2+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE differenceWith #-}+#endif++-- | /O(n+m)/. Difference with a combining function. When two equal keys are+-- encountered, the combining function is applied to the key and both values.+-- If it returns 'Nothing', the element is discarded (proper set difference). If+-- it returns (@'Just' y@), the element is updated with a new value @y@.+-- The implementation uses an efficient /hedge/ algorithm comparable with /hedge-union/.+--+-- > let f k al ar = if al == "b" then Just ((show k) ++ ":" ++ al ++ "|" ++ ar) else Nothing+-- > differenceWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (3, "B"), (10, "C")])+-- > == singleton 3 "3:b|B"++{-@ differenceWithKey :: (Ord k) => (k -> a -> b -> Maybe a) -> OMap k a -> OMap k b -> OMap k a @-}+differenceWithKey :: Ord k => (k -> a -> b -> Maybe a) -> Map k a -> Map k b -> Map k a+differenceWithKey f t1 t2 = mergeWithKey f (\_ _ x -> x) (\ _ _ _ -> Tip) t1 t2+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE differenceWithKey #-}+#endif+++{--------------------------------------------------------------------+ Intersection+--------------------------------------------------------------------}+-- | /O(n+m)/. Intersection of two maps.+-- Return data in the first map for the keys existing in both maps.+-- (@'intersection' m1 m2 == 'intersectionWith' 'const' m1 m2@).+--+-- > intersection (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "a"++{-@ intersection :: (Ord k) => x:OMap k a -> y:OMap k b+ -> {v:OMap k a | (mapKeys v) = (Set_cap (mapKeys x) (mapKeys y))}+ @-}+intersection :: Ord k => Map k a -> Map k b -> Map k a+intersection Tip _ = Tip+intersection _ Tip = Tip+intersection t1 t2 = hedgeInt NothingS NothingS t1 t2+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE intersection #-}+#endif++{-@ hedgeInt :: (Ord k) => lo: MaybeS k+ -> hi: MaybeS {v: k | (IfDefLt lo v) }+ -> OMap {v: k | (KeyBetween lo hi v) } a+ -> {v: OMap k b | (RootBetween lo hi v) }+ -> OMap {v: k | (KeyBetween lo hi v)} a @-}++hedgeInt :: Ord k => MaybeS k -> MaybeS k -> Map k a -> Map k b -> Map k a+hedgeInt _ _ _ Tip = Tip+hedgeInt _ _ Tip _ = Tip+hedgeInt blo bhi (Bin _ kx x l r) t2 = let l' = hedgeInt blo bmi l (trim blo bmi t2)+ r' = hedgeInt bmi bhi r (trim bmi bhi t2)+ in if kx `member` t2 then join' kx x l' r' else merge kx l' r'+ where bmi = JustS kx++#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE hedgeInt #-}+#endif++-- | /O(n+m)/. Intersection with a combining function.+--+-- > intersectionWith (++) (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "aA"++{-@ intersectionWith :: (Ord k) => (a -> b -> c) -> OMap k a -> OMap k b -> OMap k c @-}+intersectionWith :: Ord k => (a -> b -> c) -> Map k a -> Map k b -> Map k c+intersectionWith f m1 m2+ = intersectionWithKey (\_ x y -> f x y) m1 m2+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE intersectionWith #-}+#endif++-- | /O(n+m)/. Intersection with a combining function.+-- Intersection is more efficient on (bigset \``intersection`\` smallset).+--+-- > let f k al ar = (show k) ++ ":" ++ al ++ "|" ++ ar+-- > intersectionWithKey f (fromList [(5, "a"), (3, "b")]) (fromList [(5, "A"), (7, "C")]) == singleton 5 "5:a|A"+++{-@ intersectionWithKey :: (Ord k) => (k -> a -> b -> c) -> OMap k a -> OMap k b -> OMap k c @-}+intersectionWithKey :: Ord k => (k -> a -> b -> c) -> Map k a -> Map k b -> Map k c+intersectionWithKey f t1 t2 = mergeWithKey (\k x1 x2 -> Just $ f k x1 x2) (\ _ _ _ -> Tip) (\ _ _ _ -> Tip) t1 t2+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE intersectionWithKey #-}+#endif+++{--------------------------------------------------------------------+ MergeWithKey+--------------------------------------------------------------------}++-- | /O(n+m)/. A high-performance universal combining function. This function+-- is used to define 'unionWith', 'unionWithKey', 'differenceWith',+-- 'differenceWithKey', 'intersectionWith', 'intersectionWithKey' and can be+-- used to define other custom combine functions.+--+-- Please make sure you know what is going on when using 'mergeWithKey',+-- otherwise you can be surprised by unexpected code growth or even+-- corruption of the data structure.+--+-- When 'mergeWithKey' is given three arguments, it is inlined to the call+-- site. You should therefore use 'mergeWithKey' only to define your custom+-- combining functions. For example, you could define 'unionWithKey',+-- 'differenceWithKey' and 'intersectionWithKey' as+--+-- > myUnionWithKey f m1 m2 = mergeWithKey (\k x1 x2 -> Just (f k x1 x2)) id id m1 m2+-- > myDifferenceWithKey f m1 m2 = mergeWithKey f id (const empty) m1 m2+-- > myIntersectionWithKey f m1 m2 = mergeWithKey (\k x1 x2 -> Just (f k x1 x2)) (const empty) (const empty) m1 m2+--+-- When calling @'mergeWithKey' combine only1 only2@, a function combining two+-- 'IntMap's is created, such that+--+-- * if a key is present in both maps, it is passed with both corresponding+-- values to the @combine@ function. Depending on the result, the key is either+-- present in the result with specified value, or is left out;+--+-- * a nonempty subtree present only in the first map is passed to @only1@ and+-- the output is added to the result;+--+-- * a nonempty subtree present only in the second map is passed to @only2@ and+-- the output is added to the result.+--+-- The @only1@ and @only2@ methods /must return a map with a subset (possibly empty) of the keys of the given map/.+-- The values can be modified arbitrarily. Most common variants of @only1@ and+-- @only2@ are 'id' and @'const' 'empty'@, but for example @'map' f@ or+-- @'filterWithKey' f@ could be used for any @f@.++{-@ mergeWithKey :: (Ord k) => (k -> a -> b -> Maybe c)+ -> (lo:MaybeS k -> hi: MaybeS k -> OMap {v: k | (KeyBetween lo hi v) } a -> OMap {v: k | (KeyBetween lo hi v) } c)+ -> (lo:MaybeS k -> hi: MaybeS k -> OMap {v: k | (KeyBetween lo hi v) } b -> OMap {v: k | (KeyBetween lo hi v) } c)+ -> OMap k a -> OMap k b -> OMap k c @-}+mergeWithKey :: Ord k => (k -> a -> b -> Maybe c) -> (MaybeS k -> MaybeS k -> Map k a -> Map k c) -> (MaybeS k -> MaybeS k -> Map k b -> Map k c)+ -> Map k a -> Map k b -> Map k c+mergeWithKey f g1 g2 = go+ where+ go Tip t2 = g2 NothingS NothingS t2+ go t1 Tip = g1 NothingS NothingS t1+ go t1 t2 = hedgeMerge f g1 g2 NothingS NothingS t1 t2++{-@ hedgeMerge :: (Ord k) => (k -> a -> b -> Maybe c)+ -> (lo:MaybeS k -> hi: MaybeS k -> OMap {v: k | (KeyBetween lo hi v) } a -> OMap {v: k | (KeyBetween lo hi v) } c)+ -> (lo:MaybeS k -> hi: MaybeS k -> OMap {v: k | (KeyBetween lo hi v) } b -> OMap {v: k | (KeyBetween lo hi v) } c)+ -> lo: MaybeS k+ -> hi: MaybeS {v: k | (IfDefLt lo v) }+ -> OMap {v: k | (KeyBetween lo hi v) } a+ -> {v: OMap k b | (RootBetween lo hi v) }+ -> OMap {v: k | (KeyBetween lo hi v)} c @-}++hedgeMerge :: Ord k => (k -> a -> b -> Maybe c)+ -> (MaybeS k -> MaybeS k -> Map k a -> Map k c)+ -> (MaybeS k -> MaybeS k -> Map k b -> Map k c)+ -> MaybeS k -> MaybeS k+ -> Map k a -> Map k b -> Map k c+hedgeMerge f g1 g2 blo bhi t1 Tip+ = g1 blo bhi t1+hedgeMerge f g1 g2 blo bhi Tip (Bin _ kx x l r)+ = g2 blo bhi $ join' kx x (filterGt blo l) (filterLt bhi r)+hedgeMerge f g1 g2 blo bhi (Bin _ kx x l r) t2+ = let bmi = JustS kx+ l' = hedgeMerge f g1 g2 blo bmi l (trim blo bmi t2)+ (found, trim_t2) = trimLookupLo kx bhi t2+ r' = hedgeMerge f g1 g2 bmi bhi r trim_t2+ in case found of+ Nothing -> case g1 blo bhi (singleton kx x) of+ Tip -> merge kx l' r'+ (Bin _ _ x' Tip Tip) -> join' kx x' l' r'+ _ -> error "mergeWithKey: Given function only1 does not fulfil required conditions (see documentation)"+ Just x2 -> case f kx x x2 of+ Nothing -> merge kx l' r'+ Just x' -> join' kx x' l' r'+{-# INLINE mergeWithKey #-}++{--------------------------------------------------------------------+ Submap+--------------------------------------------------------------------}+-- | /O(n+m)/.+-- This function is defined as (@'isSubmapOf' = 'isSubmapOfBy' (==)@).+--+{-@ isSubmapOf :: (Ord k, Eq a) => OMap k a -> OMap k a -> Bool @-}+isSubmapOf :: (Ord k,Eq a) => Map k a -> Map k a -> Bool+isSubmapOf m1 m2 = isSubmapOfBy (==) m1 m2+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE isSubmapOf #-}+#endif++{- | /O(n+m)/.+ The expression (@'isSubmapOfBy' f t1 t2@) returns 'True' if+ all keys in @t1@ are in tree @t2@, and when @f@ returns 'True' when+ applied to their respective values. For example, the following+ expressions are all 'True':++ > isSubmapOfBy (==) (fromList [('a',1)]) (fromList [('a',1),('b',2)])+ > isSubmapOfBy (<=) (fromList [('a',1)]) (fromList [('a',1),('b',2)])+ > isSubmapOfBy (==) (fromList [('a',1),('b',2)]) (fromList [('a',1),('b',2)])++ But the following are all 'False':++ > isSubmapOfBy (==) (fromList [('a',2)]) (fromList [('a',1),('b',2)])+ > isSubmapOfBy (<) (fromList [('a',1)]) (fromList [('a',1),('b',2)])+ > isSubmapOfBy (==) (fromList [('a',1),('b',2)]) (fromList [('a',1)])+++-}++{-@ isSubmapOfBy :: (Ord k) => (a->b->Bool) -> OMap k a -> OMap k b -> Bool @-}+isSubmapOfBy :: Ord k => (a->b->Bool) -> Map k a -> Map k b -> Bool+isSubmapOfBy f t1 t2+ = (size t1 <= size t2) && (submap' f t1 t2)+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE isSubmapOfBy #-}+#endif++submap' :: Ord a => (b -> c -> Bool) -> Map a b -> Map a c -> Bool+submap' _ Tip _ = True+submap' _ _ Tip = False+submap' f (Bin _ kx x l r) t+ = case found of+ Nothing -> False+ Just y -> f x y && submap' f l lt && submap' f r gt+ where+ (lt,found,gt) = splitLookup kx t+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE submap' #-}+#endif++-- | /O(n+m)/. Is this a proper submap? (ie. a submap but not equal).+-- Defined as (@'isProperSubmapOf' = 'isProperSubmapOfBy' (==)@).++{-@ isProperSubmapOf :: (Ord k,Eq a) => OMap k a -> OMap k a -> Bool @-}+isProperSubmapOf :: (Ord k,Eq a) => Map k a -> Map k a -> Bool+isProperSubmapOf m1 m2+ = isProperSubmapOfBy (==) m1 m2+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE isProperSubmapOf #-}+#endif++{- | /O(n+m)/. Is this a proper submap? (ie. a submap but not equal).+ The expression (@'isProperSubmapOfBy' f m1 m2@) returns 'True' when+ @m1@ and @m2@ are not equal,+ all keys in @m1@ are in @m2@, and when @f@ returns 'True' when+ applied to their respective values. For example, the following+ expressions are all 'True':++ > isProperSubmapOfBy (==) (fromList [(1,1)]) (fromList [(1,1),(2,2)])+ > isProperSubmapOfBy (<=) (fromList [(1,1)]) (fromList [(1,1),(2,2)])++ But the following are all 'False':++ > isProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1),(2,2)])+ > isProperSubmapOfBy (==) (fromList [(1,1),(2,2)]) (fromList [(1,1)])+ > isProperSubmapOfBy (<) (fromList [(1,1)]) (fromList [(1,1),(2,2)])+++-}+{-@ isProperSubmapOfBy :: Ord k => (a -> b -> Bool) -> OMap k a -> OMap k b -> Bool @-}+isProperSubmapOfBy :: Ord k => (a -> b -> Bool) -> Map k a -> Map k b -> Bool+isProperSubmapOfBy f t1 t2+ = (size t1 < size t2) && (submap' f t1 t2)+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE isProperSubmapOfBy #-}+#endif++{--------------------------------------------------------------------+ Filter and partition+--------------------------------------------------------------------}+-- | /O(n)/. Filter all values that satisfy the predicate.+--+-- > filter (> "a") (fromList [(5,"a"), (3,"b")]) == singleton 3 "b"+-- > filter (> "x") (fromList [(5,"a"), (3,"b")]) == empty+-- > filter (< "a") (fromList [(5,"a"), (3,"b")]) == empty++{-@ filter :: (a -> Bool) -> OMap k a -> OMap k a @-}+filter :: (a -> Bool) -> Map k a -> Map k a+filter p m+ = filterWithKey (\_ x -> p x) m++-- | /O(n)/. Filter all keys\/values that satisfy the predicate.+--+-- > filterWithKey (\k _ -> k > 4) (fromList [(5,"a"), (3,"b")]) == singleton 5 "a"++{-@ filterWithKey :: (k -> a -> Bool) -> OMap k a -> OMap k a @-}+filterWithKey :: (k -> a -> Bool) -> Map k a -> Map k a+filterWithKey _ Tip = Tip+filterWithKey p (Bin _ kx x l r)+ | p kx x = join' kx x (filterWithKey p l) (filterWithKey p r)+ | otherwise = merge kx (filterWithKey p l) (filterWithKey p r)++-- | /O(n)/. Partition the map according to a predicate. The first+-- map contains all elements that satisfy the predicate, the second all+-- elements that fail the predicate. See also 'split'.+--+-- > partition (> "a") (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", singleton 5 "a")+-- > partition (< "x") (fromList [(5,"a"), (3,"b")]) == (fromList [(3, "b"), (5, "a")], empty)+-- > partition (> "x") (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3, "b"), (5, "a")])++{-@ partition :: (a -> Bool) -> OMap k a -> (OMap k a, OMap k a) @-}+partition :: (a -> Bool) -> Map k a -> (Map k a,Map k a)+partition p m+ = partitionWithKey (\_ x -> p x) m++-- | /O(n)/. Partition the map according to a predicate. The first+-- map contains all elements that satisfy the predicate, the second all+-- elements that fail the predicate. See also 'split'.+--+-- > partitionWithKey (\ k _ -> k > 3) (fromList [(5,"a"), (3,"b")]) == (singleton 5 "a", singleton 3 "b")+-- > partitionWithKey (\ k _ -> k < 7) (fromList [(5,"a"), (3,"b")]) == (fromList [(3, "b"), (5, "a")], empty)+-- > partitionWithKey (\ k _ -> k > 7) (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3, "b"), (5, "a")])++{-@ partitionWithKey :: (k -> a -> Bool) -> OMap k a -> (OMap k a, OMap k a) @-}+partitionWithKey :: (k -> a -> Bool) -> Map k a -> (Map k a, Map k a)+partitionWithKey _ Tip = (Tip,Tip)+partitionWithKey p (Bin _ kx x l r)+ | p kx x = (join' kx x l1 r1,merge kx l2 r2)+ | otherwise = (merge kx l1 r1,join' kx x l2 r2)+ where+ (l1,l2) = partitionWithKey p l+ (r1,r2) = partitionWithKey p r++-- | /O(n)/. Map values and collect the 'Just' results.+--+-- > let f x = if x == "a" then Just "new a" else Nothing+-- > mapMaybe f (fromList [(5,"a"), (3,"b")]) == singleton 5 "new a"++{-@ mapMaybe :: (a -> Maybe b) -> OMap k a -> OMap k b @-}+mapMaybe :: (a -> Maybe b) -> Map k a -> Map k b+mapMaybe f = mapMaybeWithKey (\_ x -> f x)++-- | /O(n)/. Map keys\/values and collect the 'Just' results.+--+-- > let f k _ = if k < 5 then Just ("key : " ++ (show k)) else Nothing+-- > mapMaybeWithKey f (fromList [(5,"a"), (3,"b")]) == singleton 3 "key : 3"++{-@ mapMaybeWithKey :: (k -> a -> Maybe b) -> OMap k a -> OMap k b @-}+mapMaybeWithKey :: (k -> a -> Maybe b) -> Map k a -> Map k b+mapMaybeWithKey _ Tip = Tip+mapMaybeWithKey f (Bin _ kx x l r) = case f kx x of+ Just y -> join' kx y (mapMaybeWithKey f l) (mapMaybeWithKey f r)+ Nothing -> merge kx (mapMaybeWithKey f l) (mapMaybeWithKey f r)++-- | /O(n)/. Map values and separate the 'Left' and 'Right' results.+--+-- > let f a = if a < "c" then Left a else Right a+-- > mapEither f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])+-- > == (fromList [(3,"b"), (5,"a")], fromList [(1,"x"), (7,"z")])+-- >+-- > mapEither (\ a -> Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])+-- > == (empty, fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])++{-@ mapEither :: (a -> Either b c) -> OMap k a -> (OMap k b, OMap k c) @-}+mapEither :: (a -> Either b c) -> Map k a -> (Map k b, Map k c)+mapEither f m+ = mapEitherWithKey (\_ x -> f x) m++-- | /O(n)/. Map keys\/values and separate the 'Left' and 'Right' results.+--+-- > let f k a = if k < 5 then Left (k * 2) else Right (a ++ a)+-- > mapEitherWithKey f (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])+-- > == (fromList [(1,2), (3,6)], fromList [(5,"aa"), (7,"zz")])+-- >+-- > mapEitherWithKey (\_ a -> Right a) (fromList [(5,"a"), (3,"b"), (1,"x"), (7,"z")])+-- > == (empty, fromList [(1,"x"), (3,"b"), (5,"a"), (7,"z")])++{-@ mapEitherWithKey :: (k -> a -> Either b c) -> OMap k a -> (OMap k b, OMap k c) @-}+mapEitherWithKey :: (k -> a -> Either b c) -> Map k a -> (Map k b, Map k c)+mapEitherWithKey _ Tip = (Tip, Tip)+mapEitherWithKey f (Bin _ kx x l r) = case f kx x of+ Left y -> (join' kx y l1 r1, merge kx l2 r2)+ Right z -> (merge kx l1 r1, join' kx z l2 r2)+ where+ (l1,l2) = mapEitherWithKey f l+ (r1,r2) = mapEitherWithKey f r++{--------------------------------------------------------------------+ Mapping+--------------------------------------------------------------------}+-- | /O(n)/. Map a function over all values in the map.+--+-- > map (++ "x") (fromList [(5,"a"), (3,"b")]) == fromList [(3, "bx"), (5, "ax")]++{-@ map :: (a -> b) -> OMap k a -> OMap k b @-}+map :: (a -> b) -> Map k a -> Map k b+map _ Tip = Tip+map f (Bin sx kx x l r) = Bin sx kx (f x) (map f l) (map f r)++-- | /O(n)/. Map a function over all values in the map.+--+-- > let f key x = (show key) ++ ":" ++ x+-- > mapWithKey f (fromList [(5,"a"), (3,"b")]) == fromList [(3, "3:b"), (5, "5:a")]++{-@ mapWithKey :: (k -> a -> b) -> OMap k a -> OMap k b @-}+mapWithKey :: (k -> a -> b) -> Map k a -> Map k b+mapWithKey _ Tip = Tip+mapWithKey f (Bin sx kx x l r) = Bin sx kx (f kx x) (mapWithKey f l) (mapWithKey f r)++-- | /O(n)/.+-- @'traverseWithKey' f s == 'fromList' <$> 'traverse' (\(k, v) -> (,) k <$> f k v) ('toList' m)@+-- That is, behaves exactly like a regular 'traverse' except that the traversing+-- function also has access to the key associated with a value.+--+-- > traverseWithKey (\k v -> if odd k then Just (succ v) else Nothing) (fromList [(1, 'a'), (5, 'e')]) == Just (fromList [(1, 'b'), (5, 'f')])+-- > traverseWithKey (\k v -> if odd k then Just (succ v) else Nothing) (fromList [(2, 'c')]) == Nothing+--{-# INLINE traverseWithKey #-}+--traverseWithKey :: Applicative t => (k -> a -> t b) -> Map k a -> t (Map k b)+--traverseWithKey f = go+-- where+-- go Tip = pure Tip+-- go (Bin s k v l r)+-- = flip (Bin s k) <$> go l <*> f k v <*> go r++-- | /O(n)/. The function 'mapAccum' threads an accumulating+-- argument through the map in ascending order of keys.+--+-- > let f a b = (a ++ b, b ++ "X")+-- > mapAccum f "Everything: " (fromList [(5,"a"), (3,"b")]) == ("Everything: ba", fromList [(3, "bX"), (5, "aX")])++{-@ mapAccum :: (a -> b -> (a,c)) -> a -> OMap k b -> (a, OMap k c) @-}+mapAccum :: (a -> b -> (a,c)) -> a -> Map k b -> (a, Map k c)+mapAccum f a m+ = mapAccumWithKey (\a' _ x' -> f a' x') a m++-- | /O(n)/. The function 'mapAccumWithKey' threads an accumulating+-- argument through the map in ascending order of keys.+--+-- > let f a k b = (a ++ " " ++ (show k) ++ "-" ++ b, b ++ "X")+-- > mapAccumWithKey f "Everything:" (fromList [(5,"a"), (3,"b")]) == ("Everything: 3-b 5-a", fromList [(3, "bX"), (5, "aX")])++{-@ mapAccumWithKey :: (a -> k -> b -> (a,c)) -> a -> OMap k b -> (a, OMap k c) @-}+mapAccumWithKey :: (a -> k -> b -> (a,c)) -> a -> Map k b -> (a,Map k c)+mapAccumWithKey f a t+ = mapAccumL f a t++-- | /O(n)/. The function 'mapAccumL' threads an accumulating+-- argument through the map in ascending order of keys.+mapAccumL :: (a -> k -> b -> (a,c)) -> a -> Map k b -> (a,Map k c)+mapAccumL _ a Tip = (a,Tip)+mapAccumL f a (Bin sx kx x l r) =+ let (a1,l') = mapAccumL f a l+ (a2,x') = f a1 kx x+ (a3,r') = mapAccumL f a2 r+ in (a3,Bin sx kx x' l' r')++-- | /O(n)/. The function 'mapAccumR' threads an accumulating+-- argument through the map in descending order of keys.+{-@ mapAccumRWithKey :: (a -> k -> b -> (a,c)) -> a -> OMap k b -> (a, OMap k c) @-}+mapAccumRWithKey :: (a -> k -> b -> (a,c)) -> a -> Map k b -> (a,Map k c)+mapAccumRWithKey _ a Tip = (a,Tip)+mapAccumRWithKey f a (Bin sx kx x l r) =+ let (a1,r') = mapAccumRWithKey f a r+ (a2,x') = f a1 kx x+ (a3,l') = mapAccumRWithKey f a2 l+ in (a3,Bin sx kx x' l' r')++-- | /O(n*log n)/.+-- @'mapKeys' f s@ is the map obtained by applying @f@ to each key of @s@.+--+-- The size of the result may be smaller if @f@ maps two or more distinct+-- keys to the same new key. In this case the value at the greatest of the+-- original keys is retained.+--+-- > mapKeys (+ 1) (fromList [(5,"a"), (3,"b")]) == fromList [(4, "b"), (6, "a")]+-- > mapKeys (\ _ -> 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "c"+-- > mapKeys (\ _ -> 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "c"++{-@ mapKeys :: (Ord k2) => (k1 -> k2) -> OMap k1 a -> OMap k2 a @-}+mapKeys :: Ord k2 => (k1->k2) -> Map k1 a -> Map k2 a+mapKeys f = fromList . foldrWithKey (\k x xs -> (f k, x) : xs) []+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE mapKeys #-}+#endif++-- | /O(n*log n)/.+-- @'mapKeysWith' c f s@ is the map obtained by applying @f@ to each key of @s@.+--+-- The size of the result may be smaller if @f@ maps two or more distinct+-- keys to the same new key. In this case the associated values will be+-- combined using @c@.+--+-- > mapKeysWith (++) (\ _ -> 1) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 1 "cdab"+-- > mapKeysWith (++) (\ _ -> 3) (fromList [(1,"b"), (2,"a"), (3,"d"), (4,"c")]) == singleton 3 "cdab"++{-@ mapKeysWith :: (Ord k2) => (a -> a -> a) -> (k1->k2) -> OMap k1 a -> OMap k2 a @-}+mapKeysWith :: Ord k2 => (a -> a -> a) -> (k1->k2) -> Map k1 a -> Map k2 a+mapKeysWith c f = fromListWith c . foldrWithKey (\k x xs -> (f k, x) : xs) []+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE mapKeysWith #-}+#endif+++-- | /O(n)/.+-- @'mapKeysMonotonic' f s == 'mapKeys' f s@, but works only when @f@+-- is strictly monotonic.+-- That is, for any values @x@ and @y@, if @x@ < @y@ then @f x@ < @f y@.+-- /The precondition is not checked./+-- Semi-formally, we have:+--+-- > and [x < y ==> f x < f y | x <- ls, y <- ls]+-- > ==> mapKeysMonotonic f s == mapKeys f s+-- > where ls = keys s+--+-- This means that @f@ maps distinct original keys to distinct resulting keys.+-- This function has better performance than 'mapKeys'.+--+-- > mapKeysMonotonic (\ k -> k * 2) (fromList [(5,"a"), (3,"b")]) == fromList [(6, "b"), (10, "a")]+-- > valid (mapKeysMonotonic (\ k -> k * 2) (fromList [(5,"a"), (3,"b")])) == True+-- > valid (mapKeysMonotonic (\ _ -> 1) (fromList [(5,"a"), (3,"b")])) == False+{-LIQUID argh... mapKeysMonotonic :: (x:k1 -> {v:k2 | x <= v}) -> OMap k1 a -> OMap k2 a @-}+{-@ mapKeysMonotonic :: (x:k1 -> k2) -> OMap k1 a -> OMap k2 a @-}+-- LIQUID: approximating monotonicity..+mapKeysMonotonic :: (k1->k2) -> Map k1 a -> Map k2 a+mapKeysMonotonic _ Tip = Tip+mapKeysMonotonic f (Bin sz k x l r) =+ Bin sz (f k) x (mapKeysMonotonic f l) (mapKeysMonotonic f r)++{--------------------------------------------------------------------+ Folds+--------------------------------------------------------------------}++-- | /O(n)/. Fold the values in the map using the given right-associative+-- binary operator, such that @'foldr' f z == 'Prelude.foldr' f z . 'elems'@.+--+-- For example,+--+-- > elems map = foldr (:) [] map+--+-- > let f a len = len + (length a)+-- > foldr f 0 (fromList [(5,"a"), (3,"bbb")]) == 4+{-@ foldr :: (a -> b -> b) -> b -> OMap k a -> b @-}+foldr :: (a -> b -> b) -> b -> Map k a -> b+foldr f z = go z+ where+ go z' Tip = z'+ go z' (Bin _ _ x l r) = go (f x (go z' r)) l+{-# INLINE foldr #-}++-- | /O(n)/. A strict version of 'foldr'. Each application of the operator is+-- evaluated before using the result in the next application. This+-- function is strict in the starting value.+{-@ foldr' :: (a -> b -> b) -> b -> OMap k a -> b @-}+foldr' :: (a -> b -> b) -> b -> Map k a -> b+foldr' f z = go z+ where+ go z' Tip = z'+ go z' (Bin _ _ x l r) = go (f x (go z' r)) l+{-# INLINE foldr' #-}++-- | /O(n)/. Fold the values in the map using the given left-associative+-- binary operator, such that @'foldl' f z == 'Prelude.foldl' f z . 'elems'@.+--+-- For example,+--+-- > elems = reverse . foldl (flip (:)) []+--+-- > let f len a = len + (length a)+-- > foldl f 0 (fromList [(5,"a"), (3,"bbb")]) == 4+{-@ foldl :: (a -> b -> a) -> a -> OMap k b -> a @-}+foldl :: (a -> b -> a) -> a -> Map k b -> a+foldl f z = go z+ where+ go z' Tip = z'+ go z' (Bin _ _ x l r) = go (f (go z' l) x) r+{-# INLINE foldl #-}++-- | /O(n)/. A strict version of 'foldl'. Each application of the operator is+-- evaluated before using the result in the next application. This+-- function is strict in the starting value.+{-@ foldl' :: (a -> b -> a) -> a -> OMap k b -> a @-}+foldl' :: (a -> b -> a) -> a -> Map k b -> a+foldl' f z = go z+ where+ go z' Tip = z'+ go z' (Bin _ _ x l r) = go (f (go z' l) x) r+{-# INLINE foldl' #-}++-- | /O(n)/. Fold the keys and values in the map using the given right-associative+-- binary operator, such that+-- @'foldrWithKey' f z == 'Prelude.foldr' ('uncurry' f) z . 'toAscList'@.+--+-- For example,+--+-- > keys map = foldrWithKey (\k x ks -> k:ks) [] map+--+-- > let f k a result = result ++ "(" ++ (show k) ++ ":" ++ a ++ ")"+-- > foldrWithKey f "Map: " (fromList [(5,"a"), (3,"b")]) == "Map: (5:a)(3:b)"+{-@ foldrWithKey :: (k -> a -> b -> b) -> b -> OMap k a -> b @-}+foldrWithKey :: (k -> a -> b -> b) -> b -> Map k a -> b+foldrWithKey f z = go z+ where+ go z' Tip = z'+ go z' (Bin _ kx x l r) = go (f kx x (go z' r)) l+{-# INLINE foldrWithKey #-}++-- | /O(n)/. A strict version of 'foldrWithKey'. Each application of the operator is+-- evaluated before using the result in the next application. This+-- function is strict in the starting value.+{-@ foldrWithKey' :: (k -> a -> b -> b) -> b -> OMap k a -> b @-}+foldrWithKey' :: (k -> a -> b -> b) -> b -> Map k a -> b+foldrWithKey' f z = go z+ where+ go z' Tip = z'+ go z' (Bin _ kx x l r) = go (f kx x (go z' r)) l+{-# INLINE foldrWithKey' #-}++-- | /O(n)/. Fold the keys and values in the map using the given left-associative+-- binary operator, such that+-- @'foldlWithKey' f z == 'Prelude.foldl' (\\z' (kx, x) -> f z' kx x) z . 'toAscList'@.+--+-- For example,+--+-- > keys = reverse . foldlWithKey (\ks k x -> k:ks) []+--+-- > let f result k a = result ++ "(" ++ (show k) ++ ":" ++ a ++ ")"+-- > foldlWithKey f "Map: " (fromList [(5,"a"), (3,"b")]) == "Map: (3:b)(5:a)"+{-@ foldlWithKey :: (a -> k -> b -> a) -> a -> OMap k b -> a @-}+foldlWithKey :: (a -> k -> b -> a) -> a -> Map k b -> a+foldlWithKey f z = go z+ where+ go z' Tip = z'+ go z' (Bin _ kx x l r) = go (f (go z' l) kx x) r+{-# INLINE foldlWithKey #-}++-- | /O(n)/. A strict version of 'foldlWithKey'. Each application of the operator is+-- evaluated before using the result in the next application. This+-- function is strict in the starting value.+{-@ foldlWithKey' :: (a -> k -> b -> a) -> a -> OMap k b -> a @-}+foldlWithKey' :: (a -> k -> b -> a) -> a -> Map k b -> a+foldlWithKey' f z = go z+ where+ go z' Tip = z'+ go z' (Bin _ kx x l r) = go (f (go z' l) kx x) r+{-# INLINE foldlWithKey' #-}++{--------------------------------------------------------------------+ List variations+--------------------------------------------------------------------}+-- | /O(n)/.+-- Return all elements of the map in the ascending order of their keys.+-- Subject to list fusion.+--+-- > elems (fromList [(5,"a"), (3,"b")]) == ["b","a"]+-- > elems empty == []++{-@ elems :: m:OMap k a -> {v:[a] | len v = mlen m} @-}+elems :: Map k a -> [a]+elems = foldr (:) []++-- | /O(n)/. Return all keys of the map in ascending order. Subject to list+-- fusion.+--+-- > keys (fromList [(5,"a"), (3,"b")]) == [3,5]+-- > keys empty == []++{- LIQUID: SUMMARY-VALUES: keys :: OMap k a -> [k]<{v: k | v >= fld}> @-}+{-@ keys :: m:OMap k a -> {v:[k]<{\x y -> x < y}> | len v = mlen m} @-}+keys :: Map k a -> [k]+keys = foldrWithKey (\k _ ks -> k : ks) []++-- | /O(n)/. An alias for 'toAscList'. Return all key\/value pairs in the map+-- in ascending key order. Subject to list fusion.+--+-- > assocs (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]+-- > assocs empty == []++{- LIQUID: SUMMARY-VALUES: assocs :: OMap k a -> [(k, a)]<{v: (k, a) | fst(v) >= fst(fld) }> @-}+{-@ assocs :: m:OMap k a -> {v:[(k,a)]<{\x y -> fst x < fst y}> | len v = mlen m} @-}+assocs :: Map k a -> [(k,a)]+assocs m+ = toAscList m++-- | /O(n)/. The set of all keys of the map.+--+-- > keysSet (fromList [(5,"a"), (3,"b")]) == Data.Set.fromList [3,5]+-- > keysSet empty == Data.Set.empty+-- LIQUID keysSet :: Map k a -> Set.Set k+-- LIQUID keysSet Tip = Set.Tip+-- LIQUID keysSet (Bin sz kx _ l r) = Set.Bin sz kx (keysSet l) (keysSet r)++-- | /O(n)/. Build a map from a set of keys and a function which for each key+-- computes its value.+--+-- > fromSet (\k -> replicate k 'a') (Data.Set.fromList [3, 5]) == fromList [(5,"aaaaa"), (3,"aaa")]+-- > fromSet undefined Data.Set.empty == empty+-- LIQUID fromSet :: (k -> a) -> Set.Set k -> Map k a+-- LIQUID fromSet _ Set.Tip = Tip+-- LIQUID fromSet f (Set.Bin sz x l r) = Bin sz x (f x) (fromSet f l) (fromSet f r)++{--------------------------------------------------------------------+ Lists+ use [foldlStrict] to reduce demand on the control-stack+--------------------------------------------------------------------}+-- | /O(n*log n)/. Build a map from a list of key\/value pairs. See also 'fromAscList'.+-- If the list contains more than one value for the same key, the last value+-- for the key is retained.+--+-- > fromList [] == empty+-- > fromList [(5,"a"), (3,"b"), (5, "c")] == fromList [(5,"c"), (3,"b")]+-- > fromList [(5,"c"), (3,"b"), (5, "a")] == fromList [(5,"a"), (3,"b")]++{-@ fromList :: (Ord k) => kvs:[(k,a)] -> {v:OMap k a | mlen v = len kvs} @-}+fromList :: Ord k => [(k,a)] -> Map k a+fromList xs+ = foldlStrict ins empty xs+ where+ ins t (k,x) = insert k x t+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE fromList #-}+#endif++-- | /O(n*log n)/. Build a map from a list of key\/value pairs with a combining function. See also 'fromAscListWith'.+--+-- > fromListWith (++) [(5,"a"), (5,"b"), (3,"b"), (3,"a"), (5,"a")] == fromList [(3, "ab"), (5, "aba")]+-- > fromListWith (++) [] == empty++{-@ fromListWith :: (Ord k) => (a -> a -> a) -> kvs:[(k,a)] -> {v:OMap k a | mlen v = len kvs} @-}+fromListWith :: Ord k => (a -> a -> a) -> [(k,a)] -> Map k a+fromListWith f xs+ = fromListWithKey (\_ x y -> f x y) xs+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE fromListWith #-}+#endif++-- | /O(n*log n)/. Build a map from a list of key\/value pairs with a combining function. See also 'fromAscListWithKey'.+--+-- > let f k a1 a2 = (show k) ++ a1 ++ a2+-- > fromListWithKey f [(5,"a"), (5,"b"), (3,"b"), (3,"a"), (5,"a")] == fromList [(3, "3ab"), (5, "5a5ba")]+-- > fromListWithKey f [] == empty++{-@ fromListWithKey :: (Ord k) => (k -> a -> a -> a) -> kvs:[(k,a)] -> {v:OMap k a | mlen v = len kvs} @-}+fromListWithKey :: Ord k => (k -> a -> a -> a) -> [(k,a)] -> Map k a+fromListWithKey f xs+ = foldlStrict ins empty xs+ where+ ins t (k,x) = insertWithKey f k x t+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE fromListWithKey #-}+#endif++-- | /O(n)/. Convert the map to a list of key\/value pairs. Subject to list fusion.+--+-- > toList (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]+-- > toList empty == []++{- LIQUIDTODO: toList:: OMap k a -> [(k, a)]<{v: (k, a) | fst(v) > fst(fld) }> @-}+{-@ toList :: m:OMap k a -> {v:[(k,a)]<{\x y -> fst x < fst y}> | len v = mlen m} @-}+toList :: Map k a -> [(k,a)]+toList = toAscList++-- | /O(n)/. Convert the map to a list of key\/value pairs where the keys are+-- in ascending order. Subject to list fusion.+--+-- > toAscList (fromList [(5,"a"), (3,"b")]) == [(3,"b"), (5,"a")]++{- LIQUIDTODO: toAscList :: OMap k a -> [(k, a)]<{v: (k, a) | fst(v) > fst(fld) }> @-}+{-@ toAscList :: m:OMap k a -> {v:[(k,a)]<{\x y -> fst x < fst y}> | len v = mlen m} @-}+toAscList :: Map k a -> [(k,a)]+toAscList = foldrWithKey (\k x xs -> (k,x):xs) []++-- | /O(n)/. Convert the map to a list of key\/value pairs where the keys+-- are in descending order. Subject to list fusion.+--+-- > toDescList (fromList [(5,"a"), (3,"b")]) == [(5,"a"), (3,"b")]++{- LIQUIDTODO: toAscList :: OMap k a -> [(k, a)]<{v: (k, a) | fst(v) < fst(fld) }> @-}+{-@ toDescList :: m:OMap k a -> {v:[(k,a)]<{\x y -> fst x > fst y}> | len v = mlen m} @-}+toDescList :: Map k a -> [(k,a)]+toDescList = foldlWithKey (\xs k x -> (k,x):xs) []++-- List fusion for the list generating functions.+#if __GLASGOW_HASKELL__+-- The foldrFB and foldlFB are fold{r,l}WithKey equivalents, used for list fusion.+-- They are important to convert unfused methods back, see mapFB in prelude.+{-@ foldrFB :: (k -> a -> b -> b) -> b -> OMap k a -> b @-}+foldrFB :: (k -> a -> b -> b) -> b -> Map k a -> b+foldrFB = foldrWithKey+{-# INLINE[0] foldrFB #-}+{-@ foldlFB :: (a -> k -> b -> a) -> a -> OMap k b -> a @-}+foldlFB :: (a -> k -> b -> a) -> a -> Map k b -> a+foldlFB = foldlWithKey+{-# INLINE[0] foldlFB #-}++-- Inline assocs and toList, so that we need to fuse only toAscList.+{-# INLINE assocs #-}+{-# INLINE toList #-}++-- The fusion is enabled up to phase 2 included. If it does not succeed,+-- convert in phase 1 the expanded elems,keys,to{Asc,Desc}List calls back to+-- elems,keys,to{Asc,Desc}List. In phase 0, we inline fold{lr}FB (which were+-- used in a list fusion, otherwise it would go away in phase 1), and let compiler+-- do whatever it wants with elems,keys,to{Asc,Desc}List -- it was forbidden to+-- inline it before phase 0, otherwise the fusion rules would not fire at all.+{-# NOINLINE[0] elems #-}+{-# NOINLINE[0] keys #-}+{-# NOINLINE[0] toAscList #-}+{-# NOINLINE[0] toDescList #-}+{-# RULES "Map.elems" [~1] forall m . elems m = build (\c n -> foldrFB (\_ x xs -> c x xs) n m) #-}+{-# RULES "Map.elemsBack" [1] foldrFB (\_ x xs -> x : xs) [] = elems #-}+{-# RULES "Map.keys" [~1] forall m . keys m = build (\c n -> foldrFB (\k _ xs -> c k xs) n m) #-}+{-# RULES "Map.keysBack" [1] foldrFB (\k _ xs -> k : xs) [] = keys #-}+{-# RULES "Map.toAscList" [~1] forall m . toAscList m = build (\c n -> foldrFB (\k x xs -> c (k,x) xs) n m) #-}+{-# RULES "Map.toAscListBack" [1] foldrFB (\k x xs -> (k, x) : xs) [] = toAscList #-}+{-# RULES "Map.toDescList" [~1] forall m . toDescList m = build (\c n -> foldlFB (\xs k x -> c (k,x) xs) n m) #-}+{-# RULES "Map.toDescListBack" [1] foldlFB (\xs k x -> (k, x) : xs) [] = toDescList #-}+#endif++{--------------------------------------------------------------------+ Building trees from ascending/descending lists can be done in linear time.++ Note that if [xs] is ascending that:+ fromAscList xs == fromList xs+ fromAscListWith f xs == fromListWith f xs+--------------------------------------------------------------------}+-- | /O(n)/. Build a map from an ascending list in linear time.+-- /The precondition (input list is ascending) is not checked./+--+-- > fromAscList [(3,"b"), (5,"a")] == fromList [(3, "b"), (5, "a")]+-- > fromAscList [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "b")]+-- > valid (fromAscList [(3,"b"), (5,"a"), (5,"b")]) == True+-- > valid (fromAscList [(5,"a"), (3,"b"), (5,"b")]) == False++{- LIQUIDTODO fromAscList :: (Eq k) => [(k,a)]<{v: (k, a) | fst(v) > fst(fld)}> -> OMap k a -}+{-@ fromAscList :: (Eq k) => [(k,a)]<{\h t -> fst h <= fst t}> -> OMap k a @-}+fromAscList :: Eq k => [(k,a)] -> Map k a+fromAscList xs+ = fromAscListWithKey (\_ x _ -> x) xs+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE fromAscList #-}+#endif++-- | /O(n)/. Build a map from an ascending list in linear time with a combining function for equal keys.+-- /The precondition (input list is ascending) is not checked./+--+-- > fromAscListWith (++) [(3,"b"), (5,"a"), (5,"b")] == fromList [(3, "b"), (5, "ba")]+-- > valid (fromAscListWith (++) [(3,"b"), (5,"a"), (5,"b")]) == True+-- > valid (fromAscListWith (++) [(5,"a"), (3,"b"), (5,"b")]) == False++{- LIQUIDTODO fromAscListWith :: (Eq k) => (a -> a -> a) -> [(k,a)]<{v: (k, a) | fst(v) > fst(fld)}> -> OMap k a -}+{-@ fromAscListWith :: Eq k => (a -> a -> a) -> [(k,a)]<{\h t -> fst h <= fst t}> -> OMap k a @-}+fromAscListWith :: Eq k => (a -> a -> a) -> [(k,a)] -> Map k a+fromAscListWith f xs+ = fromAscListWithKey (\_ x y -> f x y) xs+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE fromAscListWith #-}+#endif++-- | /O(n)/. Build a map from an ascending list in linear time with a+-- combining function for equal keys.+-- /The precondition (input list is ascending) is not checked./+--+-- > let f k a1 a2 = (show k) ++ ":" ++ a1 ++ a2+-- > fromAscListWithKey f [(3,"b"), (5,"a"), (5,"b"), (5,"b")] == fromList [(3, "b"), (5, "5:b5:ba")]+-- > valid (fromAscListWithKey f [(3,"b"), (5,"a"), (5,"b"), (5,"b")]) == True+-- > valid (fromAscListWithKey f [(5,"a"), (3,"b"), (5,"b"), (5,"b")]) == False++{- LIQUIDTODO fromAscListWithKey :: (Eq k) => (k -> a -> a -> a) -> [(k,a)]<{v: (k, a) | fst(v) > fst(fld)}> -> OMap k a -}+{-@ fromAscListWithKey :: (Eq k) => (k -> a -> a -> a) -> [(k,a)]<{\h t -> fst h <= fst t}> -> OMap k a @-}+fromAscListWithKey :: Eq k => (k -> a -> a -> a) -> [(k,a)] -> Map k a+fromAscListWithKey f xs+ = fromDistinctAscList (combineEq f xs)+ where+ -- [combineEq f xs] combines equal elements with function [f] in an ordered list [xs]+ combineEq _ xs'+ = case xs' of+ [] -> []+ [x] -> [x]+ (x:xx) -> combineEq' x xx++ combineEq' z [] = [z]+ combineEq' z@(kz,zz) (x@(kx,xx):xs')+ | kx==kz = let yy = f kx xx zz in combineEq' (kx,yy) xs'+ | otherwise = z:combineEq' x xs'+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE fromAscListWithKey #-}+#endif+++-- | /O(n)/. Build a map from an ascending list of distinct elements in linear time.+-- /The precondition is not checked./+--+-- > fromDistinctAscList [(3,"b"), (5,"a")] == fromList [(3, "b"), (5, "a")]+-- > valid (fromDistinctAscList [(3,"b"), (5,"a")]) == True+-- > valid (fromDistinctAscList [(3,"b"), (5,"a"), (5,"b")]) == False++{- LIQUIDTODO fromDistinctAscList :: [(k,a)]<{v: (k, a) | fst(v) > fst(fld)}> -> OMap k a -}+{-@ fromDistinctAscList :: [(k, a)]<{\h t -> fst h < fst t}> -> OMap k a @-}+fromDistinctAscList :: [(k,a)] -> Map k a+fromDistinctAscList xs+ = create const (length xs) xs+ where+ -- 1) use continuations so that we use heap space instead of stack space.+ -- 2) special case for n==5 to create bushier trees.+ create c 0 xs' = c Tip xs'+ create c 5 xs' = case xs' of+ ((k1,x1):(k2,x2):(k3,x3):(k4,x4):(k5,x5):xx)+ -> c (bin k4 x4 (bin k2 x2 (singleton k1 x1) (singleton k3 x3)) (singleton k5 x5)) xx+ _ -> error "fromDistinctAscList create"+ create c n xs' = seq nr $ create (createR nr c) nl xs'+ where nl = n `div` 2+ nr = n - nl - 1++ createR n c l ((k,x):ys) = create (createB l k x c) n ys+ createR _ _ _ [] = error "fromDistinctAscList createR []"+ createB l k x c r zs = c (bin k x l r) zs+++{--------------------------------------------------------------------+ Utility functions that return sub-ranges of the original+ tree. Some functions take a `Maybe value` as an argument to+ allow comparisons against infinite values. These are called `blow`+ (Nothing is -\infty) and `bhigh` (here Nothing is +\infty).+ We use MaybeS value, which is a Maybe strict in the Just case.++ [trim blow bhigh t] A tree that is either empty or where [x > blow]+ and [x < bhigh] for the value [x] of the root.+ [filterGt blow t] A tree where for all values [k]. [k > blow]+ [filterLt bhigh t] A tree where for all values [k]. [k < bhigh]++ [split k t] Returns two trees [l] and [r] where all keys+ in [l] are <[k] and all keys in [r] are >[k].+ [splitLookup k t] Just like [split] but also returns whether [k]+ was found in the tree.+--------------------------------------------------------------------}++data MaybeS a = NothingS | JustS a deriving (Generic, Show) -- LIQUID: !-annot-fix++instance Targetable a => Targetable (MaybeS a)+++{--------------------------------------------------------------------+ [trim blo bhi t] trims away all subtrees that surely contain no+ values between the range [blo] to [bhi]. The returned tree is either+ empty or the key of the root is between @blo@ and @bhi@.+--------------------------------------------------------------------}+-- LIQUID: EXPANDED CASE-EXPRS for lesser, greater, middle to avoid DEFAULT hassle+{-@ trim :: (Ord k) => lo:MaybeS k+ -> hi:MaybeS k+ -> OMap k a+ -> {v: OMap k a | (RootBetween lo hi v) }+ @-}+++trim :: Ord k => MaybeS k -> MaybeS k -> Map k a -> Map k a+++trim NothingS NothingS t = t+trim (JustS lk) NothingS t = greater lk t++ where greater lo t@(Bin _ k _ _ r) | k <= lo = greater lo r+ | otherwise = t+ greater _ t'@Tip = t'++trim NothingS (JustS hk) t = lesser hk t++ where lesser hi t'@(Bin _ k _ l _) | k >= hi = lesser hi l+ | otherwise = t'+ lesser _ t'@Tip = t'+trim (JustS lk) (JustS hk) t = middle lk hk t+ where middle lo hi t'@(Bin _ k _ l r) | k <= lo = middle lo hi r+ | k >= hi = middle lo hi l+ | otherwise = t'+ middle _ _ t'@Tip = t'+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE trim #-}+#endif++-- LIQUID QUALIFIER DEBUG SILLINESS+{- zoo1 :: (Ord k) => lo:k -> OMap k a -> {v: OMap k a | ((isBin(v)) => (lo < key(v)))} @-}+--zoo1 :: Ord k => k -> Map k a -> Map k a+--zoo1 = error "TODO"++{- zoo2 :: (Ord k) => lo:k -> OMap k a -> {v: OMap k a | ((isBin(v)) => (lo > key(v)))} @-}+--zoo2 :: Ord k => k -> Map k a -> Map k a+--zoo2 = error "TODO"+++-- Helper function for 'mergeWithKey'. The @'trimLookupLo' lk hk t@ performs both+-- @'trim' (JustS lk) hk t@ and @'lookup' lk t@.++-- See Note: Type of local 'go' function+-- LIQUID trimLookupLo :: Ord k => k -> MaybeS k -> Map k a -> (Maybe a, Map k a)+-- LIQUID trimLookupLo lk NothingS t = greater lk t+-- LIQUID where greater :: Ord k => k -> Map k a -> (Maybe a, Map k a)+-- LIQUID greater lo t'@(Bin _ kx x l r) = case compare lo kx of LT -> (lookup lo l, {-`strictPair`-} t')+-- LIQUID EQ -> (Just x, r)+-- LIQUID GT -> greater lo r+-- LIQUID greater _ Tip = (Nothing, Tip)+-- LIQUID trimLookupLo lk (JustS hk) t = middle lk hk t+-- LIQUID where middle :: Ord k => k -> k -> Map k a -> (Maybe a, Map k a)+-- LIQUID middle lo hi t'@(Bin _ kx x l r) = case compare lo kx of LT | kx < hi -> (lookup lo l, {- `strictPair` -} t')+-- LIQUID | otherwise -> middle lo hi l+-- LIQUID EQ -> (Just x, {-`strictPair`-} lesser hi r)+-- LIQUID GT -> middle lo hi r+-- LIQUID middle _ _ Tip = (Nothing, Tip)+-- LIQUID+-- LIQUID lesser :: Ord k => k -> Map k a -> Map k a+-- LIQUID lesser hi (Bin _ k _ l _) | k >= hi = lesser hi l+-- LIQUID lesser _ t' = t'++{-@ trimLookupLo :: (Ord k)+ => lo:k+ -> bhi:{v: MaybeS k | (isJustS(v) => (lo < fromJustS(v)))}+ -> OMap k a+ -> (Maybe a, {v: OMap k a | ((isBin(v) => (lo < key(v))) && ((isBin(v) && isJustS(bhi)) => (fromJustS(bhi) > key(v)))) }) @-}++trimLookupLo :: Ord k => k -> MaybeS k -> Map k a -> (Maybe a, Map k a)+trimLookupLo lk NothingS t = greater lk t+ where greater :: Ord k => k -> Map k a -> (Maybe a, Map k a)+ greater lo t'@(Bin _ kx x l r) = case compare lo kx of LT -> (lookup lo l, {-`strictPair`-} t')+ EQ -> (Just x, (case r of {r'@(Bin _ _ _ _ _) -> r' ; r'@Tip -> r'}))+ GT -> greater lo r+ greater _ Tip = (Nothing, Tip)+trimLookupLo lk (JustS hk) t = middle lk hk t+ where middle :: Ord k => k -> k -> Map k a -> (Maybe a, Map k a)+ middle lo hi t'@(Bin _ kx x l r) = case compare lo kx of LT | kx < hi -> (lookup lo l, {- `strictPair` -} t')+ | otherwise -> middle lo hi l+ EQ -> (Just x, {-`strictPair`-} lesser lo hi (case r of {r'@(Bin _ _ _ _ _) -> r' ; r'@Tip -> r'}))+ GT -> middle lo hi r+ middle _ _ Tip = (Nothing, Tip)++ lesser :: Ord k => k -> k -> Map k a -> Map k a+ lesser lo hi t'@(Bin _ k _ l _) | k >= hi = lesser lo hi l+ | otherwise = t'+ lesser _ _ t'@Tip = t'+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE trimLookupLo #-}+#endif+++{--------------------------------------------------------------------+ [filterGt b t] filter all keys >[b] from tree [t]+ [filterLt b t] filter all keys <[b] from tree [t]+--------------------------------------------------------------------}++{-@ filterGt :: (Ord k) => x:MaybeS k -> OMap k v -> OMap {v:k | ((isJustS(x)) => (v > fromJustS(x))) } v @-}+filterGt :: Ord k => MaybeS k -> Map k v -> Map k v+filterGt NothingS t = t+filterGt (JustS b) t = filterGt' b t++-- LIQUID TXREC-TOPLEVEL-ISSUE+filterGt' _ Tip = Tip+filterGt' b' (Bin _ kx x l r) =+ case compare b' kx of LT -> join' kx x (filterGt' b' l) r+ EQ -> r+ GT -> filterGt' b' r+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE filterGt #-}+#endif++{-@ filterLt :: (Ord k) => x:MaybeS k -> OMap k v -> OMap {v:k | ((isJustS(x)) => (v < fromJustS(x))) } v @-}+filterLt :: Ord k => MaybeS k -> Map k v -> Map k v+filterLt NothingS t = t+filterLt (JustS b) t = filterLt' b t++-- LIQUID TXREC-TOPLEVEL-ISSUE+filterLt' _ Tip = Tip+filterLt' b' (Bin _ kx x l r) =+ case compare kx b' of LT -> join' kx x l (filterLt' b' r)+ EQ -> l+ GT -> filterLt' b' l+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE filterLt #-}+#endif++{--------------------------------------------------------------------+ Split+--------------------------------------------------------------------}+-- | /O(log n)/. The expression (@'split' k map@) is a pair @(map1,map2)@ where+-- the keys in @map1@ are smaller than @k@ and the keys in @map2@ larger than @k@.+-- Any key equal to @k@ is found in neither @map1@ nor @map2@.+--+-- > split 2 (fromList [(5,"a"), (3,"b")]) == (empty, fromList [(3,"b"), (5,"a")])+-- > split 3 (fromList [(5,"a"), (3,"b")]) == (empty, singleton 5 "a")+-- > split 4 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", singleton 5 "a")+-- > split 5 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", empty)+-- > split 6 (fromList [(5,"a"), (3,"b")]) == (fromList [(3,"b"), (5,"a")], empty)++{-@ split :: (Ord k) => x:k -> OMap k a -> (OMap {v: k | v < x} a, OMap {v:k | v > x} a) @-}+split :: Ord k => k -> Map k a -> (Map k a, Map k a)+split k t = k `seq`+ case t of+ Tip -> (Tip, Tip)+ Bin _ kx x l r -> case compare k kx of+ LT -> let (lt,gt) = split k l in (lt,join' kx x gt r)+ GT -> let (lt,gt) = split k r in (join' kx x l lt,gt)+ EQ -> (l,r)+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE split #-}+#endif++-- | /O(log n)/. The expression (@'splitLookup' k map@) splits a map just+-- like 'split' but also returns @'lookup' k map@.+--+-- > splitLookup 2 (fromList [(5,"a"), (3,"b")]) == (empty, Nothing, fromList [(3,"b"), (5,"a")])+-- > splitLookup 3 (fromList [(5,"a"), (3,"b")]) == (empty, Just "b", singleton 5 "a")+-- > splitLookup 4 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", Nothing, singleton 5 "a")+-- > splitLookup 5 (fromList [(5,"a"), (3,"b")]) == (singleton 3 "b", Just "a", empty)+-- > splitLookup 6 (fromList [(5,"a"), (3,"b")]) == (fromList [(3,"b"), (5,"a")], Nothing, empty)++{-@ splitLookup :: (Ord k) => x:k -> OMap k a -> (OMap {v: k | v < x} a, Maybe a, OMap {v:k | v > x} a) @-}+splitLookup :: Ord k => k -> Map k a -> (Map k a,Maybe a,Map k a)+splitLookup k t = k `seq`+ case t of+ Tip -> (Tip,Nothing,Tip)+ Bin _ kx x l r -> case compare k kx of+ LT -> let (lt,z,gt) = splitLookup k l in (lt,z,join' kx x gt r)+ GT -> let (lt,z,gt) = splitLookup k r in (join' kx x l lt,z,gt)+ EQ -> (l,Just x,r)+#if __GLASGOW_HASKELL__ >= 700+{-# INLINABLE splitLookup #-}+#endif++{--------------------------------------------------------------------+ Utility functions that maintain the balance properties of the tree.+ All constructors assume that all values in [l] < [k] and all values+ in [r] > [k], and that [l] and [r] are valid trees.++ In order of sophistication:+ [Bin sz k x l r] The type constructor.+ [bin k x l r] Maintains the correct size, assumes that both [l]+ and [r] are balanced with respect to each other.+ [balance k x l r] Restores the balance and size.+ Assumes that the original tree was balanced and+ that [l] or [r] has changed by at most one element.+ [join' k x l r] Restores balance and size.++ Furthermore, we can construct a new tree from two trees. Both operations+ assume that all values in [l] < all values in [r] and that [l] and [r]+ are valid:+ [glue l r] Glues [l] and [r] together. Assumes that [l] and+ [r] are already balanced with respect to each other.+ [merge l r] Merges two trees and restores balance.++ Note: in contrast to Adam's paper, we use (<=) comparisons instead+ of (<) comparisons in [join'], [merge] and [balance].+ Quickcheck (on [difference]) showed that this was necessary in order+ to maintain the invariants. It is quite unsatisfactory that I haven't+ been able to find out why this is actually the case! Fortunately, it+ doesn't hurt to be a bit more conservative.+--------------------------------------------------------------------}+{--------------------------------------------------------------------+ Join'+--------------------------------------------------------------------}++{-@ join' :: k:k -> a -> OMap {v:k | v < k} a -> OMap {v:k| v > k} a -> OMap k a @-}+join' :: k -> a -> Map k a -> Map k a -> Map k a+join' kx x Tip r = insertMin kx x r+join' kx x l Tip = insertMax kx x l+join' kx x l@(Bin sizeL ky y ly ry) r@(Bin sizeR kz z lz rz)+ | delta*sizeL < sizeR = balanceL kz z (join' kx x l lz) rz+ | delta*sizeR < sizeL = balanceR ky y ly (join' kx x ry r)+ | otherwise = bin kx x l r++-- insertMin and insertMax don't perform potentially expensive comparisons.+insertMax, insertMin :: k -> a -> Map k a -> Map k a+insertMax kx x t+ = case t of+ Tip -> singleton kx x+ Bin _ ky y l r+ -> balanceR ky y l (insertMax kx x r)++insertMin kx x t+ = case t of+ Tip -> singleton kx x+ Bin _ ky y l r+ -> balanceL ky y (insertMin kx x l) r++{--------------------------------------------------------------------+ [merge l r]: merges two trees.+--------------------------------------------------------------------}+{-@ merge :: kcut:k -> OMap {v:k | v < kcut} a -> OMap {v:k| v > kcut} a -> OMap k a @-}+merge :: k -> Map k a -> Map k a -> Map k a+merge _ Tip r = r+merge _ l Tip = l+merge kcut l@(Bin sizeL kx x lx rx) r@(Bin sizeR ky y ly ry)+ | delta*sizeL < sizeR = balanceL ky y (merge kcut l ly) ry+ | delta*sizeR < sizeL = balanceR kx x lx (merge kcut rx r)+ | otherwise = glue kcut l r++{--------------------------------------------------------------------+ [glue l r]: glues two trees together.+ Assumes that [l] and [r] are already balanced with respect to each other.+--------------------------------------------------------------------}+{-@ glue :: kcut:k -> OMap {v:k | v < kcut} a -> OMap {v:k| v > kcut} a -> OMap k a @-}+glue :: k -> Map k a -> Map k a -> Map k a+glue _ Tip r = r+glue _ l Tip = l+glue kcut l r+ | size l > size r = let (km, m, l') = deleteFindMax l in balanceR km m l' r+ | otherwise = let (km, m, r') = deleteFindMin r in balanceL km m l r'++-- | /O(log n)/. Delete and find the minimal element.+--+-- > deleteFindMin (fromList [(5,"a"), (3,"b"), (10,"c")]) == ((3,"b"), fromList[(5,"a"), (10,"c")])+-- > deleteFindMin Error: can not return the minimal element of an empty map++{-@ deleteFindMin :: OMap k a -> (k, a, OMap k a)<{\k a -> true}, \a k -> {v0:Map ({v:k | v > k}) a | true}> @-}+deleteFindMin :: Map k a -> (k, a, Map k a)+deleteFindMin t+ = case t of+ Bin _ k x Tip r -> (k, x, r)+ Bin _ k x l r -> let (km, m, l') = deleteFindMin l in (km, m, balanceR k x l' r)+ Tip -> error "Map.deleteFindMin: can not return the minimal element of an empty map"++-- | /O(log n)/. Delete and find the maximal element.+--+-- > deleteFindMax (fromList [(5,"a"), (3,"b"), (10,"c")]) == ((10,"c"), fromList [(3,"b"), (5,"a")])+-- > deleteFindMax empty Error: can not return the maximal element of an empty map++{-@ deleteFindMax :: OMap k a -> (k, a, OMap k a)<{\k a -> true}, \a k -> {v0:Map ({v:k | v < k}) a | true}> @-}+deleteFindMax :: Map k a -> (k, a, Map k a)+deleteFindMax t+ = case t of+ Bin _ k x l Tip -> (k, x, l)+ Bin _ k x l r -> let (km, m, r') = deleteFindMax r in (km, m, balanceL k x l r')+ Tip -> error "Map.deleteFindMax: can not return the maximal element of an empty map"+++{--------------------------------------------------------------------+ [balance l x r] balances two trees with value x.+ The sizes of the trees should balance after decreasing the+ size of one of them. (a rotation).++ [delta] is the maximal relative difference between the sizes of+ two trees, it corresponds with the [w] in Adams' paper.+ [ratio] is the ratio between an outer and inner sibling of the+ heavier subtree in an unbalanced setting. It determines+ whether a double or single rotation should be performed+ to restore balance. It is corresponds with the inverse+ of $\alpha$ in Adam's article.++ Note that according to the Adam's paper:+ - [delta] should be larger than 4.646 with a [ratio] of 2.+ - [delta] should be larger than 3.745 with a [ratio] of 1.534.++ But the Adam's paper is erroneous:+ - It can be proved that for delta=2 and delta>=5 there does+ not exist any ratio that would work.+ - Delta=4.5 and ratio=2 does not work.++ That leaves two reasonable variants, delta=3 and delta=4,+ both with ratio=2.++ - A lower [delta] leads to a more 'perfectly' balanced tree.+ - A higher [delta] performs less rebalancing.++ In the benchmarks, delta=3 is faster on insert operations,+ and delta=4 has slightly better deletes. As the insert speedup+ is larger, we currently use delta=3.++--------------------------------------------------------------------}+delta,ratio :: Int+delta = 3+ratio = 2++{-@+measure isBalanced :: Map k a -> Prop+isBalanced (Tip) = true+isBalanced (Bin s k v l r) = ((((mlen l) + (mlen r) <= 1)+ || (((mlen l) <= 3 * (mlen r)) && ((mlen r) <= 3 * (mlen l))))+ && (isBalanced l) && (isBalanced r))+@-}++-- The balance function is equivalent to the following:+--+-- balance :: k -> a -> Map k a -> Map k a -> Map k a+-- balance k x l r+-- | sizeL + sizeR <= 1 = Bin sizeX k x l r+-- | sizeR > delta*sizeL = rotateL k x l r+-- | sizeL > delta*sizeR = rotateR k x l r+-- | otherwise = Bin sizeX k x l r+-- where+-- sizeL = size l+-- sizeR = size r+-- sizeX = sizeL + sizeR + 1+--+-- rotateL :: a -> b -> Map a b -> Map a b -> Map a b+-- rotateL k x l r@(Bin _ _ _ ly ry) | size ly < ratio*size ry = singleL k x l r+-- | otherwise = doubleL k x l r+--+-- rotateR :: a -> b -> Map a b -> Map a b -> Map a b+-- rotateR k x l@(Bin _ _ _ ly ry) r | size ry < ratio*size ly = singleR k x l r+-- | otherwise = doubleR k x l r+--+-- singleL, singleR :: a -> b -> Map a b -> Map a b -> Map a b+-- singleL k1 x1 t1 (Bin _ k2 x2 t2 t3) = bin k2 x2 (bin k1 x1 t1 t2) t3+-- singleR k1 x1 (Bin _ k2 x2 t1 t2) t3 = bin k2 x2 t1 (bin k1 x1 t2 t3)+--+-- doubleL, doubleR :: a -> b -> Map a b -> Map a b -> Map a b+-- doubleL k1 x1 t1 (Bin _ k2 x2 (Bin _ k3 x3 t2 t3) t4) = bin k3 x3 (bin k1 x1 t1 t2) (bin k2 x2 t3 t4)+-- doubleR k1 x1 (Bin _ k2 x2 t1 (Bin _ k3 x3 t2 t3)) t4 = bin k3 x3 (bin k2 x2 t1 t2) (bin k1 x1 t3 t4)+--+-- It is only written in such a way that every node is pattern-matched only once.++{-@ balance :: k:k -> a+ -> {v:OMap {v:k|v<k} a | (isBalanced v)}+ -> {v:OMap {v:k|v>k} a | (isBalanced v)}+ -> {v:OMap k a | (isBalanced v)}+ @-}+balance :: k -> a -> Map k a -> Map k a -> Map k a+balance k x l r = case l of+ Tip -> case r of+ Tip -> Bin 1 k x Tip Tip+ (Bin _ _ _ Tip Tip) -> Bin 2 k x Tip r+ (Bin _ rk rx Tip rr@(Bin _ _ _ _ _)) -> Bin 3 rk rx (Bin 1 k x Tip Tip) rr+ (Bin _ rk rx (Bin _ rlk rlx _ _) Tip) -> Bin 3 rlk rlx (Bin 1 k x Tip Tip) (Bin 1 rk rx Tip Tip)+ (Bin rs rk rx rl@(Bin rls rlk rlx rll rlr) rr@(Bin rrs _ _ _ _))+ | rls < ratio*rrs -> Bin (1+rs) rk rx (Bin (1+rls) k x Tip rl) rr+ | otherwise -> Bin (1+rs) rlk rlx (Bin (1+size rll) k x Tip rll) (Bin (1+rrs+size rlr) rk rx rlr rr)++ (Bin ls lk lx ll lr) -> case r of+ Tip -> case (ll, lr) of+ (Tip, Tip) -> Bin 2 k x l Tip+ (Tip, (Bin _ lrk lrx _ _)) -> Bin 3 lrk lrx (Bin 1 lk lx Tip Tip) (Bin 1 k x Tip Tip)+ ((Bin _ _ _ _ _), Tip) -> Bin 3 lk lx ll (Bin 1 k x Tip Tip)+ ((Bin lls _ _ _ _), (Bin lrs lrk lrx lrl lrr))+ | lrs < ratio*lls -> Bin (1+ls) lk lx ll (Bin (1+lrs) k x lr Tip)+ | otherwise -> Bin (1+ls) lrk lrx (Bin (1+lls+size lrl) lk lx ll lrl) (Bin (1+size lrr) k x lrr Tip)+ (Bin rs rk rx rl rr)+ | rs > delta*ls -> case (rl, rr) of+ (Bin rls rlk rlx rll rlr, Bin rrs _ _ _ _)+ | rls < ratio*rrs -> Bin (1+ls+rs) rk rx (Bin (1+ls+rls) k x l rl) rr+ | otherwise -> Bin (1+ls+rs) rlk rlx (Bin (1+ls+size rll) k x l rll) (Bin (1+rrs+size rlr) rk rx rlr rr)+ (_, _) -> error "Failure in Data.Map.balance"+ | ls > delta*rs -> case (ll, lr) of+ (Bin lls _ _ _ _, Bin lrs lrk lrx lrl lrr)+ | lrs < ratio*lls -> Bin (1+ls+rs) lk lx ll (Bin (1+rs+lrs) k x lr r)+ | otherwise -> Bin (1+ls+rs) lrk lrx (Bin (1+lls+size lrl) lk lx ll lrl) (Bin (1+rs+size lrr) k x lrr r)+ (_, _) -> error "Failure in Data.Map.balance"+ | otherwise -> Bin (1+ls+rs) k x l r+{-# NOINLINE balance #-}++-- Functions balanceL and balanceR are specialised versions of balance.+-- balanceL only checks whether the left subtree is too big,+-- balanceR only checks whether the right subtree is too big.++-- balanceL is called when left subtree might have been inserted to or when+-- right subtree might have been deleted from.+{-@ balanceL :: kcut:k -> a -> OMap {v:k | v < kcut} a -> OMap {v:k| v > kcut} a -> OMap k a @-}+balanceL :: k -> a -> Map k a -> Map k a -> Map k a+balanceL k x l r = case r of+ Tip -> case l of+ Tip -> Bin 1 k x Tip Tip+ (Bin _ _ _ Tip Tip) -> Bin 2 k x l Tip+ (Bin _ lk lx Tip (Bin _ lrk lrx _ _)) -> Bin 3 lrk lrx (Bin 1 lk lx Tip Tip) (Bin 1 k x Tip Tip)+ (Bin _ lk lx ll@(Bin _ _ _ _ _) Tip) -> Bin 3 lk lx ll (Bin 1 k x Tip Tip)+ (Bin ls lk lx ll@(Bin lls _ _ _ _) lr@(Bin lrs lrk lrx lrl lrr))+ | lrs < ratio+lls -> Bin (1+ls) lk lx ll (Bin (1+lrs) k x lr Tip)+ | otherwise -> Bin (1+ls) lrk lrx (Bin (1+lls+size lrl) lk lx ll lrl) (Bin (1+size lrr) k x lrr Tip)++ (Bin rs _ _ _ _) -> case l of+ Tip -> Bin (1+rs) k x Tip r++ (Bin ls lk lx ll lr)+ | ls > delta*rs -> case (ll, lr) of+ (Bin lls _ _ _ _, Bin lrs lrk lrx lrl lrr)+ | lrs < ratio*lls -> Bin (1+ls+rs) lk lx ll (Bin (1+rs+lrs) k x lr r)+ | otherwise -> Bin (1+ls+rs) lrk lrx (Bin (1+lls+size lrl) lk lx ll lrl) (Bin (1+rs+size lrr) k x lrr r)+ (_, _) -> error "Failure in Data.Map.balanceL"+ | otherwise -> Bin (1+ls+rs) k x l r+{-# NOINLINE balanceL #-}++-- balanceR is called when right subtree might have been inserted to or when+-- left subtree might have been deleted from.+{-@ balanceR :: kcut:k -> a -> OMap {v:k | v < kcut} a -> OMap {v:k| v > kcut} a -> OMap k a @-}+balanceR :: k -> a -> Map k a -> Map k a -> Map k a+balanceR k x l r = case l of+ Tip -> case r of+ Tip -> Bin 1 k x Tip Tip+ (Bin _ _ _ Tip Tip) -> Bin 2 k x Tip r+ (Bin _ rk rx Tip rr@(Bin _ _ _ _ _)) -> Bin 3 rk rx (Bin 1 k x Tip Tip) rr+ (Bin _ rk rx (Bin _ rlk rlx _ _) Tip) -> Bin 3 rlk rlx (Bin 1 k x Tip Tip) (Bin 1 rk rx Tip Tip)+ (Bin rs rk rx rl@(Bin rls rlk rlx rll rlr) rr@(Bin rrs _ _ _ _))+ | rls < ratio*rrs -> Bin (1+rs) rk rx (Bin (1+rls) k x Tip rl) rr+ | otherwise -> Bin (1+rs) rlk rlx (Bin (1+size rll) k x Tip rll) (Bin (1+rrs+size rlr) rk rx rlr rr)++ (Bin ls _ _ _ _) -> case r of+ Tip -> Bin (1+ls) k x l Tip++ (Bin rs rk rx rl rr)+ | rs > delta*ls -> case (rl, rr) of+ (Bin rls rlk rlx rll rlr, Bin rrs _ _ _ _)+ | rls < ratio*rrs -> Bin (1+ls+rs) rk rx (Bin (1+ls+rls) k x l rl) rr+ | otherwise -> Bin (1+ls+rs) rlk rlx (Bin (1+ls+size rll) k x l rll) (Bin (1+rrs+size rlr) rk rx rlr rr)+ (_, _) -> error "Failure in Data.Map.balanceR"+ | otherwise -> Bin (1+ls+rs) k x l r+{-# NOINLINE balanceR #-}+++{--------------------------------------------------------------------+ The bin constructor maintains the size of the tree+--------------------------------------------------------------------}+{-@ bin :: k:k -> a -> OMap {v:k | v < k} a -> OMap {v:k| v > k} a -> OMap k a @-}+bin :: k -> a -> Map k a -> Map k a -> Map k a+bin k x l r+ = Bin (size l + size r + 1) k x l r+{-# INLINE bin #-}++{--------------------------------------------------------------------+ Eq converts the tree to a list. In a lazy setting, this+ actually seems one of the faster methods to compare two trees+ and it is certainly the simplest :-)+--------------------------------------------------------------------}+instance (Eq k,Eq a) => Eq (Map k a) where+ t1 == t2 = (size t1 == size t2) && (toAscList t1 == toAscList t2)++{--------------------------------------------------------------------+ Ord+--------------------------------------------------------------------}++instance (Ord k, Ord v) => Ord (Map k v) where+ compare m1 m2 = compare (toAscList m1) (toAscList m2)++{--------------------------------------------------------------------+ Functor+--------------------------------------------------------------------}++-- LIQUID instance Functor (Map k) where+-- LIQUID fmap f m = map f m+-- LIQUID+-- LIQUID instance Traversable (Map k) where+-- LIQUID traverse f = traverseWithKey (\_ -> f)+-- LIQUID+-- LIQUID instance Foldable.Foldable (Map k) where+-- LIQUID fold Tip = mempty+-- LIQUID fold (Bin _ _ v l r) = Foldable.fold l `mappend` v `mappend` Foldable.fold r+-- LIQUID foldr = foldr+-- LIQUID foldl = foldl+-- LIQUID foldMap _ Tip = mempty+-- LIQUID foldMap f (Bin _ _ v l r) = Foldable.foldMap f l `mappend` f v `mappend` Foldable.foldMap f r+-- LIQUID+-- LIQUID instance (NFData k, NFData a) => NFData (Map k a) where+-- LIQUID rnf Tip = ()+-- LIQUID rnf (Bin _ kx x l r) = rnf kx `seq` rnf x `seq` rnf l `seq` rnf r++{--------------------------------------------------------------------+ Read+--------------------------------------------------------------------}+instance (Ord k, Read k, Read e) => Read (Map k e) where+#ifdef __GLASGOW_HASKELL__+ readPrec = parens $ Text.Read.prec 10 $ do+ Ident "fromList" <- lexP+ xs <- readPrec+ return (fromList xs)++ readListPrec = readListPrecDefault+#else+ readsPrec p = readParen (p > 10) $ \ r -> do+ ("fromList",s) <- lex r+ (xs,t) <- reads s+ return (fromList xs,t)+#endif++{--------------------------------------------------------------------+ Show+--------------------------------------------------------------------}+-- instance (Show k, Show a) => Show (Map k a) where+-- showsPrec d m = showParen (d > 10) $+-- showString "fromList " . shows (toList m)++-- -- | /O(n)/. Show the tree that implements the map. The tree is shown+-- -- in a compressed, hanging format. See 'showTreeWith'.+-- showTree :: (Show k,Show a) => Map k a -> String+-- showTree m+-- = showTreeWith showElem True False m+-- where+-- showElem k x = show k ++ ":=" ++ show x+--+--+-- {- | /O(n)/. The expression (@'showTreeWith' showelem hang wide map@) shows+-- the tree that implements the map. Elements are shown using the @showElem@ function. If @hang@ is+-- 'True', a /hanging/ tree is shown otherwise a rotated tree is shown. If+-- @wide@ is 'True', an extra wide version is shown.+--+-- > Map> let t = fromDistinctAscList [(x,()) | x <- [1..5]]+-- > Map> putStrLn $ showTreeWith (\k x -> show (k,x)) True False t+-- > (4,())+-- > +--(2,())+-- > | +--(1,())+-- > | +--(3,())+-- > +--(5,())+-- >+-- > Map> putStrLn $ showTreeWith (\k x -> show (k,x)) True True t+-- > (4,())+-- > |+-- > +--(2,())+-- > | |+-- > | +--(1,())+-- > | |+-- > | +--(3,())+-- > |+-- > +--(5,())+-- >+-- > Map> putStrLn $ showTreeWith (\k x -> show (k,x)) False True t+-- > +--(5,())+-- > |+-- > (4,())+-- > |+-- > | +--(3,())+-- > | |+-- > +--(2,())+-- > |+-- > +--(1,())+--+-- -}+-- showTreeWith :: (k -> a -> String) -> Bool -> Bool -> Map k a -> String+-- showTreeWith showelem hang wide t+-- | hang = (showsTreeHang showelem wide [] t) ""+-- | otherwise = (showsTree showelem wide [] [] t) ""+--+-- {- Decrease showsTree 5 @-}+-- showsTree :: (k -> a -> String) -> Bool -> [String] -> [String] -> Map k a -> ShowS+-- showsTree showelem wide lbars rbars t+-- = case t of+-- Tip -> showsBars lbars . showString "|\n"+-- Bin _ kx x Tip Tip+-- -> showsBars lbars . showString (showelem kx x) . showString "\n"+-- Bin _ kx x l r+-- -> showsTree showelem wide (withBar rbars) (withEmpty rbars) r .+-- showWide wide rbars .+-- showsBars lbars . showString (showelem kx x) . showString "\n" .+-- showWide wide lbars .+-- showsTree showelem wide (withEmpty lbars) (withBar lbars) l+--+-- {- Decrease showsTreeHang 4 @-}+-- showsTreeHang :: (k -> a -> String) -> Bool -> [String] -> Map k a -> ShowS+-- showsTreeHang showelem wide bars t+-- = case t of+-- Tip -> showsBars bars . showString "|\n"+-- Bin _ kx x Tip Tip+-- -> showsBars bars . showString (showelem kx x) . showString "\n"+-- Bin _ kx x l r+-- -> showsBars bars . showString (showelem kx x) . showString "\n" .+-- showWide wide bars .+-- showsTreeHang showelem wide (withBar bars) l .+-- showWide wide bars .+-- showsTreeHang showelem wide (withEmpty bars) r+--+-- showWide :: Bool -> [String] -> String -> String+-- showWide wide bars+-- | wide = showString (concat (reverse bars)) . showString "|\n"+-- | otherwise = id+--+-- showsBars :: [String] -> ShowS+-- showsBars bars+-- = case bars of+-- [] -> id+-- _ -> showString (concat (reverse (tail bars))) . showString node+--+-- node :: String+-- node = "+--"+--+-- withBar, withEmpty :: [String] -> [String]+-- withBar bars = "| ":bars+-- withEmpty bars = " ":bars++{--------------------------------------------------------------------+ Typeable+--------------------------------------------------------------------}++-- LIQUID #include "Typeable.h"+-- LIQUID INSTANCE_TYPEABLE2(Map,mapTc,"Map")++{--------------------------------------------------------------------+ Assertions+--------------------------------------------------------------------}+-- | /O(n)/. Test if the internal map structure is valid.+--+-- > valid (fromAscList [(3,"b"), (5,"a")]) == True+-- > valid (fromAscList [(5,"a"), (3,"b")]) == False++--valid :: Ord k => Map k a -> Bool+--valid t+-- = balanced t && ordered t && validsize t+--+--ordered :: Ord a => Map a b -> Bool+--ordered t+-- = bounded (const True) (const True) t+-- where+-- bounded lo hi t'+-- = case t' of+-- Tip -> True+-- Bin _ kx _ l r -> (lo kx) && (hi kx) && bounded lo (<kx) l && bounded (>kx) hi r+--+---- | Exported only for "Debug.QuickCheck"+--balanced :: Map k a -> Bool+--balanced t+-- = case t of+-- Tip -> True+-- Bin _ _ _ l r -> (size l + size r <= 1 || (size l <= delta*size r && size r <= delta*size l)) &&+-- balanced l && balanced r+--+--validsize :: Map a b -> Bool+--validsize t+-- = (realsize t == Just (size t))+-- where+-- realsize t'+-- = case t' of+-- Tip -> Just 0+-- Bin sz _ _ l r -> case (realsize l,realsize r) of+-- (Just n,Just m) | n+m+1 == sz -> Just sz+-- _ -> Nothing++{--------------------------------------------------------------------+ Utilities+--------------------------------------------------------------------}+foldlStrict :: (a -> b -> a) -> a -> [b] -> a+foldlStrict f = go+ where+ go z [] = z+ go z (x:xs) = let z' = f z x in z' `seq` go z' xs+{-# INLINE foldlStrict #-}+++
+ test/MapTest.hs view
@@ -0,0 +1,158 @@+{-# LANGUAGE FlexibleInstances #-}+{-# LANGUAGE MultiParamTypeClasses #-}+{-# LANGUAGE TemplateHaskell #-}+module MapTest where++import Map++import Control.Applicative+import qualified Data.List as L++import Test.Target++--------------------------------------------------------------------+-- Target+--------------------------------------------------------------------++-- The values aren't interesting in terms of the properties we want to check,+-- so treat the Map as a Set to reduce the search space+type K = Char+type V = ()+type M = Map Char ()++prop_difference_lc :: M -> M -> M+prop_difference_lc = difference++prop_delete_lc :: K -> M -> M+prop_delete_lc = delete++-- liquidTests :: [(String, Test)]+liquidTests = [ ('insert, T (insert :: K -> V -> M -> M))+ , ('delete, T (delete :: K -> M -> M))+ , ('union, T (union :: M -> M -> M))+ , ('difference, T (difference :: M -> M -> M))+ , ('intersection, T (intersection :: M -> M -> M))+ ]+++-- liquidTests_bad :: [(String, Test)]+liquidTests_bad = [ ('insert, T (insert_bad :: K -> V -> M -> M))+ , ('delete, T (delete_bad :: K -> M -> M))+ , ('union, T (union_bad :: M -> M -> M))+ , ('difference, T (difference_bad :: M -> M -> M))+ , ('intersection, T (intersection_bad :: M -> M -> M))+ ]+++insert_bad = go+ where+ go :: Ord k => k -> a -> Map k a -> Map k a+ go kx x Tip = singleton kx x+ go kx x (Bin sz ky y l r) =+ case compare kx ky of+ -- Bin ky y (go kx x l) r+ --LIQUID: swapped balanceL and balanceR to inject bug+ LT -> balanceR ky y (go kx x l) r+ GT -> balanceL ky y l (go kx x r)+ EQ -> Bin sz kx x l r+++delete_bad = go+ where+ go :: Ord k => k -> Map k a -> Map k a+ go _ Tip = Tip+ go k (Bin _ kx x l r) =+ case compare k kx of+ --LIQUID: swapped balanceL and balanceR to inject bug+ LT -> balanceL kx x (go k l) r+ GT -> balanceR kx x l (go k r)+ EQ -> glue kx l r++--LIQUID: having trouble injecting bugs here..+glue_bad :: k -> Map k a -> Map k a -> Map k a+glue_bad _ Tip r = r+glue_bad _ l Tip = l+glue_bad kcut l r+ | size l > size r = let (km, m, l') = deleteFindMax l in balanceR km m l' r+ | otherwise = let (km, m, r') = deleteFindMin r in balanceL km m l r'+++union_bad :: Ord k => Map k a -> Map k a -> Map k a+union_bad Tip t2 = t2+union_bad t1 Tip = t1+union_bad t1 t2 = hedgeUnion_bad NothingS NothingS t1 t2++hedgeUnion_bad :: Ord a => MaybeS a -> MaybeS a -> Map a b -> Map a b -> Map a b+hedgeUnion_bad _ _ t1 Tip = t1+--LIQUID: injected bug in join'+hedgeUnion_bad blo bhi Tip (Bin _ kx x l r) = join'_bad kx x (filterGt blo l) (filterLt bhi r)+hedgeUnion_bad _ _ t1 (Bin _ kx x Tip Tip) = insertR kx x t1 -- According to benchmarks, this special case increases+ -- performance up to 30%. It does not help in difference or intersection.+hedgeUnion_bad blo bhi (Bin _ kx x l r) t2 = join'_bad kx x (hedgeUnion_bad blo bmi l (trim blo bmi t2))+ (hedgeUnion_bad bmi bhi r (trim bmi bhi t2))+ where bmi = JustS kx++join'_bad kx x Tip r = insertMin kx x r+join'_bad kx x l Tip = insertMax kx x l+join'_bad kx x l@(Bin sizeL ky y ly ry) r@(Bin sizeR kz z lz rz)+ --LIQUID changed both < to > to inject bug+ | delta*sizeL > sizeR = balanceL kz z (join'_bad kx x l lz) rz+ | delta*sizeR > sizeL = balanceR ky y ly (join'_bad kx x ry r)+ | otherwise = bin kx x l r+++difference_bad :: Ord k => Map k a -> Map k b -> Map k a+difference_bad Tip _ = Tip+difference_bad t1 Tip = t1+difference_bad t1 t2 = hedgeDiff_bad NothingS NothingS t1 t2++hedgeDiff_bad :: Ord a => MaybeS a -> MaybeS a -> Map a b -> Map a c -> Map a b+hedgeDiff_bad _ _ Tip _ = Tip+hedgeDiff_bad blo bhi (Bin _ kx x l r) Tip = join'_bad kx x (filterGt blo l) (filterLt bhi r)+hedgeDiff_bad blo bhi t (Bin _ kx _ l r) = merge_bad kx (hedgeDiff_bad blo bmi (trim_bad blo bmi t) l)+ (hedgeDiff_bad bmi bhi (trim_bad bmi bhi t) r)+ where bmi = JustS kx++--LIQUID: having trouble injecting bug here+merge_bad _ Tip r = r+merge_bad _ l Tip = l+merge_bad kcut l@(Bin sizeL kx x lx rx) r@(Bin sizeR ky y ly ry)+ | delta*sizeL > sizeR = balanceL ky y (merge_bad kcut l ly) ry+ | delta*sizeR > sizeL = balanceR kx x lx (merge_bad kcut rx r)+ | otherwise = glue kcut l r+++intersection_bad :: Ord k => Map k a -> Map k b -> Map k a+intersection_bad Tip _ = Tip+intersection_bad _ Tip = Tip+intersection_bad t1 t2 = hedgeInt_bad NothingS NothingS t1 t2++hedgeInt_bad :: Ord k => MaybeS k -> MaybeS k -> Map k a -> Map k b -> Map k a+hedgeInt_bad _ _ _ Tip = Tip+hedgeInt_bad _ _ Tip _ = Tip+hedgeInt_bad blo bhi (Bin _ kx x l r) t2 = let l' = hedgeInt_bad blo bmi l (trim_bad blo bmi t2)+ r' = hedgeInt_bad bmi bhi r (trim_bad bmi bhi t2)+ in if kx `member` t2 then join' kx x l' r' else merge kx l' r'+ where bmi = JustS kx++trim_bad :: Ord k => MaybeS k -> MaybeS k -> Map k a -> Map k a+trim_bad NothingS NothingS t = t+trim_bad (JustS lk) NothingS t = greater lk t++ --LIQUID: change <= to >=+ where greater lo t@(Bin _ k _ _ r) | k >= lo = greater lo r+ | otherwise = t+ greater _ t'@Tip = t'++trim_bad NothingS (JustS hk) t = lesser hk t++ --LIQUID: change >= to <=+ where lesser hi t'@(Bin _ k _ l _) | k <= hi = lesser hi l+ | otherwise = t'+ lesser _ t'@Tip = t'+trim_bad (JustS lk) (JustS hk) t = middle lk hk t+ where middle lo hi t'@(Bin _ k _ l r) | k <= lo = middle lo hi r+ | k >= hi = middle lo hi l+ | otherwise = t'+ middle _ _ t'@Tip = t'+
+ test/RBTree.hs view
@@ -0,0 +1,239 @@+{-# LANGUAGE DeriveGeneric #-}+{-# LANGUAGE LambdaCase #-}+{-# LANGUAGE ScopedTypeVariables #-}++{-# LANGUAGE FlexibleInstances #-}+{-# LANGUAGE MultiParamTypeClasses #-}++{-@ LIQUID "--no-termination" @-}++module RBTree+ ( RBTree(..), Color(..), add, ins, remove, del, append, deleteMin+ , lbalS, rbalS, lbal, rbal, makeRed, makeBlack+ , Char, Int+ )+ where++import Debug.Trace++import GHC.Generics+import Test.Target+import Test.Target.Targetable++import Language.Haskell.Liquid.Prelude+++data RBTree a = Leaf+ | Node Color a !(RBTree a) !(RBTree a)+ deriving (Show,Generic)++data Color = B -- ^ Black+ | R -- ^ Red+ deriving (Eq,Show,Generic)++instance Targetable Color+instance Targetable a => Targetable (RBTree a)++---------------------------------------------------------------------------+-- | Add an element -------------------------------------------------------+---------------------------------------------------------------------------++{-@ add :: (Ord a) => a -> RBT a -> RBT a @-}+add x s = makeBlack (ins x s)++{-@ ins :: (Ord a) => a -> t:RBT a -> {v: ARBTN a {(bh t)} | ((IsB t) => (isRB v))} @-}+ins kx Leaf = Node R kx Leaf Leaf+ins kx s@(Node B x l r) = case compare kx x of+ LT -> let t = lbal x (ins kx l) r in t+ GT -> let t = rbal x l (ins kx r) in t+ EQ -> s+ins kx s@(Node R x l r) = case compare kx x of+ LT -> Node R x (ins kx l) r+ GT -> Node R x l (ins kx r)+ EQ -> s++---------------------------------------------------------------------------+-- | Delete an element ----------------------------------------------------+---------------------------------------------------------------------------++{-@ remove :: (Ord a) => a -> RBT a -> RBT a @-}+remove x t = makeBlack (del x t)++{-@ predicate HDel T V = (bh V) = (if (isB T) then (bh T) - 1 else (bh T)) @-}++{-@ del :: (Ord a) => a -> t:RBT a -> {v:ARBT a | ((HDel t v) && ((isB t) || (isRB v)))} @-}+del x Leaf = Leaf+del x (Node _ y a b) = case compare x y of+ EQ -> append y a b+ LT -> case a of+ Leaf -> Node R y Leaf b+ Node B _ _ _ -> lbalS y (del x a) b+ _ -> let t = Node R y (del x a) b in t+ GT -> case b of+ Leaf -> Node R y a Leaf+ Node B _ _ _ -> rbalS y a (del x b)+ _ -> Node R y a (del x b)+++{-@ append :: y:a -> l:RBT {v:a | v < y} -> r:RBTN {v:a | y < v} {(bh l)} -> (ARBT2 a l r) @-}+append :: a -> RBTree a -> RBTree a -> RBTree a+append _ Leaf r = r+append _ l Leaf = l+append piv (Node R lx ll lr) (Node R rx rl rr) = case append piv lr rl of+ Node R x lr' rl' -> Node R x (Node R lx ll lr') (Node R rx rl' rr)+ lrl -> Node R lx ll (Node R rx lrl rr)+append piv (Node B lx ll lr) (Node B rx rl rr) = case append piv lr rl of+ Node R x lr' rl' -> Node R x (Node B lx ll lr') (Node B rx rl' rr)+ lrl -> lbalS lx ll (Node B rx lrl rr)+append piv l@(Node B _ _ _) (Node R rx rl rr) = Node R rx (append piv l rl) rr+append piv l@(Node R lx ll lr) r@(Node B _ _ _) = Node R lx ll (append piv lr r)++---------------------------------------------------------------------------+-- | Delete Minimum Element -----------------------------------------------+---------------------------------------------------------------------------++{-@ deleteMin :: RBT a -> RBT a @-}+deleteMin (Leaf) = Leaf+deleteMin (Node _ x l r) = makeBlack t+ where+ (_, t) = deleteMin' x l r+++{-@ deleteMin' :: k:a -> l:RBT {v:a | v < k} -> r:RBTN {v:a | k < v} {(bh l)} -> (a, ARBT2 a l r) @-}+deleteMin' k Leaf r = (k, r)+deleteMin' x (Node R lx ll lr) r = (k, Node R x l' r) where (k, l') = deleteMin' lx ll lr+deleteMin' x (Node B lx ll lr) r = (k, lbalS x l' r ) where (k, l') = deleteMin' lx ll lr++---------------------------------------------------------------------------+-- | Rotations ------------------------------------------------------------+---------------------------------------------------------------------------++{-@ lbalS :: k:a -> l:ARBT {v:a | v < k} -> r:RBTN {v:a | k < v} {1 + (bh l)} -> {v: ARBTN a {1 + (bh l)} | ((IsB r) => (isRB v))} @-}+lbalS k (Node R x a b) r = Node R k (Node B x a b) r+lbalS k l (Node B y a b) = let t = rbal k l (Node R y a b) in t+lbalS k l (Node R z (Node B y a b) c) = Node R y (Node B k l a) (rbal z b (makeRed c))+lbalS k l r = liquidError "nein" -- Node R l k r++{-@ rbalS :: k:a -> l:RBT {v:a | v < k} -> r:ARBTN {v:a | k < v} {(bh l) - 1} -> {v: ARBTN a {(bh l)} | ((IsB l) => (isRB v))} @-}+rbalS k l (Node R y b c) = Node R k l (Node B y b c)+rbalS k (Node B x a b) r = let t = lbal k (Node R x a b) r in t+rbalS k (Node R x a (Node B y b c)) r = Node R y (lbal x (makeRed a) b) (Node B k c r)+rbalS k l r = liquidError "nein" -- Node R l k r++{-@ lbal :: k:a -> l:ARBT {v:a | v < k} -> RBTN {v:a | k < v} {(bh l)} -> RBTN a {1 + (bh l)} @-}+lbal k (Node R y (Node R x a b) c) r = Node R y (Node B x a b) (Node B k c r)+lbal k (Node R x a (Node R y b c)) r = Node R y (Node B x a b) (Node B k c r)+lbal k l r = Node B k l r++{-@ rbal :: k:a -> l:RBT {v:a | v < k} -> ARBTN {v:a | k < v} {(bh l)} -> RBTN a {1 + (bh l)} @-}+rbal x a (Node R y b (Node R z c d)) = Node R y (Node B x a b) (Node B z c d)+rbal x a (Node R z (Node R y b c) d) = Node R y (Node B x a b) (Node B z c d)+rbal x l r = Node B x l r++---------------------------------------------------------------------------+---------------------------------------------------------------------------+---------------------------------------------------------------------------++{-@ type BlackRBT a = {v: RBT a | ((IsB v) && (bh v) > 0)} @-}++{-@ makeRed :: l:BlackRBT a -> ARBTN a {(bh l) - 1} @-}+makeRed (Node B x l r) = Node R x l r+makeRed _ = liquidError "nein"++{-@ makeBlack :: ARBT a -> RBT a @-}+makeBlack Leaf = Leaf+makeBlack (Node _ x l r) = Node B x l r++---------------------------------------------------------------------------+-- | Specifications -------------------------------------------------------+---------------------------------------------------------------------------++-- | Ordered Red-Black Trees++{-@ type ORBT a = RBTree <{\root v -> v < root }, {\root v -> v > root}> a @-}++-- | Red-Black Trees++{-@ type RBT a = {v: (ORBT a) | ((isRB v) && (isBH v)) } @-}+{-@ type RBTN a N = {v: (RBT a) | (bh v) = N } @-}++{-@ type ORBTL a X = RBT {v:a | v < X} @-}+{-@ type ORBTG a X = RBT {v:a | X < v} @-}++{-@ measure isRB :: RBTree a -> Prop+ isRB (Leaf) = true+ isRB (Node c x l r) = ((isRB l) && (isRB r) && (c == R => ((IsB l) && (IsB r))))+ @-}++-- | Almost Red-Black Trees++{-@ type ARBT a = {v: (ORBT a) | ((isARB v) && (isBH v))} @-}+{-@ type ARBTN a N = {v: (ARBT a) | (bh v) = N } @-}++{-@ measure isARB :: (RBTree a) -> Prop+ isARB (Leaf) = true+ isARB (Node c x l r) = ((isRB l) && (isRB r))+ @-}++-- | Conditionally Red-Black Tree++{-@ type ARBT2 a L R = {v:ARBTN a {(bh L)} | (((IsB L) && (IsB R)) => (isRB v))} @-}++-- | Color of a tree++{-@ measure col :: RBTree a -> Color+ col (Node c x l r) = c+ col (Leaf) = B+ @-}++{-@ measure isB :: RBTree a -> Prop+ isB (Leaf) = false+ isB (Node c x l r) = c == B+ @-}++{-@ predicate IsB T = col(T) == B @-}++-- | Black Height++{-@ measure isBH :: RBTree a -> Prop+ isBH (Leaf) = true+ isBH (Node c x l r) = ((isBH l) && (isBH r) && (bh l) = (bh r))+ @-}++{-@ measure bh :: RBTree a -> Int+ bh (Leaf) = 0+ bh (Node c x l r) = (bh l) + (if (c == R) then 0 else 1)+ @-}++-- | Binary Search Ordering++--FIXME: issue with name clash "c :: Color" from auto-gen'd measures+{-@ data RBTree a <l :: a -> a -> Prop, r :: a -> a -> Prop>+ = Leaf+ | Node (cc :: Color)+ (key :: a)+ (left :: RBTree <l, r> (a <l key>))+ (right:: RBTree <l, r> (a <r key>))+ @-}++{-@ data Color = B | R @-}++-------------------------------------------------------------------------------+-- Auxiliary Invariants -------------------------------------------------------+-------------------------------------------------------------------------------++{-@ predicate Invs V = ((Inv1 V) && (Inv2 V) && (Inv3 V)) @-}+{-@ predicate Inv1 V = (((isARB V) && (IsB V)) => (isRB V)) @-}+{-@ predicate Inv2 V = ((isRB v) => (isARB v)) @-}+{-@ predicate Inv3 V = 0 <= (bh v) @-}++{-@ invariant {v: Color | (v == R || v == B)} @-}++{-@ invariant {v: RBTree a | (Invs v)} @-}++{- inv :: RBTree a -> {v:RBTree a | (Invs v)} @-}+--inv Leaf = Leaf+--inv (Node c x l r) = Node c x (inv l) (inv r)+++
+ test/RBTreeTest.hs view
@@ -0,0 +1,47 @@+{-# LANGUAGE TemplateHaskell #-}+module RBTreeTest where++import RBTree++import Test.Target++type E = Char+type T = RBTree E++-- liquidTests :: [(String, Test)]+liquidTests = [ ('add, T (add :: E -> T -> T))+ , ('remove, T (remove :: E -> T -> T))+ ]++-- liquidTests_bad :: [(String, Test)]+liquidTests_bad = [ ('add, T (add_bad :: E -> T -> T))+ , ('remove, T (remove_bad :: E -> T -> T))+ ]++remove_bad x t = makeBlack (del_bad x t)++del_bad x Leaf = Leaf+del_bad x (Node _ y a b) = case compare x y of+ EQ -> append_bad y a b+ LT -> case a of+ Leaf -> Node R y Leaf b+ Node B _ _ _ -> lbalS y (del_bad x a) b+ _ -> let t = Node R y (del_bad x a) b in t+ GT -> case b of+ Leaf -> Node R y a Leaf+ Node B _ _ _ -> rbalS y a (del_bad x b)+ _ -> Node R y a (del_bad x b)++append_bad :: a -> RBTree a -> RBTree a -> RBTree a+append_bad _ Leaf r = r+append_bad _ l Leaf = l+append_bad piv (Node R lx ll lr) (Node R rx rl rr) = case append_bad piv lr rl of+ --Node R x lr' rl' -> Node R x (Node R lx ll lr') (Node R rx rl' rr)+ lrl -> Node R lx ll (Node R rx lrl rr)+append_bad piv (Node B lx ll lr) (Node B rx rl rr) = case append_bad piv lr rl of+ --Node R x lr' rl' -> Node R x (Node B lx ll lr') (Node B rx rl' rr)+ lrl -> lbalS lx ll (Node B rx lrl rr)+append_bad piv l@(Node B _ _ _) (Node R rx rl rr) = Node R rx (append_bad piv l rl) rr+append_bad piv l@(Node R lx ll lr) r@(Node B _ _ _) = Node R lx ll (append_bad piv lr r)++add_bad x s = ins x s