WeakSets (empty) → 0.1.0.0

raw patch · 9 files changed

+706/−0 lines, 9 filesdep +WeakSetsdep +base

Dependencies added: WeakSets, base

Files

• CHANGELOG.md +5/−0
• LICENSE +165/−0
• Readme.md +46/−0
• WeakSets.cabal +88/−0
• src/HomogeneousSet.hs +202/−0
• src/PureSet.hs +136/−0
• test/AllTests.hs +10/−0
• test/TestHomogeneousSet.hs +36/−0
• test/TestPureSet.hs +18/−0

@@ -0,0 +1,5 @@+# Revision history for Sets
+
+## 0.1.0.0 -- YYYY-mm-dd
+
+* First version. Released on an unsuspecting world.

@@ -0,0 +1,165 @@+                  GNU LESSER GENERAL PUBLIC LICENSE
+                       Version 3, 29 June 2007
+
+ Copyright (C) 2007 Free Software Foundation, Inc. <http://fsf.org/>
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+ of this license document, but changing it is not allowed.
+
+
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+the terms and conditions of version 3 of the GNU General Public
+License, supplemented by the additional permissions listed below.
+
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@@ -0,0 +1,46 @@+# WeakSets++This is a Haskell package which defines sets in a more general way than Data.Set.++Data.Set only allows to create sets of types which implements the Ord typeclass, this package introduces HomogeneousSet which does not require the typeclass Ord.++The PureSet type allows to create nested sets easily. It may be used to study set theory for example.++## General info++The homogenous sets implemented in this package require the type contained in the set to implement the typeclass Eq. It ensures there are no duplicate element in the set and that the order of elements does not matter when testing equality. It is slower than Data.Set because we do not require the Ord typeclass, if you only use types which are orderable, use Data.Set instead.++The pure set type implemented in this package is a tree like structure where the order of branches does not matter. It allows arbitrary nesting of sets which is useful to do set theoretic constructions.++## Installation++`cabal install WeakSets`++## Usage++Example usage of homogenous sets :++```haskell+ghci> import HomogeneousSet+ghci> data Foo = Foo Int Char deriving (Eq) -- an arbitrary type which is not required to implement >Ord typeclass+ghci> s1 = set [Foo 3 'a', Foo 2 'c', Foo 3 'a']+ghci> s2 = set [Foo 2 'c', Foo 3 'a']+ghci> s1 == s2+True+```++Example usage of pure sets :++```haskell+ghci> import PureSet+ghci> numberToSet 3+(pureSet [(pureSet []),(pureSet [(pureSet [])]),(pureSet [(pureSet []),(pureSet [(pureSet [])])])])+ghci> putStrLn.prettify $ numberToSet 3+{{}, {{}}, {{}, {{}}}}+```++## Contribution++Any input is appreciated ! Send an email for any remark or question.+The git repository : https://gitlab.utc.fr/gsabbagh/sets+

@@ -0,0 +1,88 @@+cabal-version:      2.4
+
+-- Initial package description 'WeakSets.cabal' generated by
+-- 'cabal init'. For further documentation, see:
+--   http://haskell.org/cabal/users-guide/
+-- 
+-- The name of the package.
+name:               WeakSets
+
+-- The package version.
+-- See the Haskell package versioning policy (PVP) for standards
+-- guiding when and how versions should be incremented.
+-- https://pvp.haskell.org
+-- PVP summary:      +-+------- breaking API changes
+--                   | | +----- non-breaking API additions
+--                   | | | +--- code changes with no API change
+version:            0.1.0.0
+
+-- A short (one-line) description of the package.
+synopsis:
+    Simple set types. Useful to create sets of arbitrary types and nested sets.
+
+-- A longer description of the package.
+description: This package answers two problems : how to make sets of types which does not implement the Ord typeclass and how to make arbitrarily nested sets as set theory allows. The first problem is resolved thanks to `HomogeneousSet` which is a list where duplicates elements are not allowed and the order of elements is forgotten. The second problem is resolved thanks to `PureSet`, it is a tree structure where the order of the branches does not matter.
+
+-- URL for the project homepage or repository.
+homepage: https://gitlab.utc.fr/gsabbagh/sets
+
+-- A URL where users can report bugs.
+-- bug-reports:
+
+-- The license under which the package is released.
+license:            LGPL-3.0-or-later
+
+-- The file containing the license text.
+license-file:       LICENSE
+
+-- The package author(s).
+author:             Guillaume Sabbagh
+
+-- An email address to which users can send suggestions, bug reports, and patches.
+maintainer:         guillaumesabbagh@protonmail.com
+
+-- A copyright notice.
+-- copyright:
+category:           Math, Data
+
+-- Extra files to be distributed with the package, such as examples or a README.
+extra-source-files: 
+    CHANGELOG.md
+    Readme.md
+
+library
+    -- Modules exported by the library.
+    exposed-modules:  HomogeneousSet, PureSet
+
+    -- Modules included in this library but not exported.
+    -- other-modules:
+
+    -- LANGUAGE extensions used by modules in this package.
+    -- other-extensions:
+
+    -- Other library packages from which modules are imported.
+    build-depends:    base ^>=4.15.0.0
+
+    -- Directories containing source files.
+    hs-source-dirs:   src
+
+    -- Base language which the package is written in.
+    default-language: Haskell2010
+
+test-suite WeakSets-test
+    -- Base language which the package is written in.
+    default-language: Haskell2010
+
+    -- The interface type and version of the test suite.
+    type:             exitcode-stdio-1.0
+
+    -- Directories containing source files.
+    hs-source-dirs:   test
+
+    -- The entrypoint to the test suite.
+    main-is:          AllTests.hs
+
+    -- Test dependencies.
+    build-depends:    base ^>=4.15.0.0, WeakSets
+    
+    other-modules:    TestHomogeneousSet, TestPureSet

@@ -0,0 +1,202 @@+{-| Module  : WeakSets
+Description : Homogeneous sets are sets which can contain only one type of values. They are more flexible than Data.Set because they do not require the objects contained to be orderable.
+Copyright   : Guillaume Sabbagh 2022
+License     : LGPL-3.0-or-later
+Maintainer  : guillaumesabbagh@protonmail.com
+Stability   : experimental
+Portability : portable
+
+Homogeneous sets are sets which can contain only one type of values.
+
+They are more flexible than Data.Set because they do not require the objects contained to be orderable.
+
+The datatype only assumes its components are equatable, it is therefore slower than the Data.Set datatype.
+
+We use this datatype because most of the datatypes we care about are not orderable.
+
+Inline functions related to homogeneous sets are written between pipes @|@.
+-}
+
+module HomogeneousSet
+(
+    -- * Set datatype and smart constructor
+    Set, -- abstract type, the smart constructor is `set`
+    set, -- the smart constructor for `Set`
+    -- * Set related functions
+    setToList,
+    isIncludedIn,
+    cardinal,
+    isIn,
+    (|<$>|),
+    (|&|),
+    (|||),
+    (|*|),
+    (|+|),
+    (|-|),
+    powerSet,
+    -- * Function datatype and smart constructor
+    AssociationList(..),
+    Function, -- abstract type, the smart constructor is `function`
+    function, -- the smart constructor for `Function`
+    -- * Function related functions
+    functionToSet,
+    domain,
+    image,
+    (|$|),
+    (|!|),
+    findWithDefault,
+    (|.|),
+    memorizeFunction,
+)
+where
+    import Data.List (intercalate, nub, nubBy, intersect, union, (\\), subsequences)
+    
+    -- | A homogeneous set is a list of values.
+    --
+    -- The only differences are that we don't want duplicate elements and we don't need the order of the list elements.
+    --
+    -- To force these constraints, the `Set` constructor is abstract and is not exported. The only way to construct a set is to use the smart constructor `set` which ensures the previous conditions.
+    data Set a = Set [a]
+    
+    -- | The smart constructor of sets. This is the only way of instantiating a `Set`.
+    --
+    -- If several elements are equal, only the first element is kept.
+    set :: (Eq a) => [a] -> Set a
+    set xs = Set (nub xs)
+    
+    instance (Show a) => Show (Set a) where
+        show (Set xs) = "(set "++show xs++")"
+    
+    -- | Returns a boolean indicating if a `Set` is included in another one.
+    isIncludedIn :: (Eq a) => Set a -> Set a -> Bool
+    (Set []) `isIncludedIn` _ = True
+    (Set (x:xs)) `isIncludedIn` (Set ys)
+        | x `elem` ys = (Set xs) `isIncludedIn` (Set ys)
+        | otherwise = False
+    
+    instance (Eq a) => Eq (Set a) where
+        x == y = x `isIncludedIn` y && y `isIncludedIn` x
+        
+
+    instance (Eq a) => Semigroup (Set a) where
+        (Set xs) <> (Set ys) = set $ xs <> ys
+        
+    instance (Eq a) => Monoid (Set a) where
+        mempty = Set []
+    
+    instance Foldable Set where
+        foldr f d (Set xs) = foldr f d xs
+
+    -- | Transforms a `Set` back into a list, the list returned does not have duplicate elements, the order of the original list holds.
+    setToList :: Set a -> [a]
+    setToList (Set xs) = xs
+
+    -- | Maps a function to every element of the set. We can't instantiate `Functor` because we would have to add a contraint @Eq@ to the type parameter of `Set`.
+    (|<$>|) :: (Eq b) => (a -> b) -> Set a -> Set b
+    (|<$>|) f (Set xs) = set $ f <$> xs
+    
+    -- | Unsafe map of a funtion to every element of a set. The function should be injective. This function is not exported.
+    unsafeMap :: (a -> b) -> Set a -> Set b
+    unsafeMap f (Set xs) = Set $ f <$> xs
+    
+    -- | Size of a set.
+    cardinal :: Set a -> Int
+    cardinal = length.setToList
+    
+    -- | Returns wether an element is in a set.
+    isIn :: (Eq a) => a -> Set a -> Bool
+    isIn x = (elem x).setToList
+    
+    -- | Returns the intersection of two sets.
+    (|&|) :: (Eq a) => Set a -> Set a -> Set a
+    (|&|) (Set xs) (Set ys) = Set $ xs `intersect` ys
+    
+    -- | Returns the union of two sets.
+    (|||) ::  (Eq a) => Set a -> Set a -> Set a
+    (|||) (Set xs) (Set ys) = Set $ xs `union` ys
+    
+    -- | Unsafe union where we do not check duplicates. This function is not exported.
+    (|!) :: Set a -> Set a -> Set a
+    (|!) (Set xs) (Set ys) = Set $ xs ++ ys
+    
+    -- | Returns the cartesian product of two sets.
+    (|*|) ::  Set a -> Set b -> Set (a,b)
+    (|*|) (Set xs) (Set ys) = Set $ [(x,y) | x <- xs, y <- ys]
+    
+    -- | Returns the disjoint union of two sets.
+    (|+|) :: Set a -> Set b -> Set (Either a b)
+    (|+|) (Set xs) (Set ys) = Set $ [Left x | x <- xs] ++ [Right y | y <- ys]
+    
+    -- | Returns the difference of two sets.
+    (|-|) :: (Eq a) => Set a -> Set a -> Set a
+    (|-|) (Set xs) (Set ys) = Set $ xs \\ ys
+    
+    -- | Returns the set of all subsets of a given set.
+    powerSet :: Set a -> Set (Set a)
+    powerSet (Set xs) = Set $ Set <$> subsequences xs
+    
+    -- | A function of homogeneous sets. It is a set of pairs (key,value) such that their should only be one pair with a given key.
+    --
+    -- It is an abstract type, the smart constructor is `function`.
+    data Function a b = Function (Set (a,b)) deriving (Eq)
+    
+    instance (Show a, Show b) => Show (Function a b) where
+        show (Function al) = "(function "++show al++")"
+    
+    -- | An association list is a list of pairs (key,value).
+    type AssociationList a b = [(a,b)]
+    
+    -- | The smart constructor of functions. This is the only way of instantiating a `Function`.
+    --
+    -- Takes an association list and returns a function which maps to each key the value associated.
+    --
+    -- If several pairs have the same keys, the first pair is kept.
+    function :: (Eq a) => AssociationList a b -> Function a b
+    function al = Function $ Set $ nubBy (\x y -> (fst x) == (fst y)) al
+    
+    -- | Transforms a function back into its underlying association list.
+    functionToSet :: Function a b -> Set (a,b)
+    functionToSet (Function al) = al
+    
+    -- | Returns the domain of a function.
+    domain :: Function a b -> Set a
+    domain = (unsafeMap fst).functionToSet
+    
+    -- | Returns the image of a function. The image of a function is the set of values which are reachable by applying the function.
+    image :: Function a b -> Set b
+    image = (unsafeMap snd).functionToSet
+        
+    -- | Apply a function to a given value. If the function is not defined on the given value returns `Nothing`, otherwise returns `Just` the image.
+    --
+    -- This function is like `lookup` in Data.Map for function (the order of the argument are reversed though).
+    (|$|) :: (Eq a) => Function a b -> a -> Maybe b
+    (|$|) (Function (Set [])) _ = Nothing
+    (|$|) (Function (Set ((k,v):xs))) x
+        | x == k = Just v
+        | otherwise = (Function (Set xs)) |$| x
+    
+    -- | Unsafe version of `(|$|)`.
+    --
+    -- This function is like `(!)` in Data.Map for function.
+    (|!|) :: (Eq a) => Function a b -> a -> b
+    (|!|) (Function (Set [])) _ = error "Function applied on a value not in the domain."
+    (|!|) (Function (Set ((k,v):xs))) x
+        | x == k = v
+        | otherwise = (Function (Set xs)) |!| x
+    
+    -- | Apply a function to a given value, if the value is in the domain returns the image, otherwise return a default value.
+    --
+    -- This function is like `findWithDefault` in Data.Map for function (the order of the argument are reversed though).
+    findWithDefault :: (Eq a) => Function a b -> b -> a -> b
+    findWithDefault (Function (Set [])) d _ = d
+    findWithDefault (Function (Set ((k,v):xs))) d x
+        | x == k = v
+        | otherwise = findWithDefault (Function (Set xs)) d x
+   
+    -- | Composes two functions. If the two functions are not composable, strips the functions until they can compose.
+    (|.|) :: (Eq a, Eq b) => Function b c -> Function a b -> Function a c
+    (|.|) f2 f1 = Function $ Set [(k,(f2 |!| (f1 |!| k))) | k <- (setToList.domain $ f1), f1 |!| k `isIn` (domain f2)]
+    
+    -- | Memorize a Haskell function on a given finite domain.
+    memorizeFunction :: (a -> b) -> Set a -> Function a b
+    memorizeFunction f (Set xs) = Function $ Set [(k, f k) | k <- xs]

@@ -0,0 +1,136 @@+{-| Module  : WeakSets
+Description : Pure sets are nested sets which only contain other sets all the way down. They allow to explore basic set theory.
+Copyright   : Guillaume Sabbagh 2021
+License     : GPL-3
+Maintainer  : guillaumesabbagh@protonmail.com
+Stability   : experimental
+Portability : portable
+
+Pure sets are nested sets which only contain other sets all the way down. They allow to explore basic set theory.
+
+Every mathematical object is a set, usual constructions such as Von Neumann numbers and Kuratowski pairs are implemented.
+
+It is a tree where the order of the branches does not matter.
+
+Functions with the same name as homogeneous set functions are suffixed with the letter 'P' for pure to avoid name collision.
+-}
+
+module PureSet 
+(
+    -- * `PureSet` datatype
+    PureSet(..),
+    pureSet,
+    -- * Mathematical constructions using sets
+    emptySet,
+    singleton,
+    pair,
+    cartesianProduct,
+    numberToSet,
+    (||||),
+    (&&&&),
+    isInP,
+    isIncludedInP,
+    card,
+    powerSetP,
+    -- * Formatting functions
+    prettify,
+    formatPureSet,
+)
+where
+    import HomogeneousSet
+    import Data.List    (intersect, nub, intercalate, subsequences)
+    import Data.Maybe   (fromJust, catMaybes)
+    
+    -- | A `PureSet` is a `Set` of other pure sets.
+    data PureSet = PureSet (Set PureSet) deriving (Eq)
+    
+    instance Show PureSet where
+        show (PureSet xs) = "(pureSet "++ show (setToList xs) ++")"
+    
+    -- | Constructs a `PureSet` from a list of pure sets.
+    pureSet :: [PureSet] -> PureSet
+    pureSet = (PureSet).set
+    
+    -- | Peels a `PureSet` into a `Set`.
+    pureSetToSet :: PureSet -> Set PureSet
+    pureSetToSet (PureSet xs) = xs
+    
+    -- | Constructs the empty set.
+    emptySet :: PureSet
+    emptySet = pureSet []
+  
+    -- | Constructs the singleton containing a given set.
+    singleton :: PureSet -> PureSet
+    singleton x = pureSet $ [x]
+    
+    -- | Constructs an ordered pair from two sets according to Kuratowski's definition of a tuple.
+    pair :: PureSet -> PureSet -> PureSet
+    pair x y = PureSet $ set [singleton x, pureSet $ [x,y]]
+    
+    -- | Constructs the cartesian product of two sets.
+    cartesianProduct :: PureSet -> PureSet -> PureSet
+    cartesianProduct (PureSet xs) (PureSet ys) = pureSet $ [pair x y | x <- setToList xs, y <- setToList ys]
+    
+    -- | Union of two pure sets.
+    (||||) :: PureSet -> PureSet -> PureSet
+    (||||) (PureSet xs) (PureSet ys) = PureSet $ xs ||| ys
+    
+    -- | Intersection of two pure sets.
+    (&&&&) :: PureSet -> PureSet -> PureSet
+    (&&&&) (PureSet xs) (PureSet ys) = PureSet $ xs |&| ys
+    
+    -- | Difference of two pure sets.
+    (\\\\) :: PureSet -> PureSet -> PureSet
+    (\\\\) (PureSet xs) (PureSet ys) = PureSet $ xs |-| ys
+   
+    -- | Transforms a number into its Von Neumann construction
+    numberToSet :: (Num a, Eq a) => a -> PureSet
+    numberToSet 0 = emptySet
+    numberToSet n = (numberToSet (n-1)) |||| (singleton (numberToSet (n-1)))
+    
+    -- | Returns wether a pure set is in another one.
+    isInP :: PureSet -> PureSet -> Bool
+    isInP x (PureSet xs) = x `isIn` xs
+    
+    -- | Returns wether a pure set is included in another one.
+    isIncludedInP :: PureSet -> PureSet -> Bool
+    isIncludedInP (PureSet xs) (PureSet ys) = xs `isIncludedIn` ys
+    
+    -- | Returns the size of a pure set.
+    card :: PureSet -> Int
+    card (PureSet xs) = cardinal xs
+    
+    -- | Returns the set of subsets of a given set.
+    powerSetP :: PureSet -> PureSet
+    powerSetP (PureSet xs) = PureSet $ PureSet |<$>| powerSet xs
+    
+    -- | Prettiffies a pure set according to usual mathematical notation.
+    prettify :: PureSet -> String
+    prettify (PureSet xs)
+        | cardinal xs == 0 = "{}"
+        | otherwise = "{" ++ (intercalate ", " $ prettify <$> setToList xs) ++ "}"
+        
+    -- | Format pure sets such that if numbers are recognized, they are transformed into integer and if pairs are recognized, they are transformed into pairs.
+    formatPureSet :: PureSet -> String
+    formatPureSet x
+        | (not.null) $ toNumber x = show.fromJust $ toNumber x
+        | (not.null) $ toPair x = fromJust.toPair $ x
+        | otherwise = "{"++intercalate "," (formatPureSet <$> (setToList.pureSetToSet $ x))++"}"
+            where
+                toNumber s@(PureSet xs)
+                    | s == emptySet = Just 0
+                    | otherwise =   let
+                                        numbers = setToList $ toNumber |<$>| xs
+                                        anyMissing = null $ foldr1 (>>) numbers
+                                        maxNb = maximum $ catMaybes numbers
+                                    in 
+                                        if (not anyMissing) && (set (Just <$> [0..maxNb])) == (set numbers) then Just (maxNb + 1) else Nothing
+                toPair (PureSet xs)
+                    | cardinal xs == 2 = 
+                        case () of
+                         () | ((card $ (setToList xs) !! 0) == 1 && (card $ (setToList xs) !! 1) == 2) && ((setToList xs) !! 0) `isInP` ((setToList xs) !! 1) -> Just $ "(" ++ (formatPureSet.head.setToList.pureSetToSet $ ((setToList xs) !! 0)) ++ "," ++ (formatPureSet.head.setToList.pureSetToSet $ (((setToList xs) !! 1) \\\\ ((setToList xs) !! 0))) ++ ")"
+                            | ((card $ (setToList xs) !! 1) == 1 && (card $ (setToList xs) !! 0) == 2) && ((setToList xs) !! 1) `isInP` ((setToList xs) !! 0) -> Just $ "(" ++ (formatPureSet.head.setToList.pureSetToSet $ ((setToList xs) !! 1)) ++ "," ++ (formatPureSet.head.setToList.pureSetToSet $ (((setToList xs) !! 0) \\\\ ((setToList xs) !! 1))) ++ ")"
+                            | otherwise -> Nothing
+                    | otherwise = Nothing
+    
+    

@@ -0,0 +1,10 @@+module Main (main) where
+    import qualified TestHomogeneousSet as HSET
+    import qualified TestPureSet as PSET
+
+    main :: IO ()
+    main = do
+        putStrLn "Beginning of the test."
+        HSET.main
+        PSET.main
+        putStrLn "End of the tests."

@@ -0,0 +1,36 @@+module TestHomogeneousSet where
+    import HomogeneousSet
+    
+    -- | Tests all functions related to homogeneous sets.
+    main :: IO ()
+    main = do
+            putStrLn "Beginning of TestHomogeneousSet"
+            let s1 = set [1,2,3] 
+            let s2 = set [3,4,5] 
+            putStrLn $ show $ s1
+            putStrLn $ show $ setToList s1
+            putStrLn $ show $ s2 `isIncludedIn` s1
+            putStrLn $ show $ (set [1,2]) `isIncludedIn` s1
+            putStrLn $ show $ cardinal s1
+            putStrLn $ show $ 3 `isIn` s1
+            putStrLn $ show $ (+1) |<$>| s1
+            putStrLn $ show $ s1 |&| s2
+            putStrLn $ show $ s1 ||| s2
+            putStrLn $ show $ s1 |*| s2
+            putStrLn $ show $ s1 |+| s2
+            putStrLn $ show $ s1 |-| s2
+            putStrLn $ show $ powerSet s1
+            let f = function $ zip (setToList s1) (setToList s2)
+            let g = function $ zip (setToList s2) (setToList s1)
+            putStrLn $ show $ f
+            putStrLn $ show $ g
+            putStrLn $ show $ functionToSet $ f
+            putStrLn $ show $ domain $ f
+            putStrLn $ show $ image $ f
+            putStrLn $ show $ f |$| 1
+            putStrLn $ show $ f |!| 1
+            putStrLn $ show $ f |.| g
+            putStrLn $ show $ g |.| f
+            putStrLn $ show $ memorizeFunction (*3) (set [1..10])
+            putStrLn "End of TestHomogeneousSet"
+    

@@ -0,0 +1,18 @@+module TestPureSet where
+    import HomogeneousSet
+    import PureSet
+    import Data.List (intercalate)
+    
+    -- | Tests all functions related to pure sets.
+    main :: IO ()
+    main = do
+            putStrLn "Beginning of TestPureSet"
+            putStrLn.show $ emptySet
+            putStrLn.show $ singleton $ emptySet
+            putStrLn.show $ pair emptySet emptySet
+            putStrLn.show $ numberToSet <$> [0,3]
+            putStrLn $ intercalate "\n" $ formatPureSet <$> numberToSet <$> [0..3]
+            putStrLn.prettify $ pureSet [pureSet [pureSet []],pureSet [], pureSet [],pureSet [pureSet [],pureSet [pureSet [pureSet [pureSet [pureSet []]]]]]]
+            putStrLn.formatPureSet $ pureSet [pureSet [pureSet []],pureSet [], pureSet [],pureSet [pureSet [],pureSet [pureSet [pureSet [pureSet [pureSet []]]]]]]
+            putStrLn "End of TestPureSet"
+