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Correct-by-composition set operations that cannot branch on ordering
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{-# LANGUAGE FlexibleInstances, UndecidableInstances, LambdaCase, DeriveFunctor, MonadComprehensions #-} | |
import qualified Prelude | |
import Prelude (Show, Num, Functor, Bool(..), not, Int, Maybe(..), Either(..), Eq, (==), (.), ($), const, otherwise) | |
import Control.Monad (MonadPlus, guard, when, unless, (>>=), fmap, mzero, mplus, return) | |
import qualified Data.Foldable | |
import Data.Foldable (Foldable) | |
import Data.Maybe (maybe, isJust) | |
import Data.Functor.Foldable (Fix) | |
class (MonadPlus s) => Set s where | |
null :: s a -> Bool | |
class (Set b) => Bag b where | |
count :: b a -> Int | |
instance (Foldable s, MonadPlus s) => Set s where | |
null = Data.Foldable.null | |
instance (Foldable s, MonadPlus s) => Bag s where | |
count = Data.Foldable.length | |
or :: (Set s) => s Bool -> Bool | |
or = not . null . (>>= guard) | |
and :: (Set s) => s Bool -> Bool | |
and = not . or . fmap not | |
any :: (Set s) => (a -> Bool) -> s a -> Bool | |
any pred = or . fmap pred | |
all :: (Set s) => (a -> Bool) -> s a -> Bool | |
all pred = and . fmap pred | |
(+) :: (Set s) => s a -> s a -> s a | |
xs + ys = xs `mplus` ys | |
(-) :: (Set s, Eq a) => s a -> s a -> s a | |
xs - ys = do x <- xs | |
guard . not $ any (== x) ys | |
return x | |
catMaybes :: (Set s) => s (Maybe a) -> s a | |
catMaybes xs = [x | Just x <- xs] | |
-- How can we show the type system we're not dropping anything when we catMaybes? | |
sequence :: (Set s) => s (Maybe a) -> Maybe (s a) | |
sequence xs = if (all isJust xs) then Just (catMaybes xs) else Nothing | |
--sequence :: (Set s) => s (Maybe a) -> Maybe (s a) | |
--sequence xs = if (all isJust xs) then Just (catMaybes xs) else Nothing | |
lefts :: (Set s) => s (Either a b) -> s a | |
lefts xs = [x | Left x <- xs] | |
rights :: (Set s) => s (Either a b) -> s b | |
rights xs = [x | Right x <- xs] | |
partitionEithers :: (Set s) => s (Either a b) -> (s a, s b) | |
partitionEithers xs = (lefts xs, rights xs) | |
data Nat = Zero | Succ Nat | |
deriving Show | |
isZero :: Nat -> Bool | |
isZero Zero = True | |
isZero (Succ _) = False | |
pred :: Nat -> Maybe Nat | |
pred Zero = Nothing | |
pred (Succ n) = Just n | |
min :: (Set r) => r Nat -> Maybe Nat | |
min ns | null ns = Nothing | |
min ns | any isZero ns = Just Zero | |
| otherwise = (fmap Succ . min . catMaybes. fmap pred) ns | |
-- We use catMaybes remove the Just, but we're guarenteed that there are no Nothing | |
-- Could the type system handle this? | |
max :: (Set r) => r Nat -> Maybe Nat | |
max ns | null ns = Nothing | |
max ns | all isZero ns = Just Zero | |
| otherwise = (fmap Succ . max . catMaybes . fmap pred) ns |
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