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-- Sieve of Eratosthenes
-- Melissa O'Neill "The Genuine Sieve of Eratosthenes"
-- http://www.cs.hmc.edu/~oneill/papers/Sieve-JFP.pdf
-- priority queue from http://hpaste.org/report/1447

import Data.List

data PriorityQ k v = Lf
                   | Br !k v !(PriorityQ k v) !(PriorityQ k v)
               deriving (Eq, Ord, Read, Show)

empty :: PriorityQ k v
empty = Lf

isEmpty :: PriorityQ k v -> Bool
isEmpty Lf  = True
isEmpty _   = False

minKeyValue :: PriorityQ k v -> (k, v)
minKeyValue (Br k v _ _)    = (k,v)
minKeyValue _               = error "Empty heap!"

minKey :: PriorityQ k v -> k
minKey (Br k v _ _)         = k
minKey _                    = error "Empty heap!"

minValue :: PriorityQ k v -> v
minValue (Br k v _ _)       = v
minValue _                  = error "Empty heap!"

insert :: Ord k => k -> v -> PriorityQ k v -> PriorityQ k v
insert wk wv (Br vk vv t1 t2)
               | wk <= vk   = Br wk wv (Main.insert vk vv t2) t1
               | otherwise  = Br vk vv (Main.insert wk wv t2) t1
insert wk wv Lf             = Br wk wv Lf Lf

siftdown :: Ord k => k -> v -> PriorityQ k v -> PriorityQ k v -> PriorityQ k v
siftdown wk wv Lf _             = Br wk wv Lf Lf
siftdown wk wv (t @ (Br vk vv _ _)) Lf 
    | wk <= vk                  = Br wk wv t Lf
    | otherwise                 = Br vk vv (Br wk wv Lf Lf) Lf
siftdown wk wv (t1 @ (Br vk1 vv1 p1 q1)) (t2 @ (Br vk2 vv2 p2 q2))
    | wk <= vk1 && wk <= vk2    = Br wk wv t1 t2
    | vk1 <= vk2                = Br vk1 vv1 (siftdown wk wv p1 q1) t2
    | otherwise                 = Br vk2 vv2 t1 (siftdown wk wv p2 q2)

deleteMinAndInsert :: Ord k => k -> v -> PriorityQ k v -> PriorityQ k v
deleteMinAndInsert wk wv Lf             = error "Empty PriorityQ"
deleteMinAndInsert wk wv (Br _ _ t1 t2) = siftdown wk wv t1 t2

leftrem :: PriorityQ k v -> (k, v, PriorityQ k v)
leftrem (Br vk vv Lf Lf) = (vk, vv, Lf)
leftrem (Br vk vv t1 t2) = (wk, wv, Br vk vv t t2) where
    (wk, wv, t) = leftrem t1
leftrem _                = error "Empty heap!"

deleteMin :: Ord k => PriorityQ k v -> PriorityQ k v
deleteMin (Br vk vv Lf _) = Lf
deleteMin (Br vk vv t1 t2) = siftdown wk wv t2 t where
    (wk,wv,t) = leftrem t1
deleteMin _ = error "Empty heap!"

sieve [] = []
sieve (x:xs) = x : sieve' xs (insertprime x xs PQ.empty)
  where
    insertprime p xs table = PQ.insert (p*p) (map (* p) xs) table
    sieve' [] table = []
    sieve' (x:xs) table
      | nextComposite <= x = sieve' xs (adjust table)
      | otherwise = x : sieve' xs (insertprime x xs table)
         where
           nextComposite = PQ.minKey table
           adjust table
             | n <= x = adjust (PQ.deleteMinAndInsert n' ns table)
             | otherwise = table
                 where
                   (n, n':ns) = PQ.minKeyValue table

primes = 2 : sieve [3,5..]

main = print $ takewhile (< 100) primes

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