```--
-- Copyright (c) 2009 - 2010 Brendan Hickey - http://bhickey.net
--

module Data.Heap.Binomial
(BinomialHeap, head, tail, merge, singleton, empty, null, fromList, toList, insert)
where

import Prelude hiding (head, tail, null)
import Data.List (delete)

data (Ord a, Ord b, Eq a, Eq b) => HeapNode a b = HeapNode a {-# UNPACK #-} !Int [b]

data (Ord a, Eq a) => BinomialHeap a =
EmptyHeap
| Heap {-# UNPACK #-} ![HeapNode a (BinomialHeap a)] deriving (Eq, Ord)

instance (Ord a, Ord b, Eq a, Eq b) => Ord (HeapNode a b) where
compare (HeapNode e1 _ _) (HeapNode e2 _ _) = compare e1 e2

instance (Ord a, Ord b, Eq a, Eq b) => Eq (HeapNode a b) where
(HeapNode e1 _ _) == (HeapNode e2 _ _) = e1 == e2

rank :: (Ord a, Ord b, Eq a, Eq b) => HeapNode a b -> Int
rank (HeapNode _ n _) = n

hRank :: (Ord a, Ord b, Eq a, Eq b) => [HeapNode a b] -> Int
hRank [] = 0
hRank (hd:_) = rank hd

extract :: (Ord a, Ord b, Eq a, Eq b) => HeapNode a b -> a
extract (HeapNode n _ _) = n

empty :: (Ord a) => BinomialHeap a
empty = EmptyHeap

null :: (Ord a) => BinomialHeap a -> Bool
null EmptyHeap = True
null _         = False

-- | /O(1)/.
singleton :: (Ord a) => a -> BinomialHeap a
singleton n = Heap [HeapNode n 1 []]

-- | /O(lg n)/
insert :: (Ord a) => a -> BinomialHeap a -> BinomialHeap a
insert a = merge (singleton a)

-- | /O(lg n)/.
merge :: (Ord a) => BinomialHeap a -> BinomialHeap a -> BinomialHeap a
merge EmptyHeap n = n
merge n EmptyHeap = n
merge (Heap h1) (Heap h2) = Heap \$! mergeNodes h1 h2

mergeNodes :: (Ord a, Eq a) => [HeapNode a (BinomialHeap a)] -> [HeapNode a (BinomialHeap a)] -> [HeapNode a (BinomialHeap a)]
mergeNodes [] h  = h
mergeNodes h  [] = h
mergeNodes f@(h1 : t1) s@(h2 : t2)
| rank h1 == rank h2 =
let merged = combine h1 h2
r = rank merged
in
if r /= hRank t1 then
if r /= hRank t2 then merged : mergeNodes t1 t2 else
mergeNodes (merged : t1) t2
else
if r /= hRank t2 then mergeNodes t1 (merged : t2) else
merged : mergeNodes t1 t2
| rank h1 < rank h2 = h1 : mergeNodes t1 s
| otherwise = h2 : mergeNodes t2 f

combine :: (Ord a, Eq a) => HeapNode a (BinomialHeap a) -> HeapNode a (BinomialHeap a) -> HeapNode a (BinomialHeap a)
combine h1@(HeapNode e1 n1 l1) h2 =
if h1 <= h2
then HeapNode e1 (n1 + 1) (l1 ++ [Heap [h2]])
else combine h2 h1

-- | /O(lg n)/
head :: (Ord a) => BinomialHeap a -> a
head EmptyHeap = error "Data.Heap: empty list"
head (Heap hn) = extract \$! minimum hn

-- | /O(lg n)/
tail :: (Ord a) => BinomialHeap a -> BinomialHeap a
tail EmptyHeap = error "Data.Heap: empty list"
tail (Heap hn) =
let n@(HeapNode _ _ hd) = (minimum hn) in
foldl merge (Heap (delete n hn)) hd

-- | /O(n)/
fromList :: (Ord a, Eq a) => [a] -> BinomialHeap a
fromList =  foldl merge EmptyHeap . map singleton

-- | /O(n lg n)/
toList :: (Ord a) => BinomialHeap a -> [a]
toList EmptyHeap  = []
toList (Heap [])  = []
toList h@(Heap _) = head h : (toList \$ if null h then h else tail h)
```