content: 
  {-@ LIQUID "--no-termination"   @-}
  {-@ LIQUID "--no-totality"      @-}
  
  module RBTree_ord (add, remove, deleteMin, deleteMin') where
  
  import Language.Haskell.Liquid.Prelude
  
  data RBTree a = Leaf
                | Node { nCol   :: Color
                       , nKey   :: a
                       , nLeft  :: !(RBTree a)
                       , nRight :: !(RBTree a)
                       }
                deriving (Show)
  
  data Color = B -- ^ Black
             | R -- ^ Red
             deriving (Eq,Show)
  
  ---------------------------------------------------------------------------
  -- | Add an element -------------------------------------------------------
  ---------------------------------------------------------------------------
  
  {-@ add :: (Ord a) => a -> ORBT a -> ORBT a @-}
  add x s = makeBlack (ins x s)
  
  {-@ ins :: (Ord a) => a -> ORBT a -> ORBT a @-}
  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 -> ORBT a -> ORBT a @-}
  remove x t = makeBlack (del x t)
  
  {-@ del              :: (Ord a) => a -> ORBT a -> ORBT a @-}
  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 -> ORBT {v:a | v < y} -> ORBT {v:a | y < v} -> ORBT a @-}
  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  :: ORBT a -> ORBT a @-}
  deleteMin (Leaf)         = Leaf
  deleteMin (Node _ x l r) = makeBlack t
    where
      (_, t)               = deleteMin' x l r
  
  
  {-@ deleteMin' :: k:a -> ORBT {v:a | v < k} -> ORBT {v:a | k < v} -> (a, ORBT a) @-}
  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 (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                           = unsafeError "nein"
  
  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                           = unsafeError "nein"
  
  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 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
  
  ---------------------------------------------------------------------------
  ---------------------------------------------------------------------------
  ---------------------------------------------------------------------------
  
  makeRed (Node _ x l r) = Node R x l r
  makeRed Leaf           = unsafeError "nein"
  
  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 @-}
  
  -- | Binary Search Ordering
  
  {-@ data RBTree a <l :: a -> a -> Bool, r :: a -> a -> Bool>
                = Leaf
                | Node { nCol   :: Color
                       , nKey   :: a
                       , nLeft  :: RBTree <l, r> (a <l nKey>)
                       , nRight :: RBTree <l, r> (a <r nKey>) }
    @-}
  