liquidhaskell-0.4.0.0: tests/pos/RBTree-height.hs
{-@ LIQUID "--no-termination" @-}
module RedBlackTree where
import Language.Haskell.Liquid.Prelude
data RBTree a = Leaf
| Node Color a !(RBTree a) !(RBTree a)
deriving (Show)
data Color = B -- ^ Black
| R -- ^ Red
deriving (Eq,Show)
---------------------------------------------------------------------------
-- | Add an element -------------------------------------------------------
---------------------------------------------------------------------------
{-@ add :: (Ord a) => a -> RBT a -> RBT a @-}
add x s = makeBlack (ins x s)
{-@ ins :: (Ord a) => a -> t:RBT a -> RBTN a {(bh t)} @-}
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 -> RBT a -> RBT a @-}
remove x t = makeBlack (del x t)
{-@ predicate HDel T V = (bh V) = (if (isB T) then (bh T) - 1 else (bh T)) @-}
{-@ del :: (Ord a) => a -> t:RBT a -> {v:RBT a | (HDel t v)} @-}
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 -> l:RBT a -> r:RBTN a {(bh l)} -> RBTN a {(bh l)} @-}
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 :: RBT a -> RBT a @-}
deleteMin (Leaf) = Leaf
deleteMin (Node _ x l r) = makeBlack t
where
(_, t) = deleteMin' x l r
{-@ deleteMin' :: k:a -> l:RBT a -> r:RBTN a {(bh l)} -> (a, RBTN a {(bh l)}) @-}
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:a -> l:RBT a -> r:RBTN a {1 + (bh l)} -> RBTN a {1 + (bh l)} @-}
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 = error "nein"
{-@ rbalS :: k:a -> l:RBT a -> r:RBTN a {(bh l) - 1} -> RBTN a {(bh l)} @-}
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 = error "nein"
{-@ lbal :: k:a -> l:RBT a -> RBTN a {(bh l)} -> RBTN a {1 + (bh l)} @-}
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 :: k:a -> l:RBT a -> RBTN a {(bh l)} -> RBTN a {1 + (bh l)} @-}
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 :: l:RBT a -> RBTN a {(bh l) - 1} @-}
makeRed (Node B x l r) = Node R x l r
makeRed _ = error "nein"
{-@ makeBlack :: RBT a -> RBT a @-}
makeBlack Leaf = Leaf
makeBlack (Node _ x l r) = Node B x l r
---------------------------------------------------------------------------
-- | Specifications -------------------------------------------------------
---------------------------------------------------------------------------
-- | Red-Black Trees
{-@ type RBT a = {v: RBTree a | (isBH v) } @-}
{-@ type RBTN a N = {v: RBT a | (bh v) = N } @-}
-- | Color of a tree
{-@ measure isB :: RBTree a -> Prop
isB (Leaf) = false
isB (Node c x l r) = c == B
@-}
-- | Black Height
{-@ measure isBH :: RBTree a -> Prop
isBH (Leaf) = true
isBH (Node c x l r) = ((isBH l) && (isBH r) && (bh l) = (bh r))
@-}
{-@ measure bh :: RBTree a -> Int
bh (Leaf) = 0
bh (Node c x l r) = (bh l) + (if (c == R) then 0 else 1)
@-}
-------------------------------------------------------------------------------
-- Auxiliary Invariants -------------------------------------------------------
-------------------------------------------------------------------------------
{-@ invariant {v: RBTree a | (Invs v)} @-}
{-@ predicate Invs V = (Inv3 V) @-}
{-@ predicate Inv3 V = 0 <= (bh v) @-}
{-@ invariant {v: Color | (v = R || v = B)} @-}
{-@ inv :: RBTree a -> {v:RBTree a | (Invs v)} @-}
inv Leaf = Leaf
inv (Node c x l r) = Node c x (inv l) (inv r)