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type t = Empty | Node : node -> t
and node = { x: int; y: int; l: t; r: t; h: int }
let empty = Empty
let is_empty = function Empty -> true | _ -> false
let height = function Empty -> 0 | Node n -> n.h
let create x y l r =
let h = Int.max (height l) (height r) + 1 in
Node { x; y; l; r; h }
let rec node x y l r =
let hl = height l and hr = height r in
if hl > hr + 2 then begin
match l with
| Empty -> assert false
| Node { x= lx; y= ly; l= ll; r= lr; _ } -> (
if height ll >= height lr then node lx ly ll (node x y lr r)
else
match lr with
| Empty -> assert false
| Node { x= lrx; y= lry; l= lrl; r= lrr; _ } ->
node lrx lry (node lx ly ll lrl) (node x y lrr r))
end
else if hr > hl + 2 then begin
match r with
| Empty -> assert false
| Node { x= rx; y= ry; l= rl; r= rr; _ } -> (
if height rr >= height rl then node rx ry (node x y l rl) rr
else
match rl with
| Empty -> assert false
| Node { x= rlx; y= rly; l= rll; r= rlr; _ } ->
node rlx rly (node x y l rll) (node rx ry rlr rr))
end
else create x y l r
let rec splitMax = function
| { x; y; l; r= Empty; _ } -> (x, y, l)
| { r= Node r; _ } as n ->
let u, v, r' = splitMax r in
(u, v, node n.x n.y n.l r')
let rec splitMin = function
| { x; y; l= Empty; r; _ } -> (x, y, r)
| { l= Node l; _ } as n ->
let u, v, l' = splitMin l in
(u, v, node n.x n.y l' n.r)
let addL = function
| { l= Empty; _ } as n -> n
| { l= Node l; _ } as n ->
let x', y', l' = splitMax l in
if succ y' = n.x then { n with x= x'; l= l' } else n
let addR = function
| { r= Empty; _ } as n -> n
| { r= Node r; _ } as n ->
let x', y', r' = splitMin r in
if succ n.y = x' then { n with y= y'; r= r' } else n
let rec add x y t =
match t with
| Empty -> node x y Empty Empty
| Node n when y < pred n.x ->
let l = add x y n.l in
node n.x n.y l n.r
| Node n when succ n.y < x ->
let r = add x y n.r in
node n.x n.y n.l r
| Node n when x < n.x && y <= n.y ->
let l = add x (pred n.x) n.l in
let n = addL { n with l } in
node n.x n.y n.l n.r
| Node n when y > n.y && x >= n.x ->
let r = add (succ n.y) y n.r in
let n = addR { n with r } in
node n.x n.y n.l n.r
| Node n when x < n.x && y > n.y ->
let l = add x (pred n.x) n.l in
let r = add (succ n.y) y n.r in
let n = addL { (addR { n with r }) with l } in
node n.x n.y n.l n.r
| Node n -> Node n
let singleton x y = add x y Empty
let merge l r =
match (l, r) with
| l, Empty -> l
| Empty, r -> r
| Node l, r ->
let x, y, l' = splitMax l in
node x y l' r
let rec remove (x, y) t =
match t with
| Empty -> Empty
| Node n when y < n.x ->
let l = remove (x, y) n.l in
node n.x n.y l n.r
| Node n when n.y < x ->
let r = remove (x, y) n.r in
node n.x n.y n.l r
| Node n when x < n.x && y < n.y ->
let n' = node (succ y) n.y n.l n.r in
remove (x, pred n.x) n'
| Node n when y > n.y && x > n.x ->
let n' = node n.x (pred x) n.l n.r in
remove (succ n.y, y) n'
| Node n when x <= n.x && y >= n.y ->
let l = remove (x, n.x) n.l in
let r = remove (n.y, y) n.r in
merge l r
| Node n when y = n.y -> node n.x (pred x) n.l n.r
| Node n when x = n.x -> node (succ y) n.y n.l n.r
| Node n ->
assert (n.x <= pred x);
assert (succ y <= n.y);
let r = node (succ y) n.y Empty n.r in
node n.x (pred x) n.l r
let rec fold fn t acc =
match t with
| Empty -> acc
| Node n ->
let acc = fold fn n.l acc in
let acc = fn (n.x, n.y) acc in
fold fn n.r acc
let diff a b = fold remove b a
let inter a b = diff a (diff a b)