module type KEY = sig

    type 'a t

    val comp: 'a t -> 'b t -> ('a, 'b) Tools.cmp

    val repr: Format.formatter -> 'a t -> unit

end

module Make (El : KEY) = struct

  type 'a elem = 'a El.t

  type leaf (** Fantom type for typing the tree *)
  type node (** Fantom type for typing the tree *)

  type 'a treeVal =
    | Leaf : leaf treeVal
    | Node : _ treeVal * ('a elem * 'a) * _ treeVal -> node treeVal

  type t = T : 'a treeVal ref -> t [@@unboxed]

  let empty = T (ref Leaf)

  let isEmpty (T tree) = match !tree with
    | Leaf -> true
    | _ -> false

  let rec splay : type a. a elem -> node treeVal -> node treeVal = fun x t -> begin
    let Node (l, y, r) = t in
    begin match El.comp x (fst y)  with
      | Tools.Eq -> t
      | Tools.Lt ->
        begin match l with
          | Leaf -> t
          | Node (ll, z, rr) ->
            begin match El.comp x (fst z)  with
              | Tools.Eq -> Node (ll, z, Node (rr, y, r))
              | Tools.Lt ->
                begin match ll with
                  | Leaf -> Node (ll, z, Node (rr, y, r))
                  | Node _ as ll ->
                    let Node (newL, newV, newR) = splay x ll
                    in Node (newL, newV, Node (newR, z, Node (rr, y, r)))
                end
              | Tools.Gt ->
                begin match rr with
                  | Leaf -> Node (ll, z, Node (rr, y, r))
                  | Node _ as rr ->
                    let Node (newL, newV, newR) = splay x rr
                    in Node (Node (ll, z, newL), newV, Node (newR, y, r))
                end
            end
        end
      | Tools.Gt ->
        begin match r with
          | Leaf -> t
          | Node (ll, z, rr) ->
            begin match El.comp x (fst z)  with
              | Tools.Eq -> Node (Node (l, y, ll), z, rr)
              | Tools.Lt ->
                begin match ll with
                  | Leaf -> Node (Node (l, y, ll), z, rr)
                  | Node _ as ll ->
                    let Node (newL, newV, newR) = splay x ll
                    in Node (Node (l, y, newL), newV, Node (newR, z, rr))
                end
              | Tools.Gt ->
                begin match rr with
                  | Leaf -> Node (Node (l, y, ll), z, rr)
                  | Node _ as rr ->
                    let Node (newL, newV, newR) = splay x rr
                    in Node (Node (Node(l, y, ll), z, newL), newV, newR)
                end
            end
        end
    end
  end

  let member: type a. a elem -> t -> bool = fun x (T t) -> match !t with
    | Leaf -> false
    | Node _ as root ->
        let root' = splay x root in
        t := root';
        let Node (_, c', _) = root' in
        begin match El.comp (fst c') x  with
          | Tools.Eq -> true
          | _ -> false
        end

  let find: type a. a elem -> t -> a = fun x (T t) -> match !t with
    | Leaf -> raise Not_found
    | Node _ as root ->
        let root' = splay x root in
        t := root';
        let Node (_, c', _) = root' in
        begin match El.comp (fst c') x with
          | Tools.Eq -> snd c'
          | _ -> raise Not_found
        end

  let add: type a. a elem -> a -> t -> t = fun key value (T t) -> match !t with
    | Leaf -> T (ref (Node (Leaf, (key, value), Leaf)))
    | Node _ as root ->
        let root' = splay key root in
        let Node (l, y, r) = root' in
        begin match El.comp key (fst y)  with
          | Tools.Eq -> T (ref root')
          | Tools.Lt -> T (ref (Node (l, (key, value), Node (Leaf, y, r))))
          | Tools.Gt -> T (ref (Node (Node (l, y, Leaf), (key, value), r)))
        end

  let repr formatter (T t) = begin

    let repr_edge from formatter dest = begin
      Format.fprintf formatter "\"%a\" -> \"%a\"\n"
        El.repr from
        El.repr dest
    end in

    let rec repr': type a b. a El.t -> Format.formatter -> b treeVal -> unit = fun parent formatter -> function
    | Leaf -> ()
    | Node (l, c, r) ->
        let key = fst c in
        Format.fprintf formatter "%a%a%a"
        (repr_edge parent) key
        (repr' key) l
        (repr' key) r in

    begin match !t with
    | Leaf -> Format.fprintf formatter "digraph G {}"
    | Node (l, c, r) ->
      let key = fst c in
      Format.fprintf formatter "digraph G {\n%a%a}"
        (repr' key) l
        (repr' key) r
    end

  end

end