(********************************************************************)
(*  OCaml code from the book ``Learn Programming with OCaml''       *)
(*  See https://usr.lmf.cnrs.fr/lpo/                                *)
(*                                                                  *)
(*  Sylvain Conchon and Jean-Christophe Filliâtre                   *)
(*  Copyright 2025 Université Paris-Saclay and CNRS                 *)
(*                                                                  *)
(*  Openly licensed via CC BY SA 4.0                                *)
(*  See https://creativecommons.org/licenses/by-sa/4.0/deed.en      *)
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(* Program 37 on page 224
   Insertion and Removal in an AVL Tree *)

  let rec add x = function
    | Empty ->
        Node (Empty, x, Empty, 1)
    | Node (l, v, r, _) as t ->
        let c = X.compare x v in
        if c = 0 then t
        else if c < 0 then balance (add x l) v r
        else balance l v (add x r)

  let rec remove_min_elt = function
    | Empty -> Empty
    | Node (Empty, _, r, _) -> r
    | Node (l, v, r, _) -> balance (remove_min_elt l) v r

  let merge t1 t2 = match t1, t2 with
    | Empty, t | t, Empty -> t
    | _ -> balance t1 (min_elt t2) (remove_min_elt t2)

  let rec remove x = function
    | Empty ->
        Empty
    | Node (l, v, r, _) ->
        let c = X.compare x v in
        if c = 0 then merge l r
        else if c < 0 then balance (remove x l) v r
        else balance l v (remove x r)

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