```(***********************************************************************)
(*                                                                     *)
(*  OCaml library from the book ``Apprendre à programmer avec OCaml''  *)
(*                                                                     *)
(*  Sylvain Conchon and Jean-Christophe Filliâtre                      *)
(*  Université Paris Sud                                               *)
(*                                                                     *)
(*  file is distributed under the terms of the GNU Library General     *)
(*  Public License, with the same special exception on linking as the  *)
(*  OCaml library. See http://caml.inria.fr/ocaml/license.fr.html      *)
(*                                                                     *)
(***********************************************************************)

(* Programme 37 page 210
Insertion et suppression dans un AVL *)

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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