(* inserting in the right son *)

Lemma insert_r :
forall (n p : key) (v v' : data) (t1 t2 t'2 : btree),
Key.lt p n ->
search_tree (bnode p v' t1 t2) ->
INSERT n v t2 t'2 -> INSERT n v (bnode p v' t1 t2) (bnode p v' t1 t'2).