Hint Resolve insert_leaf: searchtrees.
(* Inserting in the left son *)

Lemma insert_l :
forall (n p : key) (v v' : data) (t1 t'1 t2 : btree),
Key.lt n p ->
search_tree (bnode p v' t1 t2) ->
INSERT n v t1 t'1 -> INSERT n v (bnode p v' t1 t2) (bnode p v' t'1 t2).