intros key update init.
apply equiv_ins_intro.
intros k data equal lookup_nil.
elimtype False.
apply (inv_lookup_nil k data lookup_nil).
intros k equal notlookup_nil.
rewrite (equal_eq k key equal).
apply Lookup_Equal.
intros k data notequal lookup_nil.
elimtype False.
apply (inv_lookup_nil k data lookup_nil).
intros k data notequal lookup_nil.
elim
(inv_lookup k key (update init) Avl_Nil Avl_Nil Balanced data lookup_nil).
intro u0.
elimtype False.
apply notequal.
elim u0; intros u00 u01.
rewrite u00.
apply equal_refl.
tauto.
Qed.

Lemma avl_ins_eq :
forall (k : Int) (d : B) (l r : avl_tree) (b : bal) (d' : B),
is_avl (Avl_Node k d l r b) -> is_avl (Avl_Node k d' l r b).