(***********************************************************************)


Inductive equiv_del (key : Int) (t0 t : avl_tree) : Prop :=
    equiv_del_intro :
      (forall k : Int, Equal k key -> forall d : B, lookup k t d -> False) ->
      (forall (k : Int) (data : B),
       ~ Equal k key -> lookup k t0 data -> lookup k t data) ->
      (forall (k : Int) (data : B),
       ~ Equal k key -> lookup k t data -> lookup k t0 data) ->
      equiv_del key t0 t.(***********************************************************************)


Inductive equiv_del (key : Int) (t0 t : avl_tree) : Prop :=
    equiv_del_intro :
      (forall k : Int, Equal k key -> forall d : B, lookup k t d -> False) ->
      (forall (k : Int) (data : B),
       ~ Equal k key -> lookup k t0 data -> lookup k t data) ->
      (forall (k : Int) (data : B),
       ~ Equal k key -> lookup k t data -> lookup k t0 data) ->
      equiv_del key t0 t.

Lemma below_equiv_below : forall (k : Int) (t0 t : avl_tree), is_below t0 k -> equiv t t0 -> is_below t k.