(*********************************************************************)
(*  equiv_ins                                                        *)

Inductive equiv_ins (key : Int) (update : B -> B) (init : B)
(t0 t : avl_tree) : Prop :=
    equiv_ins_intro :
      (forall (k : Int) (data : B),
       Equal k key -> lookup k t0 data -> lookup k t (update data)) ->
      (forall k : Int,
       Equal k key ->
       (forall data : B, lookup k t0 data -> False) ->
       lookup k t (update init)) ->
      (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_ins key update init t0 t.(*********************************************************************)
(*  equiv_ins                                                        *)

Inductive equiv_ins (key : Int) (update : B -> B) (init : B)
(t0 t : avl_tree) : Prop :=
    equiv_ins_intro :
      (forall (k : Int) (data : B),
       Equal k key -> lookup k t0 data -> lookup k t (update data)) ->
      (forall k : Int,
       Equal k key ->
       (forall data : B, lookup k t0 data -> False) ->
       lookup k t (update init)) ->
      (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_ins key update init t0 t.

Lemma equiv_ins_below : forall (key : Int) (update : B -> B) (init : B) (t0 t : avl_tree) (k0 : Int), equiv_ins key update init t0 t -> Less key k0 -> is_below t0 k0 -> is_below t k0.