(*********************************************************************)
(*  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_left : forall (key : Int) (update : B -> B) (init : B) (k0 : Int) (d0 : B) (l0 r0 : avl_tree) (b0 : bal) (l : avl_tree) (b : bal), is_avl (Avl_Node k0 d0 l0 r0 b0) -> Less key k0 -> equiv_ins key update init l0 l -> equiv_ins key update init (Avl_Node k0 d0 l0 r0 b0) (Avl_Node k0 d0 l r0 b).