simple induction 1; auto with arith pts.
simple destruct v; intros.
inversion_clear H2.
inversion_clear H3; auto with arith pts.
Qed.

(* Specifications: *)
Lemma ins_item_lt :
forall (d' : decl) (n : nat) (g e f : env),
ins_in_env g d' n e f ->
forall v : nat,
n > v -> forall d : decl, item d e v -> item (lift_decl 1 d (n - S v)) f v.