simple induction 1; intros; auto with arith pts.
inversion_clear H1; auto with arith pts.
inversion_clear H2; auto with arith pts.
Qed.

(* Specifications: *)
Lemma nth_sub_inf :
forall (t : term) (d1 : decl) (n : nat) (g e f : env),
sub_in_env g d1 t n e f ->
forall v : nat,
n > v ->
forall d : decl, item d e v -> item (subst_decl t d (n - S v)) f v.