simple induction v; simpl in |- *; intros; auto with arith pts.
elim (lt_eq_lt_dec n0 n); intros; auto with arith pts.
elim a; intros; auto with arith pts.
apply iter_R_lift with g; auto with arith pts.
Qed.

(* Specifications: *)
Lemma bd_par_red1_subst_rec :
forall (d : decl) (x y t u : term) (e g : env),
bd_par_red1 g x y ->
bd_par_red1 e t u ->
forall (n : nat) (f : env),
sub_in_env g d x n e f ->
bd_par_red1 f (subst_rec x t n) (subst_rec y u n).
Lemma beta_par_red1_subst_rec :
forall (d : decl) (x y t u : term) (e g : env),
beta_par_red1 g x y ->
beta_par_red1 e t u ->
forall (n : nat) (f : env),
sub_in_env g d x n e f ->
beta_par_red1 f (subst_rec x t n) (subst_rec y u n).