simple induction 1; simpl in |- *; intros; auto with arith pts.
rewrite distr_lift_subst; auto with arith pts.
apply par_beta; auto with arith pts.
replace (Ax (lift_rec 1 T k)) with (lift_decl 1 (Ax T) k);
auto with arith pts.
apply abs_par_red_b; auto with arith pts.
replace (Ax (lift_rec 1 T' k)) with (lift_decl 1 (Ax T') k);
auto with arith pts.
apply prod_par_red_b; auto with arith pts.
replace (Ax (lift_rec 1 M' k)) with (lift_decl 1 (Ax M') k);
auto with arith pts.
Qed.
Hint Resolve beta_par_red1_lift: pts.

(* Specifications: *)
Lemma beta_par_red1_subst_l :
forall (t u : term) (g : env),
beta_par_red1 g t u ->
forall (v : term) (n : nat) (e : env),
trunc n e g -> beta_par_red1 e (subst_rec t v n) (subst_rec u v n).