Section Beta_Whnf.
Fixpoint whnf_beta (t : term) : Prop :=
match t with
| App u v => whnf_beta u /\ (forall T t : term, u <> Abs T t)
| _ => True
end.
Hint Unfold whnf_beta: pts.
simple induction 1; simpl in |- *; intros; auto with arith pts.
generalize H1.
elim H0; simpl in |- *; auto with arith pts.
simple induction 1; simpl in |- *; intros.
inversion_clear H3.
elim H5 with T M; auto with arith pts.
intros.
inversion_clear H4.
split; auto with arith pts.
red in |- *; intros.
generalize H5.
inversion_clear H4 in H2.
generalize H6.
inversion_clear H2; simpl in |- *; intros.
generalize H7.
inversion_clear H4; simpl in |- *; intros.
inversion_clear H4.
elim H9 with T0 M; auto with arith pts.
elim H2 with M t0; auto with arith pts.
elim H2 with T M; auto with arith pts.
Qed.
red in |- *; intros.
generalize H.
elim H0.
simple induction 1.
simple induction 1; simpl in |- *; intros.
inversion_clear H3.
elim H5 with T M; auto with arith pts.
red in |- *; red in |- *; intros.
inversion_clear H5.
red in |- *; red in |- *; intros.
inversion_clear H5.
red in |- *; red in |- *; intros.
generalize H2.
inversion_clear H5.
intros.
generalize H4.
inversion_clear H5; simpl in |- *; intros.
inversion_clear H5.
generalize H7.
inversion_clear H6.
simpl in |- *; intros.
inversion_clear H5.
elim H9 with T0 M0; auto with arith pts.
inversion_clear H5.
elim H8 with M0 M; auto with arith pts.
inversion_clear H5.
elim H8 with T M0; auto with arith pts.
red in |- *; red in |- *; intros.
generalize H4.
inversion_clear H5; simpl in |- *; intros.
inversion_clear H5.
elim H7 with T M; auto with arith pts.
red in |- *; red in |- *; intros.
inversion_clear H5.
red in |- *; red in |- *; intros.
inversion_clear H5.
red in |- *; red in |- *; intros.
generalize H1.
inversion_clear H2; simpl in |- *; intros.
inversion_clear H2.
elim H4 with T M; auto with arith pts.
intros.
apply H4.
apply beta_red_whnf with e x; auto with arith pts.
Qed.

Lemma whnf_app_list :
forall (args : list term) (e : env) (t : term),
whnf_beta t ->
(forall A M : term, t <> Abs A M) -> head_normal beta e (app_list args t).