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.
simple induction args; simpl in |- *; intros; auto with arith pts.
apply whnf_head_normal; auto with arith pts.
apply H; intros.
simpl in |- *; auto with arith pts.
discriminate.
Qed.
Let f1 (a : approx) :=
match a with
| (e, (t, args)) => existT (fun _ => value) (e, app_list args t) (e, t)
end.
Definition whnf_ord (x y : approx) :=
lexprod value (fun _ => value) (R_exp beta) (fun _ => subterm)
(f1 x) (f1 y).
simple induction l; simpl in |- *; intros; auto with arith pts.
Qed.
Hint Resolve beta_app_list_ctxt: pts.
intros.
pattern e, t, args in |- *.
apply (Acc3_rec (R:=whnf_ord)).
clear H e t args.
intros e t args.
case t; intros.
exists (app_list args (Srt s)); auto with arith pts.
apply whnf_app_list; simpl in |- *; intros; auto with arith pts.
discriminate.
exists (app_list args (Ref n)); auto with arith pts.
apply whnf_app_list; simpl in |- *; intros; auto with arith pts.
discriminate.
generalize H.
case args; simpl in |- *; intros.
exists (Abs t0 t1); simpl in |- *; auto with arith pts.
apply whnf_head_normal; auto with arith pts.
simpl in |- *; auto with arith pts.
elim H0 with e (subst t2 t1) l; simpl in |- *; intros; auto with arith pts.
exists x; auto with arith pts.
apply red_trans with (app_list l (subst t2 t1)); auto with arith pts.
red in |- *; red in |- *.
apply rt_step.
auto with arith pts.
red in |- *; simpl in |- *.
left.
apply Rex_intro.
auto with arith pts.
elim H with e t0 (t1 :: args); intros; auto with arith pts.
exists x; auto with arith pts.
red in |- *; simpl in |- *; right; auto with arith pts.
exists (app_list args (Prod t0 t1)); simpl in |- *; auto with arith pts.
apply whnf_app_list; simpl in |- *; intros; auto with arith pts.
discriminate.
apply Acc_Acc3.
unfold whnf_ord in |- *.
apply (Acc_inverse_image approx (sigT (fun (_:value) => value))).
simpl in |- *.
apply acc_A_B_lexprod; intros.
elim H; intros.
apply Acc_intro; intros; auto with arith pts.
inversion_clear H2; auto with arith pts.
exact wf_subterm.
apply wf_subterm.
Qed.

Theorem beta_whnf :
forall (e : env) (t : term),
sn (ctxt beta) e t -> {u : term | red beta e t u &  head_normal beta e u}.