Inductive step : term -> term -> Prop :=
| step_beta s P Q :
step (app (abs s P) Q) (subst_term poly_var (scons Q var) P)
| step_ty_beta P s :
step (ty_app (ty_abs P) s) (subst_term (scons s poly_var) var P)
| step_appL P P' Q :
step P P' -> step (app P Q) (app P' Q)
| step_appR P P' Q :
step P P' -> step (app Q P) (app Q P')
| step_ty_app P P' s :
step P P' -> step (ty_app P s) (ty_app P' s)
| step_lam s P P' :
step P P' -> step (abs s P) (abs s P')
| step_ty_lam P P' :
step P P' -> step (ty_abs P) (ty_abs P').
Inductive sn x : Prop :=
| SNI : (forall y, step x y -> sn y) -> sn x.
Hint Constructors step normal_form head_form : core.
Ltac inv_step :=
match goal with
[ H : step ?P ?Q |- _] => inversion H; subst; clear H; try now firstorder
end.
Ltac now_asimpl := asimpl; ( (reflexivity || eapply ext_term; now intros []; repeat asimpl) ||
f_equal; (reflexivity || eapply ext_term; now intros []; repeat asimpl)).
(* Require Import Equations.Prop.DepElim. *)
Ltac inv H := inversion H; subst; clear H.
Proof with eexists; split; [eauto | now_asimpl].
intros H. dependent induction H; rename x into Eqn.
- destruct P; inv Eqn. destruct P1; inv H0...
- destruct P; inv Eqn. destruct P; inv H0...
- destruct P; inv Eqn. destruct (IHstep _ _ _ eq_refl) as (P1' & H1 & <-)...
- destruct P; inv Eqn. destruct (IHstep _ _ _ eq_refl) as (P1' & H1 & <-)...
- destruct P; inv Eqn. destruct (IHstep _ _ _ eq_refl) as (P1' & H1 & <-)...
- destruct P; inv Eqn.
edestruct (IHstep P (up_term_poly_type σ) (0 .: τ >> shift))  as (P1' & H1 & <-).
now_asimpl. exists (abs p P1'). split. eauto. now_asimpl.
- destruct P; inv Eqn.  destruct (IHstep _ _ _ eq_refl) as (P1' & H1 & <-)...
Qed.
Definition nf P := match P with abs s P => normal_form P
| ty_abs P => normal_form P | P => head_form P end.

Lemma nf_normal_form P :
nf P -> normal_form P.