Section CoqV6Beta.
Definition trm_v6 := term srt_v6.
Definition env_v6 := env srt_v6.
(* Construction du CTS *)
Definition v6 : CTS_spec srt_v6 :=
Build_CTS_spec _ axiom_v6 rules_v6 univ_v6 (beta_rule _).
(* Construction du PTS issu du CTS *)
Definition v6_pts : PTS_sub_spec srt_v6 := cts_pts_functor _ v6.
Definition le_type : red_rule srt_v6 :=
Rule _ (Le_type _ (pts_le_type _ v6_pts)).
Definition typ_v6 : env_v6 -> trm_v6 -> trm_v6 -> Prop := typ _ v6_pts.
Definition wft_v6 : env_v6 -> trm_v6 -> Prop := wf_type _ v6_pts.
Definition wf_v6 : env_v6 -> Prop := wf _ v6_pts.
Definition v6_sn := sn srt_v6 (ctxt _ (Rule _ (head_reduct _ v6))).
Hint Unfold le_type typ_v6 wft_v6 wf_v6 v6_sn: pts.
(* Algorithme de mise en forme normale de tete Decidabilite du sous-typage pour les types bien formes *)
Proof beta_whnf srt_v6.

Lemma beta_conv_hnf :
forall (e : env_v6) (x y : trm_v6),
v6_sn e x -> v6_sn e y -> decide (conv_hn_inv _ (beta_rule _) e x y).