--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/rt_transition/cpr_tdeq.ma".
+include "basic_2/rt_transition/cnh_simple.ma".
+include "basic_2/rt_transition/cnh_drops.ma".
+
+(* NORMAL TERMS FOR HEAD T-UNUNBOUND RT-TRANSITION **************************)
+
+(* Properties with context-free sort-irrelevant equivalence for terms *******)
+
+lemma cnh_dec_tdeq (h) (G) (L):
+ ∀T1. ∨∨ ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃T1⦄
+ | ∃∃n,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 & (T1 ≛ T2 → ⊥).
+#h #G #L * *
+[ #s /3 width=5 by cnh_sort, or_introl/
+| #i elim (drops_F_uni L i)
+ [ /3 width=7 by cnh_atom_drops, or_introl/
+ | * * [ #I | * #V ] #K #HLK
+ [ /3 width=8 by cnh_unit_drops, or_introl/
+ | elim (lifts_total V 𝐔❴↑i❵) #W #HVW
+ @or_intror @(ex2_2_intro … W) [1,2: /2 width=7 by cpm_delta_drops/ ] #H
+ lapply (tdeq_inv_lref1 … H) -H #H destruct
+ /2 width=5 by lifts_inv_lref2_uni_lt/
+ | elim (lifts_total V 𝐔❴↑i❵) #W #HVW
+ @or_intror @(ex2_2_intro … W) [1,2: /2 width=7 by cpm_ell_drops/ ] #H
+ lapply (tdeq_inv_lref1 … H) -H #H destruct
+ /2 width=5 by lifts_inv_lref2_uni_lt/
+ ]
+ ]
+| #l /3 width=5 by cnh_gref, or_introl/
+| #p * [ cases p ] #V1 #T1
+ [ elim (cpr_subst h G (L.ⓓV1) T1 0 L V1) [| /2 width=1 by drops_refl/ ] #T2 #X2 #HT12 #HXT2
+ elim (tdeq_dec T1 T2) [ -HT12 #HT12 | #HnT12 ]
+ [ elim (tdeq_inv_lifts_dx … HT12 … HXT2) -T2 #X1 #HXT1 #_ -X2
+ @or_intror @(ex2_2_intro … X1) [1,2: /2 width=4 by cpm_zeta/ ] #H
+ /2 width=7 by tdeq_lifts_inv_pair_sn/
+ | @or_intror @(ex2_2_intro … (+ⓓV1.T2)) [1,2: /2 width=2 by cpm_bind/ ] #H
+ elim (tdeq_inv_pair … H) -H /2 width=1 by/
+ ]
+ | /3 width=5 by cnh_abbr_neg, or_introl/
+ | /3 width=5 by cnh_abst, or_introl/
+ ]
+| * #V1 #T1
+ [ elim (simple_dec_ex T1) [ #HT1 | * #p * #W1 #U1 #H destruct ]
+ [ /3 width=5 by cnh_appl_simple, or_introl/
+ | elim (lifts_total V1 𝐔❴1❵) #X1 #HVX1
+ @or_intror @(ex2_2_intro … (ⓓ{p}W1.ⓐX1.U1)) [1,2: /2 width=3 by cpm_theta/ ] #H
+ elim (tdeq_inv_pair … H) -H #H destruct
+ | @or_intror @(ex2_2_intro … (ⓓ{p}ⓝW1.V1.U1)) [1,2: /2 width=2 by cpm_beta/ ] #H
+ elim (tdeq_inv_pair … H) -H #H destruct
+ ]
+ | @or_intror @(ex2_2_intro … T1) [1,2: /2 width=2 by cpm_eps/ ] #H
+ /2 width=4 by tdeq_inv_pair_xy_y/
+ ]
+]
+qed-.