+(**************************************************************************)
+(* ___ *)
+(* ||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 "static_2/syntax/tweq_simple.ma".
+include "basic_2/rt_computation/cpms_cpms.ma".
+include "basic_2/rt_computation/cnuw.ma".
+
+(* NORMAL TERMS FOR T-UNUNBOUND WHD RT-TRANSITION ***************************)
+
+(* Advanced forward lemma with with simple terms ****************************)
+(*
+lemma cnuw_fwd_appl_simple (h) (G) (L):
+ ∀V,T. ⦃G,L⦄ ⊢ ➡𝐍𝐖*[h] ⓐV.T → 𝐒⦃T⦄.
+#h #G #L #V #T #HT
+elim (simple_dec_ex T) [ // ] * #p #I #W #U #H destruct
+*)
+(* Advanced properties with simple terms ************************************)
+
+lemma cnuw_appl_simple (h) (G) (L):
+ ∀V,T. 𝐒⦃T⦄ → ⦃G,L⦄ ⊢ ➡𝐍𝐖*[h] T → ⦃G,L⦄ ⊢ ➡𝐍𝐖*[h] ⓐV.T.
+#h #G #L #V1 #T1 #H1T1 #H2T1 #n #X #H
+elim (cpms_inv_appl_sn … H) -H *
+[ #V2 #T2 #_ #HT12 #H destruct -H1T1
+ /3 width=2 by tweq_appl/
+| #n1 #n2 #p #V2 #T2 #HT12 #_ #_ -n -n2
+ lapply (H2T1 … HT12) -H2T1 -n1 #H
+ lapply (tweq_simple_trans … H H1T1) -H -H1T1 #H
+ elim (simple_inv_bind … H)
+| #n1 #n2 #p #V2 #W2 #W #T2 #_ #_ #HT12 #_ #_ -n -n2 -V2 -W2
+ lapply (H2T1 … HT12) -H2T1 -n1 #H
+ lapply (tweq_simple_trans … H H1T1) -H -H1T1 #H
+ elim (simple_inv_bind … H)
+]
+qed.