+(**************************************************************************)
+(* ___ *)
+(* ||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/notation/relations/predtynormal_5.ma".
+include "basic_2/syntax/tdeq.ma".
+include "basic_2/rt_transition/cpx.ma".
+
+(* NORMAL TERMS FOR UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-TRANSITION ******)
+
+definition cnx: ∀h. sd h → relation3 genv lenv term ≝
+ λh,o,G,L. NF … (cpx h G L) (tdeq h o …).
+
+interpretation
+ "normality for uncounted context-sensitive parallel rt-transition (term)"
+ 'PRedTyNormal h o G L T = (cnx h o G L T).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma cnx_inv_sort: ∀h,o,G,L,s. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃⋆s⦄ → deg h o s 0.
+#h #o #G #L #s #H
+lapply (H (⋆(next h s)) ?) -H /2 width=2 by cpx_ess/ -G -L #H
+elim (tdeq_inv_sort1 … H) -H #s0 #d #H1 #H2 #H destruct
+lapply (deg_next … H1) #H0
+lapply (deg_mono … H0 … H2) -H0 -H2 #H
+<(pred_inv_refl … H) -H //
+qed-.
+
+lemma cnx_inv_abst: ∀h,o,p,G,L,V,T. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃ⓛ{p}V.T⦄ →
+ ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃V⦄ ∧ ⦃G, L.ⓛV⦄ ⊢ ⬈[h, o] 𝐍⦃T⦄.
+#h #o #p #G #L #V1 #T1 #HVT1 @conj
+[ #V2 #HV2 lapply (HVT1 (ⓛ{p}V2.T1) ?) -HVT1 /2 width=2 by cpx_pair_sn/ -HV2
+| #T2 #HT2 lapply (HVT1 (ⓛ{p}V1.T2) ?) -HVT1 /2 width=2 by cpx_bind/ -HT2
+]
+#H elim (tdeq_inv_pair … H) -H //
+qed-.
+
+(* Basic_2A1: was: cnx_inv_abbr *)
+lemma cnx_inv_abbr_neg: ∀h,o,G,L,V,T. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃-ⓓV.T⦄ →
+ ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃V⦄ ∧ ⦃G, L.ⓓV⦄ ⊢ ⬈[h, o] 𝐍⦃T⦄.
+#h #o #G #L #V1 #T1 #HVT1 @conj
+[ #V2 #HV2 lapply (HVT1 (-ⓓV2.T1) ?) -HVT1 /2 width=2 by cpx_pair_sn/ -HV2
+| #T2 #HT2 lapply (HVT1 (-ⓓV1.T2) ?) -HVT1 /2 width=2 by cpx_bind/ -HT2
+]
+#H elim (tdeq_inv_pair … H) -H //
+qed-.
+
+(* Basic_2A1: was: cnx_inv_eps *)
+lemma cnx_inv_cast: ∀h,o,G,L,V,T. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃ⓝV.T⦄ → ⊥.
+#h #o #G #L #V #T #H lapply (H T ?) -H
+/2 width=6 by cpx_eps, tdeq_inv_pair_xy_y/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma cnx_sort: ∀h,o,G,L,s. deg h o s 0 → ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃⋆s⦄.
+#h #o #G #L #s #Hs #X #H elim (cpx_inv_sort1 … H) -H
+/3 width=3 by tdeq_sort, deg_next/
+qed.
+
+lemma cnx_sort_iter: ∀h,o,G,L,s,d. deg h o s d → ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃⋆((next h)^d s)⦄.
+#h #o #G #L #s #d #Hs lapply (deg_iter … d Hs) -Hs
+<minus_n_n /2 width=6 by cnx_sort/
+qed.
+
+lemma cnx_abst: ∀h,o,p,G,L,W,T. ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃W⦄ → ⦃G, L.ⓛW⦄ ⊢ ⬈[h, o] 𝐍⦃T⦄ →
+ ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃ⓛ{p}W.T⦄.
+#h #o #p #G #L #W #T #HW #HT #X #H
+elim (cpx_inv_abst1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
+@tdeq_pair [ @HW | @HT ] // (**) (* auto fails because δ-expansion gets in the way *)
+qed.