--- /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/grammar/term_vector.ma".
+include "basic_2/computation/csx.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERM VECTORS *************)
+
+definition csxv: ∀h. sd h → relation3 genv lenv (list term) ≝
+ λh,o,G,L. all … (csx h o G L).
+
+interpretation
+ "context-sensitive strong normalization (term vector)"
+ 'SN h o G L Ts = (csxv h o G L Ts).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma csxv_inv_cons: ∀h,o,G,L,T,Ts. ⦃G, L⦄ ⊢ ⬊*[h, o] T @ Ts →
+ ⦃G, L⦄ ⊢ ⬊*[h, o] T ∧ ⦃G, L⦄ ⊢ ⬊*[h, o] Ts.
+normalize // qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma csx_fwd_applv: ∀h,o,G,L,T,Vs. ⦃G, L⦄ ⊢ ⬊*[h, o] Ⓐ Vs.T →
+ ⦃G, L⦄ ⊢ ⬊*[h, o] Vs ∧ ⦃G, L⦄ ⊢ ⬊*[h, o] T.
+#h #o #G #L #T #Vs elim Vs -Vs /2 width=1 by conj/
+#V #Vs #IHVs #HVs
+lapply (csx_fwd_pair_sn … HVs) #HV
+lapply (csx_fwd_flat_dx … HVs) -HVs #HVs
+elim (IHVs HVs) -IHVs -HVs /3 width=1 by conj/
+qed-.