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diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_csx_vector.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_csx_vector.ma
new file mode 100644
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+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_csx_vector.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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_computation/cpxs_theq_vector.ma".
+include "basic_2/rt_computation/csx_simple_theq.ma".
+include "basic_2/rt_computation/csx_lsubr.ma".
+include "basic_2/rt_computation/csx_lfpx.ma".
+include "basic_2/rt_computation/csx_vector.ma".
+
+(* STRONGLY NORMALIZING TERM VECTORS FOR UNCOUNTED PARALLEL RT-TRANSITION ***)
+
+(* Advanced properties ************************************* ****************)
+
+(* Basic_1: was just: sn3_appls_beta *)
+lemma csx_applv_beta: ∀h,o,p,G,L,Vs,V,W,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⒶVs.ⓓ{p}ⓝW.V.T⦄ →
+                      ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⒶVs.ⓐV.ⓛ{p}W.T⦄.
+#h #o #p #G #L #Vs elim Vs -Vs /2 width=1 by csx_appl_beta/
+#V0 #Vs #IHV #V #W #T #H1T
+lapply (csx_fwd_pair_sn … H1T) #HV0
+lapply (csx_fwd_flat_dx … H1T) #H2T
+@csx_appl_simple_theq /2 width=1 by applv_simple, simple_flat/ -IHV -HV0 -H2T
+#X #H #H0
+elim (cpxs_fwd_beta_vector … o … H) -H #H
+[ -H1T elim H0 -H0 //
+| -H0 /3 width=5 by csx_cpxs_trans, cpxs_flat_dx/
+]
+qed.
+
+lemma csx_applv_delta: ∀h,o,I,G,L,K,V1,i. ⬇*[i] L ≡ K.ⓑ{I}V1 →
+                       ∀V2. ⬆*[⫯i] V1 ≡ V2 →
+                       ∀Vs. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⒶVs.V2⦄ → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⒶVs.#i⦄.
+#h #o #I #G #L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs
+[ /4 width=11 by csx_inv_lifts, csx_lref_drops, drops_isuni_fwd_drop2/
+| #V #Vs #IHV #H1T
+  lapply (csx_fwd_pair_sn … H1T) #HV
+  lapply (csx_fwd_flat_dx … H1T) #H2T
+  @csx_appl_simple_theq /2 width=1 by applv_simple, simple_atom/ -IHV -HV  -H2T
+  #X #H #H0
+  elim (cpxs_fwd_delta_drops_vector … o … HLK … HV12 … H) -HLK -HV12 -H #H
+  [ -H1T elim H0 -H0 //
+  | -H0 /3 width=5 by csx_cpxs_trans, cpxs_flat_dx/
+  ]
+]
+qed.
+
+(* Basic_1: was just: sn3_appls_abbr *)
+lemma csx_applv_theta: ∀h,o,p,G,L,V1b,V2b. ⬆*[1] V1b ≡ V2b →
+                       ∀V,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓓ{p}V.ⒶV2b.T⦄ →
+                       ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⒶV1b.ⓓ{p}V.T⦄.
+#h #o #p #G #L #V1b #V2b * -V1b -V2b /2 width=1 by/
+#V1b #V2b #V1 #V2 #HV12 #H
+generalize in match HV12; -HV12 generalize in match V2; -V2 generalize in match V1; -V1
+elim H -V1b -V2b /2 width=3 by csx_appl_theta/
+#V1b #V2b #V1 #V2 #HV12 #HV12b #IHV12b #W1 #W2 #HW12 #V #T #H
+lapply (csx_appl_theta … H … HW12) -H -HW12 #H
+lapply (csx_fwd_pair_sn … H) #HW1
+lapply (csx_fwd_flat_dx … H) #H1
+@csx_appl_simple_theq /2 width=3 by simple_flat/ -IHV12b -HW1 -H1 #X #H1 #H2
+elim (cpxs_fwd_theta_vector … o … (V2@V2b) … H1) -H1 /2 width=1 by liftsv_cons/ -HV12b -HV12
+[ -H #H elim H2 -H2 //
+| -H2 /3 width=5 by csx_cpxs_trans, cpxs_flat_dx/
+]
+qed.
+
+(* Basic_1: was just: sn3_appls_cast *)
+lemma csx_applv_cast: ∀h,o,G,L,Vs,U. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⒶVs.U⦄ →
+                      ∀T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⒶVs.T⦄ → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⒶVs.ⓝU.T⦄.
+#h #o #G #L #Vs elim Vs -Vs /2 width=1 by csx_cast/
+#V #Vs #IHV #U #H1U #T #H1T
+lapply (csx_fwd_pair_sn … H1U) #HV
+lapply (csx_fwd_flat_dx … H1U) #H2U
+lapply (csx_fwd_flat_dx … H1T) #H2T
+@csx_appl_simple_theq /2 width=1 by applv_simple, simple_flat/ -IHV -HV -H2U -H2T
+#X #H #H0
+elim (cpxs_fwd_cast_vector … o … H) -H #H
+[ -H1U -H1T elim H0 -H0 //
+| -H1U -H0 /3 width=5 by csx_cpxs_trans, cpxs_flat_dx/
+| -H1T -H0 /3 width=5 by csx_cpxs_trans, cpxs_flat_dx/
+]
+qed.