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diff --git a/matita/matita/contribs/lambda_delta/basic_2/computation/csn_lcpr_vector.ma b/matita/matita/contribs/lambda_delta/basic_2/computation/csn_lcpr_vector.ma
new file mode 100644
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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/computation/acp_cr.ma".
+include "Basic_2/computation/cprs_tstc_vector.ma".
+include "Basic_2/computation/csn_lcpr.ma".
+include "Basic_2/computation/csn_vector.ma".
+
+(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERM VECTORS **********************)
+
+(* Advanced properties ******************************************************)
+(*
+(* Basic_1: was only: sn3_appl_appls *)
+lemma csn_appl_appls_simple_tstc: ∀L,Vs,V,T1. L ⊢ ⬇* V → L ⊢ ⬇* T1 →
+                                  (∀T2. L ⊢ ⒶVs.T1 ➡* T2 → (ⒶVs.T1 ≃ T2 → False) → L ⊢ ⬇* ⓐV. T2) →
+                                  𝐒[T1] → L ⊢ ⬇* ⓐV. ⒶVs. T1.
+#L *
+[ #V #T1 #HV
+  @csn_appl_simple_tstc //
+| #V0 #Vs #V #T1 #HV #H1T1 #H2T1 #H3T1
+  @csn_appl_simple_tstc // -HV
+  [ @H2T1
+]
+qed.
+*)
+lemma csn_applv_theta: ∀L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s →
+                       ∀V,T. L ⊢ ⬇* ⓓV. ⒶV2s. T → L ⊢ ⬇* V →
+                       L ⊢ ⬇* ⒶV1s. ⓓV. T.
+#L #V1s #V2s * -V1s -V2s /2 width=1/
+#V1s #V2s #V1 #V2 #HV12 #H 
+generalize in match HV12; -HV12 generalize in match V2; -V2 generalize in match V1; -V1
+elim H -V1s -V2s /2 width=3/
+#V1s #V2s #V1 #V2 #HV12 #HV12s #IHV12s #W1 #W2 #HW12 #V #T #H #HV
+lapply (csn_appl_theta … HW12 … H) -H -HW12 #H
+lapply (csn_fwd_pair_sn … H) #HW1
+lapply (csn_fwd_flat_dx … H) #H1
+@csn_appl_simple_tstc // -HW1 /2 width=3/ -IHV12s -HV -H1 #X #H1 #H2
+elim (cprs_fwd_theta_vector … (V2::V2s) … H1) -H1 /2 width=1/ -HV12s -HV12
+[ -H #H elim (H2 ?) -H2 //
+| -H2 #H1 @(csn_cprs_trans … H) -H /2 width=1/
+]
+qed.
+
+lemma csn_applv_tau: ∀L,W. L ⊢ ⬇* W →
+                     ∀Vs,T. L ⊢ ⬇* ⒶVs. T →
+                     L ⊢ ⬇* ⒶVs. ⓣW. T.
+#L #W #HW #Vs elim Vs -Vs /2 width=1/ -HW
+#V #Vs #IHV #T #H1T
+lapply (csn_fwd_pair_sn … H1T) #HV
+lapply (csn_fwd_flat_dx … H1T) #H2T
+@csn_appl_simple_tstc // -HV /2 width=1/ -IHV -H2T
+[ #X #H #H0
+  elim (cprs_fwd_tau_vector … H) -H #H
+  [ -H1T elim (H0 ?) -H0 //
+  | -H0 @(csn_cprs_trans … H1T) -H1T /2 width=1/
+  ]
+| -H1T elim Vs -Vs //
+]
+qed.
+(*
+theorem csn_acr: acr cpr (eq …) (csn …) (λL,T. L ⊢ ⬇* T).
+@mk_acr //
+[
+|
+|
+| #L #V1 #V2 #HV12 #V #T #H #HVT
+  @(csn_applv_theta … HV12) -HV12 //
+  @(csn_abbr) //
+| /2 width=1/
+| @csn_lift
+]
+qed.
+*)
+axiom csn_acr: acr cpr (eq …) (csn …) (λL,T. L ⊢ ⬇* T).