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diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_toeq_vector.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_toeq_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 "static_2/syntax/toeq_simple_vector.ma".
+include "static_2/relocation/lifts_vector.ma".
+include "basic_2/rt_computation/cpxs_toeq.ma".
+
+(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS **************)
+
+(* Vector form of forward lemmas with outer equivalence for terms ***********)
+
+lemma cpxs_fwd_sort_vector (h) (G) (L):
+      ∀s,Vs,X2. ⦃G,L⦄ ⊢ ⒶVs.⋆s ⬈*[h] X2 → ⒶVs.⋆s ⩳ X2.
+#h #G #L #s #Vs elim Vs -Vs /2 width=4 by cpxs_fwd_sort/
+#V #Vs #IHVs #X2 #H
+elim (cpxs_inv_appl1 … H) -H *
+[ -IHVs #V1 #T1 #_ #_ #H destruct /2 width=1 by toeq_pair/
+| #p #W1 #T1 #HT1 #HU
+  lapply (IHVs … HT1) -IHVs -HT1 #HT1
+  elim (toeq_inv_applv_bind_simple … HT1) //
+| #p #V1 #V2 #V3 #T1 #HV01 #HV12 #HT1 #HU
+  lapply (IHVs … HT1) -IHVs -HT1 #HT1
+  elim (toeq_inv_applv_bind_simple … HT1) //
+]
+qed-.
+
+(* Basic_2A1: was: cpxs_fwd_delta_vector *)
+lemma cpxs_fwd_delta_drops_vector (h) (I) (G) (L) (K):
+      ∀V1,i. ⬇*[i] L ≘ K.ⓑ{I}V1 →
+      ∀V2. ⬆*[↑i] V1 ≘ V2 →
+      ∀Vs,X2. ⦃G,L⦄ ⊢ ⒶVs.#i ⬈*[h] X2 →
+      ∨∨ ⒶVs.#i ⩳ X2 | ⦃G,L⦄ ⊢ ⒶVs.V2 ⬈*[h] X2.
+#h #I #G #L #K #V1 #i #HLK #V2 #HV12 #Vs
+elim Vs -Vs /2 width=5 by cpxs_fwd_delta_drops/
+#V #Vs #IHVs #X2 #H -K -V1
+elim (cpxs_inv_appl1 … H) -H *
+[ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1 by toeq_pair, or_introl/
+| #q #W0 #T0 #HT0 #HU
+  elim (IHVs … HT0) -IHVs -HT0 #HT0
+  [ elim (toeq_inv_applv_bind_simple … HT0) //
+  | @or_intror -i (**) (* explicit constructor *)
+    @(cpxs_trans … HU) -X2
+    @(cpxs_strap1 … (ⓐV.ⓛ{q}W0.T0)) /3 width=1 by cpxs_flat_dx, cpx_beta/
+  ]
+| #q #V0 #V1 #V3 #T0 #HV0 #HV01 #HT0 #HU
+  elim (IHVs … HT0) -IHVs -HT0 #HT0
+  [ elim (toeq_inv_applv_bind_simple … HT0) //
+  | @or_intror -i (**) (* explicit constructor *)
+    @(cpxs_trans … HU) -X2
+    @(cpxs_strap1 … (ⓐV0.ⓓ{q}V3.T0)) /3 width=3 by cpxs_flat, cpx_theta/
+  ]
+]
+qed-.
+
+(* Basic_1: was just: pr3_iso_appls_beta *)
+lemma cpxs_fwd_beta_vector (h) (p) (G) (L):
+      ∀Vs,V,W,T,X2. ⦃G,L⦄ ⊢ ⒶVs.ⓐV.ⓛ{p}W.T ⬈*[h] X2 →
+      ∨∨ ⒶVs.ⓐV.ⓛ{p}W. T ⩳ X2 | ⦃G,L⦄ ⊢ ⒶVs.ⓓ{p}ⓝW.V.T ⬈*[h] X2.
+#h #p #G #L #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_beta/
+#V0 #Vs #IHVs #V #W #T #X2 #H
+elim (cpxs_inv_appl1 … H) -H *
+[ -IHVs #V1 #T1 #_ #_ #H destruct /2 width=1 by toeq_pair, or_introl/
+| #q #W1 #T1 #HT1 #HU
+  elim (IHVs … HT1) -IHVs -HT1 #HT1
+  [ elim (toeq_inv_applv_bind_simple … HT1) //
+  | @or_intror (**) (* explicit constructor *)
+    @(cpxs_trans … HU) -X2
+    @(cpxs_strap1 … (ⓐV0.ⓛ{q}W1.T1)) /3 width=1 by cpxs_flat_dx, cpx_beta/
+  ]
+| #q #V1 #V2 #V3 #T1 #HV01 #HV12 #HT1 #HU
+  elim (IHVs … HT1) -IHVs -HT1 #HT1
+  [ elim (toeq_inv_applv_bind_simple … HT1) //
+  | @or_intror (**) (* explicit constructor *)
+    @(cpxs_trans … HU) -X2
+    @(cpxs_strap1 … (ⓐV1.ⓓ{q}V3.T1)) /3 width=3 by cpxs_flat, cpx_theta/
+  ]
+]
+qed-.
+
+(* Basic_1: was just: pr3_iso_appls_abbr *)
+lemma cpxs_fwd_theta_vector (h) (G) (L):
+      ∀V1b,V2b. ⬆*[1] V1b ≘ V2b →
+      ∀p,V,T,X2. ⦃G,L⦄ ⊢ ⒶV1b.ⓓ{p}V.T ⬈*[h] X2 →
+      ∨∨ ⒶV1b.ⓓ{p}V.T ⩳ X2 | ⦃G,L⦄ ⊢ ⓓ{p}V.ⒶV2b.T ⬈*[h] X2.
+#h #G #L #V1b #V2b * -V1b -V2b /3 width=1 by or_intror/
+#V1b #V2b #V1a #V2a #HV12a #HV12b #p
+generalize in match HV12a; -HV12a
+generalize in match V2a; -V2a
+generalize in match V1a; -V1a
+elim HV12b -V1b -V2b /2 width=1 by cpxs_fwd_theta/
+#V1b #V2b #V1b #V2b #HV12b #_ #IHV12b #V1a #V2a #HV12a #V #T #X2 #H
+elim (cpxs_inv_appl1 … H) -H *
+[ -IHV12b -HV12a -HV12b #V0 #T0 #_ #_ #H destruct /2 width=1 by toeq_pair, or_introl/
+| #q #W0 #T0 #HT0 #HU
+  elim (IHV12b … HV12b … HT0) -IHV12b -HT0 #HT0
+  [ -HV12a -HV12b -HU
+    elim (toeq_inv_pair1 … HT0) #V1 #T1 #H destruct
+  | @or_intror -V1b (**) (* explicit constructor *)
+    @(cpxs_trans … HU) -X2
+    elim (cpxs_inv_abbr1_dx … HT0) -HT0 *
+    [ -HV12a #V1 #T1 #_ #_ #H destruct
+    | -V1b #X #HT1 #H #H0 destruct
+      elim (lifts_inv_bind1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct
+      @(cpxs_trans … (+ⓓV.ⓐV2a.ⓛ{q}W1.T1)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T -V2b -V2b
+      @(cpxs_strap2 … (ⓐV1a.ⓛ{q}W0.T0))
+      /4 width=7 by cpxs_beta_dx, cpx_zeta, lifts_bind, lifts_flat/
+    ]
+  ]
+| #q #V0a #Va #V0 #T0 #HV10a #HV0a #HT0 #HU
+  elim (IHV12b … HV12b … HT0) -HV12b -IHV12b -HT0 #HT0
+  [ -HV12a -HV10a -HV0a -HU
+    elim (toeq_inv_pair1 … HT0) #V1 #T1 #H destruct
+  | @or_intror -V1b -V1b (**) (* explicit constructor *)
+    @(cpxs_trans … HU) -X2
+    elim (cpxs_inv_abbr1_dx … HT0) -HT0 *
+    [ #V1 #T1 #HV1 #HT1 #H destruct
+      lapply (cpxs_lifts_bi … HV10a (Ⓣ) … (L.ⓓV) … HV12a … HV0a) -V1a -V0a /3 width=1 by drops_refl, drops_drop/ #HV2a
+      @(cpxs_trans … (ⓓ{p}V.ⓐV2a.T1)) /3 width=1 by cpxs_bind, cpxs_pair_sn, cpxs_flat_dx, cpxs_bind_dx/
+    | #X #HT1 #H #H0 destruct
+      elim (lifts_inv_bind1 … H) -H #V1 #T1 #HW01 #HT01 #H destruct
+      lapply (cpxs_lifts_bi … HV10a (Ⓣ) … (L.ⓓV0) … HV12a … HV0a) -V0a /3 width=1 by drops_refl, drops_drop/ #HV2a
+      @(cpxs_trans … (+ⓓV.ⓐV2a.ⓓ{q}V1.T1)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T -V2b -V2b
+      @(cpxs_strap2 … (ⓐV1a.ⓓ{q}V0.T0)) [ /4 width=7 by cpx_zeta, lifts_bind, lifts_flat/ ] -V -V1 -T1
+      @(cpxs_strap2 … (ⓓ{q}V0.ⓐV2a.T0)) /3 width=3 by cpxs_pair_sn, cpxs_bind_dx, cpx_theta/
+    ]
+  ]
+]
+qed-.
+
+(* Basic_1: was just: pr3_iso_appls_cast *)
+lemma cpxs_fwd_cast_vector (h) (G) (L):
+      ∀Vs,W,T,X2. ⦃G,L⦄ ⊢ ⒶVs.ⓝW.T ⬈*[h] X2 →
+      ∨∨ ⒶVs. ⓝW. T ⩳ X2
+       | ⦃G,L⦄ ⊢ ⒶVs.T ⬈*[h] X2
+       | ⦃G,L⦄ ⊢ ⒶVs.W ⬈*[h] X2.
+#h #G #L #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_cast/
+#V #Vs #IHVs #W #T #X2 #H
+elim (cpxs_inv_appl1 … H) -H *
+[ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1 by toeq_pair, or3_intro0/
+| #q #W0 #T0 #HT0 #HU elim (IHVs … HT0) -IHVs -HT0 #HT0
+  [ elim (toeq_inv_applv_bind_simple … HT0) //
+  | @or3_intro1 -W (**) (* explicit constructor *)
+    @(cpxs_trans … HU) -X2
+    @(cpxs_strap1 … (ⓐV.ⓛ{q}W0.T0)) /2 width=1 by cpxs_flat_dx, cpx_beta/
+  | @or3_intro2 -T (**) (* explicit constructor *)
+    @(cpxs_trans … HU) -X2
+    @(cpxs_strap1 … (ⓐV.ⓛ{q}W0.T0)) /2 width=1 by cpxs_flat_dx, cpx_beta/
+  ]
+| #q #V0 #V1 #V2 #T0 #HV0 #HV01 #HT0 #HU
+  elim (IHVs … HT0) -IHVs -HT0 #HT0
+  [ elim (toeq_inv_applv_bind_simple … HT0) //
+  | @or3_intro1 -W (**) (* explicit constructor *)
+    @(cpxs_trans … HU) -X2
+    @(cpxs_strap1 … (ⓐV0.ⓓ{q}V2.T0)) /2 width=3 by cpxs_flat, cpx_theta/
+  | @or3_intro2 -T (**) (* explicit constructor *)
+    @(cpxs_trans … HU) -X2
+    @(cpxs_strap1 … (ⓐV0.ⓓ{q}V2.T0)) /2 width=3 by cpxs_flat, cpx_theta/
+  ]
+]
+qed-.
+
+(* Basic_1: was just: nf2_iso_appls_lref *)
+lemma cpxs_fwd_cnx_vector (h) (G) (L):
+      ∀T. 𝐒⦃T⦄ → ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃T⦄ →
+      ∀Vs,X2. ⦃G,L⦄ ⊢ ⒶVs.T ⬈*[h] X2 → ⒶVs.T ⩳ X2.
+#h #G #L #T #H1T #H2T #Vs elim Vs -Vs [ @(cpxs_fwd_cnx … H2T) ] (**) (* /2 width=3 by cpxs_fwd_cnx/ does not work *)
+#V #Vs #IHVs #X2 #H
+elim (cpxs_inv_appl1 … H) -H *
+[ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1 by toeq_pair/
+| #p #W0 #T0 #HT0 #HU
+  lapply (IHVs … HT0) -IHVs -HT0 #HT0
+  elim (toeq_inv_applv_bind_simple … HT0) //
+| #p #V1 #V2 #V0 #T0 #HV1 #HV12 #HT0 #HU
+  lapply (IHVs … HT0) -IHVs -HT0 #HT0
+  elim (toeq_inv_applv_bind_simple … HT0) //
+]
+qed-.