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diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_theq_vector.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_theq_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/computation/gcp_cr.ma".
+include "basic_2/computation/cpxs_theq_vector.ma".
+include "basic_2/computation/csx_lpx.ma".
+include "basic_2/computation/csx_vector.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERM VECTORS *************)
+
+(* Advanced properties ******************************************************)
+
+(* Basic_1: was just: sn3_appls_lref *)
+lemma csx_applv_cnx: ∀h,o,G,L,T. 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ →
+                     ∀Vs. ⦃G, L⦄ ⊢ ⬊*[h, o] Vs → ⦃G, L⦄ ⊢ ⬊*[h, o] ⒶVs.T.
+#h #o #G #L #T #H1T #H2T #Vs elim Vs -Vs [ #_ @(cnx_csx … H2T) ] (**) (* /2 width=1/ does not work *)
+#V #Vs #IHV #H
+elim (csxv_inv_cons … H) -H #HV #HVs
+@csx_appl_simple_theq /2 width=1 by applv_simple/ -IHV -HV -HVs
+#X #H #H0
+lapply (cpxs_fwd_cnx_vector … H) -H // -H1T -H2T #H
+elim (H0) -H0 //
+qed.
+
+lemma csx_applv_sort: ∀h,o,G,L,s,Vs. ⦃G, L⦄ ⊢ ⬊*[h, o] Vs → ⦃G, L⦄ ⊢ ⬊*[h, o] ⒶVs.⋆s.
+#h #o #G #L #s elim (deg_total h o s)
+#d generalize in match s; -s @(nat_ind_plus … d) -d [ /3 width=6 by csx_applv_cnx, cnx_sort, simple_atom/ ]
+#d #IHd #s #Hkd lapply (deg_next_SO … Hkd) -Hkd
+#Hkd #Vs elim Vs -Vs /2 width=1 by/
+#V #Vs #IHVs #HVVs
+elim (csxv_inv_cons … HVVs) #HV #HVs
+@csx_appl_simple_theq /2 width=1 by applv_simple, simple_atom/ -IHVs -HV -HVs
+#X #H #H0
+elim (cpxs_fwd_sort_vector … H) -H #H
+[ elim H0 -H0 //
+| -H0 @(csx_cpxs_trans … (Ⓐ(V@Vs).⋆(next h s))) /2 width=1 by cpxs_flat_dx/
+]
+qed.
+
+(* Basic_1: was just: sn3_appls_beta *)
+lemma csx_applv_beta: ∀h,o,a,G,L,Vs,V,W,T. ⦃G, L⦄ ⊢ ⬊*[h, o] ⒶVs.ⓓ{a}ⓝW.V.T →
+                      ⦃G, L⦄ ⊢ ⬊*[h, o] ⒶVs. ⓐV.ⓛ{a}W.T.
+#h #o #a #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 … 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. ⬆[0, i + 1] V1 ≡ V2 →
+                       ∀Vs. ⦃G, L⦄ ⊢ ⬊*[h, o] (ⒶVs.V2) → ⦃G, L⦄ ⊢ ⬊*[h, o] (ⒶVs.#i).
+#h #o #I #G #L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs
+[ /4 width=12 by csx_inv_lift, csx_lref_bind, drop_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_vector … 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,a,G,L,V1b,V2b. ⬆[0, 1] V1b ≡ V2b →
+                       ∀V,T. ⦃G, L⦄ ⊢ ⬊*[h, o] ⓓ{a}V.ⒶV2b.T →
+                       ⦃G, L⦄ ⊢ ⬊*[h, o] ⒶV1b.ⓓ{a}V.T.
+#h #o #a #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 … HW12 … H) -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 … (V2@V2b) … H1) -H1 /2 width=1 by liftv_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,W,T. ⦃G, L⦄ ⊢ ⬊*[h, o] ⒶVs.W → ⦃G, L⦄ ⊢ ⬊*[h, o] ⒶVs.T →
+                      ⦃G, L⦄ ⊢ ⬊*[h, o] ⒶVs.ⓝW.T.
+#h #o #G #L #Vs elim Vs -Vs /2 width=1 by csx_cast/
+#V #Vs #IHV #W #T #H1W #H1T
+lapply (csx_fwd_pair_sn … H1W) #HV
+lapply (csx_fwd_flat_dx … H1W) #H2W
+lapply (csx_fwd_flat_dx … H1T) #H2T
+@csx_appl_simple_theq /2 width=1 by applv_simple, simple_flat/ -IHV -HV -H2W -H2T
+#X #H #H0
+elim (cpxs_fwd_cast_vector … H) -H #H
+[ -H1W -H1T elim H0 -H0 //
+| -H1W -H0 /3 width=5 by csx_cpxs_trans, cpxs_flat_dx/
+| -H1T -H0 /3 width=5 by csx_cpxs_trans, cpxs_flat_dx/
+]
+qed.
+
+theorem csx_gcr: ∀h,o. gcr (cpx h o) (eq …) (csx h o) (csx h o).
+#h #o @mk_gcr //
+[ /3 width=1 by csx_applv_cnx/
+|2,3,6: /2 width=1 by csx_applv_beta, csx_applv_sort, csx_applv_cast/
+| /2 width=7 by csx_applv_delta/
+| #G #L #V1b #V2b #HV12b #a #V #T #H #HV
+  @(csx_applv_theta … HV12b) -HV12b
+  @csx_abbr //
+]
+qed.