--- /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/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.