+lemma cpxs_fwd_sort_vector: ∀h,g,L,k,Vs,U. ⦃h, L⦄ ⊢ ⒶVs.⋆k ➡*[g] U →
+ ⒶVs.⋆k ≃ U ∨ ⦃h, L⦄ ⊢ ⒶVs.⋆(next h k) ➡*[g] U.
+#h #g #L #k #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_sort/
+#V #Vs #IHVs #U #H
+elim (cpxs_inv_appl1 … H) -H *
+[ -IHVs #V1 #T1 #_ #_ #H destruct /2 width=1/
+| #a #W1 #T1 #HT1 #HU
+ elim (IHVs … HT1) -IHVs -HT1 #HT1
+ [ elim (tstc_inv_bind_appls_simple … HT1) //
+ | @or_intror (**) (* explicit constructor *)
+ @(cpxs_trans … HU) -U
+ @(cpxs_strap1 … (ⓐV.ⓛ{a}W1.T1)) [ /2 width=1/ ] -Vs /3 width=1/
+ ]
+| #a #V1 #V2 #V3 #T1 #HV01 #HV12 #HT1 #HU
+ elim (IHVs … HT1) -IHVs -HT1 #HT1
+ [ elim (tstc_inv_bind_appls_simple … HT1) //
+ | @or_intror (**) (* explicit constructor *)
+ @(cpxs_trans … HU) -U
+ @(cpxs_strap1 … (ⓐV1.ⓓ{a}V3.T1)) [ /2 width=1/ ] -Vs /3 width=3/
+ ]
+]
+qed-.
+
+