+lemma cpxs_fwd_sort: ∀h,g,L,U,k. ⦃h, L⦄ ⊢ ⋆k ➡*[g] U →
+ ⋆k ≃ U ∨ ⦃h, L⦄ ⊢ ⋆(next h k) ➡*[g] U.
+#h #g #L #U #k #H
+elim (cpxs_inv_sort1 … H) -H #n #l generalize in match k; -k @(nat_ind_plus … n) -n
+[ #k #_ #H -l destruct /2 width=1/
+| #n #IHn #k >plus_plus_comm_23 #Hnl #H destruct
+ lapply (deg_next_SO … Hnl) -Hnl #Hnl
+ elim (IHn … Hnl) -IHn
+ [ #H lapply (tstc_inv_atom1 … H) -H #H >H -H /2 width=1/
+ | generalize in match Hnl; -Hnl @(nat_ind_plus … n) -n /2 width=1/
+ #n #_ /4 width=3/
+ | >iter_SO >iter_n_Sm //
+ ]
+]
+qed-.
+