-lemma cpxs_fwd_sort: ∀h,g,G,L,U,k. ⦃G, L⦄ ⊢ ⋆k ➡*[h, g] U →
- ⋆k ≂ U ∨ ⦃G, L⦄ ⊢ ⋆(next h k) ➡*[h, g] U.
-#h #g #G #L #U #k #H
-elim (cpxs_inv_sort1 … H) -H #n #d generalize in match k; -k @(nat_ind_plus … n) -n
-[ #k #_ #H -d destruct /2 width=1 by or_introl/
-| #n #IHn #k >plus_plus_comm_23 #Hnd #H destruct
+lemma cpxs_fwd_sort: ∀h,o,G,L,U,s. ⦃G, L⦄ ⊢ ⋆s ➡*[h, o] U →
+ ⋆s ≂ U ∨ ⦃G, L⦄ ⊢ ⋆(next h s) ➡*[h, o] U.
+#h #o #G #L #U #s #H
+elim (cpxs_inv_sort1 … H) -H #n #d generalize in match s; -s @(nat_ind_plus … n) -n
+[ #s #_ #H -d destruct /2 width=1 by or_introl/
+| #n #IHn #s >plus_plus_comm_23 #Hnd #H destruct