-lemma path_closed_depth (p) (n):
- p ϵ 𝐂❨n❩ → ♯p + n = ♭p.
-#p #n #Hn elim Hn -Hn //
-#p #n #_ #IH <nplus_succ_dx //
-qed.
+lemma path_closed_des_depth (o) (q) (n):
+ q ϵ 𝐂❨o,n❩ → ♯q + n = ♭q.
+#o #q #n #Hq elim Hq -q -n //
+#q #n #_ #IH <nplus_succ_dx //
+qed-.
+
+lemma path_closed_des_succ_depth (o) (q) (n):
+ q ϵ 𝐂❨o,↑n❩ → ♭q = ↑↓♭q.
+#o #q #n #Hq
+<(path_closed_des_depth … Hq) -Hq
+<nplus_succ_dx <npred_succ //
+qed-.