+lemma depth_A_dx (p):
+ โญp = โญ(pโ๐).
+// qed.
+
+lemma depth_S_dx (p):
+ โญp = โญ(pโ๐ฆ).
+// qed.
+
+(* Main constructions *******************************************************)
+
+theorem depth_append (p) (q):
+ (โญp)+(โญq) = โญ(pโq).
+#p #q elim q -q //
+* [ #k ] #q #IH <list_append_lcons_sn
+[ <depth_d_dx <depth_d_dx //
+| <depth_m_dx <depth_m_dx //
+| <depth_L_dx <depth_L_dx //
+| <depth_A_dx <depth_A_dx //
+| <depth_S_dx <depth_S_dx //
+]
+qed.
+
+(* Constructions with path_lcons ********************************************)
+
+lemma depth_d_sn (p) (k):
+ โญp = โญ(๐ฑkโp).
+// qed.
+
+lemma depth_m_sn (p):
+ โญp = โญ(๐บโp).
+// qed.
+
+lemma depth_L_sn (p):
+ โโญp = โญ(๐โp).
+// qed.
+
+lemma depth_A_sn (p):
+ โญp = โญ(๐โp).
+// qed.
+
+lemma depth_S_sn (p):
+ โญp = โญ(๐ฆโp).