+lemma depth_A_sn (q): โญq = โญ(๐โq).
+// qed.
+
+lemma depth_S_sn (q): โญq = โญ(๐ฆโq).
+// qed.
+
+(* Main constructions *******************************************************)
+
+theorem depth_append (p1) (p2):
+ (โญp2)+(โญp1) = โญ(p1โp2).
+#p1 elim p1 -p1 //
+* [ #n ] #p1 #IH #p2 <list_append_lcons_sn
+[ <depth_d_sn <depth_d_sn //
+| <depth_m_sn <depth_m_sn //
+| <depth_L_sn <depth_L_sn //
+| <depth_A_sn <depth_A_sn //
+| <depth_S_sn <depth_S_sn //
+]
+qed.
+
+(* Constructions with list_rcons ********************************************)
+
+lemma depth_d_dx (p) (n):
+ โญp = โญ(pโ๐ฑn).
+// qed.
+
+lemma depth_m_dx (p):
+ โญp = โญ(pโ๐บ).
+// qed.
+
+lemma depth_L_dx (p):
+ โโญp = โญ(pโ๐).
+// qed.
+
+lemma depth_A_dx (p):
+ โญp = โญ(pโ๐).
+// qed.
+
+lemma depth_S_dx (p):
+ โญp = โญ(pโ๐ฆ).