+theorem pcc_append_bi (o1) (o2) (p) (q) (m) (n):
+ p ϵ 𝐂❨o1,m❩ → q ϵ 𝐂❨o2,n❩ → p●q ϵ 𝐂❨o1∧o2,m+n❩.
+#o1 #o2 #p #q #m #n #Hm #Hn elim Hn -q -n
+/2 width=1 by pcc_m_dx, pcc_A_dx, pcc_S_dx, pcc_land_dx/
+#q #n [ #k #Ho2 ] #_ #IH
+[ @pcc_d_dx // #H0
+ elim (andb_inv_true_sn … H0) -H0 #_ #H0 >Ho2 //
+ <nplus_succ_dx <npred_succ //
+| <nplus_succ_dx /2 width=1 by pcc_L_dx/
+]
+qed.
+
+(* Inversions with path_append **********************************************)
+
+lemma pcc_false_inv_append_bi (x) (m) (n):
+ x ϵ 𝐂❨Ⓕ,m+n❩ →
+ ∃∃p,q. p ϵ 𝐂❨Ⓕ,m❩ & q ϵ 𝐂❨Ⓕ,n❩ & p●q = x.
+#x #m #n #Hx
+@(insert_eq_1 … (m+n) … Hx) -Hx #y #Hy
+generalize in match n; -n
+generalize in match m; -m
+elim Hy -x -y [|*: #x #y [ #k #_ ] #Hx #IH ] #m #n #Hy destruct
+[ elim (eq_inv_nplus_zero … Hy) -Hy #H1 #H2 destruct
+ /2 width=5 by pcc_empty, ex3_2_intro/
+| elim (IH m (n+k)) -IH // #p #q #Hp #Hq #H0 destruct -Hx
+ /3 width=5 by pcc_false_d_dx, ex3_2_intro/
+| elim (IH m n) -IH // #p #q #Hp #Hq #H0 destruct -Hx
+ /3 width=5 by pcc_m_dx, ex3_2_intro/
+| elim (eq_inv_succ_nplus_dx … (sym_eq … Hy)) -Hy * #H1 #H2 (**) (* sym_eq *)
+ [ destruct -IH
+ /3 width=5 by pcc_empty, pcc_L_dx, ex3_2_intro/
+ | elim (IH m (↓n)) -IH // #p #q #Hp #Hq #H0 destruct -Hx
+ /3 width=5 by pcc_L_dx, ex3_2_intro/
+ ]
+| elim (IH m n) -IH // #p #q #Hp #Hq #H0 destruct -Hx
+ /3 width=5 by pcc_A_dx, ex3_2_intro/
+| elim (IH m n) -IH // #p #q #Hp #Hq #H0 destruct -Hx
+ /3 width=5 by pcc_S_dx, ex3_2_intro/
+]
+qed-.
+
+
+(* Constructions with path_lcons ********************************************)
+
+lemma pcc_m_sn (o) (q) (n):
+ q ϵ 𝐂❨o,n❩ → (𝗺◗q) ϵ 𝐂❨o,n❩.
+#o #q #n #Hq
+lapply (pcc_append_bi (Ⓣ) … (𝐞◖𝗺) … Hq) -Hq
+/2 width=3 by pcc_m_dx/
+qed.
+
+lemma pcc_L_sn (o) (q) (n):
+ q ϵ 𝐂❨o,n❩ → (𝗟◗q) ϵ 𝐂❨o,↑n❩.
+#o #q #n #Hq
+lapply (pcc_append_bi (Ⓣ) … (𝐞◖𝗟) … Hq) -Hq
+/2 width=3 by pcc_L_dx/
+qed.
+
+lemma pcc_A_sn (o) (q) (n):
+ q ϵ 𝐂❨o,n❩ → (𝗔◗q) ϵ 𝐂❨o,n❩.
+#o #q #n #Hq
+lapply (pcc_append_bi (Ⓣ) … (𝐞◖𝗔) … Hq) -Hq
+/2 width=3 by pcc_A_dx/
+qed.
+
+lemma pcc_S_sn (o) (q) (n):
+ q ϵ 𝐂❨o,n❩ → (𝗦◗q) ϵ 𝐂❨o,n❩.
+#o #q #n #Hq
+lapply (pcc_append_bi (Ⓣ) … (𝐞◖𝗦) … Hq) -Hq
+/2 width=3 by pcc_S_dx/