+
+lemma unwind2_rmap_append_pap_closed (f) (p) (q) (n:pnat):
+ p = ↳[n](p●q) →
+ ♭p = ninj (▶[f](p●q)@⧣❨n❩).
+#f #p #q #n #Hn
+>tr_xap_ninj >Hn in ⊢ (??%?);
+>(unwind2_rmap_head_append_xap_closed … Hn) -Hn //
+qed.
+
+lemma tls_unwind2_rmap_plus_closed (f) (p) (q) (n) (k):
+ p = (↳[n]p)●q →
+ ⇂*[k]▶[f]q ≗ ⇂*[n+k]▶[f]p.
+#f #p elim p -p
+[ #q #n #k #Hq
+ elim (eq_inv_list_empty_append … Hq) -Hq #Hn #H0 destruct
+ <path_head_empty in Hn; #Hn
+ <(eq_inv_empty_labels … Hn) -n //
+| #l #p #IH #q #n @(nat_ind_succ … n) -n //
+ #n #_ #k cases l [ #m ]
+ [ <path_head_d_sn <list_append_lcons_sn #Hq
+ elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq <nrplus_inj_sn
+ @(stream_eq_trans … (tls_unwind2_rmap_d_sn …))
+ >nrplus_inj_dx >nrplus_inj_sn >nrplus_inj_sn <nplus_plus_comm_23
+ /2 width=1 by/
+ | <path_head_m_sn <list_append_lcons_sn #Hq
+ elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+ <unwind2_rmap_m_sn /2 width=1 by/
+ | <path_head_L_sn <list_append_lcons_sn #Hq
+ elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+ <unwind2_rmap_L_sn <nplus_succ_sn /2 width=1 by/
+ | <path_head_A_sn <list_append_lcons_sn #Hq
+ elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+ <unwind2_rmap_A_sn /2 width=2 by/
+ | <path_head_S_sn <list_append_lcons_sn #Hq
+ elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+ <unwind2_rmap_S_sn /2 width=2 by/
+ ]
+]
+qed-.
+
+lemma tls_unwind2_rmap_append_closed (f) (p) (q) (n):
+ p = ↳[n](p●q) →
+ ▶[f]q ≗ ⇂*[n]▶[f](p●q).
+/2 width=1 by tls_unwind2_rmap_plus_closed/
+qed.