+
+lemma unwind2_rmap_append_pap_closed (f) (p) (q) (h:pnat):
+ q = ↳[h]q →
+ ♭q = ninj (▶[f](p●q)@⧣❨h❩).
+#f #p #q #h #Hh
+>tr_xap_ninj >(path_head_refl_append_sn p … Hh) in ⊢ (??%?);
+>(unwind2_rmap_head_xap_closed … Hh) -Hh
+<path_head_depth //
+qed.
+
+lemma tls_unwind2_rmap_plus_closed (f) (p) (q) (n) (m):
+ q = ↳[n]q →
+ ⇂*[m]▶[f]p ≗ ⇂*[n+m]▶[f](p●q).
+#f #p #q elim q -q
+[ #n #m #Hq
+ <(eq_inv_path_empty_head … Hq) -n //
+| #l #q #IH #n @(nat_ind_succ … n) -n //
+ #n #_ #m cases l [ #k ]
+ [ <path_head_d_dx #Hq
+ elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq <nrplus_inj_sn
+ @(stream_eq_trans … (tls_unwind2_rmap_d_dx …))
+ >nrplus_inj_dx >nrplus_inj_sn >nrplus_inj_sn <nplus_plus_comm_23
+ /2 width=1 by/
+ | <path_head_m_dx #Hq
+ elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+ <unwind2_rmap_m_sn /2 width=1 by/
+ | <path_head_L_dx #Hq
+ elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+ <unwind2_rmap_L_dx <nplus_succ_sn /2 width=1 by/
+ | <path_head_A_dx #Hq
+ elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+ <unwind2_rmap_A_dx /2 width=2 by/
+ | <path_head_S_dx #Hq
+ elim (eq_inv_list_lcons_bi ????? Hq) -Hq #_ #Hq
+ <unwind2_rmap_S_dx /2 width=2 by/
+ ]
+]
+qed-.
+
+lemma tls_unwind2_rmap_closed (f) (p) (q) (n):
+ q = ↳[n]q →
+ ▶[f]p ≗ ⇂*[n]▶[f](p●q).
+/2 width=1 by tls_unwind2_rmap_plus_closed/
+qed.