- <tr_xap_succ_pos <path_head_d_sn >tr_xap_succ_pos
- <lift_path_d_dx >lift_rmap_append <reverse_rcons
- @eq_f2 // <(IH … H0) -IH -H0
- @eq_f2 // <tr_xap_plus //
+ <tr_xap_succ_nap <path_head_d_sn >tr_xap_succ_nap
+ <lift_path_d_dx >lift_rmap_append <reverse_rcons
+ <(IH … H0) -IH -H0 <tr_xap_plus //
+ | <reverse_rcons <path_head_m_sn #H0
+ elim (eq_inv_list_lcons_bi ????? H0) -H0 #_ #H0
+ <list_append_assoc <lift_rmap_m_dx <lift_path_m_dx <reverse_rcons
+ <tr_xap_succ_nap <path_head_m_sn >tr_xap_succ_nap
+ <lift_path_m_dx <reverse_rcons
+ <(IH … H0) -IH -H0 //
+ | <reverse_rcons <path_head_L_sn #H0
+ elim (eq_inv_list_lcons_bi ????? H0) -H0 #_ #H0
+ <list_append_assoc <lift_rmap_L_dx <lift_path_L_dx <reverse_rcons
+ <tr_xap_succ_nap <path_head_L_sn >tr_xap_succ_nap
+ <lift_path_L_dx <reverse_rcons
+ <(IH … H0) -IH -H0 //
+ | <reverse_rcons <path_head_A_sn #H0
+ elim (eq_inv_list_lcons_bi ????? H0) -H0 #_ #H0
+ <list_append_assoc <lift_rmap_A_dx <lift_path_A_dx <reverse_rcons
+ <tr_xap_succ_nap <path_head_A_sn >tr_xap_succ_nap
+ <lift_path_A_dx <reverse_rcons
+ <(IH … H0) -IH -H0 //
+ | <reverse_rcons <path_head_S_sn #H0
+ elim (eq_inv_list_lcons_bi ????? H0) -H0 #_ #H0
+ <list_append_assoc <lift_rmap_S_dx <lift_path_S_dx <reverse_rcons
+ <tr_xap_succ_nap <path_head_S_sn >tr_xap_succ_nap
+ <lift_path_S_dx <reverse_rcons
+ <(IH … H0) -IH -H0 //
+ ]
+ ]
+]
+qed.