+(* Main destructions with ifr ***********************************************)
+
+theorem dfr_des_ifr (f) (t1) (t2) (r): t1 Ο΅ π β
+ t1 β‘ππ[r] t2 β βΌ[f]t1 β‘π’π[βr] βΌ[f]t2.
+#f #t1 #t2 #r #H0t1
+* #p #q #n #Hr #Hn #Ht1 #Ht2 destruct
+@(ex4_3_intro β¦ (βp) (βq) (βq))
+[ -H0t1 -Hn -Ht1 -Ht2 //
+| -H0t1 -Ht1 -Ht2
+ /2 width=2 by path_closed_structure_depth/
+| lapply (in_comp_unwind2_path_term f β¦ Ht1) -Ht2 -Ht1 -H0t1
+ <unwind2_path_d_dx <tr_pap_succ_nap <list_append_rcons_sn
+ <unwind2_rmap_append_closed_nap //
+| lapply (unwind2_term_eq_repl_dx f β¦ Ht2) -Ht2 #Ht2
+ @(subset_eq_trans β¦ Ht2) -t2
+ @(subset_eq_trans β¦ (unwind2_term_fsubst_ppc β¦))
+ [ @fsubst_eq_repl [ // | // ]
+ @(subset_eq_trans β¦ (unwind2_term_irefβ¦))
+ @(subset_eq_canc_sn β¦ (lift_term_eq_repl_dx β¦))
+ [ @unwind2_term_grafted_S /2 width=2 by ex_intro/ | skip ] -Ht1
+ @(subset_eq_trans β¦ (lift_unwind2_term_after β¦))
+ @unwind2_term_eq_repl_sn
+(* Note: crux of the proof begins *)
+ <list_append_rcons_sn
+ @(stream_eq_trans β¦ (tr_compose_uni_dx_pap β¦)) <tr_pap_succ_nap
+ @tr_compose_eq_repl
+ [ <unwind2_rmap_append_closed_nap //
+ | /2 width=1 by tls_succ_unwind2_rmap_append_L_closed_dx/
+ ]
+(* Note: crux of the proof ends *)
+ | //
+ | /2 width=2 by ex_intro/
+ | //
+ ]
+]
+qed.