-lemma dfr_lift_bi (f) (p) (q) (t1) (t2): t1 Ļµ š ā
- t1 ā”šš[p,q] t2 ā ā[f]t1 ā”š[āp,āq] ā[f]t2.
-#f #p #q #t1 #t2 #H0t1
-* #b #n * #Hb #Hn #Ht1 #Ht2
-@(ex1_2_intro ā¦ (āb) (āāāqā)) @and4_intro
-[ //
-| #g <lift_rmap_structure <depth_structure
- >tr_pushs_swap <tr_pap_pushs_le //
-| lapply (in_comp_lift_bi f ā¦ Ht1) -Ht1 -H0t1 -Hb -Ht2
- <lift_path_d_empty_dx //
-| lapply (lift_term_eq_repl_dx f ā¦ Ht2) -Ht2 #Ht2
+(* Main destructions with ifr ***********************************************)
+
+theorem dfr_des_ifr (f) (t1) (t2) (r): t1 Ļµ š ā
+ t1 ā”šš[r] t2 ā ā¼[f]t1 ā”š¢š[ār] ā¼[f]t2.
+#f #t1 #t2 #r #H0t1
+* #p #q #k #Hr #H1k #Ht1 #Ht2 destruct
+@(ex4_3_intro ā¦ (āp) (āq) (āāq))
+[ -H0t1 -H1k -Ht1 -Ht2 //
+| -H0t1 -Ht1 -Ht2
+ >structure_L_sn
+ >H1k in ā¢ (??%?); >path_head_structure_depth <H1k -H1k //
+| lapply (in_comp_unwind2_path_term f ā¦ Ht1) -Ht2 -Ht1 -H0t1
+ <unwind2_path_d_dx <list_append_rcons_sn
+ lapply (unwind2_rmap_append_pap_closed f ā¦ (pāš) ā¦ H1k) -H1k
+ <depth_L_sn #H2k
+ lapply (eq_inv_ninj_bi ā¦ H2k) -H2k #H2k <H2k -H2k #Ht1 //
+| lapply (unwind2_term_eq_repl_dx f ā¦ Ht2) -Ht2 #Ht2