--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "delayed_updating/reduction/dbfr.ma".
+include "delayed_updating/reduction/ibfr.ma".
+
+include "delayed_updating/unwind/unwind2_constructors.ma".
+include "delayed_updating/unwind/unwind2_preterm_fsubst.ma".
+include "delayed_updating/unwind/unwind2_preterm_eq.ma".
+include "delayed_updating/unwind/unwind2_prototerm_lift.ma".
+include "delayed_updating/unwind/unwind2_rmap_head.ma".
+
+include "delayed_updating/substitution/fsubst_eq.ma".
+include "delayed_updating/substitution/lift_prototerm_eq.ma".
+
+include "delayed_updating/syntax/prototerm_proper_constructors.ma".
+include "delayed_updating/syntax/path_head_structure.ma".
+include "delayed_updating/syntax/path_structure_depth.ma".
+
+(* DELAYED BALANCED FOCUSED REDUCTION ***************************************)
+
+(* Main destructions with ibfr **********************************************)
+
+theorem dbfr_des_ibfr (f) (t1) (t2) (r): t1 Ļµ š ā
+ t1 ā”ššš[r] t2 ā ā¼[f]t1 ā”š¢šš[ār] ā¼[f]t2.
+#f #t1 #t2 #r #H0t1
+* #p #b #q #h #k #Hr #Hb #Hh #H1k #Ht1 #Ht2 destruct
+@(ex6_5_intro ā¦ (āp) (āb) (āq) (āb) (āāq))
+[ -H0t1 -Hb -Hh -H1k -Ht1 -Ht2 //
+| -H0t1 -Hh -H1k -Ht1 -Ht2 //
+| -H0t1 -Hb -Ht1 -H1k -Ht2
+ >Hh in ā¢ (??%?); >path_head_structure_depth <Hh -Hh //
+| -H0t1 -Hb -Hh -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 -Hb -Hh
+ <unwind2_path_d_dx >list_append_rcons_dx >list_append_assoc
+ lapply (unwind2_rmap_append_pap_closed f ā¦ (pāšāb) 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
+ lapply (path_head_refl_append_bi ā¦ Hh H1k) -Hh -H1k <nrplus_inj_sn #H1k
+ @(subset_eq_trans ā¦ Ht2) -t2
+ @(subset_eq_trans ā¦ (unwind2_term_fsubst_ppc ā¦))
+ [ @fsubst_eq_repl [ // | // ]
+ @(subset_eq_trans ā¦ (unwind2_term_iref ā¦))
+ @(subset_eq_canc_sn ā¦ (lift_term_eq_repl_dx ā¦))
+ [ @unwind2_term_grafted_S /2 width=2 by ex_intro/ | skip ] -Ht1
+ @(subset_eq_trans ā¦ (lift_unwind2_term_after ā¦))
+ @unwind2_term_eq_repl_sn
+(* Note: crux of the proof begins *)
+ <list_append_rcons_sn
+ @(stream_eq_trans ā¦ (tr_compose_uni_dx ā¦))
+ @tr_compose_eq_repl
+ [ <unwind2_rmap_append_pap_closed //
+ <depth_append <depth_L_sn
+ <nplus_comm <nrplus_npsucc_sn <nplus_succ_sn //
+ | >unwind2_rmap_A_dx
+ /2 width=1 by tls_unwind2_rmap_closed/
+ ]
+(* Note: crux of the proof ends *)
+ | //
+ | /2 width=2 by ex_intro/
+ | //
+ ]
+]
+qed.