+(**************************************************************************)
+(* ___ *)
+(* ||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/ibfr.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_closed.ma".
+
+include "delayed_updating/substitution/fsubst_eq.ma".
+include "delayed_updating/substitution/lift_prototerm_eq.ma".
+
+include "delayed_updating/syntax/path_closed_structure.ma".
+include "delayed_updating/syntax/path_structure_depth.ma".
+
+(* IMMEDIATE BALANCED FOCUSED REDUCTION *************************************)
+
+(* Constructions with unwind2 ***********************************************)
+
+lemma ibfr_unwind_bi (f) (t1) (t2) (r):
+ t1 ϵ 𝐓 → r ϵ 𝐈 →
+ t1 ➡𝐢𝐛𝐟[r] t2 → ▼[f]t1 ➡𝐢𝐛𝐟[⊗r] ▼[f]t2.
+#f #t1 #t2 #r #H1t1 #H2r
+* #p #b #q #m #n #Hr #Hb #Hm #Hn #Ht1 #Ht2 destruct
+@(ex6_5_intro … (⊗p) (⊗b) (⊗q) (♭b) (♭q))
+[ -H1t1 -H2r -Hb -Hm -Hn -Ht1 -Ht2 //
+| -H1t1 -H2r -Hm -Hn -Ht1 -Ht2 //
+| -H1t1 -H2r -Hb -Hn -Ht1 -Ht2
+ /2 width=2 by path_closed_structure_depth/
+| -H1t1 -H2r -Hb -Hm -Ht1 -Ht2
+ /2 width=2 by path_closed_structure_depth/
+| lapply (in_comp_unwind2_path_term f … Ht1) -Ht2 -Ht1 -H1t1 -H2r -Hb
+ <unwind2_path_d_dx <tr_pap_succ_nap <list_append_rcons_sn >list_append_assoc
+ <nap_unwind2_rmap_append_closed_Lq_dx_depth //
+| lapply (unwind2_term_eq_repl_dx f … Ht2) -Ht2 #Ht2
+ @(subset_eq_trans … Ht2) -t2
+ @(subset_eq_trans … (unwind2_term_fsubst_pic …))
+ [ @fsubst_eq_repl [ // | // ]
+ @(subset_eq_canc_sn … (lift_term_eq_repl_dx …))
+ [ @unwind2_term_grafted_S /2 width=2 by ex_intro/ | skip ] -Ht1
+ @(subset_eq_trans … (lift_unwind2_term_after …))
+ @(subset_eq_canc_dx … (unwind2_lift_term_after …))
+ @unwind2_term_eq_repl_sn
+(* Note: crux of the proof begins *)
+ <list_append_rcons_sn
+ @(stream_eq_trans … (tr_compose_uni_dx_pap …)) <tr_pap_succ_nap
+ @tr_compose_eq_repl
+ [ <nap_plus_unwind2_rmap_append_closed_bLq_dx_depth //
+ | >unwind2_rmap_A_dx
+ /2 width=2 by tls_succ_plus_unwind2_rmap_append_closed_bLq_dx/
+ ]
+(* Note: crux of the proof ends *)
+ | //
+ | /2 width=2 by ex_intro/
+ | //
+ ]
+]
+qed.