+(**************************************************************************)
+(* ___ *)
+(* ||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/dfr.ma".
+include "delayed_updating/reduction/ifr.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_closed.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_closed_structure.ma".
+include "delayed_updating/syntax/path_structure_depth.ma".
+
+(* DELAYED FOCUSED REDUCTION ************************************************)
+
+(* Main destructions with ifr ***********************************************)
+
+theorem dfr_des_ifr (f) (t1) (t2) (r): t1 ฯต ๐ โ
+ t1 โก๐๐[r] t2 โ โผ[f]t1 โก๐ข๐[โr] โผ[f]t2.
+#f #t1 #t2 #r #H0t1
+* #p #q #n #Hr #Hn #Ht1 #Ht2 destruct
+@(ex4_3_intro โฆ (โp) (โq) (โญq))
+[ -H0t1 -Hn -Ht1 -Ht2 //
+| -H0t1 -Ht1 -Ht2
+ /2 width=2 by path_closed_structure_depth/
+| lapply (in_comp_unwind2_path_term f โฆ Ht1) -Ht2 -Ht1 -H0t1
+ <unwind2_path_d_dx <tr_pap_succ_nap <list_append_rcons_sn
+ <nap_unwind2_rmap_append_closed_Lq_dx_depth //
+| lapply (unwind2_term_eq_repl_dx f โฆ Ht2) -Ht2 #Ht2
+ @(subset_eq_trans โฆ Ht2) -t2
+ @(subset_eq_trans โฆ (unwind2_term_fsubst_ppc โฆ))
+ [ @fsubst_eq_repl [ // | // ]
+ @(subset_eq_trans โฆ (unwind2_term_irefโฆ))
+ @(subset_eq_canc_sn โฆ (lift_term_eq_repl_dx โฆ))
+ [ @unwind2_term_grafted_S /2 width=2 by ex_intro/ | skip ] -Ht1
+ @(subset_eq_trans โฆ (lift_unwind2_term_after โฆ))
+ @unwind2_term_eq_repl_sn
+(* Note: crux of the proof begins *)
+ <list_append_rcons_sn
+ @(stream_eq_trans โฆ (tr_compose_uni_dx_pap โฆ)) <tr_pap_succ_nap
+ @tr_compose_eq_repl
+ [ <nap_unwind2_rmap_append_closed_Lq_dx_depth //
+ | /2 width=2 by tls_succ_unwind2_rmap_append_closed_Lq_dx/
+ ]
+(* Note: crux of the proof ends *)
+ | //
+ | /2 width=2 by ex_intro/
+ | //
+ ]
+]
+qed.