--- /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/ifr.ma".
+
+include "delayed_updating/unwind/unwind2_path_lift.ma".
+
+include "delayed_updating/substitution/fsubst_lift.ma".
+include "delayed_updating/substitution/fsubst_eq.ma".
+include "delayed_updating/substitution/lift_prototerm_eq.ma".
+include "delayed_updating/substitution/lift_path_head.ma".
+include "delayed_updating/substitution/lift_rmap_head.ma".
+
+(* IMMEDIATE FOCUSED REDUCTION **********************************************)
+
+(* Constructions with lift **************************************************)
+
+theorem ifr_lift_bi (f) (p) (q) (t1) (t2):
+ t1 ā”š¢š[p,q] t2 ā ā[f]t1 ā”š¢š[ā[f]p,ā[ā[pāšāš]f]q] ā[f]t2.
+#f #p #q #t1 #t2
+* #n * #H1n #Ht1 #Ht2
+@(ex_intro ā¦ ((ā[pāšāšāq]f)ļ¼ ā§£āØnā©)) @and3_intro
+[ -Ht1 -Ht2
+ <lift_rmap_L_dx >lift_path_L_sn
+ >list_append_rcons_sn in H1n; <reverse_append #H1n
+ <(lift_path_head ā¦ H1n) -H1n //
+| lapply (in_comp_lift_path_term f ā¦ Ht1) -Ht2 -Ht1 -H1n
+ <lift_path_d_dx #Ht1 //
+| lapply (lift_term_eq_repl_dx f ā¦ Ht2) -Ht2 #Ht2 -Ht1
+ @(subset_eq_trans ā¦ Ht2) -t2
+ @(subset_eq_trans ā¦ (lift_term_fsubst ā¦))
+ @fsubst_eq_repl [ // | <lift_path_append // ]
+(*
+ @(subset_eq_trans ā¦ (lift_term_iref ā¦))
+ @iref_eq_repl
+ @(subset_eq_canc_sn ā¦ (lift_term_grafted_S ā¦))
+ @lift_term_eq_repl_sn
+(* Note: crux of the proof begins *)
+ >list_append_rcons_sn in H1n; #H1n >lift_rmap_A_dx
+ /2 width=1 by tls_lift_rmap_append_closed/
+(* Note: crux of the proof ends *)
+]
+qed.
+*)
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