(* *)
(**************************************************************************)
-include "delayed_updating/reduction/ifr.ma".
include "delayed_updating/reduction/dfr.ma".
+include "delayed_updating/reduction/ifr.ma".
+include "delayed_updating/substitution/fsubst_lift.ma".
+include "delayed_updating/substitution/lift_structure_depth.ma".
+include "delayed_updating/syntax/prototerm_proper_constructors.ma".
+include "delayed_updating/syntax/path_structure_depth.ma".
+include "ground/relocation/tr_pap_pushs.ma".
(* DELAYED FOCUSED REDUCTION ************************************************)
-lemma dfr_lift_bi (f) (p) (q) (t1) (t2):
+lemma dfr_lift_bi (f) (p) (q) (t1) (t2): t1 ϵ 𝐓 →
t1 ➡𝐝𝐟[p,q] t2 → ↑[f]t1 ➡𝐟[⊗p,⊗q] ↑[f]t2.
-#f #p #q #t1 #t2
-* #b #n #Hr #Hb
+#f #p #q #t1 #t2 #H0t1
+* #b #n * #Hb #Hn #Ht1 #Ht2
+@(ex1_2_intro … (⊗b) (↑❘⊗q❘)) @and4_intro
+[ //
+| #g <lift_rmap_structure <depth_structure
+ >tr_pushs_swap <tr_pap_pushs_le //
+| lapply (in_comp_lift_bi f … Ht1) -Ht1 -H0t1 -Hb -Ht2
+ <lift_d_empty_dx //
+| lapply (lift_term_eq_repl_dx f … Ht2) -Ht2 #Ht2
+ @(subset_eq_trans … Ht2) -t2
+ @(subset_eq_trans … (lift_fsubst …))
+ [ <structure_append <structure_A_sn <structure_append <structure_L_sn
+ | //
+ | /2 width=2 by ex_intro/
+ | //
+ ]
+]