include "delayed_updating/reduction/dfr.ma".
include "delayed_updating/reduction/ifr.ma".
include "delayed_updating/substitution/fsubst_lift.ma".
+include "delayed_updating/substitution/fsubst_eq.ma".
+include "delayed_updating/substitution/lift_constructors.ma".
+include "delayed_updating/substitution/lift_preterm_eq.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_uni_compose.ma".
+include "ground/relocation/tr_pap_pushs.ma".
+
+include "ground/lib/stream_eq_eq.ma".
(* DELAYED FOCUSED REDUCTION ************************************************)
-lemma dfr_lift_bi (f) (p) (q) (t1) (t2): Ƭt1 →
+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 #Ht1
-* -t2 #b #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_path_d_empty_dx //
+| lapply (lift_term_eq_repl_dx f … Ht2) -Ht2 #Ht2
+ @(subset_eq_trans … Ht2) -t2
+ @(subset_eq_trans … (lift_fsubst …))
+ [ <lift_rmap_append <lift_rmap_A_sn <lift_rmap_append <lift_rmap_L_sn
+ <structure_append <structure_A_sn <structure_append <structure_L_sn
+ <depth_append <depth_L_sn <depth_structure <depth_structure
+ @fsubst_eq_repl [ // ]
+ @(subset_eq_trans … (lift_iref …))
+ @(subset_eq_canc_sn … (lift_term_eq_repl_dx …))
+ [ @lift_grafted_S /2 width=2 by ex_intro/ | skip ]
+ @(subset_eq_trans … (lift_term_after …))
+ @(subset_eq_canc_dx … (lift_term_after …))
+ @lift_term_eq_repl_sn -t1
+ @(stream_eq_trans … (tr_compose_uni_dx …))
+(*
+ >nrplus_inj_dx <tr_pap_plus
+*)
+ | //
+ | /2 width=2 by ex_intro/
+ | //
+ ]
+]