include "delayed_updating/substitution/fsubst_lift.ma".
include "delayed_updating/substitution/fsubst_eq.ma".
include "delayed_updating/substitution/lift_prototerm_after.ma".
-include "delayed_updating/substitution/lift_path_head.ma".
-include "delayed_updating/substitution/lift_rmap_head.ma".
+include "delayed_updating/substitution/lift_path_closed.ma".
+include "delayed_updating/substitution/lift_rmap_closed.ma".
include "ground/relocation/tr_uni_compose.ma".
include "ground/relocation/tr_compose_eq.ma".
theorem ifr_lift_bi (f) (t1) (t2) (r):
t1 โก๐ข๐[r] t2 โ โ[f]t1 โก๐ข๐[โ[f]r] โ[f]t2.
#f #t1 #t2 #r
-* #p #q #k #Hr #H1k #Ht1 #Ht2 destruct
-@(ex4_3_intro โฆ (โ[f]p) (โ[โ[pโ๐โ๐]f]q) ((โ[pโ๐โ๐โq]f)๏ผ โงฃโจkโฉ))
-[ -H1k -Ht1 -Ht2 //
+* #p #q #n #Hr #Hn #Ht1 #Ht2 destruct
+@(ex4_3_intro โฆ (โ[f]p) (โ[โ[pโ๐โ๐]f]q) ((โ[pโ๐โ๐โq]f)๏ผ ยงโจnโฉ))
+[ -Hn -Ht1 -Ht2 //
| -Ht1 -Ht2
- <lift_rmap_L_dx >lift_path_L_sn
- <(lift_path_head_closed โฆ H1k) in โข (??%?); -H1k //
-| lapply (in_comp_lift_path_term f โฆ Ht1) -Ht2 -Ht1 -H1k
+ /2 width=1 by lift_path_rmap_closed_L/
+| lapply (in_comp_lift_path_term f โฆ Ht1) -Ht2 -Ht1 -Hn
<lift_path_d_dx #Ht1 //
| lapply (lift_term_eq_repl_dx f โฆ Ht2) -Ht2 #Ht2 -Ht1
@(subset_eq_trans โฆ Ht2) -t2
@(subset_eq_canc_dx โฆ (lift_term_after โฆ))
@lift_term_eq_repl_sn
(* Note: crux of the proof begins *)
- @(stream_eq_trans โฆ (tr_compose_uni_dx โฆ))
- @tr_compose_eq_repl //
- >list_append_rcons_sn in H1k; #H1k >lift_rmap_A_dx
- /2 width=1 by tls_lift_rmap_closed/
+ @(stream_eq_trans โฆ (tr_compose_uni_dx_pap โฆ)) <tr_pap_succ_nap
+ @tr_compose_eq_repl // >nsucc_unfold
+ /2 width=1 by tls_succ_lift_rmap_append_L_closed_dx/
(* Note: crux of the proof ends *)
]
qed.