theorem dfr_lift_bi (f) (p) (q) (t1) (t2):
t1 ā”šš[p,q] t2 ā ā[f]t1 ā”šš[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
+* #k * #H1k #Ht1 #Ht2
+@(ex_intro ā¦ ((ā[pāšāšāq]f)ļ¼ ā§£āØkā©)) @and3_intro
[ -Ht1 -Ht2
<lift_rmap_L_dx >lift_path_L_sn
- <(lift_path_head ā¦ H1n) in ā¢ (??%?); -H1n //
-| lapply (in_comp_lift_path_term f ā¦ Ht1) -Ht2 -Ht1 -H1n
+ <(lift_path_head_closed ā¦ H1k) in ā¢ (??%?); -H1k //
+| lapply (in_comp_lift_path_term f ā¦ Ht1) -Ht2 -Ht1 -H1k
<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_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
+ >list_append_rcons_sn in H1k; #H1k >lift_rmap_A_dx
/2 width=1 by tls_lift_rmap_closed/
(* Note: crux of the proof ends *)
]