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14
15 include "delayed_updating/reduction/dfr.ma".
16
17 include "delayed_updating/substitution/fsubst_lift.ma".
18 include "delayed_updating/substitution/fsubst_eq.ma".
19 include "delayed_updating/substitution/lift_constructors.ma".
20 include "delayed_updating/substitution/lift_path_head.ma".
21 include "delayed_updating/substitution/lift_rmap_head.ma".
22
23 (* DELAYED FOCUSED REDUCTION ************************************************)
24
25 (* Constructions with lift **************************************************)
26
27 theorem dfr_lift_bi (f) (t1) (t2) (r):
28         t1 ➡𝐝𝐟[r] t2 → ↑[f]t1 ➡𝐝𝐟[↑[f]r] ↑[f]t2.
29 #f #t1 #t2 #r
30 * #p #q #k #Hr #H1k #Ht1 #Ht2 destruct
31 @(ex4_3_intro … (↑[f]p) (↑[↑[p◖𝗔◖𝗟]f]q) ((↑[p●𝗔◗𝗟◗q]f)@⧣❨k❩))
32 [ -H1k -Ht1 -Ht2 //
33 | -Ht1 -Ht2
34   <lift_rmap_L_dx >lift_path_L_sn
35   <(lift_path_head_closed … H1k) in ⊢ (??%?); -H1k //
36 | lapply (in_comp_lift_path_term f … Ht1) -Ht2 -Ht1 -H1k
37   <lift_path_d_dx #Ht1 //
38 | lapply (lift_term_eq_repl_dx f … Ht2) -Ht2 #Ht2 -Ht1
39   @(subset_eq_trans … Ht2) -t2
40   @(subset_eq_trans … (lift_term_fsubst …))
41   @fsubst_eq_repl [ // | // ]
42   @(subset_eq_trans … (lift_term_iref …))
43   @iref_eq_repl
44   @(subset_eq_canc_sn … (lift_term_grafted_S …))
45   @lift_term_eq_repl_sn
46 (* Note: crux of the proof begins *)
47   >list_append_rcons_sn in H1k; #H1k >lift_rmap_A_dx
48   /2 width=1 by tls_lift_rmap_closed/
49 (* Note: crux of the proof ends *)
50 ]
51 qed.