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3 (*      ||M||                                                             *)
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14
15 include "delayed_updating/reduction/dbfr.ma".
16
17 include "delayed_updating/substitution/fsubst_lift.ma".
18 include "delayed_updating/substitution/fsubst_eq.ma".
19 include "delayed_updating/substitution/lift_prototerm_constructors.ma".
20 include "delayed_updating/substitution/lift_path_structure.ma".
21 include "delayed_updating/substitution/lift_path_closed.ma".
22 include "delayed_updating/substitution/lift_rmap_closed.ma".
23
24 (* DELAYED BALANCED FOCUSED REDUCTION ***************************************)
25
26 (* Constructions with lift **************************************************)
27
28 theorem dbfr_lift_bi (f) (t1) (t2) (r):
29         t1 โžก๐๐›๐Ÿ[r] t2 โ†’ ๐Ÿ ก[f]t1 โžก๐๐›๐Ÿ[๐Ÿ ก[f]r] ๐Ÿ ก[f]t2.
30 #f #t1 #t2 #r
31 * #p #b #q #m #n #Hr #Hb #Hm #Hn #Ht1 #Ht2 destruct
32 @(ex6_5_intro โ€ฆ (๐Ÿ ก[f]p) (๐Ÿ ก[๐Ÿ ข[f](pโ—–๐—”)]b) (๐Ÿ ก[๐Ÿ ข[f](pโ—–๐—”โ—bโ—–๐—Ÿ)]q) (๐Ÿ ข[f](pโ—๐—”โ——b)๏ผ โจmโฉ) (๐Ÿ ข[f](pโ—๐—”โ——bโ—๐—Ÿโ——q)๏ผ ยงโจnโฉ))
33 [ -Hb -Hm -Hn -Ht1 -Ht2 //
34 | -Hm -Hn -Ht1 -Ht2 //
35 | -Hb -Hn -Ht1 -Ht2
36   /2 width=1 by lift_path_closed/
37 | -Hb -Hm -Ht1 -Ht2
38   /2 width=1 by lift_path_rmap_closed_L/
39 | lapply (in_comp_lift_path_term f โ€ฆ Ht1) -Ht2 -Ht1 -Hn
40   <lift_path_d_dx #Ht1 //
41 | lapply (lift_term_eq_repl_dx f โ€ฆ Ht2) -Ht2 #Ht2 -Ht1
42   @(subset_eq_trans โ€ฆ Ht2) -t2
43   @(subset_eq_trans โ€ฆ (lift_term_fsubst โ€ฆ))
44   @fsubst_eq_repl [ // | // ]
45   @(subset_eq_trans โ€ฆ (lift_term_iref_nap โ€ฆ))
46   <list_append_rcons_sn <list_append_rcons_sn >list_append_assoc
47   >(nap_plus_lift_rmap_append_closed_Lq_dx โ€ฆ Hn)
48   @iref_eq_repl
49   @(subset_eq_canc_sn โ€ฆ (lift_term_grafted_S โ€ฆ))
50   @lift_term_eq_repl_sn
51 (* ๐Ÿ ข[f]p โ‰— โ‡‚*[โ†‘(m+n)]๐Ÿ ข[f](((pโ—–๐—”)โ—b)โ—๐—Ÿโ——q) *)
52 (* Note: crux of the proof begins *)
53   >lift_rmap_A_dx
54   /2 width=2 by tls_succ_plus_lift_rmap_append_closed_bLq_dx/
55 (* Note: crux of the proof ends *)
56 ]
57 qed.