include "delayed_updating/syntax/path_head_structure.ma".
include "delayed_updating/syntax/path_structure_depth.ma".
-include "delayed_updating/syntax/path_structure_reverse.ma".
-include "delayed_updating/syntax/path_depth_reverse.ma".
(* IMMEDIATE FOCUSED REDUCTION **********************************************)
-(* Constructions with unwind ************************************************)
+(* Constructions with unwind2 ***********************************************)
lemma ifr_unwind_bi (f) (p) (q) (t1) (t2):
t1 Ļµ š ā t1ā(pāš¦) Ļµ š ā
t1 ā”š¢š[p,q] t2 ā ā¼[f]t1 ā”š¢š[āp,āq] ā¼[f]t2.
#f #p #q #t1 #t2 #H1t1 #H2t1
-* #n * #H1n #Ht1 #Ht2
+* #k * #H1k #Ht1 #Ht2
@(ex_intro ā¦ (āāq)) @and3_intro
[ -H1t1 -H2t1 -Ht1 -Ht2
- >structure_L_sn >structure_reverse
- >H1n >path_head_structure_depth <H1n -H1n //
+ >structure_L_sn
+ >H1k in ā¢ (??%?); >path_head_structure_depth <H1k -H1k //
| lapply (in_comp_unwind2_path_term f ā¦ Ht1) -Ht2 -Ht1 -H1t1 -H2t1
- <unwind2_path_d_dx
- >list_append_rcons_sn in H1n; <reverse_append #H1n
- lapply (unwind2_rmap_append_pap_closed f ā¦ H1n)
- <reverse_lcons <depth_L_dx #H2n
- lapply (eq_inv_ninj_bi ā¦ H2n) -H2n #H2n <H2n -H2n -H1n #Ht1 //
+ <unwind2_path_d_dx <list_append_rcons_sn
+ lapply (unwind2_rmap_append_pap_closed f ā¦ (pāš) ā¦ H1k) -H1k
+ <depth_L_sn #H2k
+ lapply (eq_inv_ninj_bi ā¦ H2k) -H2k #H2k <H2k -H2k #Ht1 //
| lapply (unwind2_term_eq_repl_dx f ā¦ Ht2) -Ht2 #Ht2
@(subset_eq_trans ā¦ Ht2) -t2
@(subset_eq_trans ā¦ (unwind2_term_fsubst ā¦))
@(subset_eq_canc_sn ā¦ (lift_term_eq_repl_dx ā¦))
[ @unwind2_term_grafted_S /2 width=2 by ex_intro/ | skip ] -Ht1
@(subset_eq_trans ā¦ (lift_unwind2_term_after ā¦))
- @(subset_eq_canc_dx ā¦ (unwind2_term_after_lift ā¦))
+ @(subset_eq_canc_dx ā¦ (unwind2_lift_term_after ā¦))
@unwind2_term_eq_repl_sn
(* Note: crux of the proof begins *)
- @nstream_eq_inv_ext #m
- <tr_compose_pap <tr_compose_pap
- <tr_uni_pap <tr_uni_pap <tr_pap_plus
- >list_append_rcons_sn in H1n; <reverse_append #H1n
- lapply (unwind2_rmap_append_pap_closed f ā¦ H1n) #H2n
- >nrplus_inj_dx in ā¢ (???%); <H2n -H2n
- lapply (tls_unwind2_rmap_append_closed f ā¦ H1n) -H1n #H2n
- <(tr_pap_eq_repl ā¦ H2n) -H2n //
+ <list_append_rcons_sn
+ @(stream_eq_trans ā¦ (tr_compose_uni_dx ā¦))
+ @tr_compose_eq_repl
+ [ <unwind2_rmap_append_pap_closed //
+ | >unwind2_rmap_A_dx
+ /2 width=1 by tls_unwind2_rmap_closed/
+ ]
(* Note: crux of the proof ends *)
| //
| /2 width=2 by ex_intro/