include "delayed_updating/syntax/prototerm_proper_constructors.ma".
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".
(* DELAYED FOCUSED REDUCTION ************************************************)
(* Main destructions with ifr ***********************************************)
-theorem dfr_des_ifr (f) (p) (q) (t1) (t2): t1 Ļµ š ā
- t1 ā”šš[p,q] t2 ā ā¼[f]t1 ā”š¢š[āp,āq] ā¼[f]t2.
-#f #p #q #t1 #t2 #H0t1
-* #n * #H1n #Ht1 #Ht2
-@(ex_intro ā¦ (āāq)) @and3_intro
-[ -H0t1 -Ht1 -Ht2
- >structure_L_sn >structure_reverse
- >H1n in ā¢ (??%?); >path_head_structure_depth <H1n -H1n //
+theorem dfr_des_ifr (f) (t1) (t2) (r): t1 Ļµ š ā
+ t1 ā”šš[r] t2 ā ā¼[f]t1 ā”š¢š[ār] ā¼[f]t2.
+#f #t1 #t2 #r #H0t1
+* #p #q #k #Hr #H1k #Ht1 #Ht2 destruct
+@(ex4_3_intro ā¦ (āp) (āq) (āāq))
+[ -H0t1 -H1k -Ht1 -Ht2 //
+| -H0t1 -Ht1 -Ht2
+ >structure_L_sn
+ >H1k in ā¢ (??%?); >path_head_structure_depth <H1k -H1k //
| lapply (in_comp_unwind2_path_term f ā¦ Ht1) -Ht2 -Ht1 -H0t1
- <unwind2_path_d_dx >(list_append_rcons_sn ā¦ p) <reverse_append
- lapply (unwind2_rmap_append_pap_closed f ā¦ (pāš)į“æ ā¦ H1n) -H1n
- <reverse_lcons <depth_L_dx #H2n
- lapply (eq_inv_ninj_bi ā¦ H2n) -H2n #H2n <H2n -H2n #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_trans ā¦ (unwind2_term_fsubst_ppc ā¦))
[ @fsubst_eq_repl [ // | // ]
@(subset_eq_trans ā¦ (unwind2_term_iref ā¦))
@(subset_eq_canc_sn ā¦ (lift_term_eq_repl_dx ā¦))
@(subset_eq_trans ā¦ (lift_unwind2_term_after ā¦))
@unwind2_term_eq_repl_sn
(* Note: crux of the proof begins *)
- >list_append_rcons_sn <reverse_append
+ <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_sn <reverse_rcons
+ | >unwind2_rmap_A_dx
/2 width=1 by tls_unwind2_rmap_closed/
]
(* Note: crux of the proof ends *)