include "delayed_updating/unwind/unwind2_preterm_fsubst.ma".
include "delayed_updating/unwind/unwind2_preterm_eq.ma".
include "delayed_updating/unwind/unwind2_prototerm_lift.ma".
-include "delayed_updating/unwind/unwind2_rmap_head.ma".
+include "delayed_updating/unwind/unwind2_rmap_closed.ma".
include "delayed_updating/substitution/fsubst_eq.ma".
include "delayed_updating/substitution/lift_prototerm_eq.ma".
include "delayed_updating/syntax/prototerm_proper_constructors.ma".
-include "delayed_updating/syntax/path_head_structure.ma".
+include "delayed_updating/syntax/path_closed_structure.ma".
include "delayed_updating/syntax/path_structure_depth.ma".
(* DELAYED FOCUSED REDUCTION ************************************************)
(* Main destructions with ifr ***********************************************)
-theorem dfr_des_ifr (f) (p) (q) (t1) (t2): t1 Ο΅ π β
- t1 β‘ππ[p,q] t2 β βΌ[f]t1 β‘π’π[βp,βq] βΌ[f]t2.
-#f #p #q #t1 #t2 #H0t1
-* #k * #H1k #Ht1 #Ht2
-@(ex_intro β¦ (ββq)) @and3_intro
-[ -H0t1 -Ht1 -Ht2
- >structure_L_sn
- >H1k in β’ (??%?); >path_head_structure_depth <H1k -H1k //
+theorem dfr_des_ifr (f) (t1) (t2) (r): t1 Ο΅ π β
+ t1 β‘ππ[r] t2 β βΌ[f]t1 β‘π’π[βr] βΌ[f]t2.
+#f #t1 #t2 #r #H0t1
+* #p #q #n #Hr #Hn #Ht1 #Ht2 destruct
+@(ex4_3_intro β¦ (βp) (βq) (βq))
+[ -H0t1 -Hn -Ht1 -Ht2 //
+| -H0t1 -Ht1 -Ht2
+ /2 width=2 by path_closed_structure_depth/
| lapply (in_comp_unwind2_path_term f β¦ Ht1) -Ht2 -Ht1 -H0t1
- <unwind2_path_d_dx <list_append_rcons_sn
- lapply (unwind2_rmap_append_pap_closed f β¦ (pβπ) β¦ H1k) -H1k
- <depth_L_sn #H2k
- lapply (eq_inv_ninj_bi β¦ H2k) -H2k #H2k <H2k -H2k #Ht1 //
+ <unwind2_path_d_dx <tr_pap_succ_nap <list_append_rcons_sn
+ <nap_unwind2_rmap_append_closed_Lq_dx_depth //
| lapply (unwind2_term_eq_repl_dx f β¦ Ht2) -Ht2 #Ht2
@(subset_eq_trans β¦ Ht2) -t2
- @(subset_eq_trans β¦ (unwind2_term_fsubst β¦))
+ @(subset_eq_trans β¦ (unwind2_term_fsubst_ppc β¦))
[ @fsubst_eq_repl [ // | // ]
- @(subset_eq_trans β¦ (unwind2_term_iref β¦))
+ @(subset_eq_trans β¦ (unwind2_term_irefβ¦))
@(subset_eq_canc_sn β¦ (lift_term_eq_repl_dx β¦))
[ @unwind2_term_grafted_S /2 width=2 by ex_intro/ | skip ] -Ht1
@(subset_eq_trans β¦ (lift_unwind2_term_after β¦))
@unwind2_term_eq_repl_sn
(* Note: crux of the proof begins *)
<list_append_rcons_sn
- @(stream_eq_trans β¦ (tr_compose_uni_dx β¦))
+ @(stream_eq_trans β¦ (tr_compose_uni_dx_pap β¦)) <tr_pap_succ_nap
@tr_compose_eq_repl
- [ <unwind2_rmap_append_pap_closed //
- | >unwind2_rmap_A_dx
- /2 width=1 by tls_unwind2_rmap_closed/
+ [ <nap_unwind2_rmap_append_closed_Lq_dx_depth //
+ | /2 width=2 by tls_succ_unwind2_rmap_append_closed_Lq_dx/
]
(* Note: crux of the proof ends *)
| //