include "delayed_updating/reduction/dfr.ma".
include "delayed_updating/reduction/ifr.ma".
-include "delayed_updating/substitution/fsubst_lift.ma".
-include "delayed_updating/substitution/lift_structure_depth.ma".
+
+include "delayed_updating/unwind/unwind2_constructors.ma".
+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_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_closed_structure.ma".
include "delayed_updating/syntax/path_structure_depth.ma".
-include "ground/relocation/tr_id_pap.ma".
-include "ground/relocation/tr_id_pushs.ma".
(* DELAYED FOCUSED REDUCTION ************************************************)
-lemma dfr_lift_bi (f) (p) (q) (t1) (t2): t1 Ο΅ π β
- t1 β‘ππ[p,q] t2 β β[f]t1 β‘π[βp,βq] β[f]t2.
-#f #p #q #t1 #t2 #H0t1
-* #b #n * #Hb #Hn #Ht1 #Ht2
-@(ex1_2_intro β¦ (βb) (ββqβ)) @and4_intro
-[ //
-| //
-| lapply (in_comp_lift_bi f β¦ Ht1) -Ht1 -H0t1 -Hb -Ht2 #Ht1
- <depth_structure
-| lapply (eq_lift_bi f β¦ Ht2) -Ht2 #Ht2
+(* Main destructions with ifr ***********************************************)
+
+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 <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¦ (lift_fsubst β¦))
- [ <structure_append <structure_A_sn <structure_append <structure_L_sn
+ @(subset_eq_trans β¦ (unwind2_term_fsubst_ppc β¦))
+ [ @fsubst_eq_repl [ // | // ]
+ @(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_pap β¦)) <tr_pap_succ_nap
+ @tr_compose_eq_repl
+ [ <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 *)
| //
| /2 width=2 by ex_intro/
| //
]
]
+qed.