include "delayed_updating/reduction/dfr.ma".
include "delayed_updating/reduction/ifr.ma".
-include "delayed_updating/substitution/fsubst_lift.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_head.ma".
+
include "delayed_updating/substitution/fsubst_eq.ma".
-include "delayed_updating/substitution/lift_constructors.ma".
-include "delayed_updating/substitution/lift_preterm_eq.ma".
-include "delayed_updating/substitution/lift_structure_depth.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_structure_depth.ma".
-include "ground/relocation/tr_uni_compose.ma".
-include "ground/relocation/tr_pap_pushs.ma".
-
-include "ground/lib/stream_eq_eq.ma".
+include "delayed_updating/syntax/path_structure_reverse.ma".
+include "delayed_updating/syntax/path_depth_reverse.ma".
(* DELAYED FOCUSED REDUCTION ************************************************)
-lemma dfr_lift_bi (f) (p) (q) (t1) (t2): t1 Ļµ š ā
- t1 ā”šš[p,q] t2 ā ā[f]t1 ā”š[āp,āq] ā[f]t2.
+(* 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
-* #b #n * #Hb #Hn #Ht1 #Ht2
-@(ex1_2_intro ā¦ (āb) (āāāqā)) @and4_intro
-[ //
-| #g <lift_rmap_structure <depth_structure
- >tr_pushs_swap <tr_pap_pushs_le //
-| lapply (in_comp_lift_bi f ā¦ Ht1) -Ht1 -H0t1 -Hb -Ht2
- <lift_path_d_empty_dx //
-| lapply (lift_term_eq_repl_dx f ā¦ Ht2) -Ht2 #Ht2
+* #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 //
+| 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 //
+| lapply (unwind2_term_eq_repl_dx f ā¦ Ht2) -Ht2 #Ht2
@(subset_eq_trans ā¦ Ht2) -t2
- @(subset_eq_trans ā¦ (lift_fsubst ā¦))
- [ <lift_rmap_append <lift_rmap_A_sn <lift_rmap_append <lift_rmap_L_sn
- <structure_append <structure_A_sn <structure_append <structure_L_sn
- <depth_append <depth_L_sn <depth_structure <depth_structure
- @fsubst_eq_repl [ // ]
- @(subset_eq_trans ā¦ (lift_iref ā¦))
+ @(subset_eq_trans ā¦ (unwind2_term_fsubst ā¦))
+ [ @fsubst_eq_repl [ // | // ]
+ @(subset_eq_trans ā¦ (unwind2_term_iref ā¦))
@(subset_eq_canc_sn ā¦ (lift_term_eq_repl_dx ā¦))
- [ @lift_grafted_S /2 width=2 by ex_intro/ | skip ]
- @(subset_eq_trans ā¦ (lift_term_after ā¦))
- @(subset_eq_canc_dx ā¦ (lift_term_after ā¦))
- @lift_term_eq_repl_sn -t1
+ [ @unwind2_term_grafted_S /2 width=2 by ex_intro/ | skip ] -Ht1
+ @(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
@(stream_eq_trans ā¦ (tr_compose_uni_dx ā¦))
@tr_compose_eq_repl
-(*
- >nrplus_inj_dx <tr_pap_plus
-*)
+ [ <unwind2_rmap_append_pap_closed //
+ | >unwind2_rmap_A_sn <reverse_rcons
+ /2 width=1 by tls_unwind2_rmap_closed/
+ ]
+(* Note: crux of the proof ends *)
| //
| /2 width=2 by ex_intro/
| //
]
]
+qed.